## [PS] Critical Thinking

Trying too hard:

Prentice Hall’s Algebra I: California Edition:

A hot air balloon flies at a speed of (n + 8) miles per hour. At this rate, how long will it take to fly (n2 + 5n – 24) miles?

[via Matt Vaudrey]

I encounter problems like this too frequently, and my ‘put on the spot’ knee-jerk reaction when they pop up is usually something like “Oh, that’s a stupid problem. Just skip it.” Of course, the message that students get is that “Math is stupid.”

Mike:

I love it when Nature has terms that factor so well.

It begs the question, who actually writes this?

The authors are all distinguished teachers and professors, many with PhDs. But would any of them stand by this? Was it handed over to an intern? Was it caving in to the “applications” lobby? Or do they consider it a good problem?

The problem with multiple authors is that none of them “own” the work; none of them consider it theirs. Its a project they are working on, but they’re just collaborators.

Given a choice, I will always choose a single author book. I know it will have been written with greater care.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Mr. K

January 17, 2012 - 9:20 pm -

Hey look – a real world application of rational expressions! That’ll motivate the kids!

2. #### Robert Hansen

January 18, 2012 - 3:52 am -

Wow, that’s bad. They shouldn’t have been trying at all. Sometimes you should just leave it alone and do the math.

3. #### Gary D

January 18, 2012 - 6:05 am -

What a pretty picture!

No, really, I looked at the photo and thought…

Can I estimate the altitude of the balloons by looking at the types of clouds?

What would I need to know to figure out how far one balloon is from any other balloon?

If any two balloon dropped some weight off the side at the same time, how much sooner would one hit the ground that the other balloon? (OK, maybe stretching artifically for quadratics there.)

How fast DO hot air balloons fly?

I wonder how many miles per gallon they get?

I’m sure there’s dozens of better questions out there.

4. #### Emily

January 18, 2012 - 7:20 am -

All kidding aside, this is exactly the type of problem that really gets my goat:

Putting on my student hat, the problem is from the textbook , so I pull out my bag of tricks and dive in. Distance equals rate times time, so time must be equal to distance divided by rate. I LOVE simplifying rational expressions, so it’s no problem to get down to time = (n – 3).

But seriously, WTF? I get that the speed might be variable based upon the wind conditions, but the TIME too? And why is the distance a quadratic function? Is the distance going to DECREASE? Oh, then maybe we’re talking about vertical height and not horizontal distance, in which case I should have started with a parabolic motion model. And what does n represent anyway!! ARGH.

OK, but my point is, that as the TEACHER and not the student, I encounter problems like this too frequently, and my ‘put on the spot’ knee-jerk reaction when they pop up is usually something like “Oh, that’s a stupid problem. Just skip it.” Of course, the message that students get is that “Math is stupid.” Explaining WHY the exercise is faulty is only slightly better… I’ve seen more than my share of glazed over expressions.

5. #### Michael

January 18, 2012 - 8:01 am -

Dan, you should write a book with all the bad problems that you have collected, then write about how to make them better at minimum for text and then write about your three act WCYDWT type problems. It will probably be the most useful thing to math teachers around the world!

6. #### Jason

January 18, 2012 - 8:27 am -

Reminds me of a college Stats exercise where an officer fined speeders if they were speeding in a statistically significant way. Apparently, this fictitious officer knew the mean, standard deviation, etc.

7. #### Dave

January 18, 2012 - 11:29 am -

I would buy several copies of a Psuedocontext expose/WCYDWT primer book to hand out to my teacher friends and co-workers, and not just math teachers. Just saying.

8. #### louise

January 18, 2012 - 11:57 am -

It’s so comforting to know that Pearson (proud owner of Prentice Hall) has been given the contract for assessment of the common core standards. I’m pretty sure that the emphasis on real world modeling and context is going to be obvious to us all. I know my students and I were thrilled with their NovaNet math courses (to the extent that nobody wanted to do them even if you can get 100% just by retaking the same test four times). With Pearson in charge of context based computerized tests to be answered country-wide, by all students in a consortium, several times a year, what can go wrong?
Sorry, I don’t think they’re trying at all.

9. #### James C.

January 18, 2012 - 12:25 pm -

I don’t think we need a new book with a collection of pseudo-contextual problems, any middle or high-school math book will do. The reality is that they are the norm, not the exception.

