Republic of Mathematics blog

Cognitive theft in mathematics teaching

Posted by: Gary Ernest Davis on: April 6, 2011

W. Gary Martin

The person who truly taught me to teach mathematics in a deeper and more productive way is W. Gary Martin.

He showed me, by example, how to stop talking so much about my own understanding, and listen carefully to student understanding.

In particular, he demonstrated for me how, when a student is presenting their own problem solution to the class, to pay careful and concentrated attention to whether other students were understanding.

Over the years, as a teacher educator in the UK and the United States, I have helped student teachers, as well as practicing teachers, to focus less on their own explanations and focus more on what their students understand. Less talk and more listening.

This point was raised again in the April 6, 2011 GenerationYes blog: part 4 of 4 of a series on Khan Academy.

The blog post cites Gary Stager as saying that “anytime you go to ‘help’ a learner, pause and think about whether you are taking away an opportunity for them to learn it themselves.”

Gary Stager summarizes this as “Less us, more them.”

Many years ago at a National Council of Teachers of Mathematics conference a well known mathematics educator, now a distinguished Professor and Dean of a major College of Education, showed a video of a student and herself. In the video the student showed how to find one-half and then one-fourth of a cookie.

The student paused when asked to find one-third of the cookie. The well known math educator explained to the student that it might be easier to replace the cookie by a candy bar and proceeded to approximately mark a drawing of a candy bar into thirds.

The student who to that point had been quite talkative, now remained silent.

I was so struck by the inappropriateness of this “showing” that I dubbed this type of instructional interference with a student’s thought processes as “cognitive theft”.

The math educator’s intervention  did not solve the original problem, did not even show how to find one-third of a candy bar, and took away  – stole – from the student the chance of continuing to think about the problem.

Well meaning, I suppose, but who wants a well-meaning brain surgeon?

Equally, who wants a well meaning teacher stealing from students the opportunity to think?

The GenerationYES blog discusses perceived failings in the Khan Academy’s instructional videos from many perspectives.

These videos are not directly stealing the opportunity for students to think, but they are emphasizing mathematics as a bunch of techniques that one learns by being shown by one’s teacher.

This seems, to me, to promote a passive consumer attitude to learning mathematics: a show me how to do it” attitude.

Whereas learning mathematics usefully, productively, flexibly and deeply requires active participation in solving problems.

The “show it to me” attitude is traveling dangerously close to cognitive theft.

In the long-run it is indeed cognitive theft: a student is stealing from themselves the opportunity to learn deeply, and, as it turns out, more joyfully.

10 Responses to "Cognitive theft in mathematics teaching"

Well put. Unfortunately, I can think of quite a lot of people who would prefer well-meaning to educational…

One of the most important ideas presented to me in teacher training was that of wait time. Teachers who are used to doing a lot of talking rarely give students an appropriate amount of time to think about even simple questions, much less sophisticated, complex ones. Giving students time and space to think is a perfect application of “Less us, more them.”

The same idea applies in reverse to the learner who is watching a video–if you don’t know that it’s important to pause the video and think the ideas through yourself, it’s hard to imagine this being anything more than a quick, superficial educational experience.

This morning, I waited 10 minutes for my class to figure out the answer to a question during a review of calculus. I wanted them to point out what is the fundamental property of the tangent line (which in my opinion is the fact it is intended to be going “the same direction at x as a function at x” in layman’s terms). They offered all sorts of properties of tangent lines, which themselves were incredibly useful and I know that I could have taken the easy way out and given them the answer, but that these 10 minutes were much more useful.

“Cognitive theft” even helps explain why students feel so cheated when you do give them the answer to a problem they are working hard on, and why my son feels so strongly that “he needs to do it himself.” We feel a sense of injustice when someone steals our ability to reason through a problem on our own.

[…] Davis at the Republic of Mathematics blog writes, Over the years, as a teacher educator in the UK and the United States, I have helped […]

What a great talk by Stager! A true statement to what students are capable of. The topic of cognitive theft is interesting, yet it might be a bit premature to apply to khan videos. If i ask directions to a store there is no need worry about cognitive theft simply because the process is meaningless to me. Many seek Khan because they need directions on their professors test. They may care nothing about the test or the material they are forced to learn. To them, math is like the puppy stories to the genius child who was told he can’t read in Stager’s talk

I think you’re right about why some people find the Khan Academy videos useful.

Yet these videos are nothing like a “revolution” in education – the real revolution is the medium: the internet.

A danger is hat the Khan Academy videos play to a consumer, “show me how to do it” mentality in mathematics education, which is linked with longer-term failure in mathematics.

I love the term “cognitive theft”. We must be careful to work with students when solving math problems rather than funnel students to the correct solution using our preferred strategies. I am constantly surprised by what students can do when given the chance to work through difficulties with probing questions. Asking the right questions during inquiry is crucial. Great post.

I am in my 22nd year of teaching high school math and would like to say that I’ve never been guilty of committing this crime; I would be lying. I do remember instances of going way “overboard” in explaining the solution to a problem without giving it a second thought. I did see the error of my ways and eventually turned this into a humerous situation.

The question that I now ask is “when does teacher-directed learning become cognitive theft”? I have recently created several pages of notes focusing on the Introduction to Differential and Integral Calculus. The article put forth by Gary forced me to reflect on what I was including in my notes; am I once again guilty of ‘cognitive theft’ by taking away the opportunity for students to develop these notions on their own? I view my notes as establishing a “framework” within which to work and that the notions put forth in them still leave many avenues for the students to explore. These notes are built from a pre-existing knowledge base closely-held by the students; the many facets of this knowledge base are “funneled” into a more concentrated area of study. For example, when deriving the area of a circle, the student’s knowledge of area of rectangles is called upon. Similarily, pre-existing knowledge of volumes of rectangular prisms is called upon when deriving the formula for volume of a sphere using the shell method. This is all done with specific goals in mind; these goals become the focal point from the beginning of each topic (call it a project). My concern at this point is how far can I go in creating my “framework” within which students learn before I become guilty of “cognitive theft”?

As much as I agree with the importance of being aware of cognitive theft, I certainly don’t agree on the claim that it applies to Khan Academy. The videos are there as a support and explain the fundamental rationale, but the student has to succeed at solving 10 problems in a row by himself. Each problems are unique and the student is on his own while solving it and has to “master” the problem-solving skill before moving on. This is why the Khan Academy appears revolutionary to me, not because of the videos (why, so many people did that before him!), which are nonetheless very well done. And this is why, contrary to what the actual educational system does, the Khan Academy concept is so great: it ensures that you master the basics before moving on, thus avoiding cognitive theft and swiss cheese learning.

I think his last TED speech explains that very well:

In relation to the Khan Academy I wrote: “These videos are not directly stealing the opportunity for students to think, but they are emphasizing mathematics as a bunch of techniques that one learns by being shown by one’s teacher.”

I stick to that point of view. Frankly, if I encountered mathematics as a young person through the Khan Academy videos I would have taken up something else. In other words, this is not mathematics from my perspective – it’s a bag of tricks.

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