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The Common Core State Standards
Art of Problem Solving courses are not designed to align to the Common Core State Standards ( CCSS ). However, we have many students who have easily replaced CCSS-aligned classes using our coursework. Our curriculum covers all of the Common Core Practices. Each of our classes also covers most or all of the Standards covered in the corresponding class from a CCSS-aligned school (and a good deal more!).
We've compiled some information below to help students and their schools choose the best possible education plan.
On this page
- What Is The Common Core?
Differences Between AoPS and the CCSS
Common core coverage, what is the common core.
The Common Core State Standards are a collection of Standards for Mathematical Practice combined with a collection of Standards for Mathematical Content. You can find the Common Core here .
Standards for Mathematical Practice
The Standards for Mathematical Practice ( Practices ) are eight general pedagogical goals suggested for all mathematical education:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with Mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
- Make sense of problems and persevere in solving them : AoPS does not believe that math is memorizing Trick A to solve Problem A and Trick B for Problem B. Problem solving is about understanding the problem, weighing the possible approaches, and deciding which is the best strategy for solving it. Sometimes the path is not apparent, and sometimes what at first seemed like a promising start needs to be rethought. A good problem solver has the tenacity to work through these obstacles.
- Reason abstractly and quantitatively and Model with Mathematics : Part of problem solving is the ability to translate the problem description into mathematical symbols. At the same time, it is important for students to not treat equations as a series of symbols to be manipulated but understand what their equations mean in the context of the mathematical concepts and the problem. To understand problems deeply, a mathematician needs to be able to follow paths of context and abstraction simultaneously.
- Construct viable arguments and critique the reasoning of others , Use appropriate tools strategically and Attend to precision : Many traditional schools only discuss proofs in Geometry class, and only in simplistic terms. AoPS integrates proof-writing—having the students construct rigorous logical arguments and communicate their ideas clearly—starting from our Prealgebra courses. Each week in our subject series courses, students have one or more free response writing problems. Student responses are human-graded and receive feedback on their solution, including their ability to communicate mathematically. Students learn to construct and analyze mathematical arguments in our classroom as well, where we make the students do all the work .
- Look for and make use of structure and Look for and express regularity in repeated reasoning : Recognizing patterns is an important problem solving strategy. AoPS students learn by studying relevant problems and generalizing their ideas into the underlying principles. This requires dissecting each problem into its constituent pieces and their connective structures. This also requires finding common principles inside different problems and finding ways to extract those principles and use them on the current problem or on problems down the road.
Standards for Mathematical Content
Traditional schools and the Common Core are generally concerned with getting students through the typical four-year high school curriculum, in a large part to prepare them for college-level Calculus. Art of Problem Solving serves high-performing students who learn at a faster pace and can handle more complex cognitive tasks. These students are not stretched by the traditional curriculum, so they do not grow to their full potential. They need more intellectual stimulation.
One way AoPS addresses this is by including advanced topics in Algebra and Geometry that do not appear on the Common Core State Standards. The AoPS curriculum also includes courses in discrete math, namely counting and number theory, which also are not part of the Common Core or standard high school curricula. For example, our Introductory and Intermediate Counting & Probability and Number Theory courses do not contain many Standards because much of their material lies outside the Common Core. This omission in the CCSS is unfortunate since discrete math is a key part of college-level math and the mathematics of programming, and students who have a background in these fields will have a significant advantage in those majors and in the real world. These might be the most applicable skills our students learn in their high school careers.
Furthermore, counting and number theory are accessible fields where students can further their mathematical reasoning and proof techniques. The traditional curriculum generally teaches students many tools to apply to specific straightforward problems, again in large part because the school is in a rush to provide the student all the prerequisites to enroll in Calculus. But this is not what mathematics is about. Our philosophy is to teach students problem solving which, in contrast, is about taking the tools they have and applying them to solve complex problems. Real life will throw students many problems they have not specifically been trained to solve, and the students with stronger problem solving skills will be better positioned to tackle these problems. Learning discrete math, in addition to expanding the students' horizons, gives them the opportunity to develop problem solving skills before moving onto more complex topics.
The other major difference between the Common Core State Standards and the Art of Problem Solving curriculum is the treatment of Statistics. AoPS currently does not have a Statistics course. Several AoPS courses cover standards regarding probability, specifically using problems where student calculate probabilities to illustrate interesting applications of counting techniques and geometric probability. Many Common Core standards regarding interpreting data and justifying conclusions are not covered by AoPS courses at this time. If a student is using an AoPS course to skip a class in a traditional school, she may need to spend a couple of days in the school library reading through the statistics chapter of her local textbook.
- The Common Core grades 6-8 contain numerous standards involving 3-dimensional shapes. The Geometry component in AoPS Prealgebra 2 is closer to a traditional Euclidean plane geometry course, where students apply geometric properties to solve problems. AoPS discusses 3-dimensional geometry first in its Beast Academy 5A curriculum and then defers until Introduction to Geometry , where students can extend their mastery of 2-dimensional geometry to reason about 3-dimensional configurations. We reach the depth expected in the CCSS in our 5A explorations and then wait on the material until students have the tools to solve more complex problems instead of learning extra forumulas earlier.
- The Common Core uses geometric transformations—dilation, reflection, rotation, and translation—to define congruence and similarity. In contrast, AoPS uses an informal definition of congruence and similarity that utilizes the students' intuition, saving transformations for later in the Introduction to Geometry course. AoPS also discusses geometric transformations in Precalculus using the tools of complex numbers and vectors.
- Also, AoPS and the Common Core emphasize different aspects of the Expressions & Equations domains in grades 6-8. The Common Core uses the coordinate plane to connect concepts in ratios, expressions, and geometry that deal with linear equations and their graphs. AoPS does not cover the graphs of linear equations until the high school-level Introduction to Algebra A course. Similarly, the grade 8 Functions domain and standards regarding systems of linear equations are taught in Algebra, where we can provide a more rigorous treatment. Instead, the Common Core high school cluster A.SSE (Algebra: Seeing Structure in Expressions) is pushed forward into AoPS Prealgebra as it is an important problem solving skill when manipulating algebraic expressions.
The following tables describe which AoPS courses cover the various Common Core content standards. You can find similar information for our courses in each Course Syllabus .
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How to Do Common Core Math – A Guide for Parents
During this distance learning journey, many parents are confused by the “new way” of teaching and learning math – commonly referred to as “common core math.” Although at first glance common core math seems overwhelming, this way of doing math is actually a lot simpler than you think. In short, common core math means that children are taught several ways to solve the same math problem.
Common Core Math – The Basics
By teaching kids several ways to solve math problems and giving them different skills, they’re more likely to understand math in a way that’s best for them. When you think about it that way, it makes a lot of sense. Every child is unique and has their own way of absorbing and processing information. Math has always been one of those subjects that some kids seem to take to easily, while others need more time to work out problems.
Most teachers like the idea that when a math problem is presented to a child, they can solve it the way that they want. This is helpful to those students who need to find the answer in a different way, because rote is not a one-size-fits-all approach.
Most of us parents were taught one, rote way to do math problems and calculations. That’s why common core math seems so confusing! But, if parents take a little time to understand common core math, they might even start to appreciate it.
How to Teach Common Core Math
There are a number of different teaching strategies at your disposal when you get ready to teach kids common core math skills. To help kids find out what problem-solving methods work best for them, you can give them several types of common core math activities. You can use teaching tools that give kids several methods at once. This way, they can pick ones that appeal to them naturally.
Scaffolding Math Problems
For younger children, teachers often “scaffold” math problems. Here is an example of scaffolding from our scaffolded math journal
The word problem is given, but this will also work with given equations. Children then fill out all or part of the sheet to find the answer. Solving the problem in a few different ways helps children verify correct answers.
Here’s an explanation of five different ways to scaffold to help children figure out the answer to math problems or equations:
1. Number Paths
Number paths do not have zeros. They are a row of boxes with a number in each box, in sequential order. So basically, the blocks act as a type of counter.
2. Number Lines
Number lines start at zero. The numbers are usually seen on some sort of ticked lines, and the numbers may appear above or below the line. Many times students are taught to “jump” from one number to another to help them count, add, or subtract.
3. Ten Frames
A ten frame is a tool to help children see numbers in fives and tens. Without counting, the students should be able to see that if the first row is full, that means there is at least five, then they can continue counting the second row, unless they see that full as well, which means 10.
We have a ten-frame stamp that is easy to use on children’s papers.
4. Number Bonds
A number bond is another way for children to see the “part-part-whole” of the problem. The larger number holds the whole number, while the two smaller circles hold the parts that make up the whole.
5. Draw Pictures
Many kids love to draw, and this common core method combines a favorite activity with math lessons…they can simply draw circles, stars, etc. The main thing is to keep it simple!
Drawing a math problem is a big help, especially for visual learners. Children must be reminded when drawing for math, that this isn’t art class – they’re solving a problem. Instead of cats, they can simply draw circles. The main thing is to keep it simple!
Some final thoughts on common core math for parents
So, as you can see, different children will be able to complete some or all math problems using scaffolding or common core math tools. Other than these simple tools, there are other ways that children can try to solve math problems to learn what best works for them!
Our Strategies for Math Banner (shown above) can also help with the many different ways to solve math problems.
I hope that parents see that “new math” or “common core math” benefits all students. Not only will kids find a way that works for them, but they may just be able to explain their thinking as well!
By Angela French
Angela French is the Senior Product Development and Content Manager at Really Good Stuff. She has worked for the company for nearly seven years and has created hundreds of resources for the classroom. She has a Master’s Degree in Early Childhood Education from Old Dominion University in Norfolk, VA. Her classroom experiences include teaching grade levels K–5 and inclusion, special education, literacy intervention, and gifted and talented programs in three different states .
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8 Popular Common Core Math Standards Explained with Examples in the Classroom
Did you know?
Prodigy Math Game covers many math curricula across the world, including Common Core.
- Definitions for each of the eight standards
- Examples to help you seamlessly incorporate each standard into your classroom
- Suggestions for talking to parents about the new Common Core math curriculum
The dust has finally settled, and it looks like Common Core math is here to stay.
After countless political battles (and more than one Common Core math meme floating around social media), the initiative that incorporates techniques like cooperative learning and active learning has settled into the American education system.
Prodigy offers common Core-aligned math practice that your students will love. Start today!
Beginning in 2010, the Common Core State Standards Initiative (CCSSI) aimed to change the way American students were taught English language arts and mathematics by countering low test scores, inconsistent learning standards and a curriculum that was a “mile wide and an inch deep.”
Of the 45 states (plus the District of Columbia and the Department of Defense Education Activity) that fully implemented Common Core by 2015, 24 chose to revise some aspects of the program but still remain aligned with the original standards today.
What is Common Core math?
The Common Core State Standards for Mathematical Practice were designed to reform the American education system, with three main goals:
- Provide graduating high school students with the skills they need to be successful either in the workforce or in post-secondary education
- Boost math test scores for all American students
- Smooth out the differences between individual state curriculums and practices
At the heart of Common Core math are the eight Standards for Mathematical Practice. These standards were created by education professionals at all levels, and are based on research, leading state curricula and exceptional international math programs.
- Make sense of problems and persevere in solving them
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated reasoning
These standards allow students to learn deeply instead of widely and build a solid foundation for advanced study . The traditional Common Core math provides guidelines for grade-specific concepts, but it’s up to individual school districts to implement a curriculum that’s in line with the standards.
Keep reading to find out what they mean, or download our free, condensed list of the eight standards and examples for teaching them!
Is Common Core math working?
Due to the massive overhaul of state education frameworks, many teachers are still scrambling to prepare. A study from the Center for Education Policy Research at Harvard University reported that 82% of math teachers are changing “ more than half of their instructional materials” in response to the new practice standards. The same study found that “three out of four teachers (73%) reported that they have embraced the new standards ‘quite a bit’ or ‘fully’.”
Source: Center for Education Policy Research
1. Make sense of problems and persevere in solving them
When students approach a new problem for the first time, they might be tempted to go straight for the solution. After all, isn’t that the point? The first standard directly counters this impulse.
When students rush in to immediately solve a problem, they often fail to understand the underlying concepts. Rote memorization and a quick recall are essential parts of mathematical fluency, but can often lead to greater problems. If a student doesn’t understand the underlying concept behind the facts they’ve spent time learning, they might struggle with more complicated problems or ideas.
Giving students more open-ended questions or methods allows them to work with the concepts behind the problem, instead of going straight to the solution.
