Rich Math Problem
How math should look like?
This is how math should look like:
In this photo the student is engaged, taking initiative for her learning and she is showing her proof and the steps she took to get to the answer.
She used tools like a bar graph, a math representation of a puzzle piece with a diagram (Not A Picture), and she uses algorithms.
She has effective communication because she used a statement, and she showed labels with words.
In Paying attention to Mathematics Education K-12
the range of higher-level thinking skills in mathematical processes:
– reasoning and proving
– selecting tools and computational strategies
encouraging multiple approaches for learning and actively doing mathematics ( p.4, Paying Attention to Mathematic Education K-12, Queen's Printer for Ontario, 2011)
The above points are relevant to the photo that shows the student working on a rich math problem.
The Mathematical Process, How it further Support Students Learning
The mathematical processes that were mentioned above these support effective learning in mathematics.
"The mathematical processes can be seen as the processes through which students acquire and apply mathematical knowledge and skills. These processes are interconnected. Problem solving and communicating have strong links to all the other processes. A problem-solving approach encourages students to reason their way to a solution or a new understanding. As students engage in reasoning, teachers further encourage them to make conjectures and justify solutions, orally and in writing. The communication and reflection that occur during and after the process of problem solving help students not only to articulate and refine their thinking but also to see the problem they are solving from different perspectives. " (p.11, The Ontario Curriculum Math Grades 1-8, 2005)
The Four-Step Problem Solving Model was developed by George Polya and is not recommended to be introduced any earlier than Grade 3 because if it is introduced earlier, students focus on the model rather than making sense of the problem and the mathematical concept. The Four-Step Problem helps pilot and embody the math problem for the students and when it is combined with the mathematical process it is more effective in aiding students' learning.
The Four-Step Problem Solving Model
Figure 1: A Problem-Solving Model
Understand the Problem (the exploratory stage)
➤ reread and restate the problem
➤ identify the information given and the information that needs to be determined
Communication: talk about the problem to understand it better
Make a Plan
➤ relate the problem to similar problems solved in the past
➤ consider possible strategies
➤ select a strategy or a combination of strategies
Communication: discuss ideas with others to clarify which strategy or strategies would work best
Carry Out the Plan
➤ execute the chosen strategy
➤ do the necessary calculations
➤ monitor success
➤ revise or apply different strategies as necessary
➤ draw pictures; use manipulatives to represent interim results
➤ use words and symbols to represent the steps in carrying out the plan or doing the calculations
➤ share results of computer or calculator operations
Look Back at the Solution
➤ check the reasonableness of the answer
➤ review the method used: Did it make sense? Is there a better way to approach the problem?
➤ consider extensions or variations
Communication: describe how the solution was reached, using the most suitable format, and explain the solution
(p.13, The Ontario Curriculum Math Grades 1-8, 2005)
This model helps support the developmental process and makes a connection to what is being taught for students.
The picture above summaries the mathematical process very well. It allows the student to reflect, make connections to the math problem and communicate their learning. Therefore a Rich Math Problem illustrates the steps being used, and displays the tools and strategies used through labels and diagrams. It facilitates communication of math vocabulary and encourages students to discuss with a partner or in groups and allows teachers to participate as facilitators, and instructors. It achieves a sense of accomplishment, a partnership between teachers and students, and teamwork within the whole classroom.