Perseverance in Math

A major challenge for students is not content but how to “do math” which includes perseverance. The photo above shows a table that can be used to monitor progress on perseverance. It addresses two situations involving perseverance (see below).

The focus of perseverance in math is making an informed attempt when a path or next step is unclear (and does not necessarily result in a solution). Paths can be categorized as using a strategy, e.g. drawing a picture, or following an algorithm, e.g. steps to solve an equation. (See excerpt of CCSS Standards of Mathematical Practices below).

Perseverance in math involves two situations:

• The initial entry point (strategy or algorithm) is not apparent but one is selected and implemented
• An ongoing strategy or algorithm is determined to be insufficient and an alternative strategy or algorithm is selected and implemented

From the CCSS Standards of Mathematical Practice (bold font is my emphasis on the perseverance component) 1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Scaffolding Multi-step Math Problems

This is the figure from Mrs. Olsen’s Sidewalk Problem from the CT CAPT test 2010 (released to public). The problem has four major steps: divide the figure into common shapes, use Pythagorean Theorem to find height of the resulting triangles, find area of these shapes and compute total cost for  pouring the asphalt for the sidewalk. Simply finding an entry point into this problem is a major challenge as is keeping track of the multiple steps.

Below are photos of the handouts I use to break the problem down into parts – scaffolding. Eventually the students have to learn to find an entry point and navigate the steps on their own. They learn to do this incrementally with the teacher shaping the problem-solving skills.

Shaping Critical Thinking and Self-help Skills

Below is an example of a puzzle I use to train students to make an effort and to think about problems. I have found that many students not only have a learned helplessness when it comes to math but they have been trained to follow steps mindlessly. Following a task analysis approach the first step is for students to find an entry point to a problem. They also need to feel comfortable taking risks. Slowly moving the students towards solving more complicated problems is called shaping.

Below you can see that the student made a more simplistic attempt. Given that this was done the first day of school I was very pleased with the effort.