Category Archives: The end of word problems

The comprehensive Oreo database

Here, representing many hours of data collection, including several notebook-equipped excursions to Cub Foods, I present to you the Comprehensive Oreo Database.

I would like to replace the present system initial licensure exams in secondary mathematics teaching with a single task: Design a mathematics lesson around some or all of this information, including answer key for all tasks.

Until this happens (at which point the information below will be embargoed for test-security purposes), I share it with you.

Oreo type Serving size (in cookies) Calories per serving Fat grams per serving Mass per serving Notes
Regular 3 160 7 34
Berry burst ice cream 2 150 7 30
Candy corn 2 150 7 29
Chocolate 2 150 7 30
Cookie dough 2 140 6 29 Sighted Feb. 3, 2014
Double chocolate fudge creme 3 180 9 36 One choc. wafer & one serving choc. stuf, coated in fudge
Double Stuf 2 140 7 29
Double Stuf Heads or Tails 2 140 7 29
Football 2 120 5 26 These are shaped like a football
Golden 3 160 7 34
Fudge creme 3 180 9 35 One choc. wafer & one serving van. stuf, coated in fudge
Golden chocolate 3 170 7 34
Golden Double Stuf 2 150 7 30
Halloween 2 140 7 29
Marshmallow Crispy 2 140 6 29 Spotted Feb. 3, 2014
Mega Stuf 2 180 9 36
Reduced-fat 3 150 4.5 34
Spring 2 150 7 29
Triple-double 1 100 4.5 21
Triple-double Neapolitan 1 100 4.5 21

Send your revisions, and additional Oreo sightings, my way via the comments or Twitter.

Exothermic fauna: A surface area poem

Two animals

Are both close to a cube

In shape.

The height of the first animal

Is 4 cm

The height of the second animal

Is 6 cm

Which animal

Would experience the greater

Amount

Of heat loss?

Explain.

Thanks to Cathy Yenca and Chris Robinson for the find, whose source is evidently a 2007 U.S. textbook.

mhampton‘s reformulation of my poem (in the comments below) led to some fun over on Twitter, which I collected on Storify.

Follow up on that Tabitha post

Yesterday’s post recounted a conversation with Tabitha (5) in which she asked for a “math class” problem.

I focused the initial discussion on where she learned what constitutes a “math class” problem.

But there’s lots more in there that’s interesting.

It wasn’t just her affective response (rejecting the driving in a car context, asking for a naked number problem) that matters here. Notice the way she engaged with the two problem types.

When I relented and posed the question, What is two plus three? she guessed. I know that she guessed because she (a) took no time to process, (b) asked rather than told me her answer, and (c) it was wrong despite being within her grasp.

When I posed the exact same problem in a situation she could imagine (she and Griffy had been to the arcade just that day), she engaged quite differently. Her body position changed. She paused. Her fingers moved. Each of these is an indication that she was thinking. And when she had an answer, she stated it; she did not ask.

The central tenet of CGI and an important belief underlying the IMAP work is that children can use contexts to solve problems that they cannot solve abstractly. Here it is in action. 2+3 is meaningless to Tabitha right now. But her 2 tickets combined with Griffin’s 3 tickets? That’s got meaning.

The conclusion here is obvious, right? We start with contexts kids understand and can reason about (here, combining tickets). We move to the abstract mathematical representations (2+3). We don’t save arcade tickets for after the kid understands addition. We don’t wait for symbolic mastery before doing some applications.

The lawnmower problem

David Peterson (@calcdave on Twitter) posted this lawnmower video to 101qs recently. It was love at first sight.

I needed a polar-coordinates-based assignment for my Calculus 2 students, so I pounced on it. The question they have been working on is, How long will it take to mow the lawn?

I read their work today. The following are some quotes from their writing.

“Establishing the polar function was difficult at first, until I thought about it as just a plain linear function.”

“I tried going on the treadmill to see what a comfortable walking speed for mowing would be.”

“Sorry for making this 13 pages. I really got into it.”

“Sometimes math needs a little touch up; this is when Photoshop is there to save the day.”

“The real real-world problem is how to convince your wife to upgrade mowers.”

