I love my nephews and nieces. They are great fun, and there is nothing more fun then getting them to do math. I don’t know why I do this, but I love having them do little things here and there. Count this, count that, what is this plus that? At one point on my spring break my nieces were doing addition practice all on their own and showing me their work! Seriously, what’s wrong with me? I was really intrigued by my niece’s ability to add large numbers together and it has caused me to reflect on the nature of language and our ability to reason.Ones, Tens, Hundreds, Problems?
My niece is in grade 1, and she is adept at adding single digits. With little hesitation she can do her basic addition. She even showed me that she could do things like add 100 + 100. I thought this was really neat so I asked her some questions.
Me: What’s 1+1?
Niece: That’s easy it’s two.
Me: What’s 100+100?
Niece: It’s 200 duh!
Me: What’s 1000 + 1000?
Niece: 2,000 these are easy!
Me: What’s 10 + 10?
Niece: … I don’t know.
She has not learned place value, and therefore her ability to reason this was not developed. I tried to walk her through it. I got her to draw dots, but miscounting the dots led to incorrect answers. I tried having her write out the answers to see a pattern.
1 + 1 = 2
10 + 10 = ?
100 + 100 = 200
1000 + 1000 = 2000
She found the process confusing. I did not know what to do. I’ve never taught math to someone this young. This was just some fun we were having, and then I felt that her confusion might result in a math phobia that would predominate the rest of her life (I may be a bit overdramatic, but I was worried). We decided to take a break from math for the moment, and worry about that problem later, but I find myself still thinking (months past) about that moment.Language and Thought
I remember in my intro to psychology class where we talked about what came first the ability to think or the ability to talk. Can we share thought or elaborate our thought without language? How does language affect our ability to think? Can we even think without language? Are we limited in thought by the language that we are given?
My niece gave a curious example for me to ponder upon. She was able to reason, with ease, how to add hundreds and thousands together, even though she has little understanding of place value. My hypothesis is that she can do this because the language of hundreds and thousands still uses our single digit counting; she is simply counting a unit
that happens to be named hundred. One plus one equals two, one apple plus one apple equals two apples, one hundred plus one hundred equals two hundred. For my niece hundred was just another quantity that we can add together. Ten, however, did not have the single digit number in front of it. We do not call ten one-ten, or twenty two-ten, and she was thus unable to reason it in any way. Why is that? Why do the tens have this magical new way of naming numbers? Every other number that does not require a ten type number, uses single digits (i.e. 1,000,000 - one million; 100,000 - one-hundred thousand; 1,000,000,000 - one billion). Is this a fault in our language of mathematics, or just another hurdle that we must overcome? Is this language deficiency making our sense of place value more difficult to grasp?
I don’t know, but it’s interesting to ponder...
A few weeks ago I was talking to a number of students in the hall at lunch. I cherish these moments, because when outside of my classroom students remove a certain number of proverbial layers that they bring into the class. One layer they commonly wear in class, and therefore the one that I love to see vanish most, is their fear of saying something wrong. Often in class I see students’ puppy dog eyes, fishing for the correct answer that will allow me to move along and bother them no longer. I am making students work for the gratification of being correct more and more, but that is for a different post.
My point however is that when out of the class I am no longer the “all mighty holder of all truths Mathematical, Sciencentific and French;” the stakes are low in their interaction with me beyond my door and that layer disappears. I have taught some of my favourite science, math and French lessons at lunch and afterschool simply because I have students that are a) not scared of being tested, and b) genuinely interested in the learning that is occurring. They have no reason to be there other than sheer curiosity. The other benefit of being outside of the scheduled class time is that students become much more candid. Since they are not worried about the correct answer, they therefore reveal much more of their true thinking, worldview, and understanding.
In the particular moment that prompted this post, I was chatting with a group of students made up of a number of boys who were winded, red in the face, and generally exhausted. The matter was that they had walked during their run in PE, and as a result had to run the route again. This was a BIG DEAL, because it meant that any time that they spent on this second run, was time spent away from lunch hour. The boys then exclaimed, “We got our fastest time of the whole year today!”
I replied, “Great! Now you know how fast you can really run!” This remark was met with many a disgruntled face. One girl in the group said, very sincerely, “Mr. Piccini, that’s not how it works. You see, what you do is you run just hard enough to make it look like your trying, but that you know you can improve your time later on, so on the record it looks like we are improving.” This amount of utter honesty almost shocked me.
I then asked her, “Why don’t you just try your hardest and actually improve your speed?” to which she promptly said, “There’s no point! Put me in basketball or soccer and I will run because there is a point to that game, but regular running, there is no point to that.”
This is what really hit home for me. These students were not lazy, nor were they rebellious; they saw no point in running for the sake of running, and therefore they labeled it as something that could be done with just enough effort to get the teacher off their back. I wanted to tell them about the physical and mental health benefits, the conditioning for sports, and the pleasantness a good run can have, but I realised like so many other lessons a lecture was not the way to show them the point.
This got me thinking about my class. In Science, Math, and French have I shown my students the point? Do I make it clear to them everyday why we are doing this? More importantly do I create for them moments and opportunities where they realise ‘What the crap? This is something I need to know now,’ because it is not enough simply to tell them. Necessity is the mother of all invention, and when we put students in a state of necessity, they are no longer receivers, but they are seekers and inventors. I want to nurture a generation of such students, because once they become true seekers “the point” becomes moot; when they become seekers they do not need to hear the point, because as seekers, they now have a new responsibility: they must create their own point.