For example, look at this typical Common Core math problem:
Source: The School Run While it looks complicated, a number line is a Common Core math example that teaches students several essential concepts:
- The relationship between numbers in a given problem
- The potential for more than one solution
- The foundation behind shortcuts and more complex processes
Jennifer Smith and Michelle Stephan used this question to incorporate the first standard into a seventh-grade classroom:
Source: Journal of the American Academy of Special Education Professionals
The problem was presented to the class with a quick introduction, and the teachers asked students to find the greater net worth -- without explaining how. Students worked alone or in groups to discuss the question and their process, while the teachers supervised and made note of different strategies.
The teachers spent minimal time with the students, fixing only minor mistakes and encouraging them to work with their group. Every student was actively engaged in working with the problems, and explained their thinking to the class in a follow-up discussion.
2. Reason abstractly and quantitatively
There are two parts to the second standard: decontextualization and contextualization.
Decontextualization refers to the process of understanding the symbols in a problem as separate from the whole. This is where the much-loved word problem becomes essential.
Take this question as an example:
Sarah has 5 bouquets of flowers on her desk. After lunch, Steve brings her 3 bouquets of flowers. How many bouquets of flowers does Sarah have on her desk now?
Decontextualization means a student is expected to infer from the above problem that they are to solve an equation (5 + 3 = 8) without getting distracted by any additional information.
Contextualization is the opposite: it refers to the ability to step back from the problem and view it as a whole. Students would have to understand that five bouquets of flowers represent the total amount, and the three more that Steve brings are adding to that original number.
Source: Stanford Graduate School of Education
A study conducted by the Stanford Graduate School of Education found that the same parts of the brain that compare physical size also compare the abstract worth of two numbers. By linking those two processes using modular tools , the study found that students were better equipped to learn about abstract concepts like negative numbers, negative fractions and pre-algebraic problems.
Researchers used different colored blocks to represent negative and positive numbers, and asked students to find the midpoint between two sums. In your classroom, have younger students model addition and subtraction with number blocks, or ask older students to find the dimensions and volume of an everyday object using the formulas they’ve learned.
3. Construct viable arguments and critique the reasoning of others
Are your students just repeating the steps without understanding what they’re actually doing, or are they building the strong theoretical foundations they need to tackle high-school- and college-level problems? Similar to the first standard, this standard encourages critical thinking and problem-solving.
Challenging your students to look at data, solve problems, draw conclusions and debate with their classmates is a great way for them to ask new questions and develop a solid understanding of definitions and processes.
The best way to develop the third standard is through structured classroom discussion. Before you start working on the solution to a problem with your class, brainstorm some strategies: Put the simplest answers on the board first, then move on to more advanced strategies. Talk through each strategy as a group, and discuss what was right or wrong about the approach.
Some more tips for having a great classroom discussion:
- Ask students to write down answers to questions, like “what did you find difficult about this problem?” or “what did you learn during this activity?” before sharing out loud
- For a more lively class, consider implementing a “talking stick” or other object to let students know who has the floor
- Read our post for more ideas on effective classroom management
4. Model with mathematics
Different types of learners respond best to different instruction styles, and it can be difficult to respond to the personalized learning needs of each student.
However, many different types of learners respond well to seeing their textbooks brought to life. That’s exactly what high school math teacher Dan Meyer illustrates in this TED Talk:
It’s not just teachers who can show students the real-world applications of math. Reversing the processes can have a valuable effect on how students engage with problems and the world around them. Challenge students to take a problem from the page to real life using number lines, diagrams, o r classroom technology .
This is also a great time to try out project-based learning strategies , like third grade teacher Renee McFall did with her classroom.
In order to bring math to the real world, she challenged her students to raise money for a local charity by selling bracelets. Students were responsible for making and selling bracelets, calculating the amount of supplies they needed, making a budget, and pitching their best business ideas to teachers. With real-world consequences, students were encouraged to be precise in their calculations, measurements and planning, because mistakes could cost money.
5. Use appropriate tools strategically
Students today have a huge variety of tools available to them, and knowing which to use is half the battle. Depending on the problem, students could use anything from scrap paper and a pencil to more advanced technology resources. When students know how to find what they need, they develop problem-solving skills and become more comfortable looking for new solutions in the future.
One practical way to get your students familiar with the appropriate tools is to challenge them to figure out what they need themselves. At the beginning of the lesson, ask them to make a list of the tools they need and gather them up.
Some options to include:
- Pencils and paper
- Modular tools
- Worksheets with key formulas
Ask more advanced students to brainstorm a list of sources to draw research from, like books, websites or even podcasts. Afterwards, have a discussion about the tools used. Were there any differences between what students chose? What worked and what didn’t? What tools would they consider using next time?
6. Attend to precision
Precision is one of the most important skills to develop early on in math study. Even if most first graders would rather finger paint than write numbers, it builds a solid foundation for more advanced math problems . Encouraging students to use correct symbols and challenging them to accurately communicate their process to others gets them comfortable with the “language” of math.
In younger grades, students can practice precision by explaining their thinking to classmates, using either words or modular tools. As students get older, they can begin to accurately define units and equations, both in writing and speaking about math.
Have students start a math journal to practice precision and communication. Younger students can respond to prompts of “what they did” and “what they learned.” Older students can use their journal space to engage more with the topic and ask questions about concepts they don’t quite understand yet.
Write prompts on the board to get your students started:
- Write a letter to a family member explaining your process
- Do you have any more questions you want answered?
- Where did you get stuck in this problem? Why?
- What tools did you use to solve this problem?
- What will you do differently next time?
Be aware that students might need some time to adjust to writing about math. Make sure to model it to the class and provide students with lots of prompts to get them started.
7. Look for and make use of structure
Seeing repeated patterns gives students the tools to reason through new, more challenging problems.
Sean Nank , recipient of the Presidential Awards for Excellence in Mathematics and Science Teaching, defines the understanding of patterns and structure as being key for math fluency:
“ I would define fluency as being able to recognize patterns, so people can do math quickly -- which is not to say memorization is bad. It’s still something that is needed. But, you can only memorize so many math facts. If you know the patterns behind them, you can break them down really fast ."
Structure allows students to understand that complicated equations are not whole entities, but rather composed of several smaller, more accessible objects. This understanding gives them the confidence to attempt more difficult equations.
One of the best ways to develop an understanding of structure is through a daily math practice. Prodigy is an engaging, game-based math platform aligned with Common Core math curricula. It’s an exciting online resource that challenges students to answer math questions every day as they duel characters, play with their friends and collect exotic pets.
To see an even greater impact on your classroom, use the teacher tools to set assignments that help students build confidence in a particular skill.
Other great options for building a daily math practice include challenging your students to solve a daily math problem when they arrive in the classroom, or setting aside time in your lesson for students to model problems with modular tools so they can see the patterns for themselves.
8. Look for and make use of repeated reasoning
The seventh and eighth standards are closely related, but it’s important to distinguish between them. Instead of focusing on the repeated structure of an object, the eighth standard encourages students to use past problems as a model for present ones .
When students can demonstrate repeated reasoning, it means they’re able to try different solutions for the same problem and adjust as needed. Students can see which elements stay the same and which are variable by testing different methods repeatedly. This process develops both attention to detail and oversight -- controlling the small parts of the problem while making sure that overall, they’re on the right path to a solution.
A great way to promote repeated reasoning is through the use of “ fact families .” When students write an equation, challenge them to write two or three more equations that directly relate to the original, like this:
Source: Teachers Pay Teachers
As students progress and become more advanced, this provides a solid foundation for more complicated equations that include fractions, integers and algebraic elements.
Fact families encourage students to maintain a focus on the overall equation, while also manipulating the individual numbers and examining the relationships between them. Working with fact families to express repeated reasoning in early elementary gives students the skills they need for later elementary, high-school level and post-secondary math.
Tips for explaining Common Core math to parents:
All set! You’ve seamlessly integrated Common Core in your classroom, your students are working together and discussing their ideas, and things are going smoothly. But what about their parents?
Parents want their children to get the best education possible. Common Core math is quite a large shift from how they were taught as children, and some of the processes and techniques might be unfamiliar to them.
With that in mind, here are three ways to get parents on board with new Common Core math standards:
- Send home an information sheet or link parents to a web resource that explains the standards behind Common Core math and how they work in the classroom. Parents are less likely to be alarmed by dramatically different-looking homework when they ’ve been given a head’s-up and an introduction to the reasoning behind the change.
- Let parents know that their children might ask questions and struggle while they get used to the new curriculum . A recent study from Psychological Science found that when parents expressed negative feelings about math, their children were also more likely to underperform. Encourage parents to model a positive attitude and to work through difficult problems with their children.
- Consider taking a few minutes during a parent evening to go over the most important points of the new curriculum, and encourage them to keep in touch and to reach out if they have any questions. Supporting an open dialogue with parents is a great classroom practice, no matter which subject you’re teaching.
Education doesn’t happen in isolation — in fact, one of the key indicators for students’ success is how engaged their parents are with their schooling. Keep parents in the loop to avoid major frustration and confusion and ensure a positive learning environment for all your students.
Common core math standards: Final thoughts
Such a large shift in curriculum and teaching habits is bound to have a few growing pains, and certainly won’t happen overnight. However, with a little bit of time, patience and hard work, you’ll begin to see confident and engaged learners.
The biggest strength of the Common Core math standards is their versatility -- they overlap and complement each other to ensure all children are confident in their math skills.
“I see the Common Core as a way to provide teachers with strategies,” says Sean Nank, “so that students can see the beauty of math -- the how it works and the why it works and the patterns.” Encourage your students to keep looking for the how’s and the why’s, and watch them flourish.
Create or log in to your free teacher account on Prodigy – an engaging , game-based learning platform for math that’s easy to use for educators and students alike. Prodigy is aligned with curricula across the English-speaking world and filled with powerful teacher tools for differentiation and assessment.
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Problem-Solving Strategies and Obstacles
Kendra Cherry, MS, is an author and educational consultant focused on helping students learn about psychology.
Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.
JGI / Jamie Grill / Getty Images
From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.
What Is Problem-Solving?
In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.
A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.
Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.
The problem-solving process involves:
- Discovery of the problem
- Deciding to tackle the issue
- Seeking to understand the problem more fully
- Researching available options or solutions
- Taking action to resolve the issue
Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.
Problem-Solving Mental Processes
Several mental processes are at work during problem-solving. Among them are:
- Perceptually recognizing the problem
- Representing the problem in memory
- Considering relevant information that applies to the problem
- Identifying different aspects of the problem
- Labeling and describing the problem
There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.
An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.
In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.
One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.
There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.
Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.
If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.
While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.
Trial and Error
A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.
This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.
In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.
Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .
Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.
How to Apply Problem-Solving Strategies in Real Life
If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:
- Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
- Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
- Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
- Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.
Obstacles to Problem-Solving
Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:
- Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
- Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
- Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
- Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.
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How to Improve Your Problem-Solving Skills
In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:
- Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
- Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
- Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
- Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
- Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
- Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.
You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.
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Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20
Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9
Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579
Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517
Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7
Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality . Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050
Huang F, Tang S, Hu Z. Unconditional perseveration of the short-term mental set in chunk decomposition . Front Psychol . 2018;9:2568. doi:10.3389/fpsyg.2018.02568
National Alliance on Mental Illness. Warning signs and symptoms .
Mayer RE. Thinking, problem solving, cognition, 2nd ed .
Schooler JW, Ohlsson S, Brooks K. Thoughts beyond words: When language overshadows insight. J Experiment Psychol: General . 1993;122:166-183. doi:10.1037/0096-3445.2.166
By Kendra Cherry Kendra Cherry, MS, is an author and educational consultant focused on helping students learn about psychology.
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The Fundamental Problem With Common Core Math
In 2010, a bold effort to reform math curriculum was adopted by the majority of the United States. Known as “Common Core Math,” the goal of this endeavor was to establish a common foundation of mathematics education across the country, and to help bolster not only students’ mathematical abilities , but also their mathematical intuition. The goal was to help students think about math more deeply, believing that this will help them work with mathematics better in later years.
Before discussing problems with this approach, I want to say that I appreciate the idea of helping students think more deeply about mathematics. After years and years and years of mathematics education, many students wind up thinking about mathematics as merely a set of formulas that they had to memorize to pass a class, which they will never ever use again in their lives.
In 2005, this was brought to the public’s attention when Jeb Bush was asked by a high school student about trigonometry, and Bush essentially replied that he hasn’t been in high school for a while. While the particular question was ill-posed, this illustrated concerns that students are being made subject to high-stakes tests on questions that they literally don’t need to know.
In other words, students are frustrated because they are being asked to learn formulas and equations for things that have no connection to their present or future, whose sole purpose seems to be jumping through an arbitrary hoop set up to make them fail.
What Common Core is supposed to bring to the table is a deeper understanding of mathematics, so that students recognize how mathematical thinking is part of thinking in general. While this is a worthwhile goal, common core radically misfires on several accounts.
