Getting right answer shows ‘deep understanding’

Students display their “deep understanding” of math problems by getting the right answer, argues James V. Shuls, director of education at the Show-Me Institute.

His second-grade daughter’s school is stressing “real world” word problems this year. His daughter subtracted correctly to find how far the snail crawled, but got the lowest rating, “does not meet expectations.”  She didn’t use the prescribed process.

“Interestingly, there have been other problems where she reached the wrong answer, but received a higher score,” writes Shuls.

The teacher apparently wanted students to follow an 8-step process that includes drawing a unit bar for each variable.


It’s clear his daughter didn’t just stumble on the right answer, he argues. She had a process that made sense.

“When a student can correctly identify the type of problem and can solve for the answer using some type of process, they understand the concept.”

“Knowing you’ve done something right and then getting criticized” anyhow is discouraging for students, Shuls writes.

Rather, we should celebrate correct answers and, when necessary, demonstrate more efficient methods or other ways of thinking about problems. This should be done while keeping in mind that what matters most is that our students have a method that works and is transferrable to other problems.

We tell students to “think like scientists” and “act like mathematicians,” Shuls writes. “Do you know what good scientists and mathematicians do? They get the answer correct.”

Multiplying and trivializing slavery

Integrating history and math, getting students to write math problems . . . It must have sounded like a good idea at the time. Now a fourth-grade teacher at New York City’s P.S. 59 is in hot water for  assigning “slavery word problems homework,” reports NY1.

Question 1 . . . asked:

“In a slave ship, there can be 3,799 slaves. One day, the slaves took over the ship. 1,897 are dead. How many slaves are alive?”

The second question . . .  said:

“One slave got whipped five times a day. How many times did he get whipped in a month (31 days)? Another slave got whipped nine times a day. How many times did he get whipped in a month? How many times did the two slaves get whipped together in one month?”

Aziza Harding, a student teacher, showed the homework to Charlton McIlwain, one of her professors at NYU. The professor contacted NY1, which showed the worksheet to the principal of P.S. 59, who responded: “I am appalled by this.”

Another fourth-grade teacher’s students wrote the questions in January.

“You’re ostensibly teaching or trying to teach history or call attention to a particular historical moment, yet there’s no explanation, there’s no education, there’s no teaching going on,” McIlwain said. “And so, for someone who is probably, at nine years of age, has maybe heard of slavery but probably doesn’t know what it is really like, their first, perhaps, and most lasting impression about this historical event comes in a very abstracted, nonchalant type of thing that they have no real sense of connection to.

Harding, the student teacher, fears students will be “desensitized to this type of violence” unless they’re taught to understand the history of slavery.


Textbook weirdness

Thanks, Textbooks gives examples of awkward, weird and just plain bad writing in textbooks. (Send examples to Karl at

This problem had me stumped:
Mmm… this is why I dine at my local “hamburger outlet.”  Where else am I offered a “choice” between four condiments that are added and omitted at random?  God, please let it be mustard and onions.

Via This Week in Education.

Can math scaffolding hinder learning?

When I was fourteen, we spent a year in Moscow. I attended a Soviet school that “specialized” in French–that is, it taught French from the early grades. The other subjects (math, literature, history, technical drawing, geography, physics, chemistry, and biology) were in Russian. No one expected me to participate in class, but I insisted on being added to the class list and asked teachers to treat me like a regular student. I was eventually doing the work in all of my subjects except for chemistry and biology, where I lacked the necessary background knowledge and was usually a bit lost. (I barely got by in physics, but I did learn something.)

My favorite classes were math and French. Here is a picture of the math textbook. It took us through algebra, beginning calculus, and some trigonometry. Its 220 pages contained more substance than many a hefty textbook I’ve seen since. When I returned to the U.S., I was ready for calculus but had to take a year of precalculus first, along with my classmates. (It didn’t hurt, as I got to do more trigonometry.)

Recently I have been wondering how this textbook manages to convey so much in such short space, and how I learned so much without finding it particularly difficult. To answer this question well, I would have to work my way through the textbook again, this time with pedagogy in mind. That’s a project for another time. In the meantime, I’ll toss out a few hypotheses.

Well, one obvious reason we were able to learn so much is that there was a standard curriculum through the grades. All students came to this course with similar knowledge and practice. Some were better at math than others, but it wasn’t because they had better preparation. (Of course this isn’t entirely true, as some students had additional resources at home and elsewhere.)

It could also be that the curriculum included fewer topics than math courses in the U.S. do; thus there was more time to learn them thoroughly.

But what strikes me about this little textbook is that it plunges right in. The first chapter talks about inductive proofs. The second goes into combinatorics. There are no pictures except for graphs of functions (and a few circles and rectangles). There are word problems, but they are relatively few. There are no needless “scaffolds.”

Scaffolds in instruction are temporary supports intended to bring students to the point of self-sufficiency. All good instruction uses them to some degree.  But certain kinds of “scaffolds” can actually  become barriers, complicating the student’s entry into the subject matter. In mathematics, excessive reliance on “visuals,” “manipulatives,” and “real-life” applications can stand in the way of the math itself.

This textbook, by contrast, “scaffolds” the instruction in one way only: it builds from simpler problems to more complex ones. It  lacks the “scaffolding” that plagues many a math textbook that I have seen: those colored graphics, tips and strategies, needless word problems, and so on. It has a few word problems, but there are reasons for them to be word problems. The vast majority of the problems use mathematical notation. Thus, students become fluent in it and learn to think in it.

I was recently looking at AMSCO’s Geometry–better than many in terms of presentation. Very little clutter. But even AMSCO has word problems like this: “Amy said that if the radius of a circular cylinder were doubled and the height decreased by one-half, the volume of the cylinder would remain unchanged. Do you agree with Amy? Explain why or why not.” There is no reason to bring Amy into this; Amy’s presence does nothing for the problem. Also, turning this into a matter of opinion (“do you agree or disagree”) confuses the matter. Instead, the student should be asked whether the statement is correct or incorrect.

In bending over backwards to make math “accessible,” we may actually make it inaccessible. What do you think?