# Got math?

M. J. McDermott talks about math education in Washington state.

She talks about and demonstrates some “reformed” methods for solving arithmetic problems that have been commonly taught in the 4th and 5th grades: Cluster problems (taught in the “Terc” books), and partial products and partial quotients, and the “lattice method” of multiplication (taught in books called “Everyday Math”). She says that these “reform” math curricula discourage the teaching of what’s called the “standard algorithm” for multiplication and division (called “long division”).

In my own life I’ve used a method similar to the “cluster problems” to do multiplication and division in my head if I don’t have pencil and paper handy, or a calculator. The best thing I have when I just do it in my head is my own internal multiplication table. I find it difficult to do the standard algorithms in my head, though I think they’re fine if I have pencil and paper. If I don’t have to do it in my head I prefer the standard algorithms. I wouldn’t want to write out all the different sub-answers I’d have to write with the other methods. I remember when I did math drills in elementary and jr. high school. It would’ve taken me forever to finish math problems using the “reform” methods she describes. By the way, she’s for teaching the standard algorithm.

McDermott illustrates as well how problems are posed. They don’t do math drills. Instead they plan vacations, with geographic data. Group work is emphasized, as opposed to students working alone.

She describes how when she returned to college about 10 years ago to get a degree in atmospheric science, she encountered traditional aged students, just out of high school, in her classes who couldn’t do the math. She said they had the inability to work alone. They always felt like they had to check their answers against others’. They had a lack of basic math fluency in the symbolic language of math, and they were lacking in the ability to think logically. They lacked basic math skills such as arithmetic (which I assume means things like order of operations), algebra, and trigonometry. I’m really curious about the reason behind the problems with algebra and trig, but she doesn’t explain. I wonder how they got into college, especially the atmospheric science program without these skills. She said they had a complete dependence on calculators.

She paints a rather scary picture. It illustrates something that Justin James, a tech blogger, talked about a while back (unfortunately I can no longer find the blog entry where he talked about this), that math is important for developing higher order brain functions. Even if you don’t end up using the type of math you’ve learned, you gain mental abilities through the math work that you can use in other areas of life.

I think what this presentation illustrates is it’s easy to take traditional math skills for granted, because we don’t realize what we’ve gained from them unless we use those math skills directly. If we don’t use the math directly we tend to think that we had the cognitive skills we developed as a result all along. It looks like that’s a misconception we should be wary of.

One of the areas she mentioned where the skill of thinking logically is critical is computer programming. I agree. I’ve written previously about the lack of interest in enrolling in computer science in college, and about people who are in non-computer science courses actively trying to get out of them when computer programming is made part of them. Would this explain part of the aversion?

Edit 2/14/07: As I investigated this video further I found that some people were calling the demonstrated “reform methods” constructivist education. In some ways they may be right, just as people picked up on this when “whole math” and “whole language” were the “reform du jour” of the late 1990s. However I am suspicious that what’s been tried so far is not entirely representative of this education method. Sadly, I think that constructivism has been misunderstood by its practitioners in this country.

The idea is supposed to be that the teacher is an active participant in the education of their students, and that the students are an active part of their own education as well. What I used to hear about the older attempts at constructivist reforms is they put teachers in a passive role, leaving the learning totally up to the students. Further, the teacher was not to correct the students. The curriculum shunned rote methods. Anything that smacked of memorization or repetition was excluded. Students were supposed to come up with their own answers, and the teachers were not supposed to judge their work for fear of damaging their self-esteem.

Why they insisted on excluding repetition is beyond me. When children play they do things repeatedly. Repetition is part of their natural learning process. I don’t think what the educational establishment has come up with is what the creators of constructivist theory had in mind.

My sense of this teaching style is that overall the goal is to teach students how to learn, by using the students’ innate process of experimentation and play, while at the same time teaching them about real subjects they need to learn. The teacher should be a guide to the students. Yes, the students are to explore and experiment to arrive at their own answers to problems, however the problems are posed by the teachers, giving the students a goal to achieve. If the students are coming up with answers that are off, the teacher can introduce further problems prodding the students to explore the subject more, until they arrive at a correct answer. The goal is to build understanding of the subject. Yes, students build their own model of the world through this process, but it does a disservice to the student if that constructed model is not compatible with the real world. Further, the teacher provides some concrete examples of the subject matter, giving the students a grounding in it, before they are allowed to go off and explore it.

My impression is that the goal is to help students arrive at correct answers, but process is emphasized so that it’s the students who obtain the answers, through their own process of discovery, without the teacher imposing a single way to arrive at those answers. This creates a sense of accomplishment in the students, and self-esteem, encouraging them to continue the process. I may be wrong on some points.