First of all, Common Core tries to teach the concepts first, and to incorrectly-aged students.
Younger students love memorizing and systems. That is what their brains are geared for. They want to learn how to do things. It isn’t that “why” questions aren’t appropriate (in my opinion, it is never too early to start talking about “why”), but the fact is that the “why” questions are not the most important thing, and it isn’t what they are best at learning.
This is why, historically, we taught students straightforward systems for doing mathematics calculation. We taught processes which, once learned, could be applied to any set of numbers. Crucial to this teaching methodology are (a) quick recall of math facts, and (b) a straightforward process that anyone can do. This gives students the skills they need to do problems and to recognize that the size of the problem doesn’t really matter as long as you have the process.
What Common Core advocates don’t like is that (a) the process prevents students from really thinking about what is going on with numbers, and (b) some claim that the high-pressure timed math facts tests cause lifelong student anxiety about mathematics. We will discuss (b) in a later article. In this article, I want to focus on (a).
It is true that younger students learning long forms of addition, subtraction, multiplication, and division don’t spend a lot of time thinking about why the process works. However, the stage of learning that students are at in this time is one where memorizing and learning processes come the easiest. I do think that students should eventually learn why the process works. However, usually students do well to learn a process, and then get curious why it works. This allows them early competence, and a foundation for later reflection.
The fact is, reflecting on why things work is a product of age. Disasters in math education have always come from people interchanging the needs of adults with the needs of children. In the 1950s, a similar attempt was done with “New Math.” It attempted to teach mathematics using set theory, which is the foundation stone most modern mathematicians use. However, while you can build mathematics from set theory, you almost always learn set theory as a reflection on the mathematics you have already done. While Common Core is not as radical as the New Math, both stem from the same basic flaw — they prioritize the thoughts and tendencies of adults over those of the students they are teaching. It is easier to ask why questions when you already understand the process. It is harder to even understand the question being asked when you haven’t learned any process at all.
A second problem with Common Core math is more social. Students often need more help with mathematics than is available in the classroom. Especially in schools with large class sizes, parents wind up being the default tutor when a student doesn’t understand a concept. Any radical shift in methodology, then, immediately strips parents of their ability to act in this capacity.
Educators often complain about the lack of engagement of parents. However, it is difficult to take this seriously when the education establishment goes out of its way to rewrite curriculum in a way that bears no connection to how parents understand the curriculum. If educators want parent engagement, they must consider the ways that their curriculum impacts the ability of parents to engage. If educators don’t want parent engagement, they should be happy when parents don’t engage. Instead, many educators want to have their cake and eat it too — write curriculum which parents don’t have any connection to, and then complain that parents aren’t involved.
Let me say that there is nothing ultimately wrong with the specific things that Common Core wants students to learn. Having more mathematical intuition and recognizing multiple ways to solve a problem are both very good things. However, intuition often develops after repeated exposure to concrete problems, not before. And, before recognizing that there are multiple ways to do something, it helps to learn one of them well.
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The Problems with the Common Core
By Stan Karp
Illustrator: Michael Duffy
The trouble with the Common Core is not primarily what is in these standards or what’s been left out, although that’s certainly at issue. The bigger problem is the role the Common Core State Standards (CCSS) are playing in the larger dynamics of current school reform and education politics.
Today everything about the Common Core, even the brand name the Common Core State Standards is contested because these standards were created as an instrument of contested policy. They have become part of a larger political project to remake public education in ways that go well beyond slogans about making sure every student graduates “college and career ready,” however that may be defined this year. We’re talking about implementing new national standards and tests for every school and district in the country in the wake of dramatic changes in the national and state context for education reform. These changes include:
- A 10-year experiment in the use of federally mandated standards and tests called No Child Left Behind (NCLB) that has been almost universally acknowledged as a failure.
- The adoption of test-based teacher evaluation frameworks in dozens of states, largely as a result of federal mandates.
- Multiple rounds of budget cuts and layoffs that have left 34 of the 50 states providing less funding for education than they did five years ago, and the elimination of more than 300,000 teaching positions.
- A wave of privatization that has increased the number of publicly funded but privately run charter schools by 50 percent, while nearly 4,000 public schools have been closed in the same period.
- An appalling increase in the inequality and child poverty surrounding our schools, categories in which the United States leads the world and that tell us far more about the source of our educational problems than the uneven quality of state curriculum standards.
- A dramatic increase in the cost and debt burden of college access.
- A massively well-financed campaign of billionaires and politically powerful advocacy organizations that seeks to replace our current system of public education which, for all its many flaws, is probably the most democratic institution we have and one that has done far more to address inequality, offer hope, and provide opportunity than the country’s financial, economic, political, and media institutions with a market-based, non-unionized, privately managed system.
I think many supporters of the Common Core don’t sufficiently take into account how these larger forces define the context in which the standards are being introduced, and how much that context is shaping implementation. As teacher-blogger Jose Vilson put it:
People who advocate for the CCSS miss the bigger picture that people on the ground don’t: The CCSS came as a package deal with the new teacher evaluations, higher stakes testing, and austerity measures, including mass school closings. Often, it seems like the leaders are talking out of both sides of their mouths when they say they want to improve education but need to defund our schools. . . . It makes no sense for us to have high expectations of our students when we don’t have high expectations for our school system.
My own first experience with standards-based reform was in New Jersey, where I taught English and journalism to high school students for many years in one of the state’s poorest cities. In the 1990s, curriculum standards became a central issue in the state’s long-running funding equity case, Abbott v. Burke . The case began by documenting how lower levels of resources in poor urban districts produced unequal educational opportunities in the form of worse facilities, poorer curriculum materials, less experienced teachers, and fewer support services. At a key point in the case, in an early example of arguments that today are painfully familiar, then-Gov. Christine Whitman declared that, instead of funding equity, what we really needed were curriculum standards and a shift from focusing on dollars to focusing on what those dollars should be spent on. If all students were taught to meet “core content curriculum standards,” Whitman argued, then everyone would receive an equitable and adequate education.
At the time, the New Jersey Supreme Court was an unusually progressive and foresighted court, and it responded to the state’s proposal for standards with a series of landmark decisions that speak to some of the same issues raised today by the Common Core. The court agreed that standards for what schools should teach and students should learn seemed like a good idea. But standards don’t deliver themselves. They require well-prepared and supported professional staff, improved instructional resources, safe and well-equipped facilities, reasonable class sizes, and especially if they are supposed to help schools compensate for the inequality that exists all around them a host of supplemental services like high quality preschools, expanded summer and after-school programs, health and social services, and more. In effect, the court said adopting “high expectations” curriculum standards was like passing out a menu from a fine restaurant. Not everyone who gets a menu can pay for the meal. So the court tied New Jersey’s core curriculum standards to the most equitable school funding mandates in the country.
And though it’s been a constant struggle to sustain and implement New Jersey’s funding equity mandates, a central problem with the Common Core is the complete absence of any similar credible plan to provide or even to determine the resources necessary to make every student “college and career ready” as defined by the CCSS.
Funding is far from the only concern, but it is a threshold credibility issue. If you’re proposing a dramatic increase in outcomes and performance to reach social and academic goals that have never been reached before, and your primary investments are standards and tests that serve mostly to document how far you are from reaching those goals, you either don’t have a very good plan or you’re planning something else. The Common Core, like NCLB before it, is failing the funding credibility test before it’s even out of the gate.
The Lure of the Common Core
Last winter, the Rethinking Schools editorial board held a discussion about the Common Core; we were trying to decide how to address this latest trend in the all-too-trendy world of education reform. Rethinking Schools has always been skeptical of standards imposed from above. Too many standards projects have been efforts to move decisions about teaching and learning away from educators and schools, and put them in the hands of distant bureaucracies and politicians. Standards have often codified sanitized versions of history, politics, and culture that reinforce official myths while leaving out the voices and concerns of our students and communities. Whatever potentially positive role standards might play in truly collaborative conversations about what schools should teach and children should learn has repeatedly been undermined by bad process, suspect political agendas, and commercial interests.
Although all these concerns were raised, we also found that teachers in different districts and states were having very different experiences with the Common Core. There were teachers in Milwaukee who had endured years of scripted curriculum and mandated textbooks. For them, the CCSS seemed like an opening to develop better curriculum and, compared to what they’d been struggling under, seemed more flexible and student-centered. For many teachers, especially in the interim between the rollout of the standards and the arrival of the tests a lot of the Common Core’s appeal is based on claims that:
- It represents a tighter set of smarter standards focused on developing critical learning skills instead of mastering fragmented bits of knowledge.
- It requires more progressive, student-centered teaching with strong elements of collaborative and reflective learning.
- It will help equalize the playing field by raising expectations for all children, especially those suffering the worst effects of “drill and kill” test prep.
Viewed in isolation, the debate over the Common Core can be confusing; who doesn’t want all students to have good preparation for life after high school? But, seen in the full context of the politics and history that produced it and the tests that are just around the bend the implications of the Common Core project look quite different.
Emerging from the Wreckage of No Child Left Behind
The CCSS emerged from the wreckage of NCLB. In 2002, NCLB was passed with overwhelming bipartisan support and presented as a way to close long-standing gaps in academic performance. NCLB marked a dramatic change in federal education policy away from its historic role as a promoter of access and equity through support for things like school integration, extra funding for high-poverty schools, and services for students with special needs, to a much less equitable set of mandates around standards and testing, closing or “reconstituting” schools, and replacing school staff.
NCLB required states to adopt curriculum standards and to test students annually to gauge progress toward reaching them. Under threat of losing federal funds, all 50 states adopted or revised their standards and began testing every student, every year, in every grade from 3_8 and again in high school. The professed goal was to make sure every student was on grade level in math and language arts by requiring schools to reach 100 percent passing rates on state tests for every student in 10 subgroups.
By any measure, NCLB was a failure in raising academic performance and narrowing gaps in opportunity and outcomes. But by very publicly measuring the test results against arbitrary benchmarks that no real schools have ever met, NCLB succeeded in creating a narrative of failure that shaped a decade of attempts to “fix” schools while blaming those who work in them. The disaggregated scores put the spotlight on gaps among student groups, but the law used these gaps to label schools as failures without providing the resources or supports needed to eliminate them.
By the time the first decade of NCLB was over, more than half the schools in the nation were on the lists of “failing schools” and the rest were poised to follow. In Massachusetts, which is generally considered to have the toughest state standards in the nation arguably more demanding than the Common Core 80 percent of the schools were facing NCLB sanctions. This is when the NCLB “waivers” appeared. As the number of schools facing sanctions and intervention grew well beyond the poor communities of color where NCLB had made “disruptive reform” the norm and began to reach into more middle-class and suburban districts, the pressure to revise NCLB’s unworkable accountability system increased. But the bipartisan coalition that passed NCLB had collapsed and gridlock in Congress made revising it impossible. So U.S. Education Secretary Arne Duncan, with dubious legal justification, made up a process to grant NCLB waivers to states that agreed to certain conditions.
Forty states were granted conditional waivers from NCLB: If they agreed to tighten the screws on the most struggling schools serving the highest needs students, they could ease up on the rest, provided they also agreed to use test scores to evaluate all their teachers, expand the reach of charter schools, and adopt “college and career ready” curriculum standards. These same requirements were part of the Race to the Top program, which turned federal education funds into competitive grants and promoted the same policies, even though they have no track record of success as school improvement strategies.
Who Created the Common Core?
Because federal law prohibits the federal government from creating national standards and tests, the Common Core project was ostensibly designed as a state effort led by the National Governors Association, the Council of Chief State School Officers, and Achieve, a private consulting firm. The Gates Foundation provided more than $160 million in funding, without which Common Core would not exist.
The standards were drafted largely behind closed doors by academics and assessment “experts,” many with ties to testing companies. Education Week blogger and science teacher Anthony Cody found that, of the 25 individuals in the work groups charged with drafting the standards, six were associated with the test makers from the College Board, five with the test publishers at ACT, and four with Achieve. Zero teachers were in the work groups. The feedback groups had 35 participants, almost all of whom were university professors. Cody found one classroom teacher involved in the entire process. According to teacher educator Nancy Carlsson-Paige: “In all, there were 135 people on the review panels for the Common Core. Not a single one of them was a K_3 classroom teacher or early childhood professional.” Parents were entirely missing. K_12 educators were mostly brought in after the fact to tweak and endorse the standards and lend legitimacy to the results.
College- and Career-Ready Standards?
