“The idea of a Math Circle is to not tell anyone anything.”

So began the Math Circle Summer Teacher Training Institute, led by Bob and Ellen Kaplan.

Bob continued:  “To discover and construct mathematics for yourself is to make it yours.  You can use it again and again.”  Throughout the week, new and experienced Math Circle leaders practiced telling kids nothing.

Some 10-14 year olds weren’t sure whether they had seen mathematical proofs before.  “Okay, you don’t remember,” said Bob.  “Good.  You should never remember anything in math.  You should figure it out on the spot.”

I was thrilled to hear this.  Too often, we hand kids an algorithm before they have the opportunity to struggle and figure things out for themselves.  For instance, we often demonstrate for kids a fraction division algorithm that goes like this:  “Change the division sign to multiplication, flip the second fraction, and multiply.”  Why?  Why does this work?  If a student memorizes this and can produce it on demand, does that mean that the student is good at math, or good at following instructions?  Is math ability the same thing as the ability to follow instructions?  It shouldn’t be, but that’s a topic for another day.

Instead, I’ll tell you about what happened when I withheld that fraction-division algorithm from R.

R and I have done a lot of mental and real-world math over the years.  Our current custom is for her to understand concepts and to avoid algorithms unless she creates them.  She had done so with other fraction operations, but had never divided fractions.

“What is 3/4 divided by 2/5?”  I asked her.

She first converted the question to verbal concepts.  Her verbal concepts for fractions come from Jim Tanton’s book Thinking Mathematics! (Volume 1).  In it, he presents “pies and boys” as one way to look at fractions.   A typical example goes something like this:  “If 5 pies are divided among 3 boys, how many pies would each boy get?”

So, to divide fractions, R naturally talked about pies and boys:  “You’re asking me to divide ¾ of a pie among 2/5 of a boy, right?  So, if 2/5 of a boy would get 3/4 of a pie, then 1/5 of a boy would get 3/8 of a pie.  Multiply that by 5 to get how much one boy gets.  One boy gets 15/8.  Right?”

Hmmm….  I had to think about that for a moment, but then realized that she’s exactly right.

I worried, though, because she was days away from a high-stakes test, and I wasn’t sure whether her method was efficient enough in all cases.  But I didn’t want to hand her a more efficient method on a silver platter.  We didn’t want to lose her conceptual grounding to a less-than-obvious algorithm.  So I worked with her to derive that flipping algorithm from her own understanding.

I asked her to practice some problems using the traditional method, and then some using her own method.  Her assessment:  her own algorithm was faster much of the time, and had the benefit of feeling intuitive.  But it was not always useful.  So she now is empowered with the flexibility of more than one approach.

I arrived at the Teacher Institute with those thoughts on my mind.  Whether algorithms are even necessary is a controversial subject – a subject that came up during many breaks this week.  I decided to ask Ellen Kaplan directly whether there is ever a situation in math where algorithms should be flat-out given.

“Only in an emergency,” she replied.  “Hmmm…,” I wondered aloud, “I wonder what would constitute an Algorithm Emergency?”  Ellen and I brainstormed.  She pointed out that some students need algorithms for college entrance exams, in order to obtain needed scholarships.  I remembered a few of my SAT students who had to get a certain score in order to attend college instead of join the military.  She also pointed out that some ESL students don’t have the comprehension yet to translate word problems into math language.  But that was about it.

I couldn’t stop thinking about this, so the next day, I decided to elicit opinions from students.   I asked O, a high-school student who likes math.  He replied that it’s hard to imagine having to derive every math formula you need.  How tedious it would likely be, he pointed out, to have to derive the quadratic formula.  When O said this, Ellen was standing nearby and overheard.  She strongly disagreed.  “The quadratic formula can be created in a few steps.  And if you’ve created it yourself, and don’t remember it during a test, you’ll be able to make sense of the problem anyway because you have a deep understanding of where it comes from.”

That sounds good to me, as a teacher, Math Circle leader, and homeschooling parent.  But, it also sounds challenging.  I have to admit that off the top of my head, I don’t know how to create the quadratic formula.  I think that I could figure it out with a bit of time and a pencil and some paper (and maybe some curious kids to collaborate with).  In fact, I’m about to go off and do just that.  It sounds like fun.  But before I go, I’ll posit what I now think is the real question of math education:

Are we educators willing to let go of the algorithms and instead figure things out with our children?

— Rodi

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