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Formula or No Formula

5/30/2015

2 Comments

 
Picture
Our geometry unit was drawing to a close and I felt like I had given them a good range of experiences with area and volume.  We counted squares and we counted cubes.  We derived formulas, but spent a lot of time digging into how the formulas represented what we had learned about the shapes.  All along I kept dodging any attempts to get me to tell them "how to do it" by posing questions of my own:  what are we counting?  How is that like the way we estimated the cubes in the coffee cup? How would you build that shape from cubes? I was like a ninja ... you couldn't pin me down.  And more and more students were responding with "Oh yeah, it's like layers in a cake / it's like levels / it's the area on the top times the height."  I resisted giving them a formula for volume, letting it just be part of our conversations, until late in the process when I showed them this slide as part of a summary discussion. 

I felt good.  I felt certain that this was just summarizing an idea that they had internalized, that I had heard them using the idea when they worked together, and that they got it.  So imagine my surprise when, as we were reviewing for our test on area and volume, students in each of my sections said at some time during the process, "You never gave us the formula for volume." 

... There are times, I fear, when the best response I can come up with for the current teachable moment is "Are you kidding me?"  Had I completely misread their level of understanding?  Did they not get it?  You don't just memorize formulas - you figure things out!  What had we been doing for the past two weeks, for the whole year?  Are you kidding me?

Now it is true that we did have a formula sheet for area.  I guess I thought that there are different formulas for different shapes, and I wanted to have a page that summarized all of them.  We had reasoned them all out for ourselves, and I didn't really expect them to re-derive the formula for the area of a circle on their own.  But the volume of a prism or cylinder seemed different.  How could they understand the concept if they needed the formula?

I must have been channeling Fawn Nguyen, who treats student input with such respect and reverence, because it suddenly occurred to me - this is a great opportunity to talk about this.  So I wrote on the board -
                      Formula                                        No Formula

And I asked "How many of you have ever come to me this year and asked me to tell you how to do something, but instead I said 'I think we better take another look at this'?" (About 90% of  the hands went up).  "And how many of you wanted to punch me?" (About 90% of those hands went up - and fortunately most of them laughed too).  "So talk to the person next to you - what do you think about formulas?  What are the pros and cons of having or not having formulas?"
Here's what they said:
Formula                                                                            No Formula
more complicated                                                          there are questions, sometimes confusing
non-chaos                                                                        chaos - scrambling
like cooking with a recipe
                                              like cooking on Chopped     -  either way, you just want the cake!
anthill with a system                                                      anthill on fire
hike with a plan                                                               just walking

Formulas are useful and important, but they are hard to memorize
I could figure out area without a formula
Formulas help you remember how
You have to know how in order to make the formula
If you learn the idea, you don't need a formula
You can't be given the formula right away
With a formula, you can remember it better, use it outside
Formulas are the easy way
With no formula, you learn to understand faster
You learn more yourself with no formula, and it stays longer


The students are reading Call of the Wild, and one student compared formulas to the dog's cushy life at the start of the book, and no formulas to the kind of thinking and adapting he had to do in the wild.  I really liked that.

So what do you think?  Was this a failed unit?  Formula or No Formula?  What I really got out of this day was that  sincerely asking the students what they think told me a lot more than I expected it would, and turned a stressful moment into a learning one. 

The next day, during the test, the student who started the whole thing by asking for the formula seemed to be struggling.  He asked me "When you find the perimeter of a triangle, do you have to divide by two?"  Ah.  With that and the volume problems, I encouraged him to think back over some of the things we had done in class, and he remembered finding one level for volume and the perimeter looked like it was making sense ... ah ha!  A victory for thinking it through.  And then afterward I saw that his surface area work was a jumble - "Yeah, my tutor and I did base times height ÷ 2 *2 ... or something like that."   I asked "So were the formulas a help or a hindrance?"  He said "Oh the formulas totally helped."  Ah.  Well...one step forward...

2 Comments

I Drive a Hybrid

5/9/2015

2 Comments

 
I have been inspired and educated by Jo Boaler over the years, but I think her most recent article contributes to the unnecessary polarization of the math education community.  In the article Jo writes “The lowest achieving students worldwide were those who used a memorization strategy – those who thought of math as a set of methods to remember.  The highest achieving students were those who thought of math as a set of connected, big ideas.” She seems to imply that anyone who uses a memorization strategy believes math is just a set of methods, and also that the highest achieving students do not use memorization strategies.  Both are clearly nonsense.

In my experience, learning is all about clunking around, learning something, practicing and automatizing that, and chunking it so that you can learn something more complex.  Just because we have kids practice and memorize in our classrooms doesn’t mean we “value the faster memorizers over those who think slowly, deeply and creatively.”  We should be talking about the proper balance between concepts and practice/memorization (I think it is splitting hairs to distinguish the two), and how to be most effective at both.  I think Jo may be thinking this as well, but her rhetoric tends to push people to one side or the other of this common ground. 

I drive a hybrid.  I am striving to have my math class be as conceptually based as I can at all times.  My students write linear equations from situations and data before they do any solving, and then we use bar models so they can direct the process themselves.  We learn the formulas for the area of a triangle and area of a circle by drawing and counting and reasoning.  It drives me nuts when a student puts a 1 under the 5 to multiply 5 x 2/3.

But every year, I ask my 7th graders (who were also with me in 6th grade doing that great lesson on areas) to find the area of a triangle and about half “forgot”.  If they haven’t memorized the formula for the area of a triangle or a circle, the surface area of cylinders and prisms becomes a real challenge.  Could they get along without knowing them? Sure, but with very little time investment, I can help them to memorize them.  There doesn’t have to be a sacrifice of any conceptual thinking.  What’s 7^0? Multiply 2 fractions? Solve 5x =7 or 2/3x = 3/5?  Short, frequent, painless practice sessions can make these routines useful later on, long after the conceptual work is done.

Because here is the big aha for me – does the fact that students don’t recall the formula for the area of a triangle mean I did a crummy job of teaching it last year?  I don’t think so.  During the unit they could all tell me about the related rectangle… and obviously the triangle was half.  But I think it means I didn’t finish the job.  Practice is the second half of learning, both for facts and concepts. Jill Barshay’s article describes Jo’s thinking:  “The human brain is forgetful by nature, she argues, and what she wants is students to develop the number sense to calculate 7 x 8 quickly even when their brains can’t recall the math fact instantly.”  But without practice, the students will lose those strategies as easily as they lose the facts.

Think about what happens when you solve any math problem.  There are things you have to think through, maybe use trial and error or other problem solving strategies, and things you just know or recognize.  Those are two parts of the brain at work, and as math teachers, we need to be effective in serving both.
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