The substance of the standards themselves is also, in a sense, top down. To arrive at “college- and career-ready standards,” the Common Core developers began by defining the “skills and abilities” they claim are needed to succeed in a four-year college. The CCSS tests being developed by two federally funded multistate consortia, at a cost of about $350 million, are designed to assess these skills. One of these consortia, the Partnership for Assessment of Readiness for College and Careers, claims that students who earn a “college ready” designation by scoring a level 4 on these still-under-construction tests will have a 75 percent chance of getting a C or better in their freshman composition course. But there is no actual evidence connecting scores on any of these new experimental tests with future college success.
And it will take far more than standards and tests to make college affordable, accessible, and attainable for all. When I went to college many years ago, “college for all” meant open admissions, free tuition, and race, class, and gender studies. Today, it means cutthroat competition to get in, mountains of debt to stay, and often bleak prospects when you leave. Yet “college readiness” is about to become the new AYP (adequate yearly progress) by which schools will be ranked.
The idea that by next year Common Core tests will start labeling kids in the 3rd grade as on track or not for college is absurd and offensive.
Substantive questions have been raised about the Common Core’s tendency to push difficult academic skills to lower grades, about the appropriateness of the early childhood standards, about the sequencing of the math standards, about the mix and type of mandated readings, and about the priority Common Core puts on the close reading of texts in ways that devalue student experience and prior knowledge.
A decade of NCLB tests showed that millions of students were not meeting existing standards, but the sponsors of the Common Core decided that the solution was tougher ones. And this time, instead of each state developing its own standards, the Common Core seeks to create national tests that are comparable across states and districts, and that can produce results that can be plugged into the data-driven crisis machine that is the engine of corporate reform.
Educational Plan or Marketing Campaign?
The way the standards are being rushed into classrooms across the country is further undercutting their credibility. These standards have never been fully implemented in real schools anywhere. They’re more or less abstract descriptions of academic abilities organized into sequences by people who have never taught at all or who have not taught this particular set of standards. To have any impact, the standards must be translated into curriculum, instructional plans, classroom materials, and valid assessments. A reasonable approach to implementing new standards would include a few multi-year pilot programs that provided time, resources, opportunities for collaboration, and transparent evaluation plans.
Instead we’re getting an overhyped all-state implementation drive that seems more like a marketing campaign than an educational plan. And I use the word marketing advisedly, because another defining characteristic of the Common Core project is rampant profiteering.
Joanne Weiss, Duncan’s former chief of staff and head of the Race to the Top grant program, which effectively made adoption of the Common Core a condition for federal grants, described how it is opening up huge new markets for commercial exploitation:
The development of common standards and shared assessments radically alters the market for innovation in curriculum development, professional development, and formative assessments. Previously, these markets operated on a state-by-state basis, and often on a district-by-district basis. But the adoption of common standards and shared assessments means that education entrepreneurs will enjoy national markets where the best products can be taken to scale.
Who Controls Public Education?
Having financed the creation of the standards, the Gates Foundation has entered into a partnership with Pearson to produce a full set of K_12 courses aligned with the Common Core that will be marketed to schools across the country. Nearly every educational product now comes wrapped in the Common Core brand name.
The curriculum and assessments our schools and students need will not emerge from this process. Instead, the top-down, bureaucratic rollout of the Common Core has put schools in the middle of a multilayered political struggle over who will control education policy corporate power and private wealth or public institutions managed, however imperfectly, by citizens in a democratic process.
The web-based news service Politico recently described what it called “the Common Core money war,” reporting that “tens of millions of dollars are pouring into the battle over the Common Core. . . . The Bill & Melinda Gates Foundation already has pumped more than $160 million into developing and promoting the Common Core, including $10 million just in the past few months, and it’s getting set to announce up to $4 million in new grants to keep the advocacy cranking. Corporate sponsors are pitching in, too. Dozens of the nation’s top CEOs will meet to set the plans for a national advertising blitz that may include TV, radio, and print.”
At the same time, opposing the Common Core is “an array of organizations with multimillion-dollar budgets of their own and much experience in mobilizing crowds and lobbying lawmakers, including the Heritage Foundation, Americans for Prosperity, the Pioneer Institute, FreedomWorks, and the Koch Bros.” These groups are feeding a growing right-wing opposition to the Common Core that combines hostility to all federal education initiatives and anything supported by the Obama administration with more populist sentiments.
Tests, Tests, Tests
But while this larger political battle rages, the most immediate threat for educators and schools remains the new wave of high-stakes Common Core tests.
Duncan, who once said “The best thing that happened to the education system in New Orleans was Hurricane Katrina” and who called Waiting for Superman “a Rosa Parks moment,” now tells us, “I am convinced that this new generation of state assessments will be an absolute game-changer in public education.”
The problem is that this game, like the last one, is rigged. Although reasonable people have found things of value in the Common Core standards, there is no credible defense to be made of the high-stakes uses planned for these new tests. Instead, the Common Core project threatens to reproduce the narrative of public school failure that just led to a decade of bad policy in the name of reform.
Reports from the first wave of Common Core testing provide evidence for these fears. Last spring, students, parents, and teachers in New York schools responded to new Common Core tests developed by Pearson with outcries against their length, difficulty, and inappropriate content. Pearson included corporate logos and promotional material in reading passages. Students reported feeling overstressed and underprepared meeting the tests with shock, anger, tears, and anxiety. Administrators requested guidelines for handling tests students had vomited on. Teachers and principals complained about the disruptive nature of the testing process and many parents encouraged their children to opt out.
Only about 30 percent of students were deemed “proficient” based on arbitrary cut scores designed to create new categories of failure. The achievement gaps Common Core is supposed to narrow grew larger. Less than 4 percent of students who are English language learners passed. The number of students identified by the tests for “academic intervention” skyrocketed to 70 percent, far beyond the capacity of districts to meet.
The tests are on track to squeeze out whatever positive potential exists in the Common Core:
- The arrival of the tests will pre-empt the already too short period teachers and schools have to review the standards and develop appropriate curriculum responses before that space is filled by the assessments themselves.
- Instead of reversing the mania for over-testing, the new assessments will extend it with pre-tests, interim tests, post-tests, and computer-based “performance assessments.” It’s the difference between giving a patient a blood test and draining the patient’s blood.
- The scores will be plugged into data systems that will generate value-added measures, student growth percentiles, and other imaginary numbers for what I call psychometric astrology. The inaccurate and unreliable practice of using test scores for teacher evaluation will distort the assessments before they’re even in place, and has the potential to make Common Core implementation part of the assault on the teaching profession instead of a renewal of it.
- If the Common Core’s college- and career-ready performance levels become the standard for high school graduation, it will push more kids out of high school than it will prepare for college. The most vulnerable students will be the most at risk. As FairTest put it: “If a child struggles to clear the high bar at 5 feet, she will not become a ‘world-class’ jumper because someone raised the bar to 6 feet and yelled ‘jump higher,’ or if her ‘poor’ performance is used to punish her coach.”
- The costs of the tests, which have multiple pieces throughout the year and must be given on computers many schools don’t have, will be enormous and will come at the expense of more important things. The plunging scores will be used as an excuse to close more public schools and open more privatized charters and voucher schools, especially in poor communities of color.
This is not just cynical speculation. It is a reasonable projection based on the history of the NCLB decade, the dismantling of public education in the nation’s urban centers, and the appalling growth of the inequality and concentrated poverty that remains the central problem in public education.
Common Core has become part of the corporate reform project now stalking our schools. As schools struggle with these new mandates, we should defend our students, our schools, and ourselves by pushing back against implementation timelines, resisting the stakes and priority attached to the tests, and exposing the truth about the commercial and political interests shaping this false panacea for the problems our schools face.
There are encouraging signs that the movement we need is growing. Last year in Seattle, teachers led a boycott of district testing that drew national support and won a partial rollback of the testing. In New York this fall, parents sent score reports on new Common Core tests back to the state commissioner of education with a letter declaring “This year’s test scores are invalid and provide NO useful information about student learning.” Opt-out efforts are growing daily. Even some supporters of the CCSS have endorsed a call for the moratorium on the use of tests to make policy decisions. It’s not enough, but it’s a start.
It took nearly a decade for NCLB’s counterfeit “accountability system” to bog down in the face of its many contradictions and near universal rejection. The Common Core meltdown may not take that long. Many of Common Core’s myths and claims have already lost credibility with large numbers of educators and citizens. We have more than a decade of experience with the negative and unpopular results of imposing increasing numbers of standardized tests on children and classrooms. Whether this growing resistance will lead to better, more democratic efforts to sustain and improve public education, or be overwhelmed by the massive testing apparatus that NCLB left behind and that the Common Core seeks to expand, will depend on the organizing and advocacy efforts of those with the most at stake: parents, educators, and students. As usual, organizing and activism are the only things that will save us, and remain our best hope for the future of public education and the democracy that depends on it.
The administrators stuffed in suits strut through our school clenching clipboards, nod plastic smiles Speak words like “common core,” like “standards” & “benchmarks”.
But those of us who live in these rooms, who know each other’s stories & share apples and granola bars because there was no food in the house after dad was arrested – We nod & smile back – Our secret knowing: Core is community Core is complex Core is connection.
After bullshit banter, The suits slip out, sip bad coffee, fill out rubrics on clipboards.
We close classroom doors, Proceed to spin magic uncommonly connected at the core.
– Maureen Geraghty
Maureen Geraghty teaches at Reynolds Learning Academy in Fairview, Oregon. She wrote this poem during a visit to her class by slam poet Mosley Wotta.
“Stan Karp ([email protected]) is a Rethinking Schools editor.
Illustrator Michael Duffy’s work can be found at duff-co.com.”
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5 Effective Problem-Solving Strategies
Got a problem you’re trying to solve? Strategies like trial and error, gut instincts, and “working backward” can help. We look at some examples and how to use them.
We all face problems daily. Some are simple, like deciding what to eat for dinner. Others are more complex, like resolving a conflict with a loved one or figuring out how to overcome barriers to your goals.
No matter what problem you’re facing, these five problem-solving strategies can help you develop an effective solution.
What are problem-solving strategies?
To effectively solve a problem, you need a problem-solving strategy .
If you’ve had to make a hard decision before then you know that simply ruminating on the problem isn’t likely to get you anywhere. You need an effective strategy — or a plan of action — to find a solution.
In general, effective problem-solving strategies include the following steps:
- Define the problem.
- Come up with alternative solutions.
- Decide on a solution.
- Implement the solution.
Problem-solving strategies don’t guarantee a solution, but they do help guide you through the process of finding a resolution.
Using problem-solving strategies also has other benefits . For example, having a strategy you can turn to can help you overcome anxiety and distress when you’re first faced with a problem or difficult decision.
The key is to find a problem-solving strategy that works for your specific situation, as well as your personality. One strategy may work well for one type of problem but not another. In addition, some people may prefer certain strategies over others; for example, creative people may prefer to depend on their insights than use algorithms.
It’s important to be equipped with several problem-solving strategies so you use the one that’s most effective for your current situation.
1. Trial and error
One of the most common problem-solving strategies is trial and error. In other words, you try different solutions until you find one that works.
For example, say the problem is that your Wi-Fi isn’t working. You might try different things until it starts working again, like restarting your modem or your devices until you find or resolve the problem. When one solution isn’t successful, you try another until you find what works.
Trial and error can also work for interpersonal problems . For example, if your child always stays up past their bedtime, you might try different solutions — a visual clock to remind them of the time, a reward system, or gentle punishments — to find a solution that works.
Sometimes, it’s more effective to solve a problem based on a formula than to try different solutions blindly.
Heuristics are problem-solving strategies or frameworks people use to quickly find an approximate solution. It may not be the optimal solution, but it’s faster than finding the perfect resolution, and it’s “good enough.”
Algorithms or equations are examples of heuristics.
An algorithm is a step-by-step problem-solving strategy based on a formula guaranteed to give you positive results. For example, you might use an algorithm to determine how much food is needed to feed people at a large party.
However, many life problems have no formulaic solution; for example, you may not be able to come up with an algorithm to solve the problem of making amends with your spouse after a fight.
3. Gut instincts (insight problem-solving)
While algorithm-based problem-solving is formulaic, insight problem-solving is the opposite.
When we use insight as a problem-solving strategy we depend on our “gut instincts” or what we know and feel about a situation to come up with a solution. People might describe insight-based solutions to problems as an “aha moment.”
For example, you might face the problem of whether or not to stay in a relationship. The solution to this problem may come as a sudden insight that you need to leave. In insight problem-solving, the cognitive processes that help you solve a problem happen outside your conscious awareness.
4. Working backward
Working backward is a problem-solving approach often taught to help students solve problems in mathematics. However, it’s useful for real-world problems as well.
Working backward is when you start with the solution and “work backward” to figure out how you got to the solution. For example, if you know you need to be at a party by 8 p.m., you might work backward to problem-solve when you must leave the house, when you need to start getting ready, and so on.
