Steve makes an excellent point.

Is there any less understanding in solving the problem algebraically than the discovery way?

Of course not.

In both cases you have to perform a procedure. In the algebra way, you have to reduce the problem to a set of simultaneous equatiosna and unknowns adn solve. In the discovery way, you draw a bunch of boxes representing the various amounts and manipulate the boxes to solve.

The advantage of the algebraic solution is that the procedure generalizes better and, more importantly, is easily solvable no matter how goofy the numbers are, you know, like the ones you wrok with in the real world. I dare you to draw the boxes and solve the problem when you're given 6.72 liters of a .1875 percent alcohol solution, and a final solution of 8.93 percent.

Solving the problem algrbraically requires sophisticated abstract though and manipulation of abstract variables. The discovery method keeps students in the realm of the concrete throughout.

-- KDeRosa - 13 Dec 2006

I was extremely amused once to run across a guy who was arguing that instead of teaching algebra in maths, schools should teach problem solving and communication skills.

Of course algebra is the most useful problem-solving tool I know of for maths, and communicating maths without using algebra is generally hard. (Read some translations of the Ancient Greeks and Egyptians' work).

-- TracyW - 13 Dec 2006

Steve,

Very excellent post.

I especially loved this: This technique tells students that solving math problems requires some sort of mystical understanding and each problem is completely new.

And I agree that when confronted with similar problems, I would have had a much easier time if they'd said "pure alcohol."

You didn't mention an additional assumption: does the question mean per cent by weight or per cent by volume? (By volume is implied but not completely explicit.)

Just to make things more confusing (remember, I'm a chemist) when you mix pure alcohol and pure water, the total volume is less than the sum of the separate volumes. Don't tell the math people that!

I prefer math problems that are abstract enough that physical science doesn't intrude.

This next is embarassing but I'll admit it anyway. I never had any trouble with d = rt in algebra. (Distance equals the product of rate and time.) But later I had a big conceptual hurdle to get over in order to understand what is going on when you needed to calculus to compute the distance when the rate is a function of time and in physics when rate is written as dx/dt.

-- SusanJ - 13 Dec 2006

"The goal of math is not to constantly avoid or rediscover governing equations. Equations and algebra are tools for understanding, not the results of understanding. Mastery of skills is required for understanding, not the other way around."

Very well put. I'm going to share this with my new group of dissident math parent friends.

The problem is, in our district and many others, most parents do not want to think about these things. They're busy with their lives, they spent a lot of money on their houses, and they want to assume that the people in charge know what they're doing.

But if you examine the assumptions behind the curriculum, they don't make a whole lot of sense.

-- RobynW - 13 Dec 2006

I've tried to figure out what the fuzzies think mathematical understanding is. (because they can't seem to explain it themselves) I was looking up mixture problems and came across the Dr. Math site. One question was how to solve the following problem:

If you have 6 liters of a 25 percent alcohol solution, how much alcohol do you have to add to make it a 50 percent solution?

I remember these problems growing up. I remember thinking that it wasn't exactly clear to me that "alcohol" meant a pure solution of 100 percent alcohol and that the rest was pure water.

However, the solution that was given was entirely based on a discovery or intuitive approach. They drew boxes (?) for the amount of (pure) alcohol and the amount of water. They showed graphically how many pure alcohol boxes had to be added to make it one half of all of the boxes. No equations. No algebra. It's almost as if they think it's better to solve problems this way.

Another mixture problem started out with an intuitive approach that ended up with one equation and one unknown, but it seemed to imply that the goal is to always discover governing equations. You can see this with the chicken and cow leg problems. No equations allowed, or you have to discover or create your own equations. Guess and check understanding.

What happens when the students get to problems that cannot be solved this way? This technique tells students that solving math problems requires some sort of mystical understanding and each problem is completely new.

Math is about making life easier than that. If you understand the rules of math, the rules of algebra, and the governing equations, you can solve problems without any intuitive understanding or discovery process.

The equations will teach you the understanding - not the other way around.

Given a problem with distance and time, I would probably think to use the governing equation of d = r*t. Without a whole lot of thinking or intuitive work, I would start defining variables and creating equations. I KNOW that all I have to do is to define legal equations. I know that I have to have an equal number of equations and unknowns. THIS process helps me understand the problem. I don't have to rely on some sort of fuzzy or gut-level understanding of what is going on. As the problems get more difficult, prior understanding or discovery just cannot be done.

For example, look at the mixture problem. What is the governing equation?

Percent solution = Amount of Pure / Total Amount

The percent solution I want is 50% or .5.

The starting amount of pure alcohol I have is .25 * 6 = 1.5 liters.

The amount of pure alcohol I want to add is what I have to find, so I will call that 'X'.

The amount of pure alcohol I will have is 1.5 + the 'X' amount.

The total amount of fluid I will have is 6 liters plus the 'X' amount.

Plug this into the governing equation to get

.5 = (1.5 + X) / (6 + X)

Solve to get X = 3 liters. I don't have to have an intuitive understanding about anything, at least, not beforehand.

I can't tell you how many times (in real life) my solutions are driven by finding additional (legal) equations to make sure that M = N. If I screw up, I might find that two of the equations are linearly dependent. The math (solution process) will show me the problem. The unknowns and governing equations will teach me the understanding. I know how to look at eqations to find understanding.

The goal of math is not to constantly avoid or rediscover governing equations. Equations and algebra are tools for understanding, not the results of understanding. Mastery of skills is required for understanding, not the other way around.

