Center for Teaching
Teaching problem solving.
Tips and Techniques
Expert vs. novice problem solvers, communicate.
- Have students identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
- If students are unable to articulate their concerns, determine where they are having trouble by asking them to identify the specific concepts or principles associated with the problem.
- In a one-on-one tutoring session, ask the student to work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
- When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)
- Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
- Have students work through problems on their own. Ask directing questions or give helpful suggestions, but provide only minimal assistance and only when needed to overcome obstacles.
- Don’t fear group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others
- Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing positive reinforcement to let students know when they have mastered a new concept or skill.
Encourage Thoroughness and Patience
- Try to communicate that the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.
Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills, a teacher should be aware of principles and strategies of good problem solving in his or her discipline .
The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.
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Teaching problem solving
Strategies for teaching problem solving apply across disciplines and instructional contexts. First, introduce the problem and explain how people in your discipline generally make sense of the given information. Then, explain how to apply these approaches to solve the problem.
Introducing the problem
Explaining how people in your discipline understand and interpret these types of problems can help students develop the skills they need to understand the problem (and find a solution). After introducing how you would go about solving a problem, you could then ask students to:
- frame the problem in their own words
- define key terms and concepts
- determine statements that accurately represent the givens of a problem
- identify analogous problems
- determine what information is needed to solve the problem
Working on solutions
In the solution phase, one develops and then implements a coherent plan for solving the problem. As you help students with this phase, you might ask them to:
- identify the general model or procedure they have in mind for solving the problem
- set sub-goals for solving the problem
- identify necessary operations and steps
- draw conclusions
- carry out necessary operations
You can help students tackle a problem effectively by asking them to:
- systematically explain each step and its rationale
- explain how they would approach solving the problem
- help you solve the problem by posing questions at key points in the process
- work together in small groups (3 to 5 students) to solve the problem and then have the solution presented to the rest of the class (either by you or by a student in the group)
In all cases, the more you get the students to articulate their own understandings of the problem and potential solutions, the more you can help them develop their expertise in approaching problems in your discipline.
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Teaching Problem-Solving Skills
Many instructors design opportunities for students to solve “problems”. But are their students solving true problems or merely participating in practice exercises? The former stresses critical thinking and decision making skills whereas the latter requires only the application of previously learned procedures.
Problem solving is often broadly defined as "the ability to understand the environment, identify complex problems, review related information to develop, evaluate strategies and implement solutions to build the desired outcome" (Fissore, C. et al, 2021). True problem solving is the process of applying a method – not known in advance – to a problem that is subject to a specific set of conditions and that the problem solver has not seen before, in order to obtain a satisfactory solution.
Below you will find some basic principles for teaching problem solving and one model to implement in your classroom teaching.
Principles for teaching problem solving
- Model a useful problem-solving method . Problem solving can be difficult and sometimes tedious. Show students how to be patient and persistent, and how to follow a structured method, such as Woods’ model described below. Articulate your method as you use it so students see the connections.
- Teach within a specific context . Teach problem-solving skills in the context in which they will be used by students (e.g., mole fraction calculations in a chemistry course). Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill.
- Help students understand the problem . In order to solve problems, students need to define the end goal. This step is crucial to successful learning of problem-solving skills. If you succeed at helping students answer the questions “what?” and “why?”, finding the answer to “how?” will be easier.
- Take enough time . When planning a lecture/tutorial, budget enough time for: understanding the problem and defining the goal (both individually and as a class); dealing with questions from you and your students; making, finding, and fixing mistakes; and solving entire problems in a single session.
- Ask questions and make suggestions . Ask students to predict “what would happen if …” or explain why something happened. This will help them to develop analytical and deductive thinking skills. Also, ask questions and make suggestions about strategies to encourage students to reflect on the problem-solving strategies that they use.
- Link errors to misconceptions . Use errors as evidence of misconceptions, not carelessness or random guessing. Make an effort to isolate the misconception and correct it, then teach students to do this by themselves. We can all learn from mistakes.
Woods’ problem-solving model
Define the problem.
- The system . Have students identify the system under study (e.g., a metal bridge subject to certain forces) by interpreting the information provided in the problem statement. Drawing a diagram is a great way to do this.
- Known(s) and concepts . List what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it.
- Unknown(s) . Once you have a list of knowns, identifying the unknown(s) becomes simpler. One unknown is generally the answer to the problem, but there may be other unknowns. Be sure that students understand what they are expected to find.
- Units and symbols . One key aspect in problem solving is teaching students how to select, interpret, and use units and symbols. Emphasize the use of units whenever applicable. Develop a habit of using appropriate units and symbols yourself at all times.
- Constraints . All problems have some stated or implied constraints. Teach students to look for the words "only", "must", "neglect", or "assume" to help identify the constraints.
- Criteria for success . Help students consider, from the beginning, what a logical type of answer would be. What characteristics will it possess? For example, a quantitative problem will require an answer in some form of numerical units (e.g., $/kg product, square cm, etc.) while an optimization problem requires an answer in the form of either a numerical maximum or minimum.
Think about it
- “Let it simmer”. Use this stage to ponder the problem. Ideally, students will develop a mental image of the problem at hand during this stage.
- Identify specific pieces of knowledge . Students need to determine by themselves the required background knowledge from illustrations, examples and problems covered in the course.
- Collect information . Encourage students to collect pertinent information such as conversion factors, constants, and tables needed to solve the problem.
Plan a solution
- Consider possible strategies . Often, the type of solution will be determined by the type of problem. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards.
- Choose the best strategy . Help students to choose the best strategy by reminding them again what they are required to find or calculate.
Carry out the plan
- Be patient . Most problems are not solved quickly or on the first attempt. In other cases, executing the solution may be the easiest step.
- Be persistent . If a plan does not work immediately, do not let students get discouraged. Encourage them to try a different strategy and keep trying.
Encourage students to reflect. Once a solution has been reached, students should ask themselves the following questions:
- Does the answer make sense?
- Does it fit with the criteria established in step 1?
- Did I answer the question(s)?
- What did I learn by doing this?
- Could I have done the problem another way?
If you would like support applying these tips to your own teaching, CTE staff members are here to help. View the CTE Support page to find the most relevant staff member to contact.
- Fissore, C., Marchisio, M., Roman, F., & Sacchet, M. (2021). Development of problem solving skills with Maple in higher education. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414. Springer, Cham. https://doi.org/10.1007/978-3-030-81698-8_15
- Foshay, R., & Kirkley, J. (1998). Principles for Teaching Problem Solving. TRO Learning Inc., Edina MN. (PDF) Principles for Teaching Problem Solving (researchgate.net)
- Hayes, J.R. (1989). The Complete Problem Solver. 2nd Edition. Hillsdale, NJ: Lawrence Erlbaum Associates.
- Woods, D.R., Wright, J.D., Hoffman, T.W., Swartman, R.K., Doig, I.D. (1975). Teaching Problem solving Skills.
- Engineering Education. Vol 1, No. 1. p. 238. Washington, DC: The American Society for Engineering Education.
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Teaching problem solving: Let students get ‘stuck’ and ‘unstuck’
Subscribe to the center for universal education bulletin, kate mills , km kate mills literacy interventionist - red bank primary school helyn kim helyn kim former brookings expert @helyn_kim.
October 31, 2017
This is the second in a six-part blog series on teaching 21st century skills , including problem solving , metacognition , critical thinking , and collaboration , in classrooms.
In the real world, students encounter problems that are complex, not well defined, and lack a clear solution and approach. They need to be able to identify and apply different strategies to solve these problems. However, problem solving skills do not necessarily develop naturally; they need to be explicitly taught in a way that can be transferred across multiple settings and contexts.
Here’s what Kate Mills, who taught 4 th grade for 10 years at Knollwood School in New Jersey and is now a Literacy Interventionist at Red Bank Primary School, has to say about creating a classroom culture of problem solvers:
Helping my students grow to be people who will be successful outside of the classroom is equally as important as teaching the curriculum. From the first day of school, I intentionally choose language and activities that help to create a classroom culture of problem solvers. I want to produce students who are able to think about achieving a particular goal and manage their mental processes . This is known as metacognition , and research shows that metacognitive skills help students become better problem solvers.
I begin by “normalizing trouble” in the classroom. Peter H. Johnston teaches the importance of normalizing struggle , of naming it, acknowledging it, and calling it what it is: a sign that we’re growing. The goal is for the students to accept challenge and failure as a chance to grow and do better.
I look for every chance to share problems and highlight how the students— not the teachers— worked through those problems. There is, of course, coaching along the way. For example, a science class that is arguing over whose turn it is to build a vehicle will most likely need a teacher to help them find a way to the balance the work in an equitable way. Afterwards, I make it a point to turn it back to the class and say, “Do you see how you …” By naming what it is they did to solve the problem , students can be more independent and productive as they apply and adapt their thinking when engaging in future complex tasks.
After a few weeks, most of the class understands that the teachers aren’t there to solve problems for the students, but to support them in solving the problems themselves. With that important part of our classroom culture established, we can move to focusing on the strategies that students might need.
Here’s one way I do this in the classroom:
I show the broken escalator video to the class. Since my students are fourth graders, they think it’s hilarious and immediately start exclaiming, “Just get off! Walk!”
When the video is over, I say, “Many of us, probably all of us, are like the man in the video yelling for help when we get stuck. When we get stuck, we stop and immediately say ‘Help!’ instead of embracing the challenge and trying new ways to work through it.” I often introduce this lesson during math class, but it can apply to any area of our lives, and I can refer to the experience and conversation we had during any part of our day.
Research shows that just because students know the strategies does not mean they will engage in the appropriate strategies. Therefore, I try to provide opportunities where students can explicitly practice learning how, when, and why to use which strategies effectively so that they can become self-directed learners.
For example, I give students a math problem that will make many of them feel “stuck”. I will say, “Your job is to get yourselves stuck—or to allow yourselves to get stuck on this problem—and then work through it, being mindful of how you’re getting yourselves unstuck.” As students work, I check-in to help them name their process: “How did you get yourself unstuck?” or “What was your first step? What are you doing now? What might you try next?” As students talk about their process, I’ll add to a list of strategies that students are using and, if they are struggling, help students name a specific process. For instance, if a student says he wrote the information from the math problem down and points to a chart, I will say: “Oh that’s interesting. You pulled the important information from the problem out and organized it into a chart.” In this way, I am giving him the language to match what he did, so that he now has a strategy he could use in other times of struggle.
The charts grow with us over time and are something that we refer to when students are stuck or struggling. They become a resource for students and a way for them to talk about their process when they are reflecting on and monitoring what did or did not work.
