In Chapter 1 of What’s Your Math Problem?, Linda Gojak gives some initial thoughts on learning and teaching problem solving.
She introduces the concepts of routine problems and non-routine problems.
Routine problems are what you typically see at the end of a problem set in a traditional textbook. “In solving routine problems, the learner reproduces and applies a new procedure,” Gojak writes.
Non-routine problems, or rich problems, are the way of the world. They are the things grown-ups solve everyday effortlessly, and often don’t think of them as math problems.
Is solving non-routine problems teachable?
There is a divergence between the way traditional word problems are taught to kids and how grown-ups handle the rich problems in their lives. What’s Your Math Problem? attempts to distill and label each strategy of what grown-ups naturally do, so that we can teach these strategies to our children.
To make this work, knowledge of the various strategies is important. So Gojak labels, defines and gives examples of each strategy throughout the book.
This method of teaching problem solving to children will work if an instructor is careful not to force the use of a particular strategy.
Offer a strategy, but don’t force it.
The idea is to label and clarify each problem solving strategy so it can be one of the options in the toolbox of problem-solving for each child.
As students learn a strategy, teachers shouldn’t require it be used “so they can practice it.” Instead it should be offered and encouraged, but allowed to be tossed aside if the student prefers another method.
And caution should be used to ensure problem solving using these various strategies NOT turn into another algorithm.
Read more about it…
Don’t forget to check out Math Coach’s Corner for some other thoughts on Chapter 1 of What’s Your Math Problem? Make sure to scroll to the bottom, because others are linking up their thoughts and opinions!
I love how Linda Gojak calls juicy, meaty problems “rich problems.” A good, fun thinkable is indeed a math word problem rich with problem solving challenges.
But getting started on a rich problem can leave you feeling rather poor. So Chapter 3 gives, and is called, “Getting Started Strategies.”
What’s that problem about anyway?
The first question you (or your child) should ask when given a problem is, “What’s it all about, anyway?” This is the strategy of “Restate the Problem in Your Own Words.”
Ask
What’s happening — what does it look like?
What bits of this problem are useless to me?
If a normal person were to ask the question, how would it be written?
Now what the heck does it really say?
Restating the question in your own words means understanding what’s being asked and what’s happening.
Is this a trick question?
Sometimes textbooks (and even life) give you problems without giving you all the required information. This is grownup-talk for what kids call a trick question.
If there’s missing information, call that bluff! What info do you need to calculate the final answer?
Is that information contained in the problem?
Can you find that information online or in a library?
Can you figure out that information using other stuff in the problem?
Is it just a flat-out trick question — there can’t be an answer because there’s no way to get the information needed?
Calculate the information, if you can.
Now it’s time to do a little pre-work. Gojak calls it “identifying a subgoal.”
If you’re missing some numbers in the problem but you can get these from others, then start calculating.
I walked 30 feet and then walked another 24 inches. How many feet did I walk?
The subgoal here — figure out how many feet I walked the second time.
Figure out how to show your work — or not?
One of the strategies in chapter 3 is “Select Appropriate Notation” — which means determine how you’ll show your work.
But first ask the question, “Do you want to show your work?”
Here’s the big place where classroom schoolers and homeschoolers will diverge. It isn’t really necessary to show your work. Ever. And in homeschool, children don’t have to.
If you want to show your work that’s great. If you want to share your work, you have to show your work. If you want to be a famous mathematician or even a run-of-the-mill engineer, you have to show your work.
It doesn’t hurt to learn to show your work. But it’s not required to be a great problem solver.
In fact, if your child struggles with notation, and you push this too hard, their developing problem-solving strategies could be stunted.
But should you show your work for other reasons?
Gojak writes:
“…you use notation to help you reach a solution.”
This is not necessarily true. Some people do. I don’t. The problem-solving strategy that works for me is doodling pictures and trial and error, strategies covered later in the book.
I rarely solve a problem using x and y — or even crude representations of x and y like question marks or blanks.
If you feel the need to verify that your child is thinking properly, ask them to explain it out loud. Or give them another rich problem. Don’t force them to show their work because you want to see it.
Read more about it…
Don’t forget to check out Math Coach’s Corner for some other thoughts on Chapter 3 of What’s Your Math Problem? Make sure to scroll to the bottom, because others are linking up their thoughts and opinions!