5. Means-end analysis
Means-end analysis is a problem-solving strategy that, to put it simply, helps you get from “point A” to “point B” by examining and coming up with solutions to obstacles.
When using means-end analysis you define the current state or situation (where you are now) and the intended goal. Then, you come up with solutions to get from where you are now to where you need to be.
For example, a student might be faced with the problem of how to successfully get through finals season . They haven’t started studying, but their end goal is to pass all of their finals. Using means-end analysis, the student can examine the obstacles that stand between their current state and their end goal (passing their finals).
They could see, for example, that one obstacle is that they get distracted from studying by their friends. They could devise a solution to this obstacle by putting their phone on “do not disturb” mode while studying.
Whether they’re simple or complex, we’re faced with problems every day. To successfully solve these problems we need an effective strategy. There are many different problem-solving strategies to choose from.
Although problem-solving strategies don’t guarantee a solution, they can help you feel less anxious about problems and make it more likely that you come up with an answer.
Last medically reviewed on October 31, 2022
8 sources collapsed
- Chu Y, et al. (2011). Human performance on insight problem-solving: A review. docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1094&context=jps
- Dumper K, et al. (n.d.) Chapter 7.3: Problem-solving in introductory psychology. opentext.wsu.edu/psych105/chapter/7-4-problem-solving/
- Foulds LR. (2017). The heuristic problem-solving approach. tandfonline.com/doi/abs/10.1057/jors.1983.205
- Gick ML. (1986). Problem-solving strategies. tandfonline.com/doi/abs/10.1080/00461520.1986.9653026
- Montgomery ME. (2015). Problem solving using means-end analysis. sites.psu.edu/psych256sp15/2015/04/19/problem-solving-using-means-end-analysis/
- Posamentier A, et al. (2015). Problem-solving strategies in mathematics. Chapter 3: Working backwards. worldscientific.com/doi/10.1142/9789814651646_0003
- Sarathy V. (2018). Real world problem-solving. frontiersin.org/articles/10.3389/fnhum.2018.00261/full
- Woods D. (2000). An evidence-based strategy for problem solving. researchgate.net/publication/245332888_An_Evidence-Based_Strategy_for_Problem_Solving
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Turn your team into skilled problem solvers with these problem-solving strategies
Picture this, you're handling your daily tasks at work and your boss calls you in and says, "We have a problem."
Unfortunately, we don't live in a world in which problems are instantly resolved with the snap of our fingers. Knowing how to effectively solve problems is an important professional skill to hone. If you have a problem that needs to be solved, what is the right process to use to ensure you get the most effective solution?
In this article we'll break down the problem-solving process and how you can find the most effective solutions for complex problems.
What is problem solving?
Problem solving is the process of finding a resolution for a specific issue or conflict. There are many possible solutions for solving a problem, which is why it's important to go through a problem-solving process to find the best solution. You could use a flathead screwdriver to unscrew a Phillips head screw, but there is a better tool for the situation. Utilizing common problem-solving techniques helps you find the best solution to fit the needs of the specific situation, much like using the right tools.
4 steps to better problem solving
While it might be tempting to dive into a problem head first, take the time to move step by step. Here’s how you can effectively break down the problem-solving process with your team:
1. Identify the problem that needs to be solved
One of the easiest ways to identify a problem is to ask questions. A good place to start is to ask journalistic questions, like:
Who : Who is involved with this problem? Who caused the problem? Who is most affected by this issue?
What: What is happening? What is the extent of the issue? What does this problem prevent from moving forward?
Where: Where did this problem take place? Does this problem affect anything else in the immediate area?
When: When did this problem happen? When does this problem take effect? Is this an urgent issue that needs to be solved within a certain timeframe?
Why: Why is it happening? Why does it impact workflows?
How: How did this problem occur? How is it affecting workflows and team members from being productive?
Asking journalistic questions can help you define a strong problem statement so you can highlight the current situation objectively, and create a plan around that situation.
Here’s an example of how a design team uses journalistic questions to identify their problem:
Overarching problem: Design requests are being missed
Who: Design team, digital marketing team, web development team
What: Design requests are forgotten, lost, or being created ad hoc.
Where: Email requests, design request spreadsheet
When: Missed requests on January 20th, January 31st, February 4th, February 6th
How : Email request was lost in inbox and the intake spreadsheet was not updated correctly. The digital marketing team had to delay launching ads for a few days while design requests were bottlenecked. Designers had to work extra hours to ensure all requests were completed.
In this example, there are many different aspects of this problem that can be solved. Using journalistic questions can help you identify different issues and who you should involve in the process.
2. Brainstorm multiple solutions
If at all possible, bring in a facilitator who doesn't have a major stake in the solution. Bringing an individual who has little-to-no stake in the matter can help keep your team on track and encourage good problem-solving skills.
Here are a few brainstorming techniques to encourage creative thinking:
Brainstorm alone before hand: Before you come together as a group, provide some context to your team on what exactly the issue is that you're brainstorming. This will give time for you and your teammates to have some ideas ready by the time you meet.
Say yes to everything (at first): When you first start brainstorming, don't say no to any ideas just yet—try to get as many ideas down as possible. Having as many ideas as possible ensures that you’ll get a variety of solutions. Save the trimming for the next step of the strategy.
Talk to team members one-on-one: Some people may be less comfortable sharing their ideas in a group setting. Discuss the issue with team members individually and encourage them to share their opinions without restrictions—you might find some more detailed insights than originally anticipated.
Break out of your routine: If you're used to brainstorming in a conference room or over Zoom calls, do something a little different! Take your brainstorming meeting to a coffee shop or have your Zoom call while you're taking a walk. Getting out of your routine can force your brain out of its usual rut and increase critical thinking.
3. Define the solution
After you brainstorm with team members to get their unique perspectives on a scenario, it's time to look at the different strategies and decide which option is the best solution for the problem at hand. When defining the solution, consider these main two questions: What is the desired outcome of this solution and who stands to benefit from this solution?
Set a deadline for when this decision needs to be made and update stakeholders accordingly. Sometimes there's too many people who need to make a decision. Use your best judgement based on the limitations provided to do great things fast.
4. Implement the solution
To implement your solution, start by working with the individuals who are as closest to the problem. This can help those most affected by the problem get unblocked. Then move farther out to those who are less affected, and so on and so forth. Some solutions are simple enough that you don’t need to work through multiple teams.
After you prioritize implementation with the right teams, assign out the ongoing work that needs to be completed by the rest of the team. This can prevent people from becoming overburdened during the implementation plan . Once your solution is in place, schedule check-ins to see how the solution is working and course-correct if necessary.
Implement common problem-solving strategies
There are a few ways to go about identifying problems (and solutions). Here are some strategies you can try, as well as common ways to apply them:
Trial and error
Trial and error problem solving doesn't usually require a whole team of people to solve. To use trial and error problem solving, identify the cause of the problem, and then rapidly test possible solutions to see if anything changes.
This problem-solving method is often used in tech support teams through troubleshooting.
The 5 whys problem-solving method helps get to the root cause of an issue. You start by asking once, “Why did this issue happen?” After answering the first why, ask again, “Why did that happen?” You'll do this five times until you can attribute the problem to a root cause.
This technique can help you dig in and find the human error that caused something to go wrong. More importantly, it also helps you and your team develop an actionable plan so that you can prevent the issue from happening again.
Here’s an example:
Problem: The email marketing campaign was accidentally sent to the wrong audience.
“Why did this happen?” Because the audience name was not updated in our email platform.
“Why were the audience names not changed?” Because the audience segment was not renamed after editing.
“Why was the audience segment not renamed?” Because everybody has an individual way of creating an audience segment.
“Why does everybody have an individual way of creating an audience segment?” Because there is no standardized process for creating audience segments.
“Why is there no standardized process for creating audience segments?” Because the team hasn't decided on a way to standardize the process as the team introduced new members.
In this example, we can see a few areas that could be optimized to prevent this mistake from happening again. When working through these questions, make sure that everyone who was involved in the situation is present so that you can co-create next steps to avoid the same problem.
A SWOT analysis
A SWOT analysis can help you highlight the strengths and weaknesses of a specific solution. SWOT stands for:
Strength: Why is this specific solution a good fit for this problem?
Weaknesses: What are the weak points of this solution? Is there anything that you can do to strengthen those weaknesses?
Opportunities: What other benefits could arise from implementing this solution?
Threats: Is there anything about this decision that can detrimentally impact your team?
As you identify specific solutions, you can highlight the different strengths, weaknesses, opportunities, and threats of each solution.
This particular problem-solving strategy is good to use when you're narrowing down the answers and need to compare and contrast the differences between different solutions.
Even more successful problem solving
After you’ve worked through a tough problem, don't forget to celebrate how far you've come. Not only is this important for your team of problem solvers to see their work in action, but this can also help you become a more efficient, effective , and flexible team. The more problems you tackle together, the more you’ll achieve.
Looking for a tool to help solve problems on your team? Track project implementation with a work management tool like Asana .
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- About Problem Solving
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- Problem Solving
What is Problem Solving?.
Quality Glossary Definition: Problem solving
Problem solving is the act of defining a problem; determining the cause of the problem; identifying, prioritizing, and selecting alternatives for a solution; and implementing a solution.
- The problem-solving process
- Problem solving resources
Problem Solving Chart
The Problem-Solving Process
In order to effectively manage and run a successful organization, leadership must guide their employees and develop problem-solving techniques. Finding a suitable solution for issues can be accomplished by following the basic four-step problem-solving process and methodology outlined below.
1. Define the problem
Diagnose the situation so that your focus is on the problem, not just its symptoms. Helpful problem-solving techniques include using flowcharts to identify the expected steps of a process and cause-and-effect diagrams to define and analyze root causes .
The sections below help explain key problem-solving steps. These steps support the involvement of interested parties, the use of factual information, comparison of expectations to reality, and a focus on root causes of a problem. You should begin by:
- Reviewing and documenting how processes currently work (i.e., who does what, with what information, using what tools, communicating with what organizations and individuals, in what time frame, using what format).
- Evaluating the possible impact of new tools and revised policies in the development of your "what should be" model.
2. Generate alternative solutions
Postpone the selection of one solution until several problem-solving alternatives have been proposed. Considering multiple alternatives can significantly enhance the value of your ideal solution. Once you have decided on the "what should be" model, this target standard becomes the basis for developing a road map for investigating alternatives. Brainstorming and team problem-solving techniques are both useful tools in this stage of problem solving.
Many alternative solutions to the problem should be generated before final evaluation. A common mistake in problem solving is that alternatives are evaluated as they are proposed, so the first acceptable solution is chosen, even if it’s not the best fit. If we focus on trying to get the results we want, we miss the potential for learning something new that will allow for real improvement in the problem-solving process.
3. Evaluate and select an alternative
Skilled problem solvers use a series of considerations when selecting the best alternative. They consider the extent to which:
- A particular alternative will solve the problem without causing other unanticipated problems.
- All the individuals involved will accept the alternative.
- Implementation of the alternative is likely.
- The alternative fits within the organizational constraints.
4. Implement and follow up on the solution
Leaders may be called upon to direct others to implement the solution, "sell" the solution, or facilitate the implementation with the help of others. Involving others in the implementation is an effective way to gain buy-in and support and minimize resistance to subsequent changes.
Regardless of how the solution is rolled out, feedback channels should be built into the implementation. This allows for continuous monitoring and testing of actual events against expectations. Problem solving, and the techniques used to gain clarity, are most effective if the solution remains in place and is updated to respond to future changes.
You can also search articles , case studies , and publications for problem solving resources.
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Introduction To 8D Problem Solving: Including Practical Applications and Examples
The Quality Toolbox
Root Cause Analysis: The Core of Problem Solving and Corrective Action
One Good Idea: Some Sage Advice ( Quality Progress ) The person with the problem just wants it to go away quickly, and the problem-solvers also want to resolve it in as little time as possible because they have other responsibilities. Whatever the urgency, effective problem-solvers have the self-discipline to develop a complete description of the problem.
Diagnostic Quality Problem Solving: A Conceptual Framework And Six Strategies ( Quality Management Journal ) This paper contributes a conceptual framework for the generic process of diagnosis in quality problem solving by identifying its activities and how they are related.
Weathering The Storm ( Quality Progress ) Even in the most contentious circumstances, this approach describes how to sustain customer-supplier relationships during high-stakes problem solving situations to actually enhance customer-supplier relationships.
The Right Questions ( Quality Progress ) All problem solving begins with a problem description. Make the most of problem solving by asking effective questions.
Solving the Problem ( Quality Progress ) Brush up on your problem-solving skills and address the primary issues with these seven methods.