For me, as a teacher, it is important that I create a classroom environment in which students are problem solvers. This helps tie struggles to strategies so that the students will not only see value in working harder but in working smarter by trying new and different strategies and revising their process. In doing so, they will more successful the next time around.
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Problem-solving is the ability to identify and solve problems by applying appropriate skills systematically.
Problem-solving is a process—an ongoing activity in which we take what we know to discover what we don't know. It involves overcoming obstacles by generating hypo-theses, testing those predictions, and arriving at satisfactory solutions.
Problem-solving involves three basic functions:
Generating new knowledge
Problem-solving is, and should be, a very real part of the curriculum. It presupposes that students can take on some of the responsibility for their own learning and can take personal action to solve problems, resolve conflicts, discuss alternatives, and focus on thinking as a vital element of the curriculum. It provides students with opportunities to use their newly acquired knowledge in meaningful, real-life activities and assists them in working at higher levels of thinking (see Levels of Questions ).
Here is a five-stage model that most students can easily memorize and put into action and which has direct applications to many areas of the curriculum as well as everyday life:
Here are some techniques that will help students understand the nature of a problem and the conditions that surround it:
- List all related relevant facts.
- Make a list of all the given information.
- Restate the problem in their own words.
- List the conditions that surround a problem.
- Describe related known problems.
For younger students, illustrations are helpful in organizing data, manipulating information, and outlining the limits of a problem and its possible solution(s). Students can use drawings to help them look at a problem from many different perspectives.
Understand the problem. It's important that students understand the nature of a problem and its related goals. Encourage students to frame a problem in their own words.
Describe any barriers. Students need to be aware of any barriers or constraints that may be preventing them from achieving their goal. In short, what is creating the problem? Encouraging students to verbalize these impediments is always an important step.
Identify various solutions. After the nature and parameters of a problem are understood, students will need to select one or more appropriate strategies to help resolve the problem. Students need to understand that they have many strategies available to them and that no single strategy will work for all problems. Here are some problem-solving possibilities:
Create visual images. Many problem-solvers find it useful to create “mind pictures” of a problem and its potential solutions prior to working on the problem. Mental imaging allows the problem-solvers to map out many dimensions of a problem and “see” it clearly.
Guesstimate. Give students opportunities to engage in some trial-and-error approaches to problem-solving. It should be understood, however, that this is not a singular approach to problem-solving but rather an attempt to gather some preliminary data.
Create a table. A table is an orderly arrangement of data. When students have opportunities to design and create tables of information, they begin to understand that they can group and organize most data relative to a problem.
Use manipulatives. By moving objects around on a table or desk, students can develop patterns and organize elements of a problem into recognizable and visually satisfying components.
Work backward. It's frequently helpful for students to take the data presented at the end of a problem and use a series of computations to arrive at the data presented at the beginning of the problem.
Look for a pattern. Looking for patterns is an important problem-solving strategy because many problems are similar and fall into predictable patterns. A pattern, by definition, is a regular, systematic repetition and may be numerical, visual, or behavioral.
Create a systematic list. Recording information in list form is a process used quite frequently to map out a plan of attack for defining and solving problems. Encourage students to record their ideas in lists to determine regularities, patterns, or similarities between problem elements.
Try out a solution. When working through a strategy or combination of strategies, it will be important for students to …
Keep accurate and up-to-date records of their thoughts, proceedings, and procedures. Recording the data collected, the predictions made, and the strategies used is an important part of the problem solving process.
Try to work through a selected strategy or combination of strategies until it becomes evident that it's not working, it needs to be modified, or it is yielding inappropriate data. As students become more proficient problem-solvers, they should feel comfortable rejecting potential strategies at any time during their quest for solutions.
Monitor with great care the steps undertaken as part of a solution. Although it might be a natural tendency for students to “rush” through a strategy to arrive at a quick answer, encourage them to carefully assess and monitor their progress.
Feel comfortable putting a problem aside for a period of time and tackling it at a later time. For example, scientists rarely come up with a solution the first time they approach a problem. Students should also feel comfortable letting a problem rest for a while and returning to it later.
Evaluate the results. It's vitally important that students have multiple opportunities to assess their own problem-solving skills and the solutions they generate from using those skills. Frequently, students are overly dependent upon teachers to evaluate their performance in the classroom. The process of self-assessment is not easy, however. It involves risk-taking, self-assurance, and a certain level of independence. But it can be effectively promoted by asking students questions such as “How do you feel about your progress so far?” “Are you satisfied with the results you obtained?” and “Why do you believe this is an appropriate response to the problem?”
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Key Tips On Problem Solving Method Of Teaching
Problem-solving skills are necessary for all strata of life, and none can be better than classroom problem-solving activities. It can be an excellent way to introduce students to problem-solving skills, get them prepped and ready to solve real problems in real-life settings.
The ability to critically analyze a problem, map out all its elements and then prepare a solution that works is one of the most valuable skills; one must acquire in life. Educating your students about problem-solving techniques from an early age can be facilitated with in-class problem-solving activities. Such efforts encourage cognitive and social development and equip students with the tools they will need to tackle and resolve their lives.
So, what is a problem-solving method of teaching ?
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. In a problem-solving method, children learn by working on problems. This skill enables the students to learn new knowledge by facing the problems to be solved. It is expected of them to observe, understand, analyze, interpret, find solutions, and perform applications that lead to a holistic understanding of the concept. This method develops scientific process skills. This method helps in developing a brainstorming approach to learning concepts.
In simple words, problem-solving is an ongoing activity in which we take what we know to discover what we do not know. It involves overcoming obstacles by generating hypotheses, testing those predictions, and arriving at satisfactory solutions.
The problem-solving method involves three basic functions
- Seeking information
- Generating new knowledge
- Making decisions
This post will include key strategies to help you inculcate problem-solving skills in your students.
First and foremostly, follow the 5-step model of problem-solving presented by Wood
Woods' problem-solving model
Identify the problem .
Allow your students to identify the system under study by interpreting the information provided in the problem statement. Then, prepare a list of what is known about the problem, and identify the knowledge needed to understand (and eventually) solve it. Once you have a list of known problems, identifying the unknown(s) becomes simpler. The unknown one is usually the answer to the problem; however, there may be other unknowns. Make sure that your students have a clear understanding of what they are expected to find.
While teaching problem solving, it is very important to have students know how to select, interpret, and use units and symbols. Emphasize the use of units and symbols whenever appropriate. Develop a habit of using appropriate units and symbols yourself at all times. Teach your students to look for the words only and neglect or assume to help identify the constraints.
Furthermore, help students consider from the beginning what a logical type of answer would be. What characteristics will it possess?
Think about it
Use the next stage to ponder the identified problem. Ideally, students will develop an imaginary image of the problem at hand during this stage. They need to determine the required background knowledge from illustrations, examples and problems covered in the course and collect pertinent information such as conversion factors, constants, and tables needed to solve the problem.
Plan a solution
Often, the type of problem will determine the type of solution. Some common problem-solving strategies are: compute; simplify; use an equation; make a model, diagram, table, or chart; or work backwards.
Help your students choose the best strategy by reminding them again what they must find or calculate.
Carry out the plan
Now that the major part of problem-solving has been done start executing the solution. There are possibilities that a plan may not work immediately, do not let students get discouraged. Encourage them to try a different strategy and keep trying.
Encourage students to reflect. Once a solution has been reached, students should ask themselves the following questions:
- Does the answer make sense?
- Does it fit with the criteria established in step 1?
- Did I answer the question(s)?
- What did I learn by doing this?
- Could I have done the problem another way?
Other tips include
Ask open-ended questions.
When a student seeks help, you might be willing to give them the answer they are looking for so you can both move on. But what is recommend is that instead of giving answers promptly, try using open-ended questions and prompts. For example: ask What do you think will happen if..? Why do you think so? What would you do if you get into such situations? Etc.
Emphasize Process Over Product
For elementary students, reflecting on the process of solving a problem helps them develop a growth mindset. Getting an 'incorrect' response does not have to be a bad thing! What matters most is what they have done to achieve it and how they might change their approach next time. As a teacher, you can help students learn the process of reflection.
Model The Strategies
As children learn creative problem-solving techniques, there will probably be times when they will be frustrated or uncertain. Here are just a few simple ways to model what creative problem-solving looks like and sounds like.
- Ask questions in case you don't understand anything.
- Admit to not knowing the right answer.
- Discuss the many possible outcomes of different situations.
- Verbalize what you feel when you come across a problem.
- Practising these strategies with your students will help create an environment where struggle, failure and growth are celebrated!
Grappling is not confined to perseverance! This includes critical thinking, asking questions, observing evidence, asking more questions, formulating hypotheses and building a deep understanding of a problem.
There are numerous ways to provide opportunities for students to struggle. All that includes the engineering design process is right! Examples include:
- Engineering or creative projects
- Design-thinking challenges
- Informatics projects
- Science experiments
Make problem resolution relevant to the lives of your students
Limiting problem solving to class is a bad idea. This will affect students later in life because problem-solving is an essential part of human life, and we have had a chance to look at it from a mathematical perspective. Such problems are relevant to us, and they are not things that we are supposed to remember or learn but to put into practice in real life. These are things from which we can take very significant life lessons and apply them later in life.
What's your strategy? How do you teach Problem-Solving to your students? Do let us know in the comments.
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Problem-Solving Method in Teaching
The problem-solving method is a highly effective teaching strategy that is designed to help students develop critical thinking skills and problem-solving abilities . It involves providing students with real-world problems and challenges that require them to apply their knowledge, skills, and creativity to find solutions. This method encourages active learning, promotes collaboration, and allows students to take ownership of their learning.
Table of Contents
Definition of problem-solving method.
Problem-solving is a process of identifying, analyzing, and resolving problems. The problem-solving method in teaching involves providing students with real-world problems that they must solve through collaboration and critical thinking. This method encourages students to apply their knowledge and creativity to develop solutions that are effective and practical.
Meaning of Problem-Solving Method
The meaning and Definition of problem-solving are given by different Scholars. These are-
Woodworth and Marquis(1948) : Problem-solving behavior occurs in novel or difficult situations in which a solution is not obtainable by the habitual methods of applying concepts and principles derived from past experience in very similar situations.
Skinner (1968): Problem-solving is a process of overcoming difficulties that appear to interfere with the attainment of a goal. It is the procedure of making adjustments in spite of interference
Benefits of Problem-Solving Method
The problem-solving method has several benefits for both students and teachers. These benefits include:
- Encourages active learning: The problem-solving method encourages students to actively participate in their own learning by engaging them in real-world problems that require critical thinking and collaboration
- Promotes collaboration: Problem-solving requires students to work together to find solutions. This promotes teamwork, communication, and cooperation.
