What math teachers should know about memory

Posted by: Gary Ernest Davis on: April 11, 2011

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How often do mathematics teachers say about students: “If only they would memorize the information”?

In my experience, a lot, and with considerable anguish.

What this concern shows, in my experience, is a lack of understanding of human memory systems – an aspect of brain functioning that is critical to education.

Rote memorization

This is about a useless a technique as was ever thought of for educational purposes, yet it is a plea made often by teachers: “Why won’t they just rote memorize it?”

Hermann Ebbinghaus

The definitive study on rote memorization was done by Hermann Ebbinghaus and published in 1885. He studied the rate of learning and forgetting of nonsense syllables by himself, and formulated his law of forgetting in which an exponential decay describes the rate at which rote memorized information is lost.

One might argue that mathematical terminology, facts, procedures, and formulas are not nonsense syllables, having a great deal of logical structure. This argument misses the point that to many students the mathematics they are expected to remember appears to them to be more or less nonsense syllables, devoid of any real meaning. Mathematical experience is not, for these students, something that they can fit into a broader scheme of related memories. As a result their knowledge decays exponentially, and as experience shows, with a quite small half-life.

Students often recognize this problem themselves, as the following poignant example illustrates:

Should I drop my AP Calculus Class? (from Yahoo! Answers)

“… Being a senior this year I was kind of happy to know that I made it this far to AP Calc! Wow, thats great! But my teacher is awful. I try to talk to her for help and she doesnâ€™t help. Her teaching style is really confusing and I am having a really hard time getting the concepts. She has this thing where she makes us do reviews that are about 10 questions and then she calls us up to the board to answer each question putting us on the spot. I am really slow at doing the problems and I canâ€™t always remember how to do the problems very mcuh (sic) since they are a concept from last year. She then calls me up to the board and I donâ€™t know how to do it and it makes me feel awful because she tells me what I did wrong in front of the whole class! I really try, but it makes me feel so sad inside that Iâ€™m not as fast as others or understand the problems like I should. I really struggle and I want to quit and do something else, but I could get college credit at the end of the year if I pass the AP test, but is it worth it if my teacher wonâ€™t help me and my parents canâ€™t help me either. ….”

Motor memory

“They can do it, but they can’t explain it.”

I can recall students who, after much practice, can carry out Gaussian elimination to solve a system of linear equations.

They can with extended practice do this more or less accurately.

When asked on an examination to take the next step in a half competed example most students fail miserably.

When I first observed thisÂ most of my colleagues and I did not know why.

Now I do: the memory for Gaussian elimination was in their hands.

It was stored in their brains as a motor memory – a memory of carrying out physical actions, like riding a bicycle.

Unless we are cycling champions of the caliber of Lance Armstrong, it is unlikely we can explain in words how to ride a bicycle. Probably the best we could do is hop on a bicycle and start pedaling.

A lot – perhaps a majority – of mathematics learning involves motor skills – “how to” actions.

These actions used to be calculating on paper, and are more commonly nowadays calculating using software.

Whichever, the chief memory of such actions lies in our fingers and the motor parts of our brains.

Declarative memory

Such memories are generally not available for use by our declarative memory system, which uses sounds, images and gestures to declare that we know something.

Declarative memories involve relationships between things, and the establishment of declarative memories is mediated by the hippocampus:Â quite different, in general,Â from the formation of motor memories.

One of my colleagues recently gave a very clear explanation of conditional probability.

The following class period he asked the students to explain conditional probability.

Not one could.

He despaired that the students had failed to “memorize” his earlier definition and explanation.

This is not to say the students had no memory of what took place; they simply had not used their relational memory, created via the hippocampus, to put memory of the episode explaining conditional memoryÂ into relational declarative memory space.

They needed to do something like the teacher had done for himself, but which was unknown to them, because it took place inside his brain.

They see his words, which are declarative, explaining something he knows declaratively as result of his own relational thinking.

What they do not see is how his brain got to that declarative state.

What to do?

If we want our students to remember things and be able to talk about them then we need to help students form declarative, relational memories.

Perhaps one way to begin to do this is to model out loud the processes taking place in own brains as we explain a concept.

In other words, put our own thinking out there on the table to be examined.

That way students might have a better chance of knowing what to focus on, and so begin to be able to develop relational memories, instead of simply memories of isolated episodes, or memories of carrying out activities.

A critical capacity of the hippocampus, and one vital to the formation of declarative, relational memories, is linking.

The hippocampus works to establish relationships – links – between things, particularly between episodes.

So, explicitly focusing on links and connections might be a better way to establish long-term relational memories than simply giving clear explanations.