5 Phases of Learning Math

This post is an answer to a question Michelle, a high school math teacher, sent via the Request a Tip form. She writes:

I explain a new concept then give an example and finally do an example with the class directing me on how to solve. What I have noticed is that the bulk of the questions occur during or after the chapter assessment. Most of the students’ questions are great questions. It’s just that the questions they are asking are ones they should have asked much earlier.

Think of knowledge as a bag of shapes – all of which have different colors and textures. Once you get the bag, you have to sort through them. Some people want to count them, some people want to sort them by color, some by shape. Some people need to sort by texture. Some people need to sort by all three or just lay all of them out in rows to get a good look at them.

Everyone sorts the bag of shapes differently. Likewise, everyone sorts information differently.

Turns out, the bag of shapes isn’t the knowledge. It’s merely the information. And it takes each person “sorting” it in their own way to turn the information into knowledge.

In teaching and learning math, I’ve noticed a cycle. I’ll use the bag of shapes to illustrate it:



Exposure is usually in the form of lecture and examples done by the instructor. It can be the first time the student has seen it or the first time in a long while.

This is where the student receives the closed bag of shapes.



Any activity following the exposure. This typically is in the form of homework or classwork practicing the concept and problems.

This is the “peeking into” the bag of shapes. The student gets to remove a few of them and start looking at them.



Allowing the subconscious to work. The brain does this all on its own.

Often mathematicians will go for long walks, go to the movies, hang out with their kids, talk to non-math people or do any number of non-math things to force the settling phase.

Children don’t know how to force the settling phase, nor do they need to. It just happens between when they do the homework and when they start to study for the exam.

In the bag-o-shapes analogy, this is where the students dumps all the shapes onto the floor and sorts them in various ways. It takes a while to get through all the shapes and see what kinds of sorting can be done.



This is typically in the form of studying for the test and taking the test. It can be a heightened emotional situation where the learner is under stress.

This is a revisit of the concepts. It becomes easier because the settling has occurred and the information (the bag of shapes) is already organized.

The learner at this point will attempt to modify some of the conscious thinking to best fit with what the subconscious has done. The added stress will allow them to connect with what they’ve done better – as the “feeling” state induces a different type of learning.



This is using the concepts for something else. This will often be the next class or next term of the math curriculum. If you learned graphing functions, you will likely use graphing functions in the future.

To wrap up the analogy, this might be a student realizing that the bag of shapes is needed for something – not just a random bag of shapes. Therefore he can re-organize them to be of use in the new situation.

Let the learning flow.

This flow of learning is natural. It will happen and has to happen. The only thing you can do to artificially speed it up is cycle it more often.

The students ask the good questions, as Michelle said, “during or after the chapter assessment.” This is in phase 4 – Re-engagement.

Instead of going through a single cycle, do it two or three times. Like this:

  1. Monday: Lecture, chapter 1 (exposure)
  2. Monday: Classwork and homework, chapter 1 (activity)
  3. Friday: Test, chapter 1 (re-engagement)
  4. Monday: Lecture, chapter 2 (initial exposure to chapter 2 and application of chapter 1)
  5. Monday: Classwork and homework, chapter 2 (activity)
  6. Wednesday: Test, chapter 1 (yes – chapter ONE; re-engagement again)
  7. Friday: Test, chapter 2 (re-engagement)
  8. Monday: Lecture, chapter 3 (initial exposure to chapter 3 and application of chapters 1 and 2)
  9. Monday: Classwork and homework, chapter 3 (activity)
  10. Wednesday: Test, chapters 1 and 2 (re-engagement again)
  11. Friday: Test, chapter 3 (re-engagement)
  12. Monday: Lecture, chapter 4 (initial exposure to chapter 3 and application of chapters 1, 2 and 3)
  13. Monday: Classwork and homework, chapter 2 (activity)
  14. Wednesday: Test, chapters 1, 2 and 3 (re-engagement again)
  15. Friday: Test, chapter 4 (re-engagement)

What do you think? Share your experiences with the cycle below in the comments.

Thanks to Michelle for requesting this tip. Michelle is one of only two math teachers in a rural private school. She teaches Algebra I, II, Geometry and Calculus.

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2 Responses to 5 Phases of Learning Math

  1. Hi Bon!
    I really like the way you approached the question, the bag of shapes analogy is a good one. I wonder how parents can apply this when supporting their child’s math learning?
    Perhaps encouraging their child to use varying techniques from the cycle, so instead of aalways “testing”, finding activities that they could work on together or perhaps the child “teaching” the parent the topic (a form of re-exposure?)

    • Thanks for the comment, Caroline!

      Indeed the “testing” aspect isn’t really used much in the home – even in a homeschool situation. To me, the essential part is settling and then re-exposure. How you tackle re-exposure is really up to the “teacher” – be in classroom teacher or parent. If you have time, and parents will make time, then re-exposure in the form of more activities and “teaching” the parent (as you suggested) is perfect.

      In the case of homeschooling – the re-exposure can be in the form of teaching younger siblings or others in the homeschool network.

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