- Builds critical thinking skills: The problem-solving method helps students develop critical thinking skills by providing them with opportunities to analyze and evaluate problems
- Increases motivation: When students are engaged in solving real-world problems, they are more motivated to learn and apply their knowledge.
- Enhances creativity: The problem-solving method encourages students to be creative in finding solutions to problems.
Steps in Problem-Solving Method
The problem-solving method involves several steps that teachers can use to guide their students. These steps include
- Identifying the problem: The first step in problem-solving is identifying the problem that needs to be solved. Teachers can present students with a real-world problem or challenge that requires critical thinking and collaboration.
- Analyzing the problem: Once the problem is identified, students should analyze it to determine its scope and underlying causes.
- Generating solutions: After analyzing the problem, students should generate possible solutions. This step requires creativity and critical thinking.
- Evaluating solutions: The next step is to evaluate each solution based on its effectiveness and practicality
- Selecting the best solution: The final step is to select the best solution and implement it.
Verification of the concluded solution or Hypothesis
The solution arrived at or the conclusion drawn must be further verified by utilizing it in solving various other likewise problems. In case, the derived solution helps in solving these problems, then and only then if one is free to agree with his finding regarding the solution. The verified solution may then become a useful product of his problem-solving behavior that can be utilized in solving further problems. The above steps can be utilized in solving various problems thereby fostering creative thinking ability in an individual.
The problem-solving method is an effective teaching strategy that promotes critical thinking, creativity, and collaboration. It provides students with real-world problems that require them to apply their knowledge and skills to find solutions. By using the problem-solving method, teachers can help their students develop the skills they need to succeed in school and in life.
- Jonassen, D. (2011). Learning to solve problems: A handbook for designing problem-solving learning environments. Routledge.
- Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn? Educational Psychology Review, 16(3), 235-266.
- Mergendoller, J. R., Maxwell, N. L., & Bellisimo, Y. (2006). The effectiveness of problem-based instruction: A comparative study of instructional methods and student characteristics. Interdisciplinary Journal of Problem-based Learning, 1(2), 49-69.
- Richey, R. C., Klein, J. D., & Tracey, M. W. (2011). The instructional design knowledge base: Theory, research, and practice. Routledge.
- Savery, J. R., & Duffy, T. M. (2001). Problem-based learning: An instructional model and its constructivist framework. CRLT Technical Report No. 16-01, University of Michigan. Wojcikowski, J. (2013). Solving real-world problems through problem-based learning. College Teaching, 61(4), 153-156
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5 Teaching Mathematics Through Problem Solving
In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)
What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.
According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.
There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.
Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.
Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.
Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.
Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):
- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.
Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.
Key features of a good mathematics problem includes:
- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.
Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.
But look at the b ack.
It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.
When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!
Mathematics Tasks and Activities that Promote Teaching through Problem Solving
Choosing the Right Task
Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:
- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?
Low Floor High Ceiling Tasks
By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].
The strengths of using Low Floor High Ceiling Tasks:
- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice
Examples of some Low Floor High Ceiling Tasks can be found at the following sites:
- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month
Math in 3-Acts
Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:
Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.
In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.
Act Three is the “reveal.” Students share their thinking as well as their solutions.
“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:
- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete
Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:
- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.
“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:
- Inside Mathematics Number Talks
- Number Talks Build Numerical Reasoning
Saying “This is Easy”
“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.
When the teacher says, “this is easy,” students may think,
- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.
Instead, you and your students could say the following:
- “I think I can do this.”
- “I have an idea I want to try.”
- “I’ve seen this kind of problem before.”
Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.
Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?
What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.
Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.
One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”
You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can
- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.
Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.
any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method
should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning
involves teaching a skill so that a student can later solve a story problem
when we teach students how to problem solve
teaching mathematics content through real contexts, problems, situations, and models
a mathematical activity where everyone in the group can begin and then work on at their own level of engagement
20 seconds to 2 minutes for students to make sense of questions
Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
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21 Ways to Encourage Students to Take Pride in their Assignments
Teaching students about brendan coyle, teaching students about antiquarks, teaching students about deborah in the old testament, teaching students about the examples of density, teaching students about eastern and western ghats, teaching students about sekhmet: the goddess of war, teaching students about general santander, teaching students about the author of heidi, johanna spyri, teaching students about korea on the map, strategies and methods to teach students problem solving and critical thinking skills.
The ability to problem solve and think critically are two of the most important skills that PreK-12 students can learn. Why? Because students need these skills to succeed in their academics and in life in general. It allows them to find a solution to issues and complex situations that are thrown there way, even if this is the first time they are faced with the predicament.
Okay, we know that these are essential skills that are also difficult to master. So how can we teach our students problem solve and think critically? I am glad you asked. In this piece will list and discuss strategies and methods that you can use to teach your students to do just that.
- Direct Analogy Method
A method of problem-solving in which a problem is compared to similar problems in nature or other settings, providing solutions that could potentially be applied.
- Attribute Listing
A technique used to encourage creative thinking in which the parts of a subject, problem, or task are listed, and then ways to change those component parts are examined.
- Attribute Modifying
A technique used to encourage creative thinking in which the parts of a subject, problem, or task are listed, and then options for changing or improving each part are considered.
- Attribute Transferring
A technique used to encourage creative thinking in which the parts of a subject, problem or task listed and then the problem solver uses analogies to other contexts to generate and consider potential solutions.
- Morphological Synthesis
A technique used to encourage creative problem solving which extends on attribute transferring. A matrix is created, listing concrete attributes along the x-axis, and the ideas from a second attribute along with the y-axis, yielding a long list of idea combinations.
SCAMPER stands for Substitute, Combine, Adapt, Modify-Magnify-Minify, Put to other uses, and Reverse or Rearrange. It is an idea checklist for solving design problems.
- Direct Analogy
A problem-solving technique in which an individual is asked to consider the ways problems of this type are solved in nature.
- Personal Analogy
A problem-solving technique in which an individual is challenged to become part of the problem to view it from a new perspective and identify possible solutions.
- Fantasy Analogy
A problem-solving process in which participants are asked to consider outlandish, fantastic or bizarre solutions which may lead to original and ground-breaking ideas.
- Symbolic Analogy
A problem-solving technique in which participants are challenged to generate a two-word phrase related to the design problem being considered and that appears self-contradictory. The process of brainstorming this phrase can stimulate design ideas.
- Implementation Charting
An activity in which problem solvers are asked to identify the next steps to implement their creative ideas. This step follows the idea generation stage and the narrowing of ideas to one or more feasible solutions. The process helps participants to view implementation as a viable next step.
- Thinking Skills
Skills aimed at aiding students to be critical, logical, and evaluative thinkers. They include analysis, comparison, classification, synthesis, generalization, discrimination, inference, planning, predicting, and identifying cause-effect relationships.
Can you think of any additional problems solving techniques that teachers use to improve their student’s problem-solving skills?
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A Powerful Rethinking of Your Math Classroom
We look at strategies you can reset this year—adjusting your testing regimen, tackling math anxiety, encouraging critical thinking, and fostering a mistake-friendly environment.
The beginning of school is a great time for teachers—both veteran and early career—to consider ways they can improve upon their classroom practices over the next year. This may be especially important for math teachers, who often spend the early days of the school year confronting math anxiety , convincing students that they are indeed “math people,” and coming up with engaging practices to guide students to find pleasure in math challenges. Innovating in these areas has the potential to yield big benefits for students all year long.
To help you re-imagine some of your teaching practices this year, we’ve pulled together a collection of big and small strategies aimed at:
- helping students adopt a more productive, curious mindset when they approach math,
- engaging students as soon as they walk through the door,
- rethinking how you handle testing,
- fostering a mistake-friendly environment,
- incorporating humanities-style discussions into math, and
- building what Peter Liljedahl calls a “ thinking classroom .”
Tackle Negative Math Mindsets
“Most of my work as a math teacher isn’t even math ,” former middle school math teacher José Vilson has said. “It’s helping students believe that they can also do math.”
Striking a similar note, middle school math coordinator Alessandra King writes that supporting students—particularly students from marginalized backgrounds—to develop a “ positive mathematical identity ” is crucial to fostering a sense of belonging in a larger math community, boosting “their willingness and ability to engage” in challenging work.
King recommends spending the first few days of the school year spotlighting mathematicians from the past who reflect the makeup of your classroom. In her all-girls school, for example, she has students read and respond to a play about the achievements and struggles of Maria Agnesi, Sofya Kovalevskaya, and Emmy Noether, three historical female mathematicians. You can also hang up posters of famous Black or Latino mathematicians—such as Euphemia Haynes, the first African American woman to earn a PhD in math, or Robert Luis Santos, a Latino statistician and director of the U.S. Census Bureau—and devote lessons to discussing their achievements and backgrounds.
To get students thinking about—and challenging—their own math identity, educator Rolanda Baldwin suggests asking students early in the year to write a “ math autobiography .” They might respond to questions like: “How do you feel about math? How did your relationship with math change over time?” To draw students closer together—and make them realize that many students, of all backgrounds, struggle—have them share their responses in groups, or with the entire class.
Sometimes, our best intentions can go awry. Rachel Fuhrman, a former special education math teacher, notes that teachers can sometimes dampen a student’s math identity by wheeling out phrases that might seem helpful but can actually demotivate students , like: “This is so easy.” Framing something as “easy,” she writes, can leave students feeling uncomfortable or afraid of asking crucial, clarifying questions.
Engage Students the Moment They Enter the Classroom
To set a playful tone and lower the stakes of the work so students can effectively experiment and collaborate, high school math teacher Lorenzo Robinson suggests starting off class with fun, challenging brain teasers .
For example, ask students to draw a cross on a sheet of paper (you should draw one on the board as a point of reference). Ask students to draw two straight lines that will segment or cut the cross into pieces. The goal is to cut the cross in a way that produces the most pieces .
You could also try math riddles: Ask students to imagine they have two coins that total 30 cents. Tell them that one of the coins is not a nickel, and ask them to figure out what the two coins are.
Robinson finds that brain teasers like these can get students primed for problem solving and critical thinking, without even realizing it.
Lower the Stakes of Testing
Big tests—centered around units or near the end of a grading period—are staples of many math classrooms. But that doesn’t mean they have to be the only opportunity for students to show what they know. One way to lower the stakes for students and give them more opportunities to practice, while providing yourself more of an opportunity to both teach content and check for understanding is to give short assessments on a regular schedule.
