Standards of Mathematical Practices - Part 2
Mathematical Practice #1 - Make sense of problems and persevere in solving them.
"Does this make sense?" is one question that students need to be asking themselves. This specific question is in the description of Mathematical Practice #1 . Mathematically proficient students are described as students who are able to create multiple entry points into a problem, are able to form conjectures about the appropriate form and meaning of the solution, are able to monitor their progress and make adjustments if needed, and are able to use and understand multiple representations of a solution or situation. This can take place at any age. This type of thought process takes time, practice, scaffolding, discussion, reflection, feedback, and modeling to develop over the school year. A growth mindset is important to have with your students to instill that this process takes time and all of the hard work will improve the thinking skills needed to persevere in solving problems.
As I have watched students over the years solve complex problems, many students are able to produce a mathematically correct solution but not understand the units of measure or be able to explain the meaning of the solution for the problem they are solving. On the other end of the spectrum, many students can explain the problem, but do not have any entry points on how to solve the problem. Both of these situations are representative of the fact that students are missing a piece of the puzzle. Providing different strategies, methods, and posing a variety of questions can provide a solid framework for students to persevere in solving problems.
Ideas on how to improve students work on MP#1:
1. Present the problem to the students and have them ...
- write in their own words what they are trying to solve
- write about what first step could be used to solve the problem
- underline/highlight key words and explain why they are important
- give the students a sample of a first step and have them agree or disagree with the first step
- give the students a sample first step and have them come up with a different first step
- gauge their own understanding of the problem (i.e. Likert Scale, short reflection, thumbs up/thumbs down, red/yellow/green signs, ...)
2. Steps on solving the problem, have the students ...
- work for 2 to 5 minutes individually and then work as a whole group, small group or
shoulder partner
- display students' work and do a gallery walk and reflect on others' work
- provide two sample pieces of work and have the students agree or disagree with the work of the students
- provide an additional solution path to solve the problem
3. Solution to the problem, have them ...
- write an explanation of the meaning of the solution with regard to the problem
- examine multiple solutions to the problem with different units of measure and have them explain which is correct and why
- critique samples of work for a problem
- examine one solution path and have them produce a different path to the solution
- take a correct solution to a problem and find a solution path for the solution
- take estimated solutions and explain which of the solution(s) could possibly be correct before they work the problem
One problem could be utilized as a stem for different activities from the list above. This would provide multiple entry points for discussion for one problem. This problem could be an example that you work with for an entire week. The following could be a series of activities throughout the week.
Monday - Present the problem and write in your own words what are you trying solve or the meaning of the problem.
Tuesday - Give estimated solutions and tell which ones you think are correct and why.
Wednesday - Write down two first steps that could be utilized to solve the problem.
Thursday - Solve the problem on your own and share your steps and reasoning with your group.
Friday - Give them a solution path and have them write about how their solution path was different from the one you gave them.
No matter which strategy is being used to help build perseverance in students, feedback is a key component in this process. As you identify an area that you want to improve in your students' problem solving, think about the type of feedback that you are giving to help students learn and grow in this process.
"The greater the challenge, the higher the probability that one seeks and needs feedback, but the more important it is that there is a teacher to provide feedback and to ensure that the learner is on the right path to successfully meet the challenges."
- John Hattie, 2012
Visible Learning for Teachers
Food For Thought:
As you meet in your PLC and discuss student work, think about the following question.
Did we provide productive timely feedback to our students as they were solving a problem? [PLC Essential Question #3]