10. #### Scott

January 18, 2012 - 3:47 pm -

It begs the question, who actually writes this?

The authors are all distinguished teachers and professors, many with PhDs. But would any of them stand by this? Was it handed over to an intern? Was it caving in to the “applications” lobby? Or do they consider it a good problem?

The problem with multiple authors is that none of them “own” the work; none of them consider it theirs. Its a project they are working on, but they’re just collaborators.

Given a choice, I will always choose a single author book. I know it will have been written with greater care.

11. #### Dan Meyer

January 19, 2012 - 7:30 am -

I posted this for catharsis, to open up the pressure valve on a lousy day. I didn’t anticipate the thoughtfulness of your commentary on this stupid little problem. In particular, I appreciated Gary’s list of alternate questions, Emily’s exposition of this problem from the teacher’s perspective, Mike’s best-in-show snark, and the connection Scott draws from sole authorship to higher quality curricula. Strong work, everybody.

PS. Jason: find that problem!

12. #### Bowen Kerins

January 19, 2012 - 8:44 am -

A note to Scott’s comment. In general I agree: books that go through many iterations and “updates” tend to look this way. There is a very tight budget and many requirements, especially for “state” versions like the one this problem comes from. “Hey, we need to write a Critical Thinking problem for this California lesson on rational expressions. Quick!” Updating a curriculum is more like grafting, and after enough grafts…

(I am not trying to defend the problem, because boy does it suck. “Critical Thinking” here boils down to ignoring as many words as possible. I love that their sidenote suggests a balloon traveling at 35 mph for 24 hours!)

My own experience on a writing team has been quite different. The team spent a long time building the organizing principles, goals, and outline for the entire four-year curriculum before writing began, a luxury that most writing teams don’t have. Writers then “team-wrote” sections of each book, passing lessons and investigations back and forth. This led to a consistent voice in the writing while gathering the best ideas we had.

Our books are miles better than anything I would have written on my own, and I think the other authors would say that too. The writing process helped make it possible for us all to “own” the work. There is no way any one of us could have written an entire high-school curriculum by ourselves, and we did better by writing one series with shared goals and style than four disparate books from single authors.

A follow-up: what single-author books are there these days? I don’t know of many, and of the ones I do know, the later editions of these books have the same “grafted” feel. I agree that single-author books, by and large, have a better chance of grabbing me, and learned more from Harold Jacobs than my actual Geometry teacher.

13. #### gasstationwithoutpumps

January 19, 2012 - 10:04 am -

Jason said “Reminds me of a college Stats exercise where an officer fined speeders if they were speeding in a statistically significant way. Apparently, this fictitious officer knew the mean, standard deviation, etc.”

That is not as far-fetched as you imagine. In order to give a ticket for speeding in California, police have to have done a speed survey in the past few years and can only ticket cars traveling faster than the 85%ile. (Actually, that’s not quite true, as they can always ticket for going too fast for conditions, but the tickets get thrown out in court without the speed survey and and 85%ile info.)

14. #### gasstationwithoutpumps

January 19, 2012 - 10:07 am -

The Art of Problem Solving series are single-author books, and they are well written, but they’re only suitable for kids in the top 10—15% of most math classes, since the problems are almost all challenging and there is no drill and no fluff.

15. #### Chris Hunter

January 19, 2012 - 10:49 am -

Just to show that trying too hard has no borders, here’s a Canadian example: http://bit.ly/wyYJXT

Thankfully, there has been a curriculum change so this text is no longer in use (but only as of one year ago).

Like your balloon example, this “problem” deserves to be criticized for presenting a problem that no one in the “real world” would ever have to solve.

I have as much of an issue with the “Inquire” questions that follow. Why not just ask “What do you notice?” and “Why is this true?”? Why give away that the key here is to factor the polynomials? In this text the previous chapter is all about factoring. Given that, wouldn’t many students think of giving this a try? (And what does it say about how math is taught if they don’t?)

Could the teacher not circulate and provide scaffolding to those students who need it? Something like “Can you find another way to express 2x+4?”.

Textbooks are guilty of many “pseudos”: pseudo-context, pseudo-exploration/inquiry, pseudo-thinking.

16. #### Geoff

January 19, 2012 - 3:45 pm -

You know what would be fun? A “who can create the worst math problem” contest. Maybe for April Fools Day or something.