Math coordinator Steven Goldman’s school switched from tests to checkpoints . These short assessments include a mix of current and past topics—retrieval practice is a research-backed way to support greater learning , after all—with some repetition for the most important skills students need to know. The checkpoints are given every two weeks and are not formally graded, but mistakes are noted and teachers leave feedback to guide student revisions. The change, Goldman writes, resulted in a big reduction in student stress levels—something research shows is a benefit of frequent practice tests, alongside a boost in long-term retention. Because the checkpoints happen all the time, Goldman and his colleagues don’t have to “go through all kinds of contortions to finish a unit before a break or on a Thursday so that we could give the test at the right time.”
Fresno State math instructor Howie Hua suggests lowering the stake in your classroom by allowing students to discuss a test before starting to work on it . Hua recommends having students put their pencils on the floor so they can focus on their discussion. Hand out the test and give students five minutes to discuss strategies they can use to solve the problems.
Good Mistakes, Better Mistakes
Mistakes are bound to happen in any math classroom, and how you respond to them throughout the year can make a world of difference. Making light of your own mistakes is a good first gambit, but research suggests a more advanced approach: Giving students space to make mistakes—and opportunities to analyze and discuss them with peers—can better encode information in their brains than simply providing them with the correct answer.
To use mistakes as building blocks to better solutions, math teacher Emma Chiapetta uses an ingenious, small-group activity that asks students to identify and reflect on common mistakes , and then explain the rationale behind them to their peers. Here’s how it works: Randomly separate students into groups and assign each group to a board to generate a problem and solve it incorrectly. Groups rotate so they’re looking at a problem and an incorrect solution, and have to identify the error and solve the problem correctly. After another rotation—so students are looking at a problem and both the incorrect and correct solutions—they explain to the class the mistake made by the first group and the correct solution provided by the second. The activity, Chiapetta writes, helps students think about the same content from various perspectives, which can lead to deeper understanding.
Not all mistakes are created equal, and they often conceal thoughtful, underlying work. Former math teacher Colin Seale asks students to reflect on which wrong answer to a problem is “ more right .” He suggests offering students two equations that are both incorrect—perhaps one is wrong conceptually and the other computationally. Seale notes that this exercise gets students to tease out nuances around the skills and concepts they’re learning, while also correcting their approach to similar problems moving forward.
Bring Humanities Strategies to Math Class
The discussion, analysis of reasoning, and argumentation that happen in humanities classrooms can be extremely useful in math classrooms to help students slow down and think through the work they’re doing , says middle school math teacher Connell Cloyd.
He does this in his classroom by posting four incorrectly solved math problems around the room and having students rotate around each problem in groups to discuss the error and write down (in complete sentences) a claim and supporting evidence to show why they believe the error occurred. As they rotate around, students read the arguments of peers and either support an argument or refute it with new evidence. This is like adding techniques from debate to Chiappetta’s strategy.
Math journals can also inject more writing and reasoning into your classroom. Former math teacher Nell McAnelly prompts students to reflect on concepts they’re having trouble with : They work through a problem and then write about the strategy they used to solve it, or informally journal about other approaches they could have used. Doing so, she writes, gives students a chance to “synthesize learning and address unanswered questions.”
Driving Deep, Critical Thinking
A traditional math classroom, where the teacher demonstrates a skill numerous times before students take the reins, can inhibit higher order thinking and result in students who “mimic” teachers rather than develop their own strategies to solve complex problems , says Peter Liljedahl, a researcher and professor of mathematics at Simon Fraser University.
Instead of starting lessons with direct instruction, Liljedahl says, give students novel “ thinking tasks ” to work on in groups. These are problem-solving activities and mental puzzles that should get students in the mindset of challenging themselves. Work groups should not be chosen based on ability or students’ preferences—Liljedahl’s research shows that students are more likely to contribute in randomized groups. Having students stand while they engage in this collaborative, messy thinking—as Chiappetta does in her mistake-analysis exercise—is another way to engage them.
Instead of using notebooks to compute, Liljedahl calls for groups to work at vertical, non-permanent surfaces, such as whiteboards, blackboards, or windows—surfaces that he says promote more risk-taking because students have the freedom to quickly erase false starts without feeling committed. As students work in groups, teachers bounce around the room and avoid directly answering questions such as “Is this right?” that circumvent student thinking and instead make suggestions that lead to further independent thinking.
Because the goal of this approach is to get students to develop perseverance, curiosity, and collaboration, Liljedahl suggests evaluating them in a way that prioritizes these competencies. He developed formative assessments that focus on informing students “about where they are and where they’re going in their learning.” These include observations, check-for-understanding questions, and unmarked quizzes. Summative assessments, meanwhile, should focus less on end products and more on the process of learning through both group and individual work.
Shifting to this thinking classroom model requires a fundamental shift in how you run your classroom, but it could result in large rewards throughout the year.
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30 Instructional Strategies Examples for Every Kind of Classroom
So many ways to help students learn!
Looking for some new ways to teach and learn in your classroom? This roundup of instructional strategies examples includes methods that will appeal to all learners and work for any teacher.
What are instructional strategies?
In the simplest of terms, instructional strategies are the methods teachers use to achieve learning objectives. In other words, pretty much every learning activity you can think of is an example of an instructional strategy. They’re also known as teaching strategies and learning strategies.
These strategies are sometimes broken into five types: direct, indirect, experiential, interactive, and independent. ( Learn more about the types of instructional strategies here. ) But many learning strategies examples fit into multiple categories. The more options teachers have in their tool kit, the easier it will be to help all learners achieve their goals.
Don’t be afraid to try new strategies from time to time—you just might find a new favorite! Here are some of the most common instructional strategies examples.
Source: STEM Activities for Kids
In this indirect learning method, students work their way through a problem to find a solution. Along the way, they must develop the knowledge to understand the problem and use creative thinking to solve it. STEM challenges are terrific examples of problem-solving instructional strategies.
Learn more: Teaching Problem Solving
This method gets a lot of flack these days for being “boring” or “old-fashioned.” It’s true that you don’t want it to be your only instructional strategy, but short lectures are still very effective learning tools. This type of direct instruction is perfect for imparting specific detailed information or teaching a step-by-step process. And lectures don’t have to be boring—just look at TED Talks .
Learn more: Advantages and Disadvantages of Lecturing
3. Didactic Questioning
These are often paired with other direct instruction methods like lecturing. The teacher asks questions to determine student understanding of the material. They’re often questions that start with “who,” “what,” “where,” and “when.”
Learn more: Asking Questions
In this direct instruction method, students watch as a teacher demonstrates an action or skill. This might be seeing a teacher solving a math problem step-by-step, or watching them demonstrate proper handwriting on the whiteboard. Usually, this is followed by having students do hands-on practice or activities in a similar manner.
Learn more: Preparing Effective Demonstrations for the Classroom and Laboratory (PDF)
Ever since Aesop’s fables, we’ve been using storytelling as a way to teach. Stories grab students’ attention right from the start and keep them engaged throughout the learning process. Real-life stories and fiction both work equally well, depending on the situation.
Learn more: Storytelling in the Classroom
6. Drill & Practice
If you’ve ever used flash cards to help kids practice math facts, or had your whole class chant the spelling of a word out loud, you’ve used drill & practice. It’s another one of those traditional instructional strategies examples. When kids need to memorize specific information or master a step-by-step skill, drill & practice really works.
Learn more: Drill and Practice
7. Spaced Repetition
Often paired with direct or independent instruction, spaced repetition is a method where students are asked to recall certain information or skills at increasingly longer intervals. For instance, the day after discussing the causes of the American Civil War in class, the teacher might return to the topic and ask students to list the causes. The following week, the teacher asks them once again, and then a few weeks after that. Spaced repetition helps make knowledge stick, and it is especially useful when it’s not something students practice each day but will need to know in the long term (such as for a final exam).
Learn more: Spaced Repetition: A Guide to the Technique
8. Project-Based Learning
When kids participate in true project-based learning, they’re learning through indirect and experiential strategies. As they work to find solutions to a real-world problem, they develop critical thinking skills and learn by research, trial and error, collaboration, and other experiences.
Learn more: What Is Project-Based Learning?
9. Concept Mapping
Source: Cornell University
Students use concept maps to break a subject down into its main points, and draw connections between these points.
Learn more: Concept Mapping at Cornell University
10. Case Studies
When you think of case studies, law school is probably the first thing that jumps to mind. But this method works at any age, for a variety of topics. This indirect learning method teaches students to use material to draw conclusions, make connections, and advance their existing knowledge.
Learn more: The Case Method
11. Reading for Meaning
This is different than learning to read. Instead, it’s when students use texts (print or digital) to learn about a topic. This traditional strategy works best when students already have strong reading comprehension skills. Try our free reading comprehension bundle to give students the ability to get the most out of reading for meaning.
Learn more: Reading for Meaning
12. Science Experiments
Source: Rhythms of Play
This is experiential learning at its best. Hands-on experiments let kids learn to establish expectations, create sound methodology, draw conclusions, and more.
Learn more: Hundreds of Science Experiment Ideas for Kids and Teens
13. Field Trips
Heading out into the real world gives kids a chance to learn indirectly, through experiences. They may see concepts they already know put into practice, or learn new information or skills from the world around them.
Learn more: The Big List of PreK-12 Field Trip Ideas
Teachers have long known that playing games is a fun (and sometimes sneaky) way to get kids to learn. You can use specially-designed educational games for any subject. Plus, regular board games ( like these classroom favorites ) often involve a lot of indirect learning about math, reading, critical thinking, and more.
Learn more: 10 Benefits to Playing Games in the Classroom
This strategy combines experiential, interactive, and indirect learning all in one. The teacher sets up a simulation of a real-world activity or experience. Students take on roles and participate in the exercise, using existing skills and knowledge or developing new ones along the way. At the end, the class reflects separately and together on what happened, and what they learned.
Learn more: A Simple Approach to Using Simulations in Any Classroom
16. Service Learning
This is another instructional strategies example that takes students out into the real world. It often involves problem-solving skills and gives kids the opportunity for meaningful social-emotional learning.
Learn more: What Is Service Learning?
17. Peer Instruction
It’s often said the best way to learn something is to teach it to others. Studies into the so-called “Protégé Effect” seem to prove it too. In order to teach, you first must understand the information yourself. Then, you have to find ways to share it with others—sometimes more than one way. This deepens your connection to the material, and it sticks with you much longer. Try having peers instruct one another in your classroom, and see the magic in action.
