Tag: developingthequestion

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[NCTM16] Beyond Relevance & Real World: Stronger Strategies for Student Engagement

My talk from the 2016 NCTM Annual Meeting is online. I won’t claim that this is a good talk in absolute terms or that this talk will be good for your interests. I only know that, given my interests, this is the best talk I have ever given.

My premise is that we’re all sympathetic towards students who dislike mathematics, this course they’re forced to take. We all have answers to the question, “What does it take to interest students in mathematics?” Though those answers are often implicit and unspoken, they’re powerful. They determine the experiences students have in our classes.

I lay out three of the most common answers I hear from teachers, principals, policymakers, publishers, etc., two of which are “make math real world” and “make math relevant.” I offer evidence that those answers are incomplete and unreliable.

Then I dive into research from Willingham, Kasmer, Roger & David Johnson, Mayer, et al., presenting stronger strategies for creating interest in mathematics education.

My call to action will only make sense if you watch the talk, but I hope you’ll take it seriously, give it a try, and let us know how it goes.

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BTW. I’ve already received one email asking me, “Wait? Are you saying never make math real world?” No. My principles for instructional design often lead me to design applied math tasks. But “make math real world” isn’t a great first-order principle because, as a category, “real world” is poorly defined and subjective to the student.

Featured Comment

Dan Smith:

This was a really helpful talk in illuminating why it doesn’t work to simply drop a mundane math task into some sort of “relevant” or “real-world” context. And it was great that you didn’t stop at deconstructing these unhelpful approaches, but instead went on to share specific ways to think, steps to take, and tools to use to increase engagement and thoughtfulness in our math classrooms. A very natural follow-up to the famous “Math class needs a makeover” talk.

These Horrible Coin Problems (And What We Can Do About Them)

From Pearson’s Common Core Algebra 2 text (and everyone else’s Algebra 2 text for that matter):

Mark has 42 coins consisting of dimes and quarters. The total value of his coins is $6. How many of each type of coin does he have? Show all your work and explain what method you used to solve the problem.

The only math students who like these problems are the ones who grow up to be math teachers.

One fix here is to locate a context that is more relevant to students than this contrivance about coins, which is a flimsy hangar for the skill of “solving systems of equations” if I ever saw one. The other fix recognizes that the work is fake also, that “solving a system of equations” is dull, formal, and procedural where “setting up a system of equations” is more interesting, informal, and relational.

Here is that fix. Show this brief clip:

Ask students to write down their best estimates of a) what kinds of coins there are, b) how many total coins there are, c) what the coins are worth.

The work in the original problem is pitched at such a formal level you’ll have students raising their hands around the room asking you how to start. In our revision, which of your students will struggle to participate?

Now tell them the coins are worth $62.00. Find out who guessed closest. Now ask them to find out what could be the answer – a number of quarters and pennies that adds up to $62.00. Write all the possibilities on the board. Do we all have the same pair? No? Then we need to know more information.

Now tell them there are 1,400 coins. Find out who guessed closest. Ask them if they think there are more quarters or pennies and how they know. Ask them now to find out what could be the answer – the coins still have to add up to $62.00 and now we know there are 1,400 of them.

This will be more challenging, but the challenge will motivate your instruction. As students guess and check and guess and check, they may experience the “need for computation“. So step in then and help them develop their ability to compute the solution of a system of equations. And once students locate an answer (200 quarters and 1200 pennies) don’t be quick to confirm it’s the only possible answer. Play coy. Sow doubt. Start a fight. “Find another possibility,” you can free to tell your fast finishers, knowing full well they’ve found the only possibility. “Okay, fine,” you can say when they call you on your ruse. “Prove that’s the only possible solution. How do you know?”

Again, I’m asking us to look at the work and not just the world. When students are bored with these coin problems, the answer isn’t to change the story from coins to mobile phones. The answer isn’t just that, anyway. The answer is to look first at what students are doing with the coins – just solving a system of equations – and add more interesting work – estimating, arguing about, and formulating a system of equations first, and then solving it.

This is a series about “developing the question” in math class.

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I asked for help making the original problem better on Twitter. Here is a selection of helpful responses:

2014 Oct 20. Michael Gier used this approach in class.

Featured Comments

Isaac D:

One of the challenges for the teacher is to guide the discussion back to the more interesting and important questions. Why does this technique (constructing systems of equations) work? Where else could we use similar strategies? Are there other ways to construct these equations that might be more useful in certain contexts?

Same & Different As It Ever Was

Same

In the September 2014 edition of Mathematics Teacher, reader Thomas Bannon reports that his research group has found that the applications of algebra haven’t changed much throughout history.