We could all submit our own, original, terrible math problems and Dan could either unilaterally declare the “winner” or we could have a NCAA-like 64 person tourney.

The only potential pitfall is that it would be tough to create worse problems than this one right here. It would be quite a challenge to top (bottom?) this one.

17. #### Chris Hunter

January 19, 2012 - 7:34 pm -

I, too, was thinking about a contest when I was scanning the question, but for finding not creating pseudocontexts. Not sure if I want to, or would be able to, create one that could top what’s already out there.

I was thinking there could be prizes for the final four. Maybe Dan could ask TI if they’d like to donate some?

If successful, we could follow it up with a shiniest T.U.R.D. contest.

http://christopherdanielson.wordpress.com/category/truly-unfortunate-representations-of-data/

By the way, google “Christopher Danielson turd” and @Trianglemancsd’s blog is the third result.

18. #### josh g.

January 20, 2012 - 8:49 am -

I’ve still seen problems like the one in the new Western Canadian curriculum textbooks.

I don’t know if it’s fair to blame the textbook authors entirely – the curriculum standards talk about “solve problems with multiplication and division of polynomials”, and so the authors try to come up with word problems to match. But just what kind of real-world problem-solving questions *can* you ask with only multiplication and division of polynomials?

(Honest question – I’d love to see good alternatives brought up.)

19. #### Bowen Kerins

January 21, 2012 - 12:16 am -

Just because it says “solve problems” doesn’t mean they have to be “real-world problems”. Some math topics just don’t real-world themselves well, or at all.

There are plenty of interesting problems to solve in the topic though. In dividing polynomials you can talk about what it means to divide and leave a remainder. Why is it that when I divide by (x-3), I can know that I’ll be able to get a constant remainder, and that the remainder when f(x) is divided by (x-3) is always f(3)? What is an acceptable remainder when dividing by (x^2 + 1)? How do the remainders when dividing by (x^2 + 1) relate to complex numbers?

There’s nothing real-world about it. But the analytic thinking (how is structure X like structure Y, and how are they different) is pretty universal.

The best I can come up with is the concept of Lagrange Interpolation (a method to fit a polynomial to given data) but that’s really more about the form and interpretation of expressions like 3(x-1)(x-6)(x+5) than anything to do with multiplying and dividing polynomials. Alright, multiplying maybe.

You can use Lagrange Interpolation to show that a question like “What number comes next: 1, 3, 5, ?” can be answered with any number at all, by finding a function that fits the sequence 1, 3, 5, 127 (for example). “Real world” that ain’t.

I guess then we can argue whether these are worthy topics for students given the relative lack of real-world application. I think the abstract nature of the topic makes it easier to focus on high-level thinking skills, and the understanding of things like quotient and remainder help students cement their understanding of the same concepts for numbers.

20. #### mweisburgh

January 21, 2012 - 11:13 am -

The reason this problem is stupid is that everyone knows that balloons travel at N squared + 7 miles an hour except in leap years.

21. #### Scott Farrar

January 25, 2012 - 11:18 am -

The single author books that I know (and love) are

Harold Jacobs: Geometry
Michael Serra: Discovering Geometry

@Bowen, thanks for the insight into how books are written. I have been curious about that process.

That brings me to my next question… I don’t know of any single author Algebra texts. Is Geometry a nice choice because it is a “new” system to the students? Because the author can build from the ground up? Do we not trust the students to be able to handle the abstract system of Algebra on the real numbers? Even for an Algebra 2 course? “Modern” algebra 1 and 2 texts are just “how-to” manuals, that sadly get thought of as “what math is” by students and the general public alike. Kahn Academy is the same thing.

Do english books get bogged down with such things? “you can use iambic pentameter to entertain your dog. Now lets drill.”

I suppose Foerster has an Algebra and Trigonometry book that is single author. I don’t have as much experience with it, though.

22. #### John Chase

January 25, 2012 - 1:34 pm -

Amen, amen, amen! to this post, and to all the comments.

I can’t tell you how often I feel this way about the problems in our math books. Anything that feels contrived to us also feels contrived to our students, you can bet on it.

Now here’s the thing–even things that *aren’t* contrived, like projectile motion application problems, STILL don’t apply to the lives of our students. We can be sure that 99% of our students will never need to model projectile motion after they graduate from school. For the 1% who study math and science, these application problems will be important. But I can’t think of a single skill in a typical Algebra 1 curriculum that’s needed on a daily basis after graduation.

