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?

We keep trying to re-invent math in the U.S., but over the last 30-40 years, all we’ve done is produce a generation or so of students who struggle with math concepts which were (IMO) common knowledge for students in the 50′s, 60′s, and 70′s.

Math is all or none proposition, you either know how to do it, or you don’t. I know that some people will say some kids are math challenged, or have test phobias, but in reality, the only thing which causes most of this (unless there is a proven learning disability, which ESL/ELL doesn’t qualify) is a complete lack of preparation in grades 1 through 5 (elementary school years).

If students receive poor preparation in these grades, the problem will simply compound later in life as the student gets older.

, but over the last 30-40 years, all we’ve done is produce a generation or so of students who struggle with math concepts which were (IMO) common knowledge for students in the 50?s, 60?s, and 70?s.Not true. Our average ability kids are learning a bit more today, I believe. And back in the 50s, the dropout rate was much higher, so the D-F students were of a much higher caliber.

In bending over backwards to make math “accessible,” we may actually make it inaccessible. What do you think?This is completely wrong. As a separate point, you are confusing visual aids with scaffolding–and on what planet are word problems introduced to make things easier?

It’s ridiculous to teach low ability kids algebra, geometry, and trig anyway. But if we are to do it, it is completely impossible to do it with a book that just jumps right in.

People of higher than average intelligence are, literally, clueless about how hard it is to teach math to kids with low ability.

I’m not arguing in favor of any one sort of scaffolding or technique. I’m just saying that a book such as you describe would leave all but 20% behind, and that’s the 20% that went on in years past. The rest of them took algebra and geometry much later.

I would debate whether the use of aids truly fits the strict definition of scaffolding…

I agree with Cal. The books approach may be a good approach for a certain segment of the student population but would be detrimental for the overall population. I am completely baffled by some students inability to learn and master some concepts. (I teach HS science.) Some math concepts that just seem so obvious to me and other students are incomprehensible to some students. And some of those students are really trying.

For whatever reason, it may be that some students just are not going to get it. We would be horrified to hear about someone continually trying to make their child into a soccer star when its obvious that after tons of effort, the kid just isn’t very good. Sometimes I feel like we are doing the same to some students academically. Must feel good to be those students.

Good post. I agree with you. And Amy.

The problem I see is that we try to advance kids who haven’t ‘gotten’ it yet at the same rate as the kids who have.

When I taught HS, I had a ninth grader who (after much help) was FINALLY starting to get fractions and decimals. So why was he in an Algebra class? Because that was the next math class offered by the school, so he was in it, and would take algebra 2 the next year because failure wasn’t an option.

He really needed to be in “Math Nine” or some other class where he could get more practice.

On the other end of the spectrum, I knew 7th graders who really OUGHT to be in Algebra, but couldn’t be because they were in 7th grade, and so had to plod through prealgebra with their age-mates. It was a small school, so there was less tracking than usual, but I think, in general, the problem with HS math is insufficient ability grouping, more than the textbooks.

True. And the textbooks are written to try to cover broad levels of ability and readiness, which results in too much scaffolding for some students.

NOT doing some ability grouping is holding the average kid back. Not the top tier, because they get grouped in G & T and Honors classes.

But, the average kid. Whose Algebra class contains those ready (or, nearly-ready – not that difficult to bring them up to that level) and those who are stumped by fractions, decimals, calculating the area of a triangle, and pretty much all math beyond counting and adding.

Honors and GT classes are, unfortunately, going the way of the dinosaurs. They are too often seen as elitist and insufficiently “diverse” (just check out the frequent WaPo articles- and the comments – on the TJ math magnet HS) so kids that could have, and should have, been ready for algebra two years earlier are now dumped into the mix with everyone else. Ideally, kids should not only have homogeneous groups by subject, but instructional materials should reflect their different needs. Praise be, my kindergarten grandkids are starting Singapore Math !!! – due to heavy parental pressure in an affluent town system that has only 2-3 elementary schools.

Hi Diana. Yes, I agree that “scaffolding” might not be the right word, but I get what you’re saying and I think you’re quite right. I remember a student who was struggling mightily in an advanced linear algebra course I was using and he came to me asking if there was another text he could try reading from. On a hunch I gave him my old university text, by Herstein, which is stern and classical: NO diagrams, sideboxes or colours, and straight to-the-point format: Definition – Example – Theorem – Proof – Example – Corollaries – More Examples – Exercises. He returned a few days later ecstatic: “I get it now! It’s so clear!”