-- SteveH - 13 Dec 2006

Hi Anne,

(Here's something I've probably mentioned before here but I think it applies to your question.)

Years ago when I taught high school chemistry, I got the idea of trying to help the students who were having trouble in class by figuring out what difficulties they were having in interpreting the textbook. I planned to ask them (in a private meeting, not in front of the others) to read a paragraph from the textbook aloud and then rephrase it in their own words.

What happened was that they all refused to read aloud so we never got to the next step.

It may be that for at least some of your students, their problems are not just restricted to math. That might be useful to know.

Good luck!

-- SusanJ - 13 Dec 2006

I tutor special needs students at a local community college. I am never told what their disability is. But I ask them about their high school and their math background. Most dropped out of math in high school because they couldn't pass algebra.

So now they are in college, and their major requires math up to a certain level. So they try to pass a class. But their basic skills are so poor and their knowledge is so rusty, that they end up taking the class at least two and sometimes three times before they squeak by with a C. Then they try to take the next level. The pattern is repeated until they get to College Algebra, which is the equivalent of Algebra II and some precalc in high school. They hit the wall. All of their gaps in knowledge catch up to them there and they just can't pass the class.

Does anyone know what this student can do now? I know about Kumon, but is there anything else they can look into to remediate this problem?

-- AnneDwyer - 13 Dec 2006

I loved polynomial division.

How about comparing the results of dividing:

3243 / 133

using the traditional algorithm, and the polynomial division

(3X^3 + 2X^2 + 4X +3) / (1X^2 + 3X + 3)

(for X = 10)

What level of understanding do they want? What level of understanding is meaningful at each grade level?

I wrote the following to my son's 5th grade math teacher as part of an email pushing for a curriculum that emphasizes mastery of the basics.

Understanding and conceptual thinking seem to be the holy grail of education, but for math, grade-by-grade mastery of skills reigns supreme. No student should be going into 5th grade without instant recall of all of their math facts. Math (and its understanding) is all about laws, axioms, and corolaries; in other words, rules. These rules are embodied as math skills. They are not rote skills. Master the skills and you master the understanding.

-- SteveH - 12 Dec 2006

It is studied in algebra so students know how to simplify rational expressions, which plays an important role later on in calculus when they learn integration by partial fractions. Also, if they go on in engineering, partial fractions also play important roles. Students who know how to do long division will catch on to polynomial division fairly quickly.

Very helpful - thanks Barry.

I'm going to keep this 'in a safe place' (or maybe just write it into my Saxon text).

-- CatherineJohnson - 12 Dec 2006

Dale

Here's an example of polynomial division:

-- CatherineJohnson - 12 Dec 2006

• If the "invert and multiply" rule for dividing fractions seems like rote to them, can they then explain why one DOES invert and multiply?

-- CatherineJohnson - 12 Dec 2006

• Another technique, when told about "understanding", is to ask them very nicely and persistently to define exactly what it means for manipulating fractions. I'm sure their idea of understanding for 1/2 + 2/3 won't hold up to an understanding of 1/(x+2) + 2/(x+3).

• For multiplication, ask them to prove why every algorithm requires you to multiply each digit of one number by each digit of the other number. If they want to talk about understanding, let's talk about real understanding. Have them use that information to design a new algorithm.

• Another test of understanding is to give them M < N, M = N, and M > N, and have them explain what kinds of problems fall into each group and explain various techniques for their solution.

-- CatherineJohnson - 12 Dec 2006

let the math professors define the content and year-to-year expectations, and then let the teachers figure out how to get the job done

oops - missed that!

-- CatherineJohnson - 12 Dec 2006

Many people don't bother figuring it out and use the simple method of assuming that the truth is somewhere in the middle.

Our constant theme here is going to be: content specialists.

When you're looking at the content of a mathematics program, you listen to mathematicians, engineers, computer scientists, statisticians, etc.

Then teachers.

There's no reason to "balance" or "compromise."

Find out from the mathematicians which book(s) have the best content; THEN find out from the teachers which books will work best in the classroom.

Simple.

-- CatherineJohnson - 12 Dec 2006

"But algebra is not the only reason to study long division."

Long division is also great practice for mental arithmetic and estimation. Isn't that number sense?

8624 / 15

Exactly how many times does 15 go into 86 before going over? Forgiving division, on the other hand, allows you to be bad at mental arithmetic - number sense.

But their problem isn't long division or wasting a lot of time learning skills that can be done by calculator. Their problem is the hard work required for mastery of anything. Look at Everyday Math. It is anti-mastery and shallow. Perhaps they think mastery is achieved by repeated partial learning.

-- SteveH - 11 Dec 2006

Polynomial division is this type of problem:

(x^3 + 3x^2 - 2x +6)/(x +1)

It is studied in algebra so students know how to simplify rational expressions, which plays an important role later on in calculus when they learn integration by partial fractions. Also, if they go on in engineering, partial fractions also play important roles. Students who know how to do long division will catch on to polynomial division fairly quickly.

But algebra is not the only reason to study long division. The exploration (and isn't exploration a big deal among the fuzzies?) of why certain fractions are repeating, non-terminating decimals is understood if you know how to do long division. Seeing 5/9 on a calculator expressed as 0.555555... doesn't really tell you what's going on. And partial quotient division (forgiving division) doesn't allow such exploration either.

A good paper on the subject is "The role of long division in the K-12 curriculum" by David Klein and Jim Milgram, available at http://www.csun.edu/~vcmth00m/longdivision.pdf.

Send