All that being said, it’s disingenuous to tell kids, “you are going to use this someday,” no matter how important the application may or may not be. Most likely, it’s NOT true that they will use it.

A better motivator is to remind students that they are being liberally educated–prepared for society, so they can talk intelligently with all kinds of people about all kinds of subjects. And it keeps they’re possibilities open, because some of them just might use it. Most of all, math is beautiful and fun and should always be treated as such. What student doesn’t love a good puzzle/problem that’s the perfect level of difficulty?

THIS hot air balloon problem, however, can go out with the trash.

23. #### Bowen Kerins

January 26, 2012 - 3:40 am -

Hi Scott, those are great choices for single-author books. I really like Serra’s 1st edition, which feels much more like a book with a single focus. More recent editions feel a lot more like basal textbooks, which makes me think it’s no longer a single-author text. I feel the same way about the Foerster books: his awesome wacked-out word problems and gags seem less prevalent in later editions. Foerster books are still single-author with “consulting editors” and “reviewers”.

As for Algebra 1, I don’t know of a good single-author book and I’m not sure why there isn’t one. Key Press’ “Discovering Algebra” had the feel of the Serra book but I didn’t like the content much.

As usual I am ridiculously biased toward CME Project (the series I worked on). The series is much more focused on mathematical habits of mind like generalizing from repeated calculation than on Khan-style process work. Process matters too, but I feel that learning the “why” helps kids understand and remember the “how”.

Iambic pentameter entertains not just dogs but all sorts of animals! So do quadratics. Try throwing your dog a quadratic, he’ll bring it right back.

24. #### Jason Dyer

January 26, 2012 - 7:38 am -

But just what kind of real-world problem-solving questions *can* you ask with only multiplication and division of polynomials?

@josh: I have seen it used for calculating running time of computer programs. (i.e. we know a program takes (5x^2+2x-1) / (2x-3) steps, but we only care about the largest polynomial term after division) That requires a LOT of excess background to explain what the problem is about, so I don’t know if I could write it in an Algebra-book friendly format. I’ll think about it though.

25. #### gasstationwithoutpumps

January 26, 2012 - 8:46 pm -

The Foerster books are pretty good in a classic math text way. Personally, I like Richard Rusczyk’s books better, though they may only be suitable for the top 10% of most classes.

26. #### Hao

February 26, 2012 - 8:52 pm -

@Jason:
Multiplication of polynomials might make sense for some algorithms (where you have to do processing of all combinations of (n+1) by (n^2) cases). It makes much less sense for division of polynomials (you only have to check every (2n-3)rd case, which scales to the size of the input, but not in a good logarithmic way?)

In general, division of polynomials with clean answers isn’t going to make a whole lot of sense, because if the answer is a simple polynomial, it almost guarantees that the numerator was derived as a product to begin with…

Following from Bowen, you could have some problem where a cubic polynomial is given, and from a plot, you can see that it has a root at x = 1. Then you could use division to find the remaining roots (if real, or show that the remaining roots are complex).

27. #### John Chase

February 27, 2012 - 5:55 am -

@Jason, @Hao, @Bowen

If the question is “Are there useful applications for polynomial division?” the answer is a definite yes! The hot air balloon problem isn’t one them, but there are other good applications, like both of you mention. I love the running time application (Jason). I also love the Lagrange interpolation application (Bowen).

I thought I’d add to what Bowen said about Lagrange interpolation, and show that it really is intimately related to polynomial division. Some people maybe haven’t thought about this before: Lagrange interpolation is equivalent to solving the Generalized Chinese Remainder Theorem problem for a set of equivalencies, like this simple problem:

Find f(x) such that f(3)=2, f(-10)=1, f(6)=25.

is equivalent to the set of congruencies:

f(x)=2 mod (x-3)
f(x)=1 mod (x+10)
f(x)=25 mod (x-6)

This is now a Chinese Remainder Theorem problem, which is phrased in terms of polynomial division and remainders. This is a topic usually reserved for an Abstract Algebra class, so you probably wouldn’t bring it up in a high school setting. Perhaps you could mention it in passing, at least.

Solving this problem is of obvious value to any application requiring curve fitting. But over a finite field, it also has applications in cryptography.

All that being said–no real world application is really necessary at all. Like Bowen said, just discussing good problems that highlight algebraic structure is of value in and of itself.