kitchen table math, the sequel: higher order thinking test

Friday, February 16, 2007

higher order thinking test

Supposedly the goal of education is higher order thinking which can be defined as:
A complex level of thinking that entails analyzing and classifying or organizing perceived qualities or relationships, meaningfully combining concepts and principles verbally or in the production of art works or performances, and then synthesizing ideas into supportable, encompassing thoughts or generalizations that hold true for many situations
It's commonly thought that these higher order thinking skills can be taught directly apart from the relevant domain knowledge (i.e., a narrow portion of knowledge that deals with the specific topic of interest). The thought is that you don't need to learn (i.e., memorize) all those messy facts because you can use your fancy higher order thinking skills to figure out whatever you need to know. Thus, instructional time is concentrated on higher order thinking skills and the learning of facts is downplayed.

Let's put that theory to the test.

No doubt if you enjoy reading (or at least take the time to read) an obscure education blog you went to college, are highly educated, and are smarter than the average bear. In other words, you have higher order thinking skills in spades. Let's test how well you can use them.

Consider the following:

You have two identical glasses, both filled to exactly the same level. One contains red dye, the other water. You take exactly one spoonful of red dye and put it in the water glass. Then you take one spoonful of the mixture from the water glass and return it to the red dye glass.

Question: Is there more red dye in the water glass than water in the red dye glass? Or is there more water in the red dye glass than red dye in the water glass? In other words, the percentage of foreign matter in each glass has changed. Has the percentage changed more in one of the glasses, or is the percentage change the same for both glasses?

Use your superior higher order thinking skills and intuit an answer. First try to do it without resorting to outside sources. Then try answering it using whatever reference source is handy, such as google.

NB: This only works if you don't know the scientific principle involved. If you happen to know the right scientific principle, you're relying on your domain knowledge to answer the question, not your higher order thinking skills. Also, no fair if you know the source of this problem.

I'll let you stew on it for a while. I'll update later today.

Partial Update: Hint--Instead of water and red dye, think of red balls and white balls. Assume that each glass starts out with 100 balls of a single color. Now remove a number of red balls from the red-ball glass and put them in the white-ball glass. Then return the same number of balls from the glass with the “mixture” and put them in the red-ball glass. Do this with different numbers of red and white balls.


KDeRosa said...

BTW, the answer can be derived mathematically as well.

SteveH said...

If you can't do it mathematically, then it's meaningless.

Why? Because there is no justification for assuming that if you solved it any other way that you have the skills to solve any other sort of higher-order thinking problem. If you are able to solve many of these sorts of problems (assuming that, like Dick Feynman, you haven't studied them ahead of time), then there is no implication that this skill can be directly taught or learned.

On the other hand, if, like Feynman, you study all sorts of problems like these, does that make you good in math? No. But why does my son's school have an after-school math club that is devoted to brain teasers?

The whole point of math is to avoid the need to pull the magical answer rabbit out of the hat. If you know math, then you don't have to think higher. If you can do the math in an instant, is that higher ordered thinking?

It's all relative. Assuming that there is some sort of skill that can be called higher-ordered thinking, don't you think that you could go much higher with more content and skills?

P.S. I've seen this one before.

Catherine Johnson said...



I had no idea that in grades 6-12 the dragon I have to slay is "developmentalism," which is, I gather, the belief that a course taught "conceptually" - which turns out to mean a course only students scoring in the top 10% of the entire country can manage - is a superior course.

Students not scoring in the top 10% of the country will be able to take this course as they "mature."

"Maturity" is the new IQ.

Nobody talks IQ here.

Nobody talks about intelligence.

Nobody talks about faster and slower learners.

In fact, it's pretty much illegal to talk about these things.

In their place?


Some students are mature enough to take a "conceptual" course.

Other students are not mature enough to take a "conceptual" course.

It's taken for granted that a 25-year old teacher is mature enough to teach a "conceptual"course.

Catherine Johnson said...

The whole point of math is to avoid the need to pull the magical answer rabbit out of the hat.

I love it.

Catherine Johnson said...

Steve's right.

The core content of constructivist education is the brain teaser.

Catherine Johnson said...

This is an extremely helpful post, because this is the core issue I'm now going to have to tackle here.

The Irvington district is doing a certain amount of gatekeeping on the basis of brainteaser course content.

TurbineGuy said...

Does anyone know where I can buy 200 marbles?

Catherine Johnson said...

amazingly enough, I do not know where you can buy 200 marbles OFFHAND

I'm sure I've purchased marbles in the past...

Anonymous said...


Yeah, that's what one of the guys from the conference I told you about said...

"I am against giving brain teasers to 15 year olds. It's okay to get them interested (when they are 10 yo). We must promote the idea of mathematical proof."

And Soifer said,

"I am fighting discrimination of young people based on their young age. We feed them puzzles and I believe we shold be feeding them real mathematics."

Content issue or pedagogy? Sounds like content to me.

Anonymous said...

“The whole point of math is to avoid the need to pull the magical answer rabbit out of the hat. If you know math, then you don't have to think higher. If you can do the math in an instant, is that higher ordered thinking?”

But at some point, math problems become higher order problems.

Warning: Purely anecdotal evidence ahead. :-) I was always terrible at math. In elementary school, when we had to do hundreds of addition/multiplication problems, I’d usually have the lowest score in the class. In middle school and beyond, I noticed that the people who scored at the top of the Math Olympiad, Putnam Exam etc. were the people who were could pull the magical answer out of a hat, instead of using some complicated technique they’d learned, so spent most of the time I would have spent doing homework doing random problems out of interesting problem collections instead.

In high school, my grades weren’t so great. Since I didn’t spend much time doing basic problems, I’d make basic arithmetic errors, leave off a +C when doing an integration, or what have you. HS teachers were pretty strict about that sort of thing, and would usually give me 0 when I did that. My grades got a lot better in college. I didn’t get any better at avoiding dumb mistakes, but Professors would give me 80-90% of the credit for the problems if I obviously knew how to do them but just forgot a step. In grad school, I aced almost everything; I still made the same mistakes I always did, but Professors would just make a note that I missed something silly, and not deduct any points.

Most people’s scores had the opposite progression. They spent years learning how to do problems where you can tell what technique to use by looking at the section header, so they could ace everything in high school. I’d get 80s on most exams and the average grade in the class would be about 90. In college, there were more novel problems, and Professors didn’t care as much about the details, so the scores were reversed: I’d get 90s and the class average would be in the 80s. In grad school, where most of the problems required coming with a novel, “magic”, solution, the average grade on most exams was somewhere between 20 and 60, and almost everyone would gripe about how none of the material on the exams was taught in class. I was used to being forced to come up with magical answers, so I mostly got 100s.

Whether or not you want to spend time learning how to do magic depends on the sort of problems you want to be able to solve. I never did come close to winning a Math Olympiad or becoming a Putnam fellow, and I still make dumb arithmetic errors all the time, but I don’t have problems doing graduate level math. Personally, I like that tradeoff.


Anonymous said...


Fascinating. Too bad you are anonymous. The people that I know with advanced degrees in math tell me the same thing.

I haven't convinced myself that it has to be a trade off. (But I'm not expert) Isn't there anything we can tweak to make both outcomes possible?