Republic of Mathematics blog

Avoiding the low road in learning mathematics

Posted by: Gary Ernest Davis on: April 9, 2011

I wrote recently about cognitive theft in mathematics, an act that teachers – including parents – often carry out that short circuits a student’s possibility of working through a problem themselves.

There’s another side to cognitive theft – an act of omission – in which students themselves often engage.

Students often take away their own possibility of deeper, more rewarding, engagement with mathematics by asking: ” Show me how to do it.”

There are myriad reasons they might say this: not engaged, don’t care, don’t have time, are possibilities.

Yet anther, more deep seated, issue might be that students who think like this are simply taking what they see as a path of least effort: a low road to engaging with mathematics.

To a student not used to the joy of thinking through a mathematical problem, using sustained and concentrated thought can seem quite an effort.

How much easier to just ask the teacher:” Show me how to do it.”

Wallace Delois Wattles (1860–1911) (as eccentric as some people may view him and his writings) seems to me to sum up this tendency perfectly:

“There is no labor from which most people shrink as they do from that of sustained and consecutive thought; it is the hardest work in the world.” (The Science of Getting Rich, Wallace D. Wattles, p. 15)

Educators ought, in my view, to be “drawing out” their students.

“Education” is derived from the root “educare” to draw out.

Education, again in my view, has little to do with instruction (commonly conflated with teaching).

Rather, education in mathematics has more to do with a teacher promoting student knowledge and growth in mathematical reasoning and ability to calculate accurately (including by use of appropriate technology).

There are two major traps in this process.

One is cognitive theft on the part of a teacher – stealing from a student an opportunity to think a problem through themselves.

The second is a tendency of a student to take the low road of asking a teacher to engage in an act of cognitive theft.

Neither of these acts are ill-intentioned, in my view.

A teacher who steals an opportunity for a student to think is usually just trying to be helpful, as in: “Here, let me show you …”

I have done this many times as a teacher of mathematics. I never meant to be bad to a student, but I did steal from them an opportunity to engage more deeply.

A student who takes the low road of asking a teacher to show them how to solve a problem is usually just seeking to minimize discomfort, as in: “It’s easier for me if I can copy what you do.”

The longer-term problem with both these acts of cognitive theft – on of commission, the other of omission – is that a student is not “lead out” of themselves into a different realm – a realm in which the power of their thinking brain is evident to them.

I think Wallace Wattles was right about our aversion to sustained thought.

Yet it is our task as teachers to awaken students to the enormous power of their thoughts, of their ability to solve even very difficult problems by “sustained and consecutive thought.”

When I was engaged in elementary teacher education in Washington state a young woman said to me at the end of semester how much she had learned from me.

When I asked what, in particular, she replied that I had taught her to dig deeper.

I was surprised, and asked her how so.

She told me that one day when she was working on a mathematics problem and asked me what I thought of her efforts, I had replied: “You could dig a little deeper.”

I had no idea that such a simple phrase could have such a profound effect.

Years later I believe in this phrase as much as, if not more than, I did then: “Dig a little deeper.”

And have fun!

That, I think is how we teachers of mathematics can avoid falling into the trap of cognitive theft, and how we can help our students to avoid the low road of asking us to show them how its done.








3 Responses to "Avoiding the low road in learning mathematics"

Thank you for this. Your words have once again caused me to reflect deeply on my teaching.

My students often say to me “You always answer our questions with more questions. Can’t you just give us the answer.” My response: “But then you wouldn’t get to think about it, and that is much more satisfying then the answer.” However, it can be really hard not to respond with the answer when the gap between their understandings and the problem they are trying to solve is a long reach.

I’m not so sure that, “Show me how to do it.” is the problem. After all, we can learn by watching. A lot of literacy instruction involves demonstration. However, there are two aspects that accompany these demonstrations that is often missing in math: (1) Before saying, “Watch what I do and listen to what I say – keep a record of it so we can discuss it later.” in order to focus learners’ attention; and (2) After asking, “What did you see and hear?” to assess what they attended to during the demonstration.

I think the problem is when students say, “Tell me what to do.” This leads to learners becoming disempowered: an unsustainable position that I discussed in my TEDx talk. It relates to Seth Godin’s point in “Linchpin” that the lizard brain wants a scapegoat. If you tell me what to do and I get it wrong, then I cannot be blamed.

I do like the idea of suggesting that learners need to dig deeper. Ellin Oliver Keene (a literacy guru) suggests adding “What else?” to any classroom discussion in order to foster understanding. In either case, it means turning over responsibility to the learner.

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