I think there are times when rote methods are best for gaining a grounding in a subject. One of the things I thought was insane about “whole language” was they banned phonics from the reading curriculum. Phonics has consistently worked well as a way to teach children about reading. I think the reason it’s essential is that language in large part is standardized in our modern civilization. It provides a way of conveying information that is logically consistent from the disseminator, to the receiver. It’s essential that people’s understanding of language be consistent. It’s how we make connections with each other, and conduct transactions. Students who don’t get this are going to be handicapped.

I think the goal of reform efforts has been noble. The problem has been in their execution. The goal was to get the schools out of the old Industrial Age model of assuming that the students were blank slates upon which the teacher would impress their knowledge. The teacher was pretty much the only one who was active in the process. The students would just passively receive the information (hopefully) and then either regurgitate it later or synthesize it into something new.

The goal with constructivist methods was to create active learners and explorers. That sounds good to me. The problem seems to be that the ways in which this method is being used leads to students who do not fully understand the subject matter. It seems they end up behind other students who were taught using the older methods. I think the old methodology provides a useful benchmark. If using a new methodology results in students who are behind grade level I think that indicates that the methodology needs to be revised and some assumptions about it need to be questioned.

## 15 thoughts on “Got math?”

1. Mark –

Here’s the link to the articles you remember (it was a 3 part series):
http://blogs.techrepublic.com.com/programming-and-development/?p=43
http://blogs.techrepublic.com.com/programming-and-development/?p=44
http://blogs.techrepublic.com.com/programming-and-development/?p=45

This first one carries the main thrust that you remember. But this is often a theme I harp upon. In fact, I was talking today to a developer from India about just this. Tomorrow morning, I am going to pump him for information on the Indian education system, since today I told him a lot about the US education system. Indeed, maybe I will get more information on the subject soon, since I will be in India in a few weeks.

I saw that video as well, if it is the “weather woman” video. It is interesting, because many of the techniques she trashed were techniques that I use mentally when paper & pen are not handy, particularly “clustering”. Also interesting is that I was only taught “old math”, those techniques were things I figured out on my own. There is some merit in them, but I do agree that if you are only going to teach math one way, the old way is the best.

J.Ja

2. Hi Justin. Good to see you!

I wonder if any studies have been done to show why “the old way” enhances mathematical literacy and logical thinking. It seems to me the reason they’re being thrown away is because what you talked about is being taken for granted.

The video I have here is the “weather woman”. As I said in my post, what was surprising to me was she said the entering high school students had trouble with the basics of algebra and trig. I would think that the standardized tests (SAT, ACT, etc.) would catch that and that those students would not be admitted, particularly to an atmospheric sciences school. The only reason I can think of is they must’ve been lowering their admission standards to keep enrollment up. She heard professors complain, nevertheless.

Thanks for the links to those old posts.

3. I think with the “old way”, the beauty of it was, it did not tie you up with knowing the low-level stuff, and just got you doing the basic stuff. It’s like starting with BASIC instead of Assembly or binary. The “new techniques” spend too much time moneying about with the underlying principles.

I do not recall the SAT having anything other than the most basic algebra on it (if 2 + x = 4, what is x?) on it, and there certainly was not trig that I can recall. I was also one of the last groups of test takers in the pre-calculator days, and the second or third group after the first “rebalancing” of it when they moved from a straight score to a curved score.

That is one reason why the SATs are not so hot, they have a bit of a curve in the scoring. I beleive you get compared to the last group who took it. If you take the test after one of those times where “everyone” takes it, instead of just the people who are really interrested in scoring high, you get a softer curve. At least that’s how I remember it. I literally slept through 50% of the time of my SATs (I forgot about them until 4 hours before, and I been up playing a video game until 4 AM when I remembered; to compound it, my “pep talk” was, “honey, I was tripping on acid when I took my SATs, so I at least expect you to do better than I did”). I vaguely remember the SAT II’s, I had a severe bout of bronchitus and could barely see the paper…

J.Ja

4. Huh. I took the ACT. I don’t remember what the math was like on it. I know there was algebra, at least, like you were saying. I remember taking a math entrance exam that all freshmen had to take when I went to college. They tested for higher math like Algebra 3 (aka “pre-Calc”), trig., and linear algebra. I know, because I ended up taking a remedial course on trig, and matrices & linear equations when I entered. I had trig. in high school, but apparently not enough…

What I was commenting on was it seemed odd that her fellow, traditional-aged students were having trouble with these basic concepts. I was wondering, “If they’re having trouble, why are they in there?” Prerequisites should’ve kept them out until they learned it, unless the math tests they took only required them to give hard answers like if you have a right triangle and angle ABC is 30 degrees, what is sin(ABC)? That’s the stuff they can figure out with a calculator easily without having to think about it.