Learn more: Want Students To Remember What They Learn? Have Them Teach It
Source: The New York Times
Some teachers shy away from debate in the classroom, afraid it will become too adversarial. But learning to discuss and defend various points of view is an important life skill. Debates teach students to research their topic, make informed choices, and argue effectively using facts instead of emotion.
Learn more: 100 High School Debate Topics To Engage Every Student
19. Class or Small-Group Discussion
Class, small-group, and pair discussions are all excellent interactive instructional strategies examples. As students discuss a topic, they clarify their own thinking and learn from others’ experiences and opinions. Of course, in addition to learning about the topic itself, they’re also developing valuable active listening and collaboration skills.
Learn more: 13 Strategies to Improve Classroom Discussions
Take your classroom discussions one step further with the fishbowl method. A small group of students sits in the middle of the class. They discuss and debate a topic, while their classmates listen silently and make notes. Eventually, the teacher opens the discussion to the whole class, who offer feedback and present their own assertions and challenges.
Learn more: How I Use Fishbowl Discussions to Engage Every Student
Rather than having a teacher provide examples to explain a topic or solve a problem, students do the work themselves. Remember the one rule of brainstorming: Every idea is welcome. Ensure everyone gets a chance to participate, and form diverse groups to generate lots of unique ideas.
Learn more: Brainstorming in the Classroom
Role-playing is sort of like a simulation but less intense. It’s perfect for practicing soft skills and focusing on social-emotional learning . Put a twist on this strategy by having students model bad interactions as well as good ones and then discussing the difference.
Learn more: How To—and How Not To—Teach Role Plays
This structured discussion technique is simple: First, students think about a question posed by the teacher. Pair students up, and let them talk about their answer. Finally open it up to whole-class discussion. This helps kids participate in discussions in a low-key way and gives them a chance to “practice” before they talk in front of the whole class.
Learn more: Think-Pair-Share and 10 Fun Alternatives
24. Learning Centers
Source: Pocket of Preschool
Foster independent learning strategies with centers just for math, writing, reading, and more. Provide a variety of activities, and let kids choose how they spend their time. They often learn better from activities they enjoy.
Learn more: 6 Important Benefits of Learning Centers in the Classroom
25. Computer-Based Instruction
Once a rarity, now a daily fact of life, computer-based instruction lets students work independently. They can go at their own pace, repeating sections without feeling like they’re holding up the class. Teach students good computer skills at a young age so you’ll feel comfortable knowing they’re focusing on the work and doing it safely.
Learn more: Computer-Assisted Instruction and Reading
Writing an essay encourages kids to clarify and organize their thinking. Written communication has become more important in recent years, so being able to write clearly and concisely is a skill every kid needs. This independent instructional strategy has stood the test of time for good reason.
Learn more: The Big List of Essay Topics for High School
27. Research Projects
Here’s another oldie-but-goodie! When kids work independently to research and present on a topic, their learning is all up to them. They set the pace, choose a focus, and learn how to plan and meet deadlines. This is often a chance for them to show off their creativity and personality too.
Learn more: Tips and Ideas for Research Projects in the Classroom
Personal journals give kids a chance to reflect and think critically on topics. Whether responding to teacher prompts or simply recording their daily thoughts and experiences, this independent learning method strengthens writing and intrapersonal skills.
Learn more: The Benefits of Journaling in the Classroom
29. Graphic Organizers
Source: Around the Kampfire
Graphic organizers are a way of organizing information visually to help students understand and remember it. A good organizer simplifies complex information and lays it out in a way that makes it easier for a learner to digest. Graphic organizers may include text and images, and help students make connections in a meaningful way.
Learn more: Graphic Organizers 101: Why and How To Use Them
Jigsaw combines group learning with peer teaching. Students are assigned to “home groups.” Within that group, each student is given a specialized topic to learn about. They join up with other students who were given the same topic, then research, discuss, and become experts. Finally, students return to their home group and teach the other members about the topic they specialized in.
Learn more: Why Use Jigsaw?
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Plus, check out the 7 things the best instructional coaches do, according to teachers ..
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Teachers’ pedagogical content knowledge in teaching word problem solving strategies
- Original Article
- Open Access
- Published: 13 December 2019
- volume 52 , pages 165–178 ( 2020 )
You have full access to this open access article
- Csaba Csíkos ORCID: orcid.org/0000-0003-3328-5535 1 &
- Judit Szitányi ORCID: orcid.org/0000-0003-3266-5966 1
Cite this article
This research addressed Hungarian pre-service and in-service (both elementary and lower secondary) teachers’ pedagogical content knowledge concerning the teaching of word problem solving strategies. By means of a standardized interview protocol, participants (N = 30) were asked about their judgement on the difficulty of teaching word problems, the factors they find difficult, and their current teaching practice. Furthermore, based on a comparative analysis of Eastern European textbooks, we tested how teachers’ current beliefs and views relate to the word problem solving algorithm described in elementary textbooks. The results suggest that in the teachers’ opinion, explicit teaching of a step-by-step algorithm is feasible and desirable as early as in the 1st school grade. According to our results, two approaches (namely, paradigmatic- and narrative-oriented) concerning how to teach the process of word problems solving, originally revealed by Chapman, were found. Furthermore, teachers in general agreed with the approach taken in the textbooks on the subject of what kinds of word problems should be used, and that explicit teaching of word problem solving strategies should be introduced by using simple, routine word problems as examples.
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In this research we aimed to reveal teachers’ views and pedagogical content knowledge on teaching elementary students to solve word problems. Teaching how to solve word problems has long been an integral part of mathematics teaching and teacher education in Hungary. The importance of teaching word problems and teaching how students should solve them have not really been questioned or challenged, and the unfavourable results of international educational surveys (IEA, PISA) and internationally comparable results in scientific investigations on word problems (see Csíkos et al. 2012 ; Verschaffel et al. 2010 ) drew attention to the role word problems might play in mathematics education in Hungary. Our research focused on pre-service and in-service teachers’ pedagogical content knowledge concerning word problems at a time when explicit teaching of the solution steps of a mathematical problem usually starts as early as in the first grade of schooling. In an attempt to redefine how and why word problems are taught, it is very important to find out how in-service and pre-service mathematics teachers think about word problems.
As Pollak ( 1969 , p. 393) pointed out, students are involved in the application of mathematics “mainly through … ‘word’ problems”. Word problems are often considered a separate genre of mathematical tasks as an addendum to the learning material. According to the nationwide survey of the Committee of Mathematics Education (Hungarian Academy of Sciences 2016 ), many teachers would eliminate word problems from the first and second grades of public schooling. This view is based on a common sense understanding of what a word problem is and how word problem solution steps are usually introduced in the schools. One reason repeatedly mentioned is that solving word problems requires a relatively high level of reading skills and reading comprehension, and it is not in the mathematics class that such skills and abilities should be improved and assessed.
“Word problems can be defined as verbal descriptions of problem situations wherein one or more questions are raised the answer to which can be obtained by the application of mathematical operations to numerical data available in the problem statement” (Verschaffel et al. 2010 , p. ix.) This definition involves several types of word problems from mere routine tasks to more sophisticated realistic word problems. Among the several possible roles word problems may play in the classroom, there seems to be an overemphasis on two purposes. Word problems are often used in a way that Palm ( 2006 ) described as “many of them are … merely ordinary school mathematics tasks ‘dressed up’ with an out-of-school figurative context” (p. 42). Maybe students’ superficial strategies in word problem solving are rooted in both our teaching practice and the cultural traditions displayed in the textbooks? Another usual aim in the mathematics classroom (especially in the lower elementary grades) is teaching an algorithm to be followed step by step. Overreliance on either of these two purposes may overshadow other possible functions word problems can (and should) fulfil, e.g., developing self-explanation (see Fonseca and Chi 2011 ) or shaping a positive attitude towards mathematics.
2 Word problem solving strategies
There has been ample evidence that students in their first years of schooling tend to follow a superficial word problem solving strategy (Verschaffel et al. 1997 ), i.e., a process comprising the steps of finding two or more numbers in the text and selecting and executing arithmetic operation(s); and the numerical results are claimed to be the answer. The truth is that in the vast majority of word problems presented in lower elementary school grades this superficial strategy works fairly well. Furthermore, the roots of mastering such a strategy originate in teachers’ instructional strategy, a part of which is teaching word problem solving algorithms.
Word problem solving algorithms are more or less explicitly taught to elementary school children. In Hungarian teacher training textbooks and students’ textbooks, several steps of word problem solving are listed and expected to be followed in the classroom. One such example is borrowed from a widely accessible on-line platform containing educational materials for both teachers and students: Footnote 1 reading, understanding, planning (and estimation), drawing (if needed), calculations, verification, answering the question . These steps represent a kind of common understanding of what a word problem solution should look like in written form and how the solution steps are to be scored. While some of the seven (or even eight, if estimation is separated from planning) steps are necessarily present in all kinds of word problems, others are more specific to certain types of word problems only. For example, sometimes the solution comes not from executing a single arithmetic operation, and often there is no sense in giving any estimation before making calculations. The last two steps may be intended to cover what, otherwise, researchers would label as the interpretation of the solution, but these steps usually prove to be just mechanical activities performed, and in the case of failing to accomplish them students will get a penalty in the form of losing points in the assessment.
Since in their textbook for pre-service elementary teachers Neményi and Szendrei ( 1997 ) present a scientifically based description of four general steps of word problem solving, the steps listed above have to be considered as a distillation of pedagogical practice still living among us. Neményi and Szendrei describe four main steps of word problem solving, namely understanding the problem, seeking for a mathematical model and turning the original problem into a mathematical problem, solving the mathematical problem, and interpretation of the mathematical solution . This general four-step model is echoing that of Pólya’s ( 1945 ) problem solving strategy phases. Understanding the text, preparing a plan for the solution, implementing the plan and checking the results, have become a dominant part of the teaching of mathematics in Eastern Europe.
How and why a general and scientifically-based description of word problem solving (as provided by Neményi and Szendrei 1997 ) has been transformed into an ordinary and practical to-do list claimed to be ‘ the ’ steps of word problem solving is in itself a complex question. It requires the consideration of issues of instructional and assessment methods, dilemmas of equity and even deep philosophical questions of mathematics education. Some of these issues concerning teachers’ pedagogical content knowledge are discussed later in this paper, and other issues can be understood from a societal perspective.
