Beehive Slides

To find the beehive slides from my September 28th presentation, please go to Annie's blog:

annieeduc315.blogspot.com

Mathematical understanding and multiple representations

I was largely convinced by the argument that the authors made. I could identify with many of the examples of cognitive representation that they gave, having had experiences which I would describe as either internalization or externalization. I was particularly motivated by their inclusion of language as an external representation. Many of my own "internal" thoughts occur in English; I know that only some people have an internal dialogue, but I certainly do. I makes me wonder in what ways my supposedly "internal" understanding is influenced by this external representation. 

As an example, I think about emotional intelligence. Many people struggle to understand and process their own emotions because they do not have the representational tools to grapple with them. Therapists can help people by providing language and other representations (people speaking to you, different states of being) that help manage the more ambiguous actual internal experience. This seems to me like a powerful tool, but also something to bear in mind when communicating with others. It's extremely difficult (perhaps impossible) to communicate our own internal thoughts without modifying them slightly to fit a shared representation. There's some interesting connections to the philosophy of Wittgenstein here, but perhaps I'll save that for discussion so as to keep focused.

I was also reminded of some research that I did on honeybee numerical understanding wherein the bees could be trained to understand symbolic representations of numbers. In the process of doing research on this, I learned about number-sensitive and number-selective neurons, which help to understand the actual neurological grounding of "twoness" as opposed to "threeness." 

Something else that I found persuasive about the article was how it spoke to my own experience teaching students who struggle to navigate between representations. In particular, there is great difficulty navigating between language and symbols. Students do almost all of their mathematical learning in symbols, not language, and thus are left stumped when asked what "13% of 54" is. If you were to ask them to calculate "0.13x54" I am certain they would be entirely capable of figuring it out (perhaps with a calculator, but still).

Finally, I agreed strongly with the assertion that students are not given enough time with new forms of representation before being instructed in their conventional use. To me, so much of the process of mathematics involves developing new ways to represent ideas that you are struggling to communicate. inventing language and frameworks is a crucial creative aspect of math. It is much more empowering, in my experience, to create such a framework as a student and then learn that one already exists. It makes the student part of a community of creative individuals, as opposed to an external observer.

As for things to use in a classroom, I am reminded of a game I learned at summer camp, a "Never-can-tell." Essentially, the purpose of the game is to determine what the hidden form of representation is. The game works by rolling some number of dice and declaring how many roses are showing, with how many total petals. As an example, a roll of 5 and 3 (on two dice) would be two roses and 6 petals. This is less a game, really, and more a puzzle for students to work out. The trick (and it's supposed to be a "Never-can-tell" so you can't tell anyone that I gave it away!) is that any central pip on the die is a rose and any pip surrounding a central one is a petal. The "roses" are an alternative visual understanding of the face of the die. Thus, a 1 is a rose with no petals, a 2 is no rose at all, a 3 is a rose with two petals, and so on.

I think that this could be a useful activity with younger students, as it demonstrates an alternative interpretation of a visual representation and also demonstrates how we can use groupings to count in different ways, much like the example of visually representing 20 that was given in the reading.

Letters from former students

Dear Jacob,

I'm writing to keep you updated on what I am doing with my passion for math, like you asked back when I graduated. Right now, I'm working at a non-profit that does modelling work for public policy workers. It's a little dry, but I really love being able to make a difference with math! The really cool thing about the job is that it's all about trying to translate real world issues into numbers, just like the problems in your class. I introduced my coworker to Fermi problems the other day and now we're giving them to each other for fun. Pop quiz: how many barbers in London? 

Anyways, I wanted to thank you for always believing in me and trying to show how math can be useful in unexpected ways. It's really served me well, and I hope you're still teaching and working hard to sell tired teenagers on the beauty of trig functions. Your class was a pleasure to be part of, and so grateful to have been part of it.

Siempre,

Alyssa

Dear Mr. Richardson,

I'm writing to you because I told you that I would, and I always keep my promises. Back when I was your student, I asked you when I would need to know this math you were teaching and you told me that I wouldn't. That part was right, at least. But then you told me that "thinking mathematically" was going to be really important, even if it didn't seem like it. I hate to let you down, but I don't do much thinking mathematically these days. The only time I do is when I'm playing games with my daughter (I know, right! She's 6!) and I break out Nim to keep her occupied. It's not the most interesting thing in the world, but it's great for little kids. Honestly, I didn't learn much in your class. I don't really know why I thought it was a good idea to write to you. I'm not trying to be mean, I guess I'm just... processing. 

You always talked about how math was beautiful, and I never really saw it that way. Math's just not for me, it never has been. I appreciate that you were trying to get a room full of bored kids to focus on algebra, but it always felt sort of disingenuous, like you were trying to trick us into having fun. I guess I hope that I'm wrong and you really believed in this stuff. Maybe you've gotten worn out and grumpy by now like every other math teacher :)

Write me back if you want, I can't imagine why you would.

Sam Geltsinger

My Own Math History

I feel very fortunate to be able to easily recall my favorite math teachers and struggle to bring to mind a least favourite. In reflecting on all of my experiences, though, I can certainly identify some qualities and methods that I appreciated seeing and some that I disliked.