5. Sadly, pre-prequisites and entrance tests don’t cut it. The tests rarely reflect either the student’s actualy knowledge, or the true requirements for the class. Pre-requisites in collegew are usually a mess; most classes seem to never quite get as far as the syllabus says it will (ever go to a class that got ahead of the syllabus? Neither have I…) so while the two courses should adjoin, there is a gap. Even worse is that 4 week break for winter, or worse, the 3 month gap overr the summer. We all know how 3 months of time off will erase knowlege that was learned but never really used.

Finally, just because you earned a passing grade in a class does not mean you know it! Even without a grading curve, a 71 is a C in many colleges! That means that if you understood the basics of a class but failed the last porrtion of it miserably, you still look great on paper. Add in a curve, where only 50% comprehension might be C work, and you have a recipe for disaster.

As always, I am reminded of the essay, “Making the grade”. (http://www.cs.technion.ac.il/~eyalk/makegrad.html)

J.Ja

6. It’s was so long ago, but my memory is it was rare if I heard the teacher say “we were going to cover this but we don’t have time.”

I disagree that a C looks “great”. C’s were considered so-so in my mind. Nothing to be proud of. I’d say that was a common perception. The only thing I could say for it was “At least I didn’t fail the class.”

I can see your point about grading curves distorting the perception of how much a student knows. I saw some of that happen with professors who didn’t know how to teach and were not motivated to do better. I took a class where half the class was failing. The teacher was terrible. So he graded on a massive curve. As you can guess we didn’t learn too much. Most of my classes graded on a curve, though it was usually slight enough that it didn’t mean much. If you were borderline between grades it could make a difference, but if you had a mid-C it would only raise you to a C+.

Here’s another story that might shed light on the discussion. I took matrices & linear equations which was supposed to be a pre-requisite for a course called Linear Algebra. I did well on M&LE, but did terribly in Linear Algebra. I was hung up on the Cartesian coordinate system. The teacher kept telling me that the origin was relative but that concept was foreign to me. I didn’t finally get it until I took physics. I had physics in high school, but I don’t think we covered vectors rigorously. There was no curve for M&LE, but the course material was shallow. They taught me the process of solving the problems, but very little about what was really going on. I had little understanding of it. This was the only experience I had where the prerequisite was not adequate preparation for the next class.

By the way, what did you learn from your Indian friend about how they teach math?

I saw a little of this when I was in college. There were some students I saw who were uptight about always getting A’s in their courses. In one course I saw a student approach the teacher after class and say, “I noticed I’m getting a B. Is there some way I could raise it?” This was towards the end of the class and I guess she knew that by going through the normal course of actions she wouldn’t be able to raise it enough to an A. I think she was asking for extra credit work. I didn’t think it was a big deal because the course was a half-semester, 1 credit. It would’ve had a negligable impact on her GPA.

I’ve heard about what Wiesenfeld is talking about in a different sense: students cheating. What’s been reported in the last several years is alarming. Every once in a while the news covers this. In the last student survey *half* of the students admit to cheating on assignments and tests. I don’t remember if they covered this while I was going to college. I imagine it was less, though I could be deluding myself. It gets to the issue that Wiesenfeld is talking about: quality of workmanship in society, and I think it’s worse than the problem of students asking for a change in grade. At least there the teacher has control over it. With cheating the students are deliberately trying to undermine the educational system itself.

This gets into a totally different topic, but I have one concern with the issue of vouchers that can be used in public and private schools. With so much money involved, less scrupulous people can infect the system with their desire that the student just get good grades, no matter what. With such a system there needs to be a counterveiling force that puts pressure on schools to grade honestly and not cave to the pressures of parents and students to grade dishonestly. I would think state-sponsored standardized testing would need to be used to make sure grades reasonably match with results. It would kill two birds with one stone, too, getting rid of massive curves.

I had a conversation with a friend several years ago that bears mentioning. I said that when I was in school my mom (I was raised by a single parent) thought it was okay if I got a B in a course. She was a bit critical if I got a C. It was “okay”, but there was room for improvement. After having worked in my field for a few years I discovered that employers and their clients expected “A” work from me *every time*, and I did that, but only because I could see that was a clear expectation from them and I hate letting people down. It was hard work to achieve that, but I always found work easier than school.

I had an experience in high school where in hindsight a teacher was trying to tell me something about this, but I didn’t get it. My trig. teacher passed out progress reports on our grades. Mine showed I was getting a B so far. The teacher continued past me. I said to a friend sitting next to me, “I’m satisfied with that,” showing him my grade. My teacher overheard me, made a point of backing up, giving me a stern look and said, “You shouldn’t be.” I was bit taken aback. No one had told me that before, but I didn’t take his word for it.

