Wednesday, March 27, 2013

On chunking numbers

For the next couple of days I get to spend time at Discovery Ed with a bunch of bright folks who are not "educational" experts, or "cognitive science" experts,  or "business" experts--they are here because they're interested in the world around them, pay attention, and take advantage of our brain's plasticity. I hope to take pick some brains.

The brain is a funny animal. What we know influences what we do; what we do influences what we know. That was true back in the day, and it's true now, and it will remain true long after the current crop of educational experts fade from the scene.

I have always had an irrational love of numbers, but not in a Sesame Street fetishistic way, where performance matters more than purpose. Numbers help sort patterns, and help us see the world. Humans love patterns, for whatever reasons, and numbers intensify what we can see. (Some people smoke pot to intensify their experiences, I play with numbers.)

In the past few years, I have seen more and more kids who are functionally innumerate. Some of these students do fine in algebra yet struggle with science. Numbers, already a level of abstraction, become symbols without meaning, punched into machines programmed by others, to give results that will lead to faint promises of success. (Get the correct answer, your grade goes up, and eventually you get to heaven. Or something like that. I lost track of the story decades ago.)

Somewhere along the way, numbers lost their connection to the patterns they represent. We no longer need them. No need to make change. No need to figure out how long it will take to walk to the diner to meet up with others. No need to build tree forts. None of this is news to teachers.

What may be news, though, is my recent (and anecdotal) recognition that one reason kids struggle with numbers is because they cannot "feel" them anymore. Show a reasonably bright adult a few objects clumped together on a desk, ask her how many are there, and she will immediately know it is "five" or "seven" or whatever the number happens to be. We chunk numbers, nothing a few other mammals can't do, and we do it well.

Clever Hans, the counting horse.

Or at least we used to. I am seeing more and more kids needing to count out what should be automatic. I see students flying through algebra without a decent grasp of arithmetic.

Are others seeing this to?
Does it matter? 
Can we fix it?

I do confess I need to count my quahogs individually once I get past a dozen or so....


Jenny said...

I don't know if this is a widely recognized problem, but I do know that many primary math curriculums address it. In my school we practice subitizing with dot cards, ten frames, domino cards. But I think we stop doing so after 1st or 2nd grade and not every kid has truly internalized this yet. We need to keep it going.

Kathryn J said...

Absolutely. I have learned to just be quiet when a student insists on multiplying 3.2 x 1 using a calculator. Seriously, the first few years of teaching I would try to convince them that they didn't need a calculator to do x1 and I found that took way longer than just letting them do it and then be amazed that it was the same number.

It is very sad. I think it happened when kids stopped learning math facts because it was thought to be "boring". Memorizing addition, subtraction, multiplication, and division facts teaches you things about numbers that you can't learn from a calculator. Students also cannot estimate.

Every year on a test in September, I give a problem that reads something like "A 133.6 g rock is 10.8% tin. How many grams of tin are in the rock?" Invariably I get answers that are much more than the mass of the entire rock. I use it as an example of needing to estimate to make sure your calculator answer is in the approximate range. The lesson never sticks.

One feature of the much-maligned common core - at least the examples that I have seen from math teachers in my district - is that the numbers will no longer be disconnected from the real world. I have long been frustrated teaching students who are "doing polynomial equations in math" yet cannot solve a density problem when given the formula.

Enough of a tirade, sorry. Can you tell this is a hot button for me as well?

Barbara said...

I agree totally with Jenny. But believe it or not, they are starting to get away from "manipulative" math in pre-school and Kindergarten as well, never mind continuing it in 1st or 2nd grade. Math and science are not internalized because they are not materialized in hands on activities. Kathryn says memorization of facts teaches you things you can't learn from a calculator,which is true, but seeing sets in a unifix cube activity sets a math concept it in a different part of your brain (although don't ask me why or how ;) Science concepts become memorable when we interact with them in more than one way as well.
I also feel for all teachers because you aren't working with a level field. Some students haven't come from hands on learning experiences and certainly cannot think outside the box. Many early childhood teachers haven't been properly taught how to effectively use manipulatives in the classroom and also don't have time to teach outside the box. We are forgetting that play is learning, and we learn through play. This stuff should be FUN, and when it is fun they will learn.

Okay, I will step off my soapbox now... (talk about hot buttons Kathryn ;)

Jeffrey Michals-Brown said...

I like to play with numbers a bit myself. Here's a nifty back-of-the-envelope question for someone: how much water does the average adult create by respiration each day? If I recall correctly, I made it out to be about a cup.

doyle said...

Dear Jenny,

Amen--I see far too many adolescents who have not internalized it. It's not something that is practiced much beyond the elementary years.

Dear Kathryn,

We had a lot of discussions the past two days about these issues--I feel a tad better that I'm not the only one seeing this.

I think it gets down to number sense--I hope so anyway, would hate to have my view on evolution destroyed in a generation. I agree with a few things in the CC, including that. I have no issues with promoting literacy of any sort.

Hang in there--not sure how you can teach chemistry if relationships are not understood.

doyle said...

Dear Jeffrey,

Sounds about right--one source I found suggests 13% of our daily 2600 ml, about 11-12 ounces.

I know you know this already, but some folks might not know that te kangaroo rat can get by with only this metabolic water.

cope said...

I throw the occasional math problem at my astronomy and earth/space students all the time. I feel that I almost always make them easy enough to do in their heads but they never can...out come the calculators or cell phones and their nimble fingers poke their way to an answer. Sigh...

Example (just given to my earth/space kids): "If a neutron star is 100 trillion times denser than water and a gallon of water weighs 8 pounds, how much does a gallon of neutron star weigh?". Maybe one students per class got the correct answer. They few who ask for help (which I always provided when asked) usually begin by asking me how many zeros in 100 trillion.

doyle said...

Dear Cope,

Sad, sad, sad--but I am at the point where I can no longer just shake my head. We're doing something wrong.

I love the question though--may use it when I get back.

Unknown said...

My students will do long division, carry the decimals wrong and then come out with a completely unreasonable number. The simple act of asking, "Does this make sense?" isn't there.

It's why I spend time on estimation, reasonableness, mental math, number sense.

doyle said...

Dear John,

Is it us (teachers and schools) missing this when the children are younger?

Is it lack of interest?

Does number sense matter in an electronic universe defined by other humans?

Is number sense an essential part of being human?

Unknown said...

I have a hunch that it isn't number sense that they're missing so much as sense. Every sense. All five of them. The tactile, real, experiential sense that is number sense - that's what they're missing. And they're missing it, because they've learned to trust the process as a Truth above what they experience.

doyle said...

Dear John,

I think that there's something to that, but I've dived into distributions and arrays the last couple of days (literally dreaming of arrays as I drifted off to sleep last night)--there's something peculiar about mathematical reasoning, and we're gliding over the "peculiar" (or interesting) in order to sate the algorithmaticians among us.

I need to ponder some more--probably the last word on this.