Learning Mathematics: The Cognitive Science Approach to Mathematics Education by Robert B. Davis

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The Essence

An in-depth look into the problem solving behind mathematical thinking. By considering how we use computation in computer science, we can uncover lessons regarding our own human information processing system (the brain). The system that represents information within our brains has identifiable patterns that when considered, reveal to us our deficiencies when approaching mathematical problems. Davis book also has implications for how we instruct students when teaching math, as well as how artificial intelligence will impact cognitive science.

Learning Mathematics Journal Entry Notes:

This is my book summary of Learning Mathematics. My notes are a reflection of the journal write up above. Written informally, the notes contain a mesh and mix of quotes and my own thoughts on the book. Sometimes, to my own fault, quotes are interlaced with my own words. Though rest assured, I am not attempting to take any credit for the main ideas below. The Journal write up includes important messages and crucial passages from the book.

 

• How do people think about Math? University students show they know far less than everyone had assumed they did about mathematics (even the majors).

• A lot of the time, the long way is wrongly spent….
Commonly-shared frames:
You are consistent with yourself.
Different people doing similar things.

• Considering human thought and computer information side by side can be extremely valuable, if only because computers operate in a highly explicit way that forces us into greater clarity in analyzing information processing.

• Anything is easy if can assimilate it to your collection of mental models.

• A problem may be quite easy if you have an effective representation of the problem itself, and effective representations for the relevant areas of knowledge. If not, the problem may be difficult indeed impossible.

• Start looking for the forest, not the trees.

• By using a broader range of mathematical topics, one improves the odds of getting a reasonably representative picture of the kind of mental information processing math requires.

• In order to say why you must interpret the ‘facts’ in terms of an appropriate theory.

• We do not see mathematics as a collection of algorithms to be memorized by rote and practice.
Nor do we see math as something to be ‘tough’ to students, with control in the hands of the teacher.
Instead, we see math as a collection of ideas and methods which a student builds up in his own head.

• Meaningful > Rote Mathematics

• KRS: Knowledge representation systems allows us to talk about representation itself, without compelling us to commit ourselves to any assumptions about the internal structure of the KRS.

• When a student is learning science, he is learning certain mythology that by no means matches commonplace experience all the time. The truth can be different based on how we define it. Words, experience, and KRS are often very different truths.

• Whenever you want to search permanent long-term memory, you MUST have a clue or cue to guide you to the correct part of memory, without a guide, you will inevitably be lost.

• Representations are fundamental to mathematical thought
 How are you representing the problem?
 How do you represent relevant knowledge that you have learned in the past?

• A single ‘piece of knowledge’ in the mind is the cognitive equivalent of a collage.

• One of our most powerful tools for ‘knowing’ something is the metaphor.

• To memorize verbatim without exception is to experience great difficulty in learning mathematics.

• Educate your intuition

• Math has always been in the background. All great ideas agree with our minds when pondered under the number tree.

• Any problem is impossible if you are unable to recognize its key terms. In these terms, math becomes less calculation and more of the correct assimilation paradigm retrieval.

 

If you liked what you saw. Here are 3 titles that I recommend based on what was discussed in Learning Mathematics.

1. A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) by Barbara Oakley
2. Algorithms to Live By: The Computer Science of Human Decisions by Brian Christian, Tom Griffiths
3. Why Science Needs Art: From Historical to Modern Day Perspectives by Richard Roche, Sean Commins, Francesca Farina

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