In my favourite math classes, each class was a journey of discovery. This sounds overwhelmingly cliché, but it's true. Part of the reason that I love math is that a math problem has always felt like a puzzle to me. I never had much patience as a student for repetitive practice and much preferred complex problems that forced me to develop new ideas or ways of using a technique. I'm also a big fan of vertical classrooms, working in visibly random groups to solve problems so that I can get insight into how other people approach the problem as well as cement my own understanding by explaining to the rest of the group how I approached the task. 

Many of the negative experiences I had as a learner of math came from a year in which I studied independently through modules (without a teacher, just someone supervising progress). I still have the modules from all those years ago and was idly looking through them the other day, in fact. My biggest frustration was when the module would insist that I approach a problem in a set way (i.e. solve geometrically). It felt as though the module didn't understand the way that my brain was working and, since it wasn't a person, I couldn't explain anything to it. It's very important to me as a teacher and a learner that everyone's ideas are heard and valued, even if they go against a conventional strategy or might not be the most efficient solution. 

"The Locker Problem"

 I first became familiar with the so-called locker problem in the first year of my undergraduate studies. I had the opportunity to revisit it in my final year, this time from the perspective of a teacher professional development day that I assisted with. The leader of the workshop was Richard Hoshino, of whom I am an undying fan. He used a pack of playing cards to simulate the first twelve lockers, which I think is a good idea for two pedagogical reasons. First, it allows the problem to be visualized and even walked through by doing the exercise of turning cards. Second, it suggests to students the value of taking a toy version of a large problem. Even just in this small decision, there's a lot to be learned.

Beyond that, he actively engaged the group of teachers by assigning them a number. We walked through the problem, each flipping our relevant cards. At the end, we could all clearly see that the square numbers (1,4,9) were closed. In this case, that was enough for us all to observe the pattern. However, we still hadn't actually solved the problem to a relational learning standard.

To push us further, we looked at the specific histories of a few lockers. Who opened them? Who closed them? We then observed that we would of course touch a locker if our student number was a factor of the locker number. Again, this was a group of teachers, so that observation came easily, but I can imagine that it might require some teasing out in a classroom setting. 

As a final step to reach an understanding of why the square number lockers were closed, we once again did an activity to involve everyone. We picked an open locker and everyone who had touched it raised their hand. Then, since we knew that we were factors, we paired up with the other part of our factor pair. Naturally, for a non-square, everyone had a partner. When we did a square number, somebody was left all by themselves. Why was that, we asked? Of course, it was because the other part of the factor pair was them! They were, as Emma Watson would put it, self-partnered.

Witnessing all of this engagement and discussion on a problem that could easily have turned into a dozen people sitting at notebooks working individually was very inspiring. It's something I'd be interested to try in a classroom myself.

Skemp Response

I ran a little long on this one, but I was just really vibing with the reading!

When Skemp talks about students being satisfied with instrumental understanding, I am reminded of tutees of mine asking me to “just tell me how to get the right answer.” We have a system which teaches children that, while relational understanding might be superior, we just don’t have time for it in the midst of all the things that must be instrumentally understood. I have largely found myself in the first of Skemp’s two hypothetical situations, with a teacher (myself) who wants relational understanding and pupils who seek the instrumental variety. What a joy it is, though, to have students eager to find the deeper reasonings!

What he is getting at is that our desire to teach in a time-efficient manner has led us to lean too heavily on instrumental understanding, which can never account for the broad array of real-life problems. While it seems easier to teach instrumentally, we would actually take far more time explaining every possible exception and minute variation on a rule than if we explained the base reasoning, at least in my experience.

I think that there is (especially in mathematics for early education) an assumption that instrumental learning is the most that can be achieved within the various constraints a teacher has (time, student interest, teacher competence). Years and years of asking relational questions and being given instrumental answers leads students to stop asking those questions. I’m reminded of a book called “The Game of School” by Robert Fried which talks about how obvious it is that students will take to behavior that is not optimal for learning when learning is so rarely the goal of school as seen by students. Instead, they have goals like “getting a good grade” or “not getting in trouble” or even “being popular.” While some of these certainly could be directly related to learning, they by no means have to be. Anyone who has spent time in a school must know how possible it is to get a good grade in most classes without truly learning much at all…

When Skemp talks about how difficult it is to assess for specifically relational understanding (without some larger, in-depth conversation) I am reminded of my own frustration as a student with being asked over and over to “show my work” on questions which I deemed too simple for anything beyond instrumental understanding. For example, what work does one show for 5x4? I suppose I could have written out a long addition statement, but that felt cumbersome and disingenuous. I hadn’t actually added four fives. I had memorized it!

When Skemp quotes Bondi, I find myself almost leaping out of my seat in excitement. My undergraduate thesis was on how popular media and parental influences affect math attitudes! So much of our paradoxical reverence and fear of mathematics comes from a place outside of the classroom. Students (and indeed adults) are bombarded with constant messages that math is only for the few, a subject known from birth or not at all. It would be a great task for a teacher to correct them in a mere handful of hours a week.

Hello World

It is a good day for mushrooms. A good day for laughing as rain soaks through your skin and cools all the heat that has been quietly burning your brain. You’re not wearing a jacket. What would your mother say? But it doesn’t matter, because nothing matters. You’re standing in the rain, and there are mushrooms growing, and nothing matters at all.