Refreshing Louisville Metro’s Problem-Solving System ( Journal for Quality and Participation ) Organization-wide transformation can be tricky, especially when it comes to sustaining any progress made over time. In Louisville Metro, a government organization based in Kentucky, many strategies were used to enact and sustain meaningful transformation.
Quality Improvement Associate Certification--CQIA
Certified Quality Improvement Associate Question Bank
Lean Problem-Solving Tools
Problem Solving Using A3
NEW Root Cause Analysis E-Learning
Making the Connection In this exclusive QP webcast, Jack ReVelle, ASQ Fellow and author, shares how quality tools can be combined to create a powerful problem-solving force.
Adapted from The Executive Guide to Improvement and Change , ASQ Quality Press.
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Rich, problem-solving supplements are organized by Common Core domain and standard.
Each standard offers 5 (or more) open-ended performance tasks that connect both the Standards for Mathematical Content and Mathematical Practice. These may be used for classroom instruction, exploration, formative assessment, and summative assessment.
Our DOK 3 problems naturally elicit the Mathematical Practices and are designed to engage students and develop their abilities to reason and communicate mathematically. Spanish translations are available.
Material supports the Concrete Representational Abstract (CRA) instructional approach for teaching mathematics.
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This resource assists with lesson preparation and assessing student work.
The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria to determine student performance. Our rubric criteria strongly support the Common Core Standards for Mathematical Practice . There are 4 performance levels – Novice, Apprentice, Practitioner (meets the standard), and Expert.
This assessment tool is designed to identify what is important, define what meets the standard, and distinguish between different levels of performance. It also provides teachers with guidelines for giving meaningful feedback to their students.
Student rubrics , written in kid-friendly language, are also included. These may be used to develop a child's ability to self- and peer-assess.
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Models & Strategies for Two-Digit Addition & Subtraction
Second grade is a very important year where students develop fluency with two-digit addition and subtraction . It is the year that we work on a multitude of addition and subtraction strategies that students can use to solve problems. We spend a lot of time discussing a variety of strategies, using many different models, and doing mental math.
Why? To develop students’ flexibility when solving math problems with addition and subtraction concepts .
The Common Core Standard for two-digit addition & subtraction is:
CCSS.MATH.CONTENT.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
And, the standard for three-digit addition and subtraction, to show where we’re headed:
CCSS.MATH.CONTENT.2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
Nowhere in those two standards does it say anything about the standard algorithm that we all learned in school (most likely with the language of “carry” and “borrow”), nor is the standard algorithm directly addressed in the Second Grade Common Core Standards. Read to the end to find out how I address the standard algorithm in our classroom.
Are you interested in a free sampler of some of my Two-Digit Addition and Subtraction Products ?
Strategies vs. Models
If you are familiar with my Addition & Subtraction Word Problems, you may have noticed that I make a big distinction between the strategies used when solving problems and the models students employ with those strategies.
Strategies are usually how students approach and manipulate the numbers. Models are how the strategies are organized on paper so that students can explain or see the strategy.
When looking at the standards above, I can see that the strategies are clearly noted in the standard:
In 2.NBT.B.5 and the strategies are:
- place value
- properties of operations
- relationship between addition and subtraction
Standard 2.NBT.B.7 even notes that the models or drawings (which I also call models) are separate from the strategies that are based on:
As you can see, the strategies are clearly outlined in the standards. Now within each of the above general strategy categories, there really are many different strategies that students can use and you can label them whatever you’d like in your classroom. I like to label them with students’ names as an easy reference. That way, we can refer to Samantha’s strategy when solving a problem. Or you can label the strategy with the action that the student takes in the problem (for example: Add Tens First).
However, I still make a distinction between the strategy and the model. Why? Because students can use multiple strategies with one model. There’s no one right way to use the model, as long as the student can explain his or her thinking. The models (or drawings) merely give students a tool to explain their thinking on paper or with manipulatives. The thinking, or what students do with the numbers, is the strategy. What they use to show it to you is the model.
In all honesty, I’m not always consistent in labeling something a strategy or a model. I try to be, but like you, I’m human and sometimes mix them up, especially when I’m in the moment with students. It’s a learning process and something I’m continually reflecting on throughout the years. All that to say, you may see a few things labeled one way and question its label. Go ahead and question it, think about it, mull it over and figure out whether it’s accurate or not. All of this is still new to many of us.
Here are some adding and subtracting charts that I’ve used the past couple years that illustrate some of the below models and strategies.
The above image demonstrates some second-grade math subtraction strategies that I have gone through with students.
Models for Two-Digit Addition
Below are a few models that we use with two-digit addition or subtraction. Are these the only models you can use? No, this is not an exhaustive list. They are what I have found useful in the classroom for students to practice and use to build conceptual understanding and number sense.
Number Lines for Two-digit Addition and Subtraction
I usually start with number lines when I introduce students to paper / pencil models. An open number line is very flexible. Students can make jumps of one or ten (or more) and easily manipulate it to show their mathematical thinking.
I usually help students get to the nearest 10, or friendly or benchmark number when using a number line because it is easier to make jumps of 10. That is an example of the difference between a model and a strategy. The model is the number line. The strategy is making jumps of 10.
Teaching how to use number lines when using 10 to add +9 and +8 facts , solidifies this strategy when students are adding larger two-digit numbers.
Remember, the number line is the model and it can be used with a variety of strategies. Modeling and practicing using a number line with easier problems will help students when using a number line with more difficult problems.
One of the daily activities that we do with numbers lines is our Daily Math . This is a whiteboard sheet that we go through daily. The number line at the bottom helps students solidify their understanding of both how to use a number line and how “make 100 or make 1000”.
Here are a few more examples of how we use number lines in the classroom.
This is from my Roll & Spin Math Stations . In this activity, students practice making jumps of 10 and 100 up a number line.
There are also versions where students subtract 10 and 100 down a number line, too. One of the skills students need to be successful on number lines is the ability to make jumps of 10 and 100.
This is an example from one of our Addition & Subtraction Word Problems where students had to figure out a separate start unknown problem. This student started at 15 and counted 35 jumps and then took one away at the end. This is also a great example of compensation (see below) because the student added one to the 34 to make easier jumps and then took it away at the end.
This is from my Second Grade Cut & Paste Math Activities . In this activity, students are practicing how to add up, starting at the smallest number and figuring out who tot get to the larger number by jumping to the friendly numbers. This student started at 19, jumped to 20, then made jumps of 10 to 60 and made a jump of 3. The student added their jumps together to get 44.
The above are a few examples from my Two-Digit Addition Math Stations . My students needed more direct practice with number lines and making jumps, despite all of our whole group practice. So, I gave them the directions and students followed them on the number lines.
A more recent resource that I developed to help students develop number fluency is the Make 100 and Make 1000 resource. This resource has MANY activities where students practice making 100 and making 1000. Number lines are one of the activities.
I also have a whole blog post on how to use a number line with even more examples of how to develop number line fluency in the classroom.
Base-10 blocks are another model I teach students to use; however, I generally teach students to draw the base-10 blocks. We do use real foam blocks in class, but I try to move away from them as quickly as possible.
Why? Students will always have pencil and paper to solve problems, but they won’t always have manipulative available to them. Using base-10 blocks also takes a lot of time. I don’t mind spending the time on them, for students who need them, but I also want to push students toward more efficient tools.
Here are a few examples of how we use base-10 blocks:
The above two are using base-10 blocks by drawing out the tens as “sticks” as we refer to them in our classroom. These particular students were having difficult counting over 100 by tens, so I had them draw each number in tens, then count by tens until they got to 100, then start over counting by 10s again. Not only did this help them add up numbers beyond 100, but it also gave them more expense with our base-10 number system.
The above example is from my Two-Digit Addition Math Stations again and is just a basic problem – answer matching with base-10 block representations.
The Number Line blog post also has an interesting visual activity to help students transition from base-10 blocks to number lines.
Strategies for Two-Digit Addition
As noted above, the main three strategies stated in the standards are:
Below are a few strategies that we use to solve two-digit addition problems. Most of them are based on place value strategies as I find those tend to be easier for students to understand and apply. Again, these are how students manipulate the numbers in the problem to make it easier to solve.
No one strategy is the “right” strategy for every student for every problem. Some problems lend themselves to certain strategies because of the numbers. Students may also switch between strategies within the same problem, depending on how they’re manipulating the numbers. The key thing to look for is if the student can explain his or her thinking when solving a problem.
Break Apart or Ungroup (Place Value)
This strategy requires a bit more mental math practice, but it can be so powerful. The basic idea is that the number is broken apart into tens and ones and then, either using a number line, base-10 blocks or just numbers, students manipulate the pieces to add or subtract the numbers.
Breaking the number part or ungrouping it helps students see the value of place value. The tens place is not just 4. Its value is 40 or 4 tens.
One resource that helps develop this strategy is the Number Talks book ( affiliate link ). We do number talks through the year, starting with addition facts and moving into two-digit addition and subtraction by the end of the year. I love seeing the strategies that my students can come up! The Number Talk book is also a great book that helps develop listening skills.
Think about the problem 64-47. Students break apart the problem into 50+14-7-40 and take away the parts by place value. I’d probably start with the 14-7, but students could start anywhere that makes sense for them.
The above examples come from my Two-Digit Addition Math Stations and illustrate how students can break apart numbers and add up each place value. Breaking apart is also called ungrouping or decomposing, depending on the math program you use.
Did you notice that in one of the problems above, the student added 60 +40 and got 106, yet he wrote the correct answer to the problem? What do you think was going on with this student? Do you then he couldn’t add 60+40, made a silly mistake, or is there another reason he wrote the 106? Seeing students interact with these types of strategies will give you a place start conversations with them about their mathematical thinking.
One more example from some Addition Task Cards where students only break apart the second number then make jumps of 10 and 1 using 100s and 1000s charts. Although we give plenty of practice using a 100s chart in first grade, I find that students don’t necessarily transfer their learning to larger numbers in second grade.
Add Tens to Tens and Ones to Ones (Place Value)
This is very similar to the break part strategies, except without breaking apart the numbers. Students can add the parts of the number (the tens or the ones) together mentally because they know their addition facts. We basically use a v-model to draw lines connecting the tens and adding or subtracting those parts.
Here is one example of how we’ve used it in the classroom:
Subtract Tens, Subtract Ones (Place Value)
Similar to add tens to tens and ones to ones, students subtract each place value separately and then subtraction the ones from the tens (or add it). There are basically two ways to use this strategy. Students can decompose a the ten or students can use negative numbers.
One way that I use this strategy with students is with negative numbers. I know we don’t teach negative numbers in second grade, but for some students, this is really a way that they understand and can hold onto more than the other strategies. You can see examples of this in the second and third anchor charts above.
Think about 64-47. If I subtract 4-7, I get -3. I tell students that the bigger number has the minus sign in front of it and so it still has more that needs to be taken away. Students then subtract 60-40, get 20 and subtract there more to get 17.
Count Down / Think Addition (Count Up) / Add Up (Relationship between Addition & Subtraction or Place Value)
I’m not exactly sure whether this strategy is about the relationship between addition and subtraction or place value. The Think Addition Strategy is similar (if not the same as) Count Up or Add Up. This strategy is also very similar to the Break Apart Strategy, in that students need to break at least one of the numbers apart to sound up or down by the parts of the number.
Although students can count by ones, I highly encourage you to help them move toward more efficient strategies and count by tens and then ones. Using a hundreds chart gives students practice moving by 10s up and down the chart. A hundreds chart is sort of like a compressed number line. See the above photo with the 100s and 1000s charts.
Here are a few examples of counting up:
The above two examples are just ones we did on the whiteboard and I had students write down in their notebooks.
This is a page from my Two-Digit Subtraction Flap Books . These Flap Books go through several different models and strategies and give students practice with vocabulary and explaining their thinking.
The thing I LOVE about these flap books are that students can dive deep into one aspect of two-digit subtraction and attach language to the numbers and processes that they use.
Use Compensation (Properties of Operations)
This last strategy is unlike any of the previous ones. It basically has you make sure that the numbers are balanced within the problem and that you’re accounting for all of the parts. It’s a precursor to algebra and a great strategy for mental math.
There are a couple different ways to use compensation, but the basic idea is that you add or subtract some of one number and add it to the other number to create a friendly number. You have to keep track of what was added or taken away and account for it somehow in the problem.
Compensation is especially useful for numbers that are close to friendly numbers, although it can be used for any number. For example, 68 – 39 could be transformed into 69 – 40. I’ve added one to each number. The value of a +1 and -1 is 0, so I haven’t changed the problem at all.
Here’s another example: 53 + 38. I might add 53 + 40 and get 93, but because I added two to the 38 to get to 40, I’ll need to subtract two from 93 to get 91.
