Friday, January 26, 2018

Word-Problem-Phobia Relief (Nuts and Bolts)



Word problems are the anti-reese’s-peanut-butter-cup of homework--a surprising combination that can make you feel bad about two subjects at once.

For many students, the concrete and discrete right-or-wrongness of math is a refuge from the more complex gray areas of other subjects, a refuge which is rudely breached by the introduction of written language into the work.

For others, a language challenge may make word problems much harder than just the conceptual/applied thinking challenge they are intended to be.

For still others, who have learned to cope with struggles in math and struggles with deep-reading separately, putting them together is a cruel trick that topples both houses of cards, and can quickly max-out cognition or resilience or both.


From early grades, many math problem-sets progress from simple to more difficult, and then repeatedly affirm that word problems are the uber-challenge because they are at the end of the set. This makes math homework feel like descending into quick-drying cement: slower...slower...annnnd stuck. That is a seriously demotivating experience. And when it happens daily or weekly....phewf!




Helping students tackle word problems, I think the first order of business is overcoming that (legitimate, experience-proven) revulsion, and the certainty of the inevitability of failure. Admittedly the practical strategies I use are not unusual, but the starting place seems to be. We need a light to break through those clouds before we can do much else. So we ignore the words.

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This is how I've quickly described the process to parents (more involved dialogue-version below):




To be very clear, this technique does not actually overcome any of the learning challenges described at the top of this article, really, at all. It merely seeks to cut through the emotional baggage that tends to compound these strugges over time, so they can be addressed directly.


Here’s an expanded conversation about a different word problem:


(First we put a data box on the scratch paper or board, and I ask the student...)



What are the numbers you see?


(student generally starts reading the problem)


Wait, wait! Sorry,  don’t read it yet. Those words are just annoying noise right now. I can even cover up the words if that will help. We don’t even know if it’s useful information, but we’re pretty sure the numbers are, so just ignore the blah blah blah around them and tell me the numbers.


....8 and 12 and 4?


Any other numbers?


...no?


There might be?


No.


Okay, so 8 and 12 and 4. So, I dunno about you, but that paragraph looks a little intimidating to me--like uh-oh, how are they gonna try to confuse me or whatever--but 8, 12, and 4? Not so much. Could you add those?


Yeah, um it’s...


No no, I was just asking if you could. I don’t care what the answer is.


Yeah, it’s...


Seriously, you don’t need to do that math. I totally believe you can.

Could you multiply or divide them? Or like add two and divide by the other? Or subtract or whatever? Is that getting too hard?


No, I can do that.


So we could put a plus, minus, times, or divide sign between those numbers any which way and you could do it? That’s a pretty tall order. Is that too much? Be honest; I won’t judge. We’re after the facts.


I think I can do that.


I want you to be super confident before we move on. Should I stick some operations symbols between those numbers?


OK.

(I do that and note if we need to review math facts, order of operations, etc. later for retention, after talking about them in this context. Or/Until...)

I’m kinda super confident...


Well, let me know if you change your mind about that--we can always backtrack. Nothing wrong with that. BUT “super confident,” eh? Awesome.  So we put those numbers in our data box. Stack em up. Boomp boomp boomp.  Okay, now we need to know what they are...


(starts reading the problem)


Not yet! That’s still blah blah blah mostly. Where would the words be that tell us what these numbers are counting?


By the numbers?


Yep. Probably. So what are they? What are our units?


Um, 8 um, bags?

OK, what else? (At this point I am only looking at the student and our scratch paper, so I’m explicitly not looking over her shoulder at the word problem and evaluating her or the quality of her work. We are a team; she is dictating and I’m recording. I need the info from her.)


Um, 12 pieces.


12 pieces? Does it say pieces of what? It’s hard to picture 12 pieces.


It says pieces. Oh! Pieces of bubblegum. Sorry.


No need to apologize! I actually said not to read the other words, so you were just trying to follow instructions. That’s good! Good job making me try to imagine random pieces. (write pbg next to 12)

O...k (generally bemused at this point)


What’s the 4?


Um (worried)... Bags?


Is it bags?


It says bags.


So 8 bags and 4 bags? Are they different colors or something?


...It just says bags.



Ok. Perfect.

Here’s a hard question; you can answer or I can tell you. I see this stuff because I’ve done these this way a lot, but maybe you can too. Either way is fine. You know how we talked about the operations you could do with those numbers?

Operations?

Plus, minus, times, divide...

Yeah?

Well, can you tell that now some of those would make sense but others wouldn’t--because we know what the numbers are counting? Um, am I making sense or should I explain what I mean?

I’m not sure.

OK, I’ll explain. If I had 8 bags could I take away 4 pieces of bubble gum?


Maybe, If there was gum in them.