I was being a bit critical of my mother in hindsight. She didn’t expect enough of me in understanding how the real world works. I think I turned out okay nevertheless.

8. I never did get to asking him, I will have to rememberr to ask him tomorrow about that.

Cheating is a huge problem. One of the fantasy company ideas I have is a system that would do IP infringement checks on the Internet. Some people (like prrofessorrs) could upload items and say, “does this already exist?” Companies with compyrights could go the other way, dumping their documents on us, and we would notify them if anything out there resembles it.

J.Ja

Hmm. I never really thought that was necessary. I guess I have heard of cases where two different people come up with papers that sound the same. I don’t hear about this happening that often, though. The excuse that was given was that when approaching specific subjects it’s possible for two people to have the same analysis of it, and even use the same words in some spots, without ever seeing each other’s work, just because the subject is so narrow. Both get led down the same path.

The technology to do this already exists. There are services professors can use to check for plagiarism in student papers. If you used the same technique on their papers it would work.

What I was talking about went beyond this. It’s an age-old problem, but students continue to find ways to cheat on tests as well. That’s something that only vigilance will address. I’ve heard about how they’ve used smartphones (text messaging), calling their friends who had the same test in an earlier section (before class–this is an old technique), even writing the answers on the backsides of water bottle labels and then sticking the labels back on the bottles. Professors have to insist that students bring hardly anything with them to tests, just the clothes on their backs. Even calculators can be a problem, because now they can send/receive, store and display text messages.

Good talking to you. If you find out the information from your Indian friend, let me know. Okay? 🙂

10. I did indeed find out the information… I’ll be blogging about it soon, but here’s a sneak preview: a student in India gets educated with no electrronics allowed in math until university, and even then it is “frowned upon”, they are not allowed in tests, and the calculators rarely have a function more advanced than what Window’s calculator has.

J.Ja

11. Hmm. Sounds similar to the kind of math education I had. I remember that allowing calculators in the classroom was a controversial thing. By and large they were not allowed. I didn’t see that restriction relaxed until my junior year in high school (1986-87). I had an old TI programmable calculator that I used. Programming it was like programming in assembly language, almost. My math teacher allowed me to use it. Some students complained that it gave me an unfair advantage, but she said that if I knew how to program it to do the problems properly, then it meant I understood the processes, and that’s what mattered. I think at the time she was pretty avant garde with that view.

I’ll be watching for your article.

12. hi mark,

Part of the M. J. McDermott video prompted me to look in Liping Ma’s book , which has been sitting unopened on my desk.

In McDermott’s demonstration of the standard algorithm she at one point says, “Sometimes we put a zero here”, and then adds a zero at the end of the 78. No explanation of why the zero is added, not to mention why it is only added “sometimes”!!

This point is discussed at some length by Liping Ma who interviews both American and Chinese teachers about their understanding of the multiplication algorithm.

There are two groups.

Procedural group who do what McDermott does and who usually don’t understand the underlying rationale for place value. ie. the zero is just added as a placeholder (some other symbol would be just as good according to these teachers) or sometimes not added (because they think this will alter the correct value)

Conceptual group who do understand the place value implications (we are multiplying by 30 not 3) and so can actually explain to their students what is happening.

So the important distinction is more one of which teachers understand and can explain arithmetic rather than which particular algorithm is used

13. @Bill:

When I was taught arithmetic I’m fairly sure we talked about the significance of place value in addition and subtraction, but I don’t remember it being discussed for multiplication and division. This may have been because up until I was in 6th grade I switched schools rather often. So I’m sure I was exposed to a bunch of different teaching styles.

I think you can see why Alan Kay has complained about the state of math education in this country, because a lot of times it’s taught procedurally, not conceptually. Depending on the quality of the teacher you have, it’s entirely possible to go through math courses understanding the steps to take to solve specific types of problems, but not understand a wit of the math involved.

In terms of planning and giving the presentation itself I thought McDermott did a good job. What got lost on me though is the relationship between the freshmen she saw who were unable to handle college math to her demonstration of the “reform” methods for doing arithmetic. I can see her complaint about the methods used to teach arithmetic, but she didn’t get into how the students were taught algebra, for example. It left me wondering if perhaps they were not taught it at all, and that’s the problem. I have reason to be suspicious about that, because shortly after I watched this video for the first time I “listened in” on a discussion on the internet between some people who like the “reform” methods and those who advocate for a return to core curriculum. The “reformers” were saying things like, “Why teach the students math they’ll never use?”, and relating it to their experience of not using a lot of the math they were taught in school. I was getting the impression that the “reformers” were missing the point about math education. Those who advocate a “back to basics” approach I think are missing a part of the point, too, but they at least want students to learn more than how to do arithmetic. If I was forced to pick between the two I’d rather side with the “core” folks.