The basic idea with compensation is that you are adjusting one part of the number into a friendly number to make it easier to add or subtract. However, when you adjust one number, you have to keep track of what you’ve adjusted and compensate for it.
What do students need to know before using these strategies?
The above strategies are very powerful if students can add them to their toolkit when approaching two-digit addition and subtraction. However, to effectively use the above strategies, students need a few things in place.
Addition and Subtraction Facts – Students need pretty good fluency with their addition and subtraction facts. Do they need to have all of them memorized with speed? No. However, if students are spending too much time trying to figure out an addition fact and it’s keeping them from focusing on the strategy because they forget what they were doing, then they need more fluency with their addition and subtraction facts. My Automaticity Assessments help students practice their facts by strategy.
Ability to find friendly numbers – At the beginning of the year, we spend a long time developing fluency with 10 as a benchmark number . Although we do it at the beginning of the year to help with our math fact fluency, it is also beneficial when students begin their journey with adding and subtracting two-digit numbers. Students need to know how to get to the next friendly number, which is essentially their 10s facts but applying them to two-digit numbers to find the next ten.
Adding 10 to a number – We start our two-digit addition unit with a lot of practice adding and subtracting ten from a number. This is a foundation skill in both my two-digit addition products as well as my two-digit subtraction products . Students must see the pattern of adding 10 to a number.
Place Value – To do two-digit addition, students need a strong foundation in the concept of ones and tens and what it means to break a number apart into ones and tens. From the first day of school, we are doing Daily Math exercises that build fluency with place value as well as skip counting by 10s from any number.
Do I teach the traditional algorithm?
Yes and no. Yes, I teach the concept of regrouping and yes, I do teach students to move toward efficiency when adding and subtracting. That could include the traditional algorithm if they can understand the meaning behind it.
Students do not need to use the standard algorithm until fourth grade ( according to the Common Core Standards ). Can they do it earlier? Maybe.
I expose them to it in second grade as a model they could use; however, we don’t spend a lot of time focused on it, because I want students to develop strategies for solving problems, not be tied to one model.
When we do work with the traditional algorithm, we attach a lot of language and meaning to it, generally tying it to work we’ve already done, like our work with base-10 blocks. Here are a few examples of I teach students the traditional algorithm by linking it to models we’ve already used and giving students accurate language to use to explain their thinking.
Here are a few examples of how I give students experience with the traditional algorithm.
Did you notice that should say 7 tens and 11 ones? The student didn’t pay attention to the base-10 blocks!
These come from my Decompose a Ten packet, which balances work the traditional algorithm with base-10 models and gives students the language of decomposing numbers.
Whew – that’s a lot of information to digest! There’s many different models and strategies a student can use to solve two-digit addition and subtraction problems. What I outlined above are a few that I have found especially helpful for students. They help students develop a solid foundation with two-digit addition and subtraction, create a bridge to three-digit addition and subtraction, as well as emphasize the idea of using strategies and models to solve problems, not just following steps in a process.
If you teach second grade, you might like a few pages from some of my two-digit addition and subtraction products. I’ve compiled this PDF of resources as a sampler from several different products that really emphasize all the work we do in our classroom to develop these strategies in depth.
Different components of the sampler can be used whole group or small group and are perfect for helping your students think outside the box when it comes to solving multi-digit addition and subtraction.
Two-Digit Resources Mentioned Above
Here is a list with links of all of the two-digit addition and subtraction resources mentioned above. They can be purchased on my website or on Teachers Pay Teachers .
- Roll and Spin Math Stations
- Cut and Paste Math Activities for Second Grade ( TpT )
- Two-Digit Addition Math Centers ( TpT )
- Two-Digit Subtraction Math Centers ( TpT )
- Addition Task Cards Using 100s Charts ( TpT )
- Two-Digit Subtraction Flap Books ( TpT )
- Decompose a Ten Task Cards ( TpT )
Many of the above are also included in a Two-Digit Addition and Subtraction BUNDLE ( TpT ).
Additional Two-Digit Addition & Subtraction Resources
- Two-Digit Addition Number Puzzles ( TpT )
- Two-Digit Subtraction Number Puzzles ( TpT )
- Decompose Two-Digit Numbers Number Puzzles ( TpT )
- Two-Digit Addition No Prep Printables / Worksheets ( TpT )
- Two-Digit Subtraction No Prep Printables / Worksheets
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This entire blog has made my life easier. I am a student teacher to second graders and I have been having a difficult time figuring out how to teach my students double digit addition and subtraction with and without regrouping. A lot of kids this year have a weak number sense and I know that is contributing to their struggle. Your booklets and packets have made their struggle nearly disappear.
Is awesome to me as well and i think it will really help my students as well
Hello, I’m curious what math program you are working with? We have just adopted EverydayMath and much of your approach reminds me of that program – but is more friendly! Thanks for your reply.
After Common Core came out we were in transition and didn’t really use one math program, but a variety of resources that meet the standards. These resources are not based on any program, but processes students can use with any program.
OMG!. Thank you so much for putting this together, and making it visible. I am a parent of a 4th grader who is still struggling with understanding this material. I’ve seen many worksheets sent home with a combination of strategies this is the first time, I’ve understood how they are all connected. I love your simple explanations, breakdown of the standards, and real-world examples. I bet you are an AMAZING teacher, and truly, truly, appreciate you.
Great explanation for some of us parents that feel kind of lost like my self right now. I grow up with other math methods, and I feel frustrated that my 2nd grader comes home with different types of strategies that I don’t know. But Thankfully, searching on the web I came along with this amazing page that really explains all those strategies that new generations are learning today. Thank you so much for this strategies, it is well appreciated from myself and I bet many parents out there. I can tell you are an amazing teacher. Thanks a lot!
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20 Best Common Core Math Teaching Strategies 
Are you looking for the best Common Core Math Teaching Strategies? Based on expert reviews, we ranked them. We've listed our top-ranked picks, including the top-selling Common Core Math Teaching Strategies.
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- 96 Pages - 01/01/2014 (Publication Date) - Scholastic Teaching Resources (Teaching Strategies) (Publisher)
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- Powerful visualizations for meaningful, connected learning
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- Set Children Up for Success in School; Grab your kids' attention with these 11x17 inch math educational posters, whether you teach in a classroom,...
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- 98 Pages - 02/23/2020 (Publication Date)
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- 188 Pages - 09/23/2015 (Publication Date) - Whole Brain Teaching Press (Publisher)
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- 216 Pages - 09/29/2021 (Publication Date) - Ulysses Press (Publisher)
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- 576 Pages - 03/15/2018 (Publication Date) - Pearson (Publisher)
- ✅ MATH MANIPULATIVES DESIGNED BY TEACHERS - Colorful and differentiated Magnetic Base Ten Blocks Set is designed by teachers for the best learning...
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- Best Common Core Math Wall
- Best Math Your Common Core Edition Volume 2
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- Best Go Math Assessment Guide Grade 3: Common Core Edition
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Q: How to do ‘Common Core’ math?
A: 1. Number Paths. Number paths do not have zeros. They are a row of boxes with a number in each box,in sequential order. So basically,the blocks act … 2. Number Lines. 3. Ten Frames. 4. Number Bonds. 5. Draw Pictures.
Q: What are the Common Core Standards in math?
A: There are two types of Mathematical Standards in the Common Core State Standards: The Standards for Mathematical Practice describe a set of intellectual habits and behaviors that all students should develop as they engage in mathematics. Content Standards – These are the mathematics skills and proficiencies that students are expected to learn.
Q: What are the eight standards for mathematical practice?
A: The Eight Practices are: Figure 1. Common Core Standards for Mathematical Practices. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision.
Q: What are common core teaching standards?
A: The Common Core is a set of academic standards for what every student is expected to learn in each grade level, from kindergarten through high school. They cover math and English language arts (ELA). The standards are publicly available to read.
Math Problem Solving Resources
Laura Candler’s Math Problem Solving page is where you’ll find great resources for teaching problem solving as well as a variety of math problem activity pages. Many of the items on this page are free and do not come with directions. For complete problem solving lessons, check out the Daily Math Puzzlers , a series of four leveled books that include information on how to teach problem solving as well as mixed-problem activity pages for students.
Free Daily Math Puzzler Sample Pages
Each page below is a free sample from one level of the Daily Math Puzzler program.
Links to Free Problem Solving Strategies
- Cooperative Math Problem Solving
- Differentiating with Daily Math Puzzlers
- 6 Reasons to Teach Calculators Skills
- How to Recharge Mathematical Thinking
- Common Core Math Problem Solving
Free Problem Solving Assessments
Math mindset challenges: editable multi-step word problems.
Recharge & Write Problem Solving Strategy (Video Clip)
Recharge & Write is a terrific cooperative learning strategy that gets kids talking about math problems but holds them accountable for understanding how to solve each problem. I recorded a short video to explain how it works, and I provided even more detail my blog post, How to Recharge Mathematical Thinking .
Weekly Math Challenge Freebies
If you’re looking for the Weekly Math Challenge printables shown below, they’re free for my newsletter subscribers. To find them, sign up for Candler’s Classroom Connections and follow the links to the Best Freebies page. You can find many more problems like this in the Math Mindset Challenges collection!
Math Problem Solving: Mindsets Matter Webinar
If you missed the live webinar, you can watch a free replay here .
Common Core Standards for Mathematical Practice
The Common Core State Standards include eight mathematical practices that should be included as a regular part of math instruction. I created the Standards for Mathematical Practice chart shown below to use as you plan instruction to ensure you are meeting these standards throughout the week.
Candler’s Classroom Connections
4 Main problem-solving strategies
In Psychology, you get to read about a ton of therapies. It’s mind-boggling how different theorists have looked at human nature differently and have come up with different, often somewhat contradictory, theoretical approaches.
Yet, you can’t deny the kernel of truth that’s there in all of them. All therapies, despite being different, have one thing in common- they all aim to solve people’s problems. They all aim to equip people with problem-solving strategies to help them deal with their life problems.
Problem-solving is really at the core of everything we do. Throughout our lives, we’re constantly trying to solve one problem or another. When we can’t, all sorts of psychological problems take hold. Getting good at solving problems is a fundamental life skill.
What problem-solving does is take you from an initial state (A) where a problem exists to a final or goal state (B), where the problem no longer exists.
To move from A to B, you need to perform some actions called operators. Engaging in the right operators moves you from A to B. So, the stages of problem-solving are:
- Initial state
The problem itself can either be well-defined or ill-defined. A well-defined problem is one where you can clearly see where you are (A), where you want to go (B), and what you need to do to get there (engaging the right operators).
For example, feeling hungry and wanting to eat can be seen as a problem, albeit a simple one for many. Your initial state is hunger (A) and your final state is satisfaction or no hunger (B). Going to the kitchen and finding something to eat is using the right operator.
In contrast, ill-defined or complex problems are those where one or more of the three problem solving stages aren’t clear. For example, if your goal is to bring about world peace, what is it exactly that you want to do?
It’s been rightly said that a problem well-defined is a problem half-solved. Whenever you face an ill-defined problem, the first thing you need to do is get clear about all the three stages.
Often, people will have a decent idea of where they are (A) and where they want to be (B). What they usually get stuck on is finding the right operators.
Initial theory in problem-solving
When people first attempt to solve a problem, i.e. when they first engage their operators, they often have an initial theory of solving the problem. As I mentioned in my article on overcoming challenges for complex problems, this initial theory is often wrong.
But, at the time, it’s usually the result of the best information the individual can gather about the problem. When this initial theory fails, the problem-solver gets more data, and he refines the theory. Eventually, he finds an actual theory i.e. a theory that works. This finally allows him to engage the right operators to move from A to B.
These are operators that a problem solver tries to move from A to B. There are several problem-solving strategies but the main ones are:
- Trial and error
When you follow a step-by-step procedure to solve a problem or reach a goal, you’re using an algorithm. If you follow the steps exactly, you’re guaranteed to find the solution. The drawback of this strategy is that it can get cumbersome and time-consuming for large problems.
Say I hand you a 200-page book and ask you to read out to me what’s written on page 100. If you start from page 1 and keep turning the pages, you’ll eventually reach page 100. There’s no question about it. But the process is time-consuming. So instead you use what’s called a heuristic.
Heuristics are rules of thumb that people use to simplify problems. They’re often based on memories from past experiences. They cut down the number of steps needed to solve a problem, but they don’t always guarantee a solution. Heuristics save us time and effort if they work.
You know that page 100 lies in the middle of the book. Instead of starting from page one, you try to open the book in the middle. Of course, you may not hit page 100, but you can get really close with just a couple of tries.
If you open page 90, for instance, you can then algorithmically move from 90 to 100. Thus, you can use a combination of heuristics and algorithms to solve the problem. In real life, we often solve problems like this.