Ha. Yep. Assume no gum. I could take away 4 from 8, but if I put 8 bags in front of you with nothing in them, could you take four pieces of bubblegum from them?


No. Not really.


Nope, so see we are probably not going to need to subtract 4 from 8. Can you see what we could subtract or should I tell you?


...I’m not sure.


So, imagine those 8 bags in front of you again. What could I subtract from them, of the things in our data box?


4 bags.


Right! Usually plus or minus in a word problem will have to have the same units, the same things being counted... bags or ducks or vampires or munchkins, but all just one of those things. Multiply and divide could have different units, or the same, just to be confusing. Can you imagine what the problem might be about 8 bags, 4 bags and 12 pieces of bubble gum?

Maybeeee.... you have one piece in each bag because there are 12 bags?


Oh! That's cool, I hadn't even thought of it that way. I like it! Like if there was a bubble gum shortage and the government subsidized bag companies or something...


...I guess?


I was thinking something more like if there are 12 pieces of bubble gum in a bag and someone gave me four more than the 8 I already had, how much would I have. I dunno. Which one of ours do you think is more likely?


Yours.


Because I'm the teacher?


No, because mine is too easy.


Oh, yes! That's smart. Yeah, the first thing you should expect would be it to be some kind of operation like you guys are actually doing in class right now. That doesn't work on like achievement tests, but day-to-day it's smart. If you do a word problem and it seems super easy, like something you learned two years ago, its totally smart to be suspicious of whether you read it right. Nice!

Two more things before we read the whole problem.


Seriously? 

I know; don't worry. I know it seems super slow, but it gets much faster as this stuff becomes automatic and it makes these SO much easier. Seriously. Trust the crazy man with the dry erase markers. Is there a question mark in the problem? 


Yes. At the end.


Okay can you read the sentence before it?


"How much bubblegum do you have left?" 


OK so I'll write, "how many pieces of bubblegum left?" as the question in our data box. Just a quick check, does the sentence with "12 pieces of bubblegum" end there or does it have more words?


It says "in each bag."


Ok, so I'm gonna write "per bag" with a line like this after pbg. Does that make sense to you? 12 pieces of bubblegum per bag?


I think so.

And I need to ask a favor. When I get all involved in a math problem, and I feel like I know what to do and it's such a relief... super satisfying, you know? Or at least, better than being stuck. I can get to my answer--like the numbers--and I forget to make sure it actually answers the question they asked. Can you remind me to reread that question in the data box to see if my answer makes sense before we decide we are done? I mean, I should check my answer anyway, but I should also reread that question.


OK.


Thanks! I think you've proven that no matter what this question is, you can do the math. Do you feel like that?


Kind of.


OK. Maybe I'm over confident in you. You just inspire confidence somehow. Nice talent!


....


Anyhoo. Why don't you read the problem and see if we can figure out what math to do.

"You are running the coin toss game at the school carnival, but you are worried that you are going to run out of prizes so you want to take inventory, and maybe you can make more prizes. You started with 8 bags with 12 pieces of bubblegum in each bag. You have given away four bags. How much bubble gum do you have left?"

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Again, at this stage the teaching goal is to see what the student can do when she is not coming at the problem from a place of insecurity. Does she know that how much is "left" probably implies subtraction? Does he know that "per" implies multiplication (sometimes division)? There are lots of ways to further problem-solve the stuck-points once they are less painful and overwhelming, as this approach hopefully enables.

My experience with this technique is that often students have trouble realizing that the approach and pre-thinking are making problems easier. They often do better but they think it is because problems have gotten easier. This can make it worth while to have them go back to their old way and read a whole problem before creating the data-box, to experience how much harder that still is, so they have a little more buy-in to what can seem like "extra" work--pulling out data and considering it before reading word problems.
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I am fairly bursting with "Oh, and...!" thoughts, and perhaps over time I will better discern which ones will be instructive to include. I think it is probably important to add that I am always monitoring teacher-talk vs student talk. If I am doing all the talking (as this might seem to suggest), it means either
a. I am teaching very poorly
or
b. we are in the very early stages of an I-We-You teaching pattern.
I think it is simplistic to assume that lots of teacher talk is bad. Similarly, I think it is mistake for tutors to assume they should never simply "show" the student what to do. It does feel odd for a student to be very passive while a teacher is very active, but as long as both know that there will be many repetitions of the task, with the student doing more and more each time until, hopefully, the teacher's role is basically to cheer and clap for a wonderfully independent worker, showing (modeling!) is a vital first step, not to be skipped! If you do skip this step you are pushing the birdy off the branch without it ever even having stretched its wings yet--that's mean.

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The real joy with this word-problem approach is watching students systematically disassemble their own Reese's cup of Confusion, into two separate, and therefore much more manageable, collections of information, before re-assembling it, knowing it intimately from the inside out, into a confection they can sink their teeth into with gusto.

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