Hungarian mathematics education has several facets that can be interpreted in a wider Eastern European context (Csíkos et al. 2019 ). According to Howson ( 1980 ), mathematics education in socialist countries (this is the group to which Eastern European countries belonged for long decades) has three remarkable characteristics: central curricula, word problem content stressing industrial and societal topics, emphasis on talent development and competitions. Here word problems served not only as a training field for arithmetic skills and for following some solution step algorithm, but word problems themselves and especially the way they are instructed were shaped by political and sociological factors. In a Hungarian follow-up study, Vidákovich and Csapó ( 1998 ) could use only 64 out of more than 300 word problems that were worded in the 70s because of the politically motivated content elements.
Other possible and potential functions of solving word problems, such as facilitating general problem solving skills, took a back seat. As a further consequence, teaching and assessing word problems failed to give place for alternative views in either the solvability of a problem or the interpretation of the data and the calculations executed. Naturally, a uniform method of how teaching and assessing (and ‘surviving’ for both teachers and students) should be done, has developed. The textbooks used currently and in the previous decades bring the implicit and often an explicit message about an expected word problem solving strategy.
3 Word problem solving strategies in elementary school textbooks
According to the TIMSS 2011 survey, which explicitly asked fourth grade teachers about their textbook usage habits, the great majority of elementary teachers (international average: 75%) used textbooks as a basis for instruction. In Hungary, the dominant role of textbooks was marked with a figure of 88% in this respect (Balázsi et al. 2012 ). According to van Zanten and van den Heuvel-Panhuizen ( 2018 ) the case is similar in lower grades in The Netherlands, and supposedly in Hungary as well. Furthermore, there are some results available on how teachers judge the mathematics textbooks from different aspects (as in a study with Hungarian teachers in Romania by Baranyai and Stark 2011 ), and there are well-documented differences among word problem types in the textbooks used in different countries (see e.g., Olkun and Toluk 2002 ; Stigler et al. 1986 ).
Nevertheless, how word problems are presented in mathematics textbooks, and the way in which the solution process is shown and illustrated, indicate both the current practices teachers are thought (or expected) to follow and textbook writers’ suggestions about how they would present and solve those word problems.
At the start of teaching school mathematics, teachers use situations expressed in words to represent an arithmetic operation. In this period the basic purpose of introducing word problems is only understanding the structure of an operation, as illustrated in Fig. 1 , demonstrating the idea of multiplication as repeated addition.
Example of a word problem illustration for visualizing a simple arithmetic operation (Lehotanová 2013 , p. 23)
Later when pupils are already acquainted with arithmetic operations, the main purpose of using word problems is to make children recognize and understand what the text describes and select a suitable model for problem solving. At the same time, explicit steps of a word problem solving method are provided in a popular Russian textbook.
The way in which the steps of the word problem solution are provided in Fig. 2 raises several issues. The explicitness of writing the solution steps in bold, and listing them in a consecutive order, have a strong message for both the teacher and the students. One strong message is that besides practicing an arithmetic operation dressed-up in familiar content, teachers and students are invited to do some extra work beyond mere calculation, thereby acting as required in the solution process of a word problem. An even stronger message may be that teachers and students are invited to solve all arithmetic word problems by following all the steps displayed in the textbook. Such a linear solution process may at the same time dissuade the teachers and students from applying a modelling cycle as described by Borromeo Ferri ( 2006 ).
Explicit teaching of the steps of word problem solving in a Russian textbook (Moro, Volkova and Stepanova 2011 , p. 88). Expressions on the left-hand side of the figure in bold, from the top to the bottom, are as follows: conditions of the task, question (of the task), problem solving, and answer. The word problem itself is presented as follows: “It is known that a task consists of conditions and an answer. We will learn how to solve problems and how to answer them. There are 6 pencils in the box and 2 others on the table. How many pencils are there together?”
The metacognitive components of students’ thinking necessarily develop while following the prescribed steps of the solution. They are invited and required to plan, monitor and evaluate their own thinking processes. These metacognitive processes have for long decades been recognized and addressed in mathematics education, mainly due to the pioneering work of the Hungarian György Pólya ( 1945 ). Pólya has a real cult among Hungarian mathematics educators, but, as we hypothesize, his maxims concerning problem solving may be misinterpreted when teaching and learning mathematical word problems in the elementary school years.
Having reviewed Eastern European textbooks of Romania, Russia, Slovakia, Croatia and Hungary, we found similarities in these books; they are as follows.
As it can be observed in Fig. 2 , usually two birds are killed with one stone. It is not only the actual solution of the word problem that is presented, but also the demonstration of the algorithm which can later be used in general in solving word problems.
When teaching the algorithm, two important factors must be taken into consideration. Firstly, it seems to be of doubtful validity to combine the processes of understanding and solving the word problem with teaching and memorizing an algorithm of the solution process. Secondly, choosing a suitable problem is essential, since if the task is too easy, the child may have difficulty in understanding why following a prescribed algorithm is necessary (there are 8 pencils and that’s all).
In most of the textbooks, a component of the solution algorithm of a word problem is the depiction of the text in some manner, e.g., by making a drawing. Making a drawing can be essential in solving word problems. The capability of making a drawing that suits the task is also the result of the learning process. However, the manner and the extent to which help is provided by the teacher are crucial points. As suggested by Csíkos et al. ( 2012 ), it is not making a drawing that is in itself fruitful but building metacognitive knowledge components on a possible repertory of drawings and about the conditions under which those drawings support the solution process.
An essential part of teaching the solution algorithm is checking the result.
Checking the result can be accomplished in two ways when solving a word problem. On the one hand, it provides an opportunity to check the operation setup based on the model by means of executing the inverse arithmetic operation, and on the other hand checking may refer to a different understanding of the text.
When we examined the textbooks, we found that the algorithm of the solution of the word problem can be observed as early as in the first school grade in the countries examined.
We could see a number of word problems in the textbooks which may be understood only with difficulty by the children, without the teacher’s help and conducting a discussion in the classroom. The first- and second-grade students may not possess the reading comprehension required to understand and solve the word problems by means of individual work.
Since textbook word problems are of a specific kind, teachers have an important task in handling those word problems and in fulfilling a desirable word problem teaching practice. Therefore, it is worth looking at teachers’ pedagogical content knowledge concerning the teaching of word problems.
4 The role of teachers’ pedagogical content knowledge in teaching word problem solving strategies
Studying the issue of when and how to introduce word problem solving strategies in elementary school, it is important to find out what teachers think and know about these issues. They may agree or disagree with what textbooks intend to propose, and it is the teachers’ experience and pedagogical content knowledge that certainly influence their textbook usage.
Teachers’ pedagogical content knowledge (PCK) is usually described by means of the Shulmanian sense of the term (Shulman 1986 ). Among the knowledge components necessary for anyone to be a successful teacher, one form of teachers’ knowledge is of utmost importance, namely the knowledge form that is built on the content knowledge but goes beyond that by powerful analogies, illustrations, demonstrations, etc., in order to make the content to be learnt comprehensible to others. Therefore, mathematics teachers must possess—beyond a high level of mathematical common content knowledge (Ball et al. 2008 )—further knowledge components: an “armamentarium … of representations some of which derive from research whereas others originate in the wisdom of practice” (Shulman 1986 , p. 9). According to Hill et al. ( 2008 ), teachers’ knowledge about what makes a problem difficult for a student highly depends on the teacher’s level of mathematical subject-matter knowledge.
As for word problems, it is clear that teachers must not only have the necessary knowledge to solve the tasks, but they must have further knowledge components, a repertoire of word problem solving strategies that are somehow comprehensible to the students. As Shulman ( 1986 , p. 9.) rightly claimed, “there are no single powerful forms of representations”, and mathematics teachers’ pedagogical content knowledge regarding word problems should adopt different solution strategies and techniques (and the corresponding specialized content knowledge). An example of how different teaching strategies evolve during a mathematics lesson among first grade students can be found in the paper by Rowland et al. ( 2005 ). Their observation with a pre-service teacher nicely illustrates how important flexibly changing the current teaching strategy can be.
Teachers’ classroom practices for teaching word problems were studied by Chapman ( 2006 ), who claimed that there had been only a few studies conducted on pre-service teachers’ difficulties with word problems, and there is a lack of research focus on how pre-service teachers handle the contextual factors of word problems, i.e., the extent to which realistic considerations are to be taken into account. Verschaffel et al. ( 1997 ) found a serious lack of realistic considerations among pre-service teachers when solving realistic tasks. This lack of realistic considerations was detected by means not only of their solutions to such problems, but also by their evaluation of different possible solution types. Therefore, pre-service teachers may not always be right in assessing the difficulty of different word problems.
According to Shulman ( 1986 ), finding out why learning a topic is easy or difficult is an integral part of the PCK. Due to Chapman’s ( 2006 ) study, two types of orientations can be distinguished with respect to how teachers view word problems. Teachers who adopt a paradigmatic-oriented approach will usually blame the realistic context of word problems as a main barrier to providing a mathematical solution. In contrast, teachers who follow the so-called narrative-oriented approach pay much attention to the psychological factors of the solution. They focus on students’ attitudes, motivation and potentially different critical interpretations of the text. According to them, failures should not be attributed mainly to a lack of mathematical skills but rather to a lack of meaningfulness.
It has been reassured in several empirical and comparative studies that drawings (both the type of drawing and the process of making drawings in the classroom) play a very important role in shaping students’ beliefs about and solution strategies for word problems (see e.g., Willis and Fuson 1988 ; Depaepe et al. 2010 ). As for the two main teaching approaches defined by Chapman ( 2006 ), the main challenge facing the pragmatic-oriented approach is to find the appropriate (and almost always schematic) drawing that should definitely display some kind of mathematical content, whereas teachers following the narrative-oriented approach may seek a variety of possible drawings each with their own possible advantages in the solution process. In a design experiment among 3rd grade students, Csíkos et al. ( 2012 ) found the latter approach beneficial in terms of both yielding better performance and inducing possible long-term effects on students’ metacognitive knowledge.
Finally, another important aspect of teachers’ pedagogical content knowledge is addressed in the current study: how they assess students’ word problem solving activities. As Verschaffel et al. ( 2010 ) emphasized, how students’ mathematical performance is assessed has a strong influence on teaching practice. They revealed that current summative evaluation practices have several characteristics that are not in line with what researchers would find fruitful for assessing higher-order thinking skills. It is highly probable that teachers’ formative assessment techniques applied in classroom settings are in line with how their students will be assessed in high-stakes tests. Independently of the current high-stakes testing practices, it is worth finding out how teachers in the classroom assess (more concretely, score) the students’ solutions provided for word problems.