“Rather than dealing with negatives and reciprocals, this paper will assume the lawnmower ‘un-mows’ the lawn from inside to out.”

“After realizing that the point on the outer ede of a circle has to cover more linear distance than a point near the center, angular velocity seems like it might have some flaws.”

I see in these excerpts students making mathematical connections that result from their struggles with the problem. I see them posing and refining mathematical models based on correspondence to the real world. I see them looking at this small slice of the world through a mathematical lens.

I am so proud of them.

NOTE: In original post, I did not know who had posted the video to 101qs. David Cox came through for me on Twitter. Credit given in revised post.

What is the chain rule? (heh)

A 10 m chain with total mass of 80 kg is laid out on the ground. How much work is required to lift one end of the chain 6 m off the ground?

This is a standard problem from a standard textbook, in the requisite section on work.

Before we solve the problem, we need to draw a picture. So I asked my students to do that the other night.

Can you see the tension?

versus

So. Who’s right?

Surely you object that things would be different if my chain actually weighed 80 kg. Perhaps you are right. Perhaps they would be different.

But here’s the thing…

My 10 ft of chain is a bit more than half a kilogram. So 10 meters of this chain is less than 2 kg. So the chain in the problem is 40 times more massive than the one from my local hardware store. How big is this chain anyway?

Is this inappropriate?

Cider price rules

I was feeling guilty the other day about spending $7.00 on a gallon of cider. We are at the end of cider season here in Minnesota. It’s a seasonal treat that doesn’t come cheaply.

But we are on a budget (community college salaries only go so far!) and I was feeling bad about the expense-guilty for giving in to the sweet, sweet temptation of delicious fresh cider.

And then my daughter asked for a juice box.

There they were, side by side…a $7.00 gallon of cider and a $2.39 pack of eight 4.23 boxes.

And suddenly I didn’t feel so guilty anymore.

Unit rates can have that effect sometimes.

Christopher cries “Uncle!” (but doesn’t give up the fight)

Last week I called out this problem (whose existence was implied by a web search that brought a reader to my blog):

In how many ways can 7 peanuts be shared among 3 people?

In particular, I argued,

Any problem that uses everyday language [such as "share"] or imagery that will mislead if taken seriously is a bad one in my view.

Readers took me to task for too narrow a view of the verb share.

I concede.

Not all sharing is equal sharing. I probably use the word sharing to mean equal sharing far too often. This makes my point while simultaneously implicating me. Sweet.

But no way am I going to let that crummy problem off the hook.

Chris Hunter argues in the comments that the problem has gotten the implicit stamp of approval from the National Council of Teachers of Mathematics (about which more in the next couple of weeks), by way of being in an article published in Teaching Children Mathematics:

Danny, Connie, and Jane have eight cookies to share among themselves. They decide that they each do not need to get the same number of cookies, but each person should get at least one cookie. If the children do not break any of the cookies, in how many different ways can they share the cookies?

But that’s not the peanut problem.

Danny, Connie and Jane are likely to be satisfied with their share of eight cookies. Indeed (equal sharing aside), it is likely possible to find some way to share these cookies so that everyone’s appetite is sated.

Were they sharing peanuts, it would be tougher going. When was the last time you stopped at the seventh peanut?

And by the way, what’s the unit here? Is this one peanut or two?

The sharing will proceed differently in each case, I would imagine.

Here’s what I’m saying. Context matters. Dan Meyer will argue that context matters for motivation and for intellectual honesty. Karim Ani will argue that context matters for motivation and so that kids understand that math is power.

All true.

But I want to argue that context matters because people bring intuitive mathematical ideas to class. More often than not in K-12 schooling (and beyond), those intuitive ideas are based on their experiences in the real world. If we don’t build on those ideas, then we alienate students from mathematics.

If we do build on those ideas, then we’re helping students to make their ideas better. There is efficiency in this, but also an opportunity to avoid the well-documented effects of instruction that doesn’t connect to everyday experience. Namely, that said instruction has absolutely no effect on people’s views of the world, nor on their ways of operating in it.

It’s not so much that students end up choosing not to use their mathematics education in their lives, it’s that it never occurs to them to do so.

Because no one shares seven peanuts among three people.