Two views of mathematics

Beautiful fractal image
(Image credit: DanCentury)

As usual, there is an argument going on Reddit on mathematics education. There is a statement from that argument that I would like to highlight here, and a related discussion on Reddit with a related comment.

"I solemnly declare that no kid ever learned math by watching a video OR by reading a paragraph, since math is an action [emphasis mine], not an exposure." deadletter

In a related discussion on resources for a 4th grade student wishing to explore mathematics further, a commenter made this bold claim, in response to someone suggesting that the Khan Academy would not be a good resource:

"Why not? If it’s to pass state Core Standards, it’s more than enough. If it’s to give his daughter who loves math more to learn, it’s more than enough. If it’s to showcase the power and beauty of math, it does that, too." misplaced_my_pants 

(Aside: I struggle to see how the Khan Academy "showcase[s] the power and beauty of math." At best, this is a side show on their site, although the introduction of Vi Hart and her videos to their team at least indicates that they are aware of this deficit.)

These are very different views about what it means to learn mathematics.  One person holds that mathematics is an activity that people undertake, the other believes that mathematics is a specific set of knowledge that ones gains through exposure. These are very different definitions of mathematics, and have very different consequences of what would be required to learn mathematics.

I tend to lean toward the mathematics as activity definition, but understand that my responsibility as a teacher is to ensure that my students also know some specific set of mathematics. It’s not a line I’m particularly comfortable straddling and I feel a lot of tension as a teacher as a result. Whenever I have some freedom from the curriculum, I lean heavily toward explorations of math, either as independent activities, or as a group activity.

At the very least, I want my students to know that there are two views of mathematics (which some may consider to be opposing views), and that they should have the ability to make an informed choice between them (or to choose both, if that is even possible).




  • David,

    I think, and feel, it’s both – an activity that people undertake, and a specific set of knowledge that ones gains through exposure. I both DO mathematics, as an activity, sometimes creating new ideas and new proofs, new techniques, and sometimes – more often – I learn what others are doing, and have done. It feels like both to me.

    When I’m teaching mathematics – helping others to learn and mentoring them to create – what’s that? It’s an activity I undertake in conjunction with students. I access specific sets of knowledge. I expect students to come to terms with that knowledge, and I expect them to be active and contribute their own ideas and ways of thinking.

    Thanks for a stimulating post


  • This is a very interesting post! I think that I straddle this line a lot as well. In many ways, I tend to lean towards the “math as an activity definition,” but sometimes I find the need to reinforce this knowledge. I think it can also depend on individual students too.

    When I was in elementary school, I was identified with a learning disability. I really struggled with spatial skills in math: I still do. I could play with shapes, move them around, and spend hours exploring them, but I still couldn’t see what I needed to see. Eventually I needed to just learn what was required. I tried to memorize the concepts when I could, and really get exposed to the concepts as often as possible, so that I could learn the skills to at least make it through the unit. This isn’t true for every unit and every concept, but for me, it was true for the geometry unit. For other students, this may be true for different units. Maybe this is also a case of differentiated instruction. What do you think? When do you find yourself straddling this line the most?


  • David Wees wrote:

    The question is: are you now more capable of working with shapes than you were before you started the unit?  If not, then what if anything did you really learn from the unit? Yes, you "got through it" but that’s not the purpose, the purpose is to extend yourself with the learning you do through the unit. If that didn’t happen, then it wasn’t a success.

    Maybe the problem with the unit was that it didn’t meet you where your needs were as a learner. Our brains are plastic enough that although you may have been pretty challenged by activities to do with shape and space, there is probably some activity you could have done that would have benefited you more than the geometry unit, at least at that time. One of the problems with our system is that we do not allow for sufficient differentiation, especially when students are on either end of the spectrum for a specific unit.

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