In the current investigation we focus on pre-service and in-service teachers’ pedagogical content knowledge concerning word problems, including whether they teach uniform solution strategies or algorithms, how they assess students’ solutions, what is their insight about making drawings during the solution process, and what they would do if students were ‘ruining’ a word problem by finding the content of the word problem incomplete or unclear. Besides exploring teachers’ PCK, further thinking components of similar importance (such as beliefs, views) were addressed. According to Pehkonen and Pietilä ( 2003 ), beliefs and views are often implicit knowledge components that may have an emotional component as well. In-service and pre-service teachers’ knowledge of word problems can often be implicit in the sense that they have not necessarily elaborated their views verbally before being asked about them. Our research questions concern pre-service and in-service teachers’ pedagogical content knowledge about teaching word problems.
According to teachers, when is it suitable to start teaching an algorithm for solving a word problem?
Which steps do teachers focus on during the solution of the word problem? Do they follow a paradigmatic-oriented or narrative-oriented approach?
How do teachers judge the role of drawings when solving word problems?
How do teachers take realistic considerations into account in elementary textbook word problems?
In this research, 30 pre-service and in-service teachers were involved. The subsample of elementary teachers was recruited from among both novice and expert teachers, and lower secondary mathematics teachers comprised the fourth subset. Therefore, our respondents belong to four groups: elementary teachers with less than 5 years of experience, elementary teachers with more than 5 years of experience, lower secondary mathematics teacher in grades 5–8, and pre-service elementary teachers. Respondents were recruited from the capital of Hungary and from a countryside village. Basic demographic characteristics of the sample are presented in Table 1 .
The students were second year students of Eötvös Loránd University Faculty of Primary and Pre-School Education. Their studies already included one semester of mathematics teaching methodology, where the solution of word problems was part of the curriculum. In their case, the Budapest/countryside distinction is not relevant.
In this research we used the method of standardized, individual interviews. Each interview was carried out based on a strict protocol that contained 26 questions, and lasted for 12–27 min per person. The interviews based on the protocol were prepared and supervised by the same person throughout in order to avoid any possible anomalies due to questioning and meta-communicative style. Footnote 2
Based on a comparative document analysis of several Eastern European textbooks, an interview-protocol was developed. We were especially curious about the very first appearances of word problems in the textbooks. Having observed that there seems to be a rather uniform introduction of word problems as a genre in eastern European textbooks, we aimed to reveal teachers’ views about teaching word problems. We aimed to explore teachers’ views about the first appearance of word problems in the textbooks, and with this intention we formulated more general questions about word problems (definition, attitude and perceived difficulty) as well as some more specific questions concerning the teaching methods, including how they assess students’ problem solving.
During the interview, specific word problems from textbooks were analysed, and we also asked questions regarding the teaching habits of the respondents. Besides, we asked for the solution of an open-ended word problem (“Friends”, see Sect. 6.5 and the “ Appendix ” for a detailed description of the task) from the often-replicated Flemish study of ‘problematic’ word problems (Verschaffel et al. 1994 ).
The questions of the interview were grouped around the following topics:
Approaches to the definition of word problems.
Attitude towards and judged difficulty of teaching word problems.
The first steps of teaching how to solve word problems.
Methods and techniques used in teaching how to solve word problems.
Participants’ own solutions to a word problem.
The questions of the interview refer to (1) the optimal school grade when explicit word problem solving strategy should be introduced, (2) the debate of how difficult a word problem should be in order to justify the use and teaching of explicit multi-step solution strategies, (3) the role of visual aids in the textbooks, and (4) how teachers react when students provide unexpectedly realistic solutions to a word problem that was seemingly designed as a pseudo-realistic routine word problem (as described by Csíkos and Verschaffel 2011 ).
The questions of the interview protocol are presented in the Appendix. In the evaluation procedure, several quantitative variables have been defined according to and in line with the interview questions. Most of them are of dichotomous nature, e.g., whether the respondent always requires students to formulate the answer in a full sentence or not. In some cases, answer codes with more than two responses were developed, as in the case of the Friends task at the end of the interview. By means of defining mainly dichotomous quantitative items, we aimed to assure the objectivity of our data analysis and interpretation.
6.1 Approaches to the definition of word problems
At the very beginning of the interview, we wished to clarify the definition of a word problem and narrow down the scope of word problems to routine word problems. Therefore, first we asked the respondents to give a definition of the concept of word problem.
Respondents approached the question in two distinctively different manners. On the one hand, they might have thought that each problem we give to pupils expressed in words, is considered a word problem.
Bori, elementary teacher: Anything, which also contains letters, not only numbers. When there are several instructions and an exclamation mark.
The other approach also contained the expectations concerning the solution algorithm of word problems.
Edit, elementary teacher: A calculation task where the task is preceded by text information. The tasks and text must be understood and written down in signs of the mathematical language. At the end of the solution process, provide an answer to the problem.
These two types of approaches are present in all groups as presented in Table 2 .
Besides the overall categories according to Chapman, further dichotomous variables were derived from the interviews. One such dichotomous variable refers to whether participants considered mathematical word problems and mathematical problems given in words to be the same. Thirteen of the respondents felt there is a sharp distinction between a word problem and a problem given in words. Another aspect among which participants can be split into two groups is whether the term ‘operation’ appeared in their definition or not. In 13 cases (43.3%) the word ‘operation’ appeared in the definition, which might indicate that these respondents considered only word problems that can be solved with an operation.
We made clear that in the following we would focus on the solution of routine word problems.
6.2 Attitude towards and judged difficulty of teaching word problems
Seven questions of the interview concerned this topic. Participants usually considered teaching word problems difficult. 23 respondents said this topic was or would be very difficult to teach. The difficulty was attributed to two factors. On the one hand, they thought that students’ everyday conceptual understanding is problematic, and on the other hand they did not consider this problem as either being mathematical or to be developed in mathematics.
Erzsi, elementary teacher: The problem is they have no everyday experience, and they also have conceptual deficiencies.
Detti, elementary teacher: The problem rather lies in comprehension, the pupils freeze when they hear they have to solve a word problem. Girls are more prone to getting scared…Children do not know tales, therefore texts from tales do not mean anything to them. In addition, there are many unknown words in the texts.
Teachers see several difficulties in how to code the word problem in the language of mathematics.
Gabi, elementary teacher: I think children understand this with great difficulty, for them this is an abstract thing to prepare a plan for the solution, write down the calculation and check the answer. These restrict children, they are confused, get scared and panic, and they mess up or miss something.
Gabi, lower secondary teacher: My class has just written a test in word problems, and I believe we were well prepared, as we had exercised typical tasks. However, when the pupils had to face the same problem with different numbers, they made mistakes and could not apply the techniques they had learnt.
András, elementary teacher: There is a problem that we cannot influence: word problems are often a testing tool in competitions and entrance examinations. There is huge pressure in teaching that pupils must be good at that. It is very difficult to provide universal models which work in all kinds of text environments.
Judit, elementary teacher: When the algorithm is clear and well-prepared, there is no problem.
It is worth mentioning that according to seven participants the solution of word problems is not difficult. Six of them are elementary teachers and one of them is a lower secondary mathematics teacher. Six of them gave a definition in answer to the question “How do you define the concept of word problem?”—an answer that restricted the meaning of word problem to arithmetic operations dressed up in a text. They probably restrict their teaching duties to word problems with straight wording that can be solved by executing one arithmetic operation.
6.3 The first steps of teaching how to solve word problems
The third part of the interview dealt with an analysis of a word problem. The problem illustrated in Fig. 3 is the very first word problem in the 1st graders’ new generation textbook in Hungary which intends to introduce the solution algorithm for word problems.
The very first word problem in a Hungarian first grade students’ textbook (OFI 2016 , p. 102). Translation of the word problem text: The titmouse has found 4 wheat grains in one of the bird feeders, and 6 sunflower seeds in the other. How many seeds has the titmouse found altogether?”. Translation of the instructions written in bold from the top to the bottom are as follows: make a drawing, write down with figures, write down with an operation, and answer the question
We consider the wording of the problem challenging, as it does not define whether there can be other seeds than wheat or sunflower in one or both of the feeders, or whether the titmouse has found seeds elsewhere, not only in the feeders. This kind of ‘ruining’ of the word problem is similar to that of Freudenthal’s idea as cited by Greer ( 1997 ). The phenomenon of “didactic contract” created by Brousseau (see Verschaffel et al. 2010 ) is continuously shaped and strengthened by such tasks in the world of the classroom social environment.
The instructions on the left of the drawing (prepare a drawing, put down in numbers, describe with an operation, answer the question) can be followed with difficulty, posing new problems beyond the word problem, which the children have probably solved by heart.
Ágnes, elementary teacher: I don’t like it. It is confusing in 1st grade that there are two texts next to each other. I think the instructions such as draw, write it down with an operation, answer, should not be written there, because this is my (the teacher’s) request. I would not start teaching the algorithm. The pupil still cannot make a distinction between the task and the instructions belonging to the algorithm.
Drawing the seeds in the feeder seems unnecessary, as in the task pupils have to depict the data and then ‘read back’ the numbers from the drawing.
András, elementary teacher: I think this problem does not require a drawing, especially since the number, the plus sign are there, the result is 10 and that’s it. I think it is unnecessary to draw here, it is not a challenge for pupils.
Another objection to the task is that its reality is questionable.
András, elementary teacher: I think most of the word problems are useless. This is why I use word problems created by myself, still it is important that the pupils enjoy it. Who cares how many this and that in total, and that the titmouse first eats this and then that? The child is not motivated. There are enough problems the children have to solve in their own lives.
According to one interpretation of realistic tasks, besides that the things in the problems should be ordinary and realistic, they also need to be meaningful and experience based, compared to tasks that are only “cover stories” for irrelevant drills (English 2003 ). However, participants’ opinions were somewhat different. 20 persons (66.7%) considered the task totally suitable for teaching the algorithm of word problems. However, eight of them noted that it might be necessary to clarify the meaning of wheat, sunflower and seed beforehand.
One third of the respondents considered the task unsuitable. Four of them said it contained too much information. Five of them were of the view that the words wheat-sunflower-seed might not be familiar to children. Nonetheless, their reasons given for their views are not always the ones with which researchers would agree.
Gabi, lower secondary mathematics teacher: How shall I add wheat and sunflower? We always say we do not add apple with pear. In 5–8 grades we can only add identical expressions. We cannot add sunflower to wheat, because they are not identical.
Only one teacher said that the wording of the task is not accurate.
Judit, elementary teacher: It is not written that there are altogether two bird feeders. The question is not accurate.
