Friday, February 17, 2012

This Is What Math Learning Looks Like

I stopped taking math courses after my sophomore year in college, so I'm aware that there is a lot about math I don't know, and even more that I don't know I don't know. By the time I hit those final years of formal math education, my main strategy was to sit next to the prettiest girl in class, under the assumption that I'd need help from my peers to survive, and being a hormonally driven teen, figured I'd have the most fun with an attractive study mate.

That's how I met my friend Renae, the girl who got me through calculus, and who has since made a career serving as chief financial officer for several profitable enterprises. Renae was, in fact, the best math teacher I'd had since elementary school because she was smart, helpful, and I had fun being with her. 

Out of sheer stubbornness, I continued to register for math courses that I didn't need for my degree: refusing to be defeated by the cliche, "Math is hard." That said, I saw the beginning of the end coming when I rearranged my entire schedule to follow Renae into the "Calculus for Business Majors" course series, rather than stick with the straight-ahead, pure mathematics of hard core calculus, where a typical day in the classroom involved a TA with a poor command of English, solving problems on the blackboard. I might not have been willing to admit that math was hard, but that action of following Renae convinced me that math without smart, pretty girls, was no fun. I later tried a discrete mathematics course on my own. It hurt my head and I dropped the course after 3 mind-numbing sessions.

I can't speak for college level math education, but as a preschool teacher, I know that there is no reason that math shouldn't be fun, even without smart, pretty girls helping you. All it is, after all, is a process of learning increasingly complex and wonderful ways to do things that give us great pleasure as human animals: patterning, classifying, and sorting. At bottom, when we boil it down, that's the entirety of math -- patterning, classifying, and sorting -- which is ultimately the foundation of analytical thinking.

We enter the world as mathematicians, exploring all the ways we can order our world, craving an understanding of the logic of things.

We repeat our mathematical inquiries over and over. 

This boy spent 10 minutes repeating the pattern of putting a small, pink detergent bottle lid into the mailbox, closing the mailbox, opening it, removing the pink lid, then putting it back inside, repeating the pattern over and over.

Tom Hunter wrote a brilliant, simple song, which he later turned into a children's book, entitled Build It Up and Knock It Down:

Build it up
And knock it down
And build it up again.
Knock it down
And build it up
And knock it down again.

Subsequent verses echo the same circular, two-step pattern, so familiar to the natural play of young children. Turn it on and turn it off and turn it on again. Pick it up and put it down and pick it up again. Put it in and take it out and put it in again. It might drive us crazy as adults, it might seem to us like they're stuck, but really the children are simply testing their formula, practicing it until it's second nature: A-B-A-B-A-B . . .

Young children, in the course of their play, go on to discover increasingly complex patterns all around them.

And then using those discoveries to do important things like take turns in a board game . . .

Or engage in a meaningful process of many steps.

Play itself is impossible without the ability to think logically. That's why we're driven to mathematical play. These are things we really must know in order to satisfy our curiosities. There is no greater motivator than the prospect of discovery. They're all ponies -- it's a discovery! Let's put them together, and to make sure we understand this, let's put blocks around them.

And even within this classification of ponies, it gets more complex as we notice that some of them have "warm" manes and tails, while others have "cool" manes and tails. It's another discovery! Let's group them within their group, two at this end and three at the other.

When we make a study of a shape . . .

. . . or put blocks in a box . . .

. . . or only the cars . . .

. . . when we line things up . . .

. . . or create one-to-one correspondence . . .

. . . we are mathematicians discovering for ourselves the classifications and patterns of the world, and using them for our own pleasure be it great beauty, great truth, or just horsing around with friends.

Math is not hard, but at some point, for many of us, it stops being about discovery, which makes it no fun. 

The opposite of play isn't work, it's rote. ~Edward Hallowell

I don't know if we, as adults, need to know more math than we already know. Maybe higher math turns us off, not because its hard, but because its boring to those of us with a particular learning style. Maybe this kind of complex patterning, classifying, and sorting can really only live on paper and doesn't have a place in the rest of the world where we can use all of our intelligences to play with it. Maybe at that level it's just a playground for a certain type of human brain. And that would be okay with me, although I hope I'm wrong.

But I don't know. These are undoubtedly the musings of an idiot because of all the things I admittedly don't know I don't know.

Maybe we already typically know enough math to live our gratifying lives. And still I'm certain that the number of exciting mathematical discoveries that I could potentially make is a number approaching infinity.

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Anonymous said...

I'm not sure I think that all learning should be "fun" since sometimes children confuse "fun" with "entertaining." In fact, when I work with students they often complain that the teacher does not keep them entertained, therefore they cannot "learn." I also think that older students confuse fun with easy. When something is challenging it ceases being fun, therefore they check out. Learning is complex.

Meagan said...

I enjoyed most math from Algeba beyond, exactly because it was a challenge that I was capable of meeting with effort. I don't see it as all that different from puzzles we seek out as adults, for fun. Now, I can't remember much beyond algebra, but every once in a while, I'll wonder something about space, time, motion, and I'll think *there's a formula for that.* I may not know it, but I like the idea of things I learned a long time ago actually being real and important.

@anon There is a big difference between edu-tainment and open ended play which allows constant accidental learning. It's the difference between feeding a child vegetables covered in cheese vs gardening with the child.

Jeanne Zuech said...

Spot on, Tom. I'll be sharing this post with my student-teachers in the Inquiry-based Math/Science ECE course I am teaching this term. Well done :) Have a great day!

Juliet Robertson said...

Maths is another tool we have for interpreting and learning to understand the world around us. Hooray!

I think the problem happens later on when we formalise maths and arguably dumb it down. It's all about experimentation, investigation, discovery and play. Mathematicians play like few other professions.

Thanks for another lovely post, Tom.

kathy said...

Thanks for a fantastic post - I am researching mathematics in early childhood for my PhD and so agree with all you have said.
Hope I can express it as eloquently as you have.

Randi Albertsen said...

Great post, Tom! Math for young children is all about the discovery process and repetition. You summed it up nicely, patterns, sorting, classifying, repeat! It's interesting that the common core standards for math have scaled back the typical push for more and higher learning, and instead focus on fewer objectives and more functional understanding of basic math concepts.

Michelle @ The Parent Vortex said...

This post is really timely for me today... I struggle with feeling like I should do formal, pencil on paper math work with my homeschooling 5yo, but the truth is that she really doesn't like to do math that way. She will, however, happily sort ponies, match chocolate chips into pairs, string beads in symmetrical patterns, count cookies and tap a tambourine in time with some music.

More play, less rote!

Unknown said...

great article- all learning may not be fun- but in preschool it certainly should be & as a homeschool mom of 3 boys- I strive to teach them discipline on necessary things which need to be done while creating a learning environment that is FUN, engaging and helps them develop a desire to learn more! balance is important - it's a part of the pattern :) they aren't going to LOVE writing that paper, but they must learn to do it- so we'll make it a paper about surfing ;) same theory for math- right? thank you for this post- very well done! (we do school outside all the time - btw- love it!) nice to"meet" you!

Anonymous said...

Tom, this was posted in good time for me. Thank you for the reminder to keep things simple. I am a teacher in a competitive Montessori school and I am often under a silent pressure to "keep the children learning". Your post today reminded me to slow down and observe that no matter how simple the adults find the exercise to be for the child, the children are already learning from that exercise on so many levels.
I really enjoy how refreshing your blog is. Thank you again