reasoning of Asked many construct the argument of “How people are wondering why they chose their others. that now?” This question, as well as encouraged students to critiquing the questions critique the ideas that that others came up with. others had come up with. Modeling and Using Tools 4. Model with The teacher chose an Once the information engaging problem that was communicated, the mathematics. could be modeled students used that mathematically. information to develop a The teacher monitored mathematical model. the students developing The model involved the model, but allowed creating a sum of terms them to develop the that expressed the value model on their own. of each layer of the pyramid. 5. Use appropriate The teacher monitored Once the model was the use of but did developed, different tools tools strategically. not interfere. were used to find the answer: In the the Numerical expression “debrief”, teacher discussed the Summation different use of tools. By Excel spreadsheet doing this, students are program provided with more tools in their toolbox for future problem solving. Seeing Structure and Generalizing 7. Look for and During the The students had to “debrief “of the problem, the teacher develop an understanding make use of facilitated a discussion of the physical structure structure. about the structure of the in to develop a arithmetic expression. mathematical model that What numbers were had a numerical structure constant in each term of of its own. The student the sum? Can we use the had to make the distributive property? Do connection between the we see a pattern? physical structure and the numerical structure of the mathematical model. 8. Look for and Early in the discussion of Once the base of 40 by 40 the problem, the teacher was given, the students express regularity helped students develop had to determine how
reasoning of others. Asked “How many people are wondering that now?” This encouraged students to critique the ideas that others had come up with. construct the argument of why they chose their question, as well as critiquing the questions that others came up with. Modeling and Using Tools 4. Model with mathematics. The teacher chose an engaging problem that could be modeled mathematically. The teacher monitored the students developing the model, but allowed them to develop the model on their own. Once the information was communicated, the students used that information to develop a mathematical model. The model involved creating a sum of terms that expressed the value of each layer of the pyramid. 5. Use appropriate tools strategically. The teacher monitored the use of but did not interfere. In the “debrief”, the teacher discussed the different use of tools. By doing this, students are provided with more tools in their toolbox for future problem solving. Once the model was developed, different tools were used to find the answer: Numerical expression Summation Excel spreadsheet program Seeing Structure and Generalizing 7. Look for and make use of structure. During the “debrief “of the problem, the teacher facilitated a discussion about the structure of the arithmetic expression. What numbers were constant in each term of the sum? Can we use the distributive property? Do we see a pattern? The students had to develop an understanding of the physical structure in to develop a mathematical model that had a numerical structure of its own. The student had to make the connection between the physical structure and the numerical structure of the mathematical model. 8. Look for and express regularity Early in the discussion of the problem, the teacher helped students develop Once the base of 40 by 40 was given, the students had to determine how