Two thirds of the respondents did not question the truthfulness of the text. This ratio elevates to 76% among the countryside respondents.
Anna, elementary teacher: When the class loves nature and feeds the birds, this is realistic. I think it is realistic, because they have seen all kinds of seeds in the feeder here. It is not strange to them.
16 respondents (53%) believed the drawing was helpful in the solution of this task. Meanwhile 73% of respondents said a drawing is always necessary in the solution of a word problem. In light of the differences between 53% and 73%, one may conclude that there should have been a better drawing attached to the very first word problem children encounter in their first school grade.
83% of respondents believe a full sentence answer is always necessary, while only 68% think it is necessary for this task as well.
Ágnes, elementary teacher: In general, it is necessary, because it strengthens the feeling that this is a real problem. Then I abstract it, and then place it back in the real environment. Then I check whether it is realistic. Here I do not think it is good that the pupils only have to insert the numbers.
Anna, elementary teacher: In the beginning I do not expect a full textual answer.
In Hungary, and based on the textbook comparisons, in Eastern Europe, too, answering in complete sentences is an important tradition. Only five respondents said that they would not expect a textual answer from the children, and four of these five respondents were pre-service teachers. It seems this tradition is strongly related to high-stakes assessment tests introduced in Hungary, where many points are lost when there is no textual answer provided. It seems that in-service teachers consider preparation for assessment to be important, while pre-service teachers do not (yet).
6.4 Methods and techniques used in teaching how to solve word problems
The next group of questions concerned teaching experience. These questions were not asked of the pre-service teachers. It is important when and how we start teaching any algorithm relevant to word problems. In our view, explicit teaching of solution strategies is questionable, since young children may mix up two things: solving a problem and knowing by heart the steps of the solution process. Typically, word problems in first grade can be solved in one step by means of executing one arithmetic operation. It is difficult to explain why a step-by-step algorithm is needed when the child already knows the answer almost immediately after having read the text.
79% of the respondents reported that they start teaching an explicit algorithm already in the first grade, and for this purpose 91% of them choose a simple word problem with straightforward wording, which can be solved with one operation, calculated mentally.
Ildikó, elementary teacher: Although I teach it already in the second semester of the first grade, we usually play and do not calculate. E.g., they stand up when they have to add.
Only two teachers formulated a radically different opinion.
Ágnes, elementary teacher: I start with a more complex problem, in higher grades. It only makes sense then, because before that they do not understand why they have to do it.
András, elementary teacher: When I started teaching I thought I would have to start with the simple ones, but I realized that a word problem really has to be a problem. When the pupil looks at it and they can solve it immediately, they do not feel that any algorithm makes sense. When they look at it and say 10, but for the teacher’s sake I will write an open sentence, the whole thing becomes an end in itself.
We asked by means of what instructional techniques they help their students in word problem solving. Two things could be distinguished clearly: text highlighting (48% indicated) and the support of reading comprehension.
Ildikó, elementary teacher: They underline essential data with a pencil.
We asked the respondents to map their assessment habits by scoring the solution of the following task as if it was in a test: “Betti and Dóri invite their friends for a party. On each tray there are 3 cheese, 5 salami and 3 ham sandwiches. How many sandwiches are altogether on the 5 trays?” Participants were free to use any scoring system from dichotomous 0–1 to a more refined scoring system.
The respondents gave 5.2 points on average for the correct solution. The minimum was 3 (two for the calculation, one for the textual answer), the maximum was 10. Two thirds of the respondents gave the points according to the steps of an expected algorithm. From the steps of the algorithm, they considered the following especially important: writing down the data, writing an open sentence, calculation, textual answer. The open sentence as a model was highlighted in half of the answers.
Bogi, elementary teacher: In the solution plan we always write an open sentence, and there might be several solutions. We write it down in a manner that allows for several solutions.
The question of textual answer proved to be especially important. We also asked whether they gave a point for the textual answer even when the calculation and the solution of the task were wrong. 19 respondents said they would give a point. Their reasons were as follows. They either do not want to punish the student twice for a mistake, or as one elementary teacher dared to confess her confusion, “It is a dilemma, because in principle it would be fair to give a point. Therefore I do not know.”
We asked what other models they showed children to solve word problems. Our question caused a little confusion, and not many ideas arose. Drawing sections were mentioned by three elementary teachers, trial and error by two lower secondary math teachers, and charts by one elementary teacher as possible models in solving word problems.
6.5 Participants’ own solutions to a word problem
The interview ended with the solution of the word problem from Verschaffel et al. ( 1994 ). “Karcsi and Gyuri organize their birthday party together. Karcsi invited 5 children, Gyuri invited 6 children. How many children were there at the party?”
This task is a typical open-ended task, as the solution is not one determined number. There are several factors you can take into consideration during the solution, which shows that the “look for the number data and then link them with the appropriate operation” strategy is not effective. Were there any children who were invited by both organizers? Did Karcsi and Gyuri participate in the party? Did all the invited children actually go to the party?
We asked the respondents to solve the word problem in a manner they would expect their 2nd grade pupils to employ. We wondered how the participants would interpret this task. If they did not recognize the trap in the task, we asked whether they would modify the solution in higher school grades. With this instruction, we softly guided them to reconsider the problem. Table 3 presents the solution according to teaching grades and experience. There were four categories defined, and all respondents’ answers fell into one of them. The most sophisticated, realistic answer went further than merely executing arithmetic operations with the numbers of the text, and opened up the possibility of different possible solutions. So the ‘complete solution’ contained one or more numerical answers (as in the other three answer categories), and additionally a remark or hesitation on the existence of a one-and-only numerical answer.
According to Table 3 , all subgroups of participants provided different answer patterns to this question. Please note these solution patterns are not inherently theirs but these are the kinds of answers they would expect from 2nd grade students. Unfavourably, from these answers it can be hypothesized how their students will behave when confronting realistic word problems in school.
We explored Hungarian pre-service and in-service teachers’ pedagogical content knowledge on how textbooks may have shaped and are still influencing the practice of teaching word problem solving strategies. While our study can be considered in some sense the replication of results from both the Leuven researchers and Olive Chapman, we think that some new explorations took place here. First, we revealed that, according to the great majority of the participants, explicit teaching of word problem solving strategies should start not only rather early in the elementary school, but also by means of using simple, one-step routine tasks for illustrating the main steps of an expected algorithm. Second, several possible scoring methods were revealed, and almost all of them included the need for scoring the mere existence of a textual answer, more or less independently of its mathematical content. Third, a variety of helping techniques appeared in the answers from underlining essential parts of the text—leaving the question open of a possible vicious circle: how do we know which parts are essential?—to playing activities and making drawings.
Having addressed the possible socio-cultural context of our study, we nevertheless consider that although the participants are Hungarian, the results may be generalized to a wider population. Textbooks have their prime role in elementary mathematics education, and as stated by Stigler et al. ( 1986 ), citing a Soviet-American textbook comparison study, “there is a bias in the American textbooks toward presenting the problems that American children find easiest to solve” (p. 166) Word problems are selected and presented in the textbooks in accordance with the language and cultural constraints (Fan et al. 2018 ) as well as reflecting the traditions of a country’s mathematics education. Nonetheless, the challenges and dilemmas elementary teachers face when teaching word problems may be more or less global with respect to improving students’ mathematical thinking.
The interview method formed another constraint regarding the sample size, taking account of our aim that it was the same interviewer who conducted all the interviews. Our data analysis is overwhelmingly of qualitative nature aiming to address some important phenomenon. Another stratum of our sampling (capital-village dichotomy) was found to be important in one aspect only, i.e., the suitability of the content element in the word problem children encounter for the very first time in their first grade of schooling.
7.3 Practical conclusions
How word problems and word problem solving strategies are introduced in the textbooks is necessarily built on an assumed consensus on these questions. In general, Hungarian elementary teachers use textbooks as a basis for their instructional practice, therefore, the consensus on teaching of word problems and word problem solution strategies can be further reassured by the instructional practice. There are at least two components of this consensus that should be reconsidered in order to activate and promote teachers’ specialized content knowledge (Ball et al. 2008 ) on teaching word problems. One such knowledge component is handling linear versus cycled solution processes. Teachers’ current views support the linear solution model of word problems where several consecutive steps are to be followed. Teachers generally agree that such a linear solution method should be introduced with simple routine tasks and at an early stage of learning mathematics. Another important specialized content knowledge component is making a distinction between checking the solution by means of executing an inverse operation or by means of retelling the story of the word problem, substituting the proposed numerical answer. There is also a pedagogical content element that should be reconsidered: the assessment practice of word problem solutions. Teachers have diverse opinions about how to score word problem solutions, and they seem to emphasize in their scoring habits how students followed the prescribed linear solution method.
7.4 Further research questions
We consider it crucially important to further study the debate concerning the question of whether any explicit teaching of a word problem solving strategy should use simple, one-step routine word problems as illustrations, or more complex and realistic word problems, which may provide justification for introducing explicit metacognitive knowledge components in word problem solving. Another dilemma raised by the current investigation concerns the appropriate timing of introducing explicit word problem solving strategies. Besides the timeline issue, another possible question would address which steps of a traditional, linear solution methods should be introduced first and which ones later.
We wish to thank teaching assistants Fanni Birtalan, Borbála Károlyi, Tímea Varga and Anna Rácz for the transcription of the interviews.
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Open access funding provided by Eötvös Loránd University (ELTE). This research was supported by the Research Center on Early Childhood Education at the Faculty of Primary and Pre-School Education, ELTE Eötvös Loránd University, Hungary. We are grateful—for their altruistic help in the collection of Eastern European mathematics textbooks—to Szilárd András, Zuzana Nagyová Lehocká, Anna Shvarts and Gordana Stankov.
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Csíkos, C., Szitányi, J. Teachers’ pedagogical content knowledge in teaching word problem solving strategies. ZDM Mathematics Education 52 , 165–178 (2020). https://doi.org/10.1007/s11858-019-01115-y
Accepted : 05 December 2019
Published : 13 December 2019
Issue Date : April 2020
DOI : https://doi.org/10.1007/s11858-019-01115-y
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8 Ways You Can Improve Your Communication Skills
Your guide to establishing better communication habits for success in the workplace.
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A leader’s ability to communicate clearly and effectively with employees, within teams, and across the organization is one of the foundations of a successful business.
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Communication nearly always involves two or more individuals.
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Emerson is a Digital Content Producer at Harvard DCE. She is a graduate of Brandeis University and Yale University and started her career as an international affairs analyst. She is an avid triathlete and has completed three Ironman triathlons, as well as the Boston Marathon.
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