310:

Demochares has lived a fourth of his life as a boy; a fifth as a youth; a third as a man; and has spent 13 years in his dotage; how old is he?

1896:

A man bought a horse and carriage for $500, paying three times as much for the carriage as for the horse. How much did each cost?

1910:

The Panama Canal will be 46 miles long. Of this distance the lower land parts on the Atlantic and Pacific sides will together be 9 times the length of the Culebra Cut, or hill part. How many miles long will the Culebra Cut be? Prove answer.

2013:

Shandra’s age is four more than three times Sherita’s age. Write an equation for Shandra’s age. Solve if Sherita is 3 years old.

I’m grateful for Bannon’s research but his conclusion is, in my opinion, overly sunny:

Looking through these century-old mathematics book can be a lot of fun. Challenging students to find and solve what they consider the most interesting problem can be a great contest or project.

My alternate reading here is that the primary application of school algebra throughout history has been to solve contrived questions. Instead of challenging students to answer the most interesting of those contrived questions, we should ask questions that aren’t contrived and that actually do justice to the power of algebra. Or skip the whole algebra thing altogether.

Different

If you told me there existed a book of arithmetic problems that didn’t include any numbers, I’d wonder which progressive post-CCSS author wrote it. Imagine my surprise to find Problems Without Figures, a book of 360 such problems, published in 1909.

For example, imagine the interesting possible responses to #39:

What would be a convenient way to find the combined weight of what you eat and drink at a meal?

That’s great question development. Now here’s an alternative where we rush students along to the answer:

Sam weighs 185.3 pounds after lunch. He weighed 184.2 before lunch. What was the weight of his lunch?

So much less interesting! As the author explains in the powerful foreword:

Adding, subtracting, multiplying and dividing do not train the power to reason, but deciding in a given set of conditions which of these operations to use and why, is the feature of arithmetic which requires reasoning.

Add the numbers back into the problem later. Two minutes later, I don’t care. But subtracting them for just two minutes allows for that many more interesting answers to that many more interesting questions.

[via @lucyefreitas]

This is a series about “developing the question” in math class.

“You Can Always Add. You Can’t Subtract.” Ctd.

Bryan Anderson and Joel Patterson simply subtracted elements from printed tasks, added them back in later, and watched their classrooms become more interesting places for students.

Anderson took a task from the Shell Centre and delayed all the calculation questions, making room for a lot of informal dialog first.

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Patterson took a Discovering Geometry task and removed the part where the textbook specified that the solution space ran from zero to eight.

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“It turns out that by shortening the question,” Joel Patterson said, “I opened the question up, and the kids surprised themselves and me!”

I believe EDC calls these “tail-less problems.” I call it being less helpful.

BTW. These are great task designers here. I spent the coldest winter of my life at the Shell Centre because I wanted to learn their craft. Discovering Geometry was written by friend-of-the-blog Michael Serra. This only demonstrates how unforgiving the print medium is to interesting math tasks, like asking Picasso to paint with a toilet plunger. You have to add everything at once.

Can Sports Save Math?

A Sports Illustrated editor emailed me last week:

I’d like to write a column re: how sports could be an effective tool to teach probability/fractions/ even behavioral economics to kids. Wonder if you have thoughts here….

My response, which will hopefully serve to illustrate my last post:

I tend to side with Daniel Willingham, a cognitive psychologist who wrote in his book Why Students Don’t Like School, “Trying to make the material relevant to students’ interests doesn’t work.” That’s because, with math, there are contexts like sports or shopping but then there’s the work students do in those contexts. The boredom of the work often overwhelms the interest of the context.

To give you an example, I could have my students take the NBA’s efficiency formula and calculate it for their five favorite players. But calculating – putting numbers into a formula and then working out the arithmetic – is boring work. Important but boring. The interesting work is in coming up with the formula, in asking ourselves, “If you had to take all the available stats out there, what would your formula use? Points? Steals? Turnovers? Playing time? Shoe size? How will you assemble those in a formula?” Realizing you need to subtract turnovers from points instead of adding them is the interesting work. Actually doing the subtraction isn’t all that interesting.

So using sports as a context for math could surely increase student interest in math but only if the work they’re doing in that context is interesting also.

Featured Email

Marcia Weinhold:

After my AP stats exam, I had my students come up with their own project to program into their TI-83 calculators. The only one I remember is the student who did what you suggest — some kind of sports formula for ranking. I remember it because he was so into it, and his classmates got into it, too, but I hardly knew what they were talking about.

He had good enough explanations for everything he put into the formula, and he ranked some well known players by his formula and everyone agreed with it. But it was building the formula that hooked him, and then he had his calculator crank out the numbers.