I think what you’re getting at is “clutter”. That gets in the way. All of the things you mention might be described as “scaffolding” and are effective to the extent that they help bridge understanding. But to the extent that they muddy the presentation and/or interrupt the main flow of ideas, they are a hindrance. Flipping through most modern texts — regardless of whether they are K-12 or University level — I see more clutter than content. Most of what is pitched today as “helps” really are just clutter.

Well-structured and sequenced ideas, with an ebb-and-flow of skill development, problem solving and knowledge, provide all the scaffolding a good student needs. The rest can and should be done by the instructor in class, as s/he sees fit — in keeping with his/er own style and perception of the students’ needs.

I think your basic question — does this stuff (“clutter” or “scaffolding”) get in the way of instruction/learning? I think the answer is that it can and often does, though not always. Completely spartan instruction can be joyless. If you think back to your Moscow experience you may find that the teacher, and your fellow classmates’ attitudes, had a lot to do with the learning experience. The clutter is unnecessary given a proper learning environment and the right kind of “live” assistance.

. I remember a student who was struggling mightily in an advanced linear algebra course I was using and he came to me asking if there was another text he could try reading from.Yeah, because this example is entirely relevant to scaffolding instruction for low-skilled students.

Hear that whizzing sound? It’s the point, a couple miles over your head.

You sure about that?

His anecdote, in summary: Student struggling with cluttered text succeeds with uncluttered text.

There’s a difference between teachers selecting appropriate scaffolds for their students and placing many possible scaffolds in the textbook along with the basic idea. Low performing students, in my experience, are most likely to be overwhelmed with too much information. Cluttering textbooks with many scaffolds increases the risk of that occurring.

Three words: Advanced Linear Algebra.

Sure. That’s the population we’re talking about.

Consider my comment above in conjunction with this paragraph from Diana’s original post:

Scaffolds are tools. Use a hammer to drive a nail, it’s good. Use a hammer to clean a TV screen, it’s bad. Same goes with different types of scaffolds.

For scaffolds to be effective, the *right* ones have to be selected to help the student. Different students have different prior knowledge and experiences and learn at different rates. That’s life. Students also don’t know what they don’t know. Looking back at the tool metaphor, the student might grab the hammer to drive a nail or clean the TV, not knowing why one works and one doesn’t.

Given this, it makes sense that selecting the right scaffolds for different students should be done by someone who knows how they work as well as what learning they are meant to support. That sounds a lot like a teacher, doesn’t it? What doesn’t make sense is to throw a book cluttered with scaffolds at a student.

R. Craigen’s anecdote was about an advanced linear algebra student. This person has likely spent a lot of time in math classes over the years. It can be assumed he’s a proficient learner, able to process a significant amount of information. This increases his odds of being able to piece things together despite the disruptions in the textbook. He still failed to do so.

Now, imagine a slow learner who’s faced with an arithmetic textbook similarly cluttered with scaffolds. By definition, he’s not able to process information as proficiently. In addition, he’s probably got far less background knowledge in the first place. I can’t think of any scenario in which he’d be more likely to succeed with the cluttered textbook than the advanced linear algebra student.

It all comes down to the right tool for the right job. Bright kids can succeed despite having the entire toolbox thrown into the textbook. Slower kids will have a harder time of it because they need the focus of having only the right tools in their hands.

Bright kids can survive, maybe, with a cluttered textbook (or cluttered pedagogy), but they’re likely to find it agonizing. Why should anyone be subjected to that stuff?

Thank you for all of the comments. I have some responses and additional thoughts, which will probably take the form of another post.

I dunno. Is Amy cute? Seems story problems with too much information could be a diversion for the ADD.

OTOH, out in the real world, everything is a story problem. You may know add, subtract, divide, multiply, but do you know when to do which?

I’ve had a hard time with math over more than half a century and it’s tempting to relate it to a b444h third grade teacher. But I think some folks will get it and some won’t.

By the time you pare off the less-competent, like me, the remainder will probably do pretty well. Which proves…?