When police are looking for suspects in an investigation, they try to narrow down the problem similarly. Knowing the suspect is 6 feet tall isn’t enough, as there could be thousands of people out there with that height.
Knowing the suspect is 6 feet tall, male, wears glasses, and has blond hair narrows down the problem significantly.
3. Trial and error
When you have an initial theory to solve a problem, you try it out. If you fail, you refine or change your theory and try again. This is the trial-and-error process of solving problems. Behavioral and cognitive trial and error often go hand in hand, but for many problems, we start with behavioural trial and error until we’re forced to think.
Say you’re in a maze, trying to find your way out. You try one route without giving it much thought and you find it leads to nowhere. Then you try another route and fail again. This is behavioural trial and error because you aren’t putting any thought into your trials. You’re just throwing things at the wall to see what sticks.
This isn’t an ideal strategy but can be useful in situations where it’s impossible to get any information about the problem without doing some trials.
Then, when you have enough information about the problem, you shuffle that information in your mind to find a solution. This is cognitive trial and error or analytical thinking. Behavioral trial and error can take a lot of time, so using cognitive trial and error as much as possible is advisable. You got to sharpen your axe before you cut the tree.
When solving complex problems, people get frustrated after having tried several operators that didn’t work. They abandon their problem and go on with their routine activities. Suddenly, they get a flash of insight that makes them confident they can now solve the problem.
I’ve done an entire article on the underlying mechanics of insight . Long story short, when you take a step back from your problem, it helps you see things in a new light. You make use of associations that were previously unavailable to you.
You get more puzzle pieces to work with and this increases the odds of you finding a path from A to B, i.e. finding operators that work.
No matter what problem-solving strategy you employ, it’s all about finding out what works. Your actual theory tells you what operators will take you from A to B. Complex problems don’t reveal their actual theories easily solely because they are complex.
Therefore, the first step to solving a complex problem is getting as clear as you can about what you’re trying to accomplish- collecting as much information as you can about the problem.
This gives you enough raw materials to formulate an initial theory. We want our initial theory to be as close to an actual theory as possible. This saves time and resources.
Solving a complex problem can mean investing a lot of resources. Therefore, it is recommended you verify your initial theory if you can. I call this pilot problem-solving.
Before businesses invest in making a product, they sometimes distribute free versions to a small sample of potential customers to ensure their target audience will be receptive to the product.
Before making a series of TV episodes, TV show producers often release pilot episodes to figure out whether the show can take off.
Before conducting a large study, researchers do a pilot study to survey a small sample of the population to determine if the study is worth carrying out.
The same ‘testing the waters’ approach needs to be applied to solving any complex problem you might be facing. Is your problem worth investing a lot of resources in? In management, we’re constantly taught about Return On Investment (ROI). The ROI should justify the investment.
If the answer is yes, go ahead and formulate your initial theory based on extensive research. Find a way to verify your initial theory. You need this reassurance that you’re going in the right direction, especially for complex problems that take a long time to solve.
Getting your causal thinking right
Problem solving boils down to getting your causal thinking right. Finding solutions is all about finding out what works, i.e. finding operators that take you from A to B. To succeed, you need to be confident in your initial theory (If I do X and Y, they’ll lead me to B). You need to be sure that doing X and Y will lead you to B- doing X and Y will cause B.
All obstacles to problem-solving or goal-accomplishing are rooted in faulty causal thinking leading to not engaging the right operators. When your causal thinking is on point, you’ll have no problem engaging the right operators.
As you can imagine, for complex problems, getting our causal thinking right isn’t easy. That’s why we need to formulate an initial theory and refine it over time.
I like to think of problem-solving as the ability to project the present into the past or into the future. When you’re solving problems, you’re basically looking at your present situation and asking yourself two questions:
“What caused this?” (Projecting present into the past)
“What will this cause?” (Projecting present into the future)
The first question is more relevant to problem-solving and the second to goal-accomplishing.
If you find yourself in a mess , you need to answer the “What caused this?” question correctly. For the operators you’re currently engaging to reach your goal, ask yourself, “What will this cause?” If you think they cannot cause B, it’s time to refine your initial theory.
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Nursing students will be expected to have or develop strong problem-solving skills. Problem solving is centered on your ability to identify critical issues and create or identify solutions. Well-developed problem solving skills is a characteristic of a successful student. Remember, problems are a part of everyday life and your ability to resolve problems will have a positive influence on your future.
6 Steps of Problem Solving
Step 1: Identify and Define the Problem
It is not difficult to overlook the true problem in a situation and focus your attention on issues that are not relevant. This is why it is important that you look at the problem from different perspectives. This provides a broad view of the situation that allows you to weed out factors that are not important and identify the root cause of the problem.
Step 2: Analyze the Problem
Break down the problem to get an understanding of the problem. Determine how the problem developed. Determine the impact of the problem.
Step 3: Develop Solutions
Brainstorm and list all possible solutions that focus on resolving the identified problem. Do not eliminate any possible solutions at this stage.
Step 4: Analyze and Select the Best Solution
List the advantages and disadvantages of each solution before deciding on a course of action. Review the advantages and disadvantages of each possible solution. Determine how the solution will resolve the problem. What are the short-term and long-term disadvantages of each solution? What are the possible short-term and long-term benefits of each solution? Which solution will help you meet your goals?
Step 5: Implement the Solution
Create a plan of action. Decide how you will move forward with your decision by determining the steps you must take to ensure that you move forward with your solution. Now, execute your plan of action.
Step 6: Evaluate the Solution
Monitor your decision. Assess the results of your solution. Are you satisfied with the results? Did your solution resolve the problem? Did it produce a new problem? Do you have to modify your solution to achieve better results? Are you closer to achieving your goal? What have you learned?
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subtract with in 1000 using models and strategies
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2 & 3 digit addition and subtraction activities lesson plans & math centers
Common Core Math Assessments for 2nd Grade - Number and Operations in Base 10
Common Core Strategies for Adding and Subtracting
3 Digit Addition & Subtraction Word Problems with & without Regrouping
- Easel Activity
Also included in: Third Grade Math & Reading YEAR LONG BUNDLE 1500+ Pages
Area Model Multiplication Worksheets (3.NBT.2 and 4.NBT.5)
Also included in: Math Worksheet Bundle (addition, subtraction, multiplication, & division)
Using Part-Part-Whole Models to Solve Addition and Subtraction Word Problems
Math Worksheet Bundle (addition, subtraction, multiplication, & division)
Multiplication Math Worksheet Bundle (area model, lattice, & standard algorithm)
Multi-Digit Addition & Subtraction Worksheets (3.NBT.A.2 & 4.NBT.B.5)
3RD GRADE ADDITION & SUBTRACTION: 8 Skills-Boosting Practice Worksheets
Also included in: 3RD GRADE MATH WORKSHEETS/TEST PREP | Skills-Boosting, Scaffolded Savings Bundle
Lattice Multiplication Worksheets (3.NBT.2 and 4.NBT.5)
NUMBER NOTES | 3-Digit Addition Strategies Worksheets (With Regrouping) 3.NBT.2
Also included in: NUMBER NOTES | 3rd Grade Math Doodle Worksheets | Super-Fun Practice Activities
Add and Subtract Within 1000 | Illustrative Math Unit 7
Also included in: 2nd Grade Math Centers | BUNDLE
NUMBER NOTES 3-Digit Subtraction Strategies Activity ★ 3.NBT.2 Doodle Worksheets
One step word problems with unknown and missing numbers
3rd - Differentiated Add Up to Subtract within 1000 Pixel Art - 3.NBT.2
Also included in: 3rd & 4th - Differentiated Multi-Digit Subtraction Pixel Art Bundle
3rd - Differentiated Subtraction within 1000 Pixel Art Activity - 3.NBT.2
3-DIGIT SUBTRACTION PUZZLES - Subtraction Practice - Print & Digital
Also included in: MATH PUZZLES BUNDLE - Math Centres - Math Printables
Elementary Math (3rd & 4th grade) - Halloween Themed Fun Activity Worksheets
Grade 3, CCS: Math Bundle- Frac, Geo, Alg, M&D, Base 10 for Student w/ ADD/ ADHD
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Make sense of problems and persevere in solving them: AoPS does not believe that math is memorizing Trick A to solve Problem A and Trick B for Problem B. Problem solving is about understanding the problem, weighing the possible approaches, and deciding which is the best strategy for solving it.
The thread of literacy found in the Common Core State Standards (CCSS) suggests a way to get to the heart of problem solving. When you say that someone is literate, you are not saying that they know how to read; you are saying that they are well read or have read a lot.
To help kids find out what problem-solving methods work best for them, you can give them several types of common core math activities. You can use teaching tools that give kids several methods at once. This way, they can pick ones that appeal to them naturally. Scaffolding Math Problems For younger children, teachers often "scaffold" math problems.
Daily problem solving is a highly effective way to help your students master the Common Core mathematical practice standards. Fortunately, when you have a plan in place, it can also be the easiest way to motivate your students to become proficient problem-solvers. I've always recognized the importance of daily math problem solving, but in the ...
Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
A problem-solving strategy is a plan used to find a solution or overcome a challenge. Each problem-solving strategy includes multiple steps to provide you with helpful guidelines on how to resolve a business problem or industry challenge.
A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.
Crucial to this teaching methodology are (a) quick recall of math facts, and (b) a straightforward process that anyone can do. This gives students the skills they need to do problems and to recognize that the size of the problem doesn't really matter as long as you have the process.
The bigger problem is the role the Common Core State Standards (CCSS) are playing in the larger dynamics of current school reform and education politics. Today everything about the Common Core, even the brand name the Common Core State Standards is contested because these standards were created as an instrument of contested policy. They have ...
Means-end analysis is a problem-solving strategy that, to put it simply, helps you get from "point A" to "point B" by examining and coming up with solutions to obstacles. When using means-end...
Browse the Khan Academy math skills by Common Core standard. With over 50,000 unique questions, we provide complete coverage. ... Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number ...
Implement common problem-solving strategies There are a few ways to go about identifying problems (and solutions). Here are some strategies you can try, as well as common ways to apply them: Trial and error Trial and error problem solving doesn't usually require a whole team of people to solve.
A common mistake in problem solving is that alternatives are evaluated as they are proposed, so the first acceptable solution is chosen, even if it's not the best fit. If we focus on trying to get the results we want, we miss the potential for learning something new that will allow for real improvement in the problem-solving process. 3.
Tasks Rich, problem-solving supplements are organized by Common Core domain and standard. Each standard offers 5 (or more) open-ended performance tasks that connect both the Standards for Mathematical Content and Mathematical Practice. These may be used for classroom instruction, exploration, formative assessment, and summative assessment.
The Common Core Standard for two-digit addition & subtraction is: CCSS.MATH.CONTENT.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. And, the standard for three-digit addition and subtraction, to show where we're headed:
After days of researching and comparing all models on the market 2021 , JohnHarvards find Best Common Core Math Teaching Strategies of January 2023. If you want to know more about finding the perfect product for you, including our expert's tips and tricks, read this article
Common Core Math Problem Solving Free Problem Solving Assessments Use the simple pretests in this assessment pack to learn how skilled your students are at solving problems and where to start them in the Daily Math Puzzler program.
Problem-solving strategies These are operators that a problem solver tries to move from A to B. There are several problem-solving strategies but the main ones are: Algorithms Heuristics Trial and error Insight 1. Algorithms When you follow a step-by-step procedure to solve a problem or reach a goal, you're using an algorithm.
the main points, analyzing possible effective strategies and procedures, and identifying bridges to span from the mundane into the creative math applications. The success of the problem solving procedure depends on the building of connections and demonstrating proficiency. The students practice multiple and varied experiences to
Problem Solving Steps Example SOLUTION OUTCOME RATING (+ OR -) 1. Ignore him He would keep it up - 2. Insult him back He'd probably try to hit me - 3. Walk away He'd leave me alone but might think I'm a "chicken" +/- 4. Hit him We'd both be sent to the Principal - 5.
Using the Common Core State Standards, math can be taught in a different way than most accepted in schools. Take a deeper look at math and the Common Core, the problem-solving process ...
Common Core's Eight Standards of Mathematical Practice may be the answer. Let's take a look at each one. 1. Make sense of problems and persevere in solving them. This standard seeks to...
Strategies for Problem Solving. Nursing students will be expected to have or develop strong problem-solving skills. Problem solving is centered on your ability to identify critical issues and create or identify solutions. Well-developed problem solving skills is a characteristic of a successful student. Remember, problems are a part of everyday ...
There are 2 activities. Both go with standards: MCC3.1.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 3.OA.8 Solve word problems using currently mastered operations.