The bride wanted to have all the tables labeled with prime numbers. She used all the primes through 43. Each table was set for 10 people. How many guests could come to the wedding?
I’ve learned of a thing called “What can you do with this?” from dy/dan. This teacher sets up a situation so that students can ask questions.
I’ve been pondering the effectiveness of this for a while.
The thought is that if you allow students to observe something interesting and ask them “What can you do with this?” then they’ll create their own word problems.
This is in response to the fake or “made up” word problems from a textbook which mostly don’t work for teaching thinking skills.
But the issue remains the same. If someone presents a student with a video of Coke vs. Sprite and the student lacks curiosity about that subject, then it’s still a contrived problem. Or a contrived situation.
The only reason to do a word problem is if you’re emotionally attached to it.
Husband and I were talking about word problems the other night. After my demonstration about using to teach math, he said he wished he learned math that way. He needed something to hold on to. A reason for doing it.
He’s a set dresser in Hollywood for part of each year. Which means that he has to hang pictures on movie sets. And they have to be 55″ above the ground – at the center of the picture.
Not hard to measure, but there’s also the wire on the back to consider. Is the wire dead center? No. It’s probably above the center of the picture.
It becomes one giant word problem. But it isn’t written in a book. And it isn’t videoed by a teacher. It isn’t fake. There’s a real reason for him to do it.
Which made me realize that there’s only one reason to do word problems: if you’re emotionally attached to it.
If you need an answer to a question, you attach to it emotionally.
Parent: You’ve got 45 minutes to clean the kitchen before we leave for softball practice.
Kid: If I finish the kitchen before we leave, can I watch TV?
Parent: Sure, but the kitchen better be spotless.
Most likely the kid has a plan for TV – like watching his favorite cartoon on DVR that takes about 30 minutes. So he works out how fast he needs to clean the kitchen so he can get in his cartoon before leaving.
This is a real problem. His problem.
Watch your kids intently. See where they are doing word problems in their heads. Ask them to explain them. Give credit for work done – especially when self-created.
If someone else needs an answer, you attach to it.
Being helpful is a powerful motivator. Try this: with a pencil and paper sit in a public place. Act like you’re writing something important. Then ask out loud, “What’s 87 minus 13?” $5 says that at least four people will chime in to be helpful.
Let your kid help with balancing the checkbook or creating the budget. If you’re a classroom teacher, let the kids help figure out what teacher supplies to buy. Give them a limit on what to spend and the catalog and some guidelines.
If someone you like wants an answer, you attach to it.
I couldn’t have given a feathery duck’s tail about biology, but the teacher was crazy cute. So I wanted to please him. So I worked. Hard. And had a 100 average.
I suspect this is why the teacher at dy/dan is so successful. He’s cute, compelling and cool. Who wouldn’t want to engage with him?
If you have carisma and charm, use it. This might not work as a parent but will definitely work as a classroom teacher – at least for some students.
Give it a try. Tap into the emotion. And share your success below!
Her question to me was, “Why wouldn’t you just rewrite the problem to focus on the appropriate concept?” She suggested that having students restate the problem in their own words without numbers would have them demonstrate that they know what is being asked of them.
Here is the original problem:
The Beebo bird lives in two places in the world. Some live in Texas and some live in Greece. Greek Beebos are about 20 inches high and weigh around seven pounds. There are about thirty-nine thousand Greek Beebos. The total weight of all the beebos in the world is 500,000 pounds. How much do the Texas Beebos weigh altogether?
Here is her suggested rewrite without numbers:
There are only two types of Beebos in the world, Greek and Texan. I know the weight of one Greek Beebo and I know how many Greek Beebos there are in the world. I need to find out how much the Texan Beebos weigh altogether.
On the outset, this seems great. If your kid does this:
It’s more likely your student will do this (especially if they’re struggling or you’re a hired tutor):
Notice the struggle and strain? And notice that both videos show the same thing – the “student” (me) just reading the problem and replacing the numbers with “I know how much…”
Watch students carefully. Listen to their intonations, watch their faces, watch their bodies. Whether you’re in a classroom or one on one, watch! If they got it, you can see it. If their little foreheads are wrinkled and they are tense – stop. They don’t have it. They are guessing. Go back. Try something else.
