Tag Archives: shaping

Perseverance in Math

perseverance checklist

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.

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Prompt Dependence


Great blog post on prompt dependence

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Authentic Learning and CCSS



SBAC and PARC problems used to test CCSS are challenging and often draw upon context unfamiliar to students. This means students must navigate the content, problem solving and deciphering context. Below is an SBAC problem dealing with photo albums…PHOTO ALBUMS. Do kids today understand this? In the subsequent pictures you will see the work of one of my students on handouts I created that develop an understanding of the SBAC problem – note the “x-2” at the end. The idea is to shape their ability to do such problems.

photo album sbac problem photo album pizza equation part 1 photo album pizza equation part 2 photo album pizza equation part 3photo album pizza equation part 3

photo album pizza equation part 4


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Scaffolding Multi-step Math Problems

sidewalk problem

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.

sidewalk problem scaffolded


side walk problem PT

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Shaping Behavior

Shaping is a term in special ed which basically means to train a student to incrementally follow through on a sequence of substeps to accomplish a task. B.F. Skinner coined a different name for this which he called operant conditioning. The top photo below shows “Skinner’s box.” The pigeon was placed in the box and when it pecked at the metal wall a pellet of food was presented to it. Eventually the reward was given as the pigeon pecked closer and closer to the disk in the wall where its beak is in the photo. Finally, it learned that by pecking this disk was the only means of getting the pellet.

In special education, this shaping is used to train a student to achieve specific outcomes. In the bottom photo below my son Gabriel is playing with his favorite all time toy, Legos. As is the case with many with autism, he would not look at people in the eye. His therapists trained him to make eye contact by first holding his Legos in the air until he requested them. Eventually the Legos were held incrementally closer to the therapist’s eyes. The second to last step was to hold them next to the eyes and finally he had to look into the therapist’s eyes before getting the Legos.

This same approach can be used to train students to attempt problems, think critically, follow classroom norms such as the appropriate steps for starting classs and any other desired behavior.

skinner box

lucas and Legos

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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.

How many squares problem

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Addressing Homework Completion Problems

Addressing Homework Completion Problems

Nice article Dr. Gary Brannigan (@GaryBrannigan) with 9 suggestions for helping a student with homework completion problems. #3 is especially pertinent to math:

Initially assign homework with which the struggling learner is unlikely to have difficulty. Mark the homework for punctual submission and content. Gradually increase difficulty but never beyond the struggling learner’s ability to succeed with moderate effort.

This is called shaping behavior. By starting slowly and moving a student towards the desired outcome. Simply getting the homework turned in on a regular basis is often a major task.

I also posted about notebooks. My approach is to have students use a simple folder for storing homework and to leave their notebook in class. The initial focus is on completing homework and submitting it on a regular basis. As the homework becomes more challenging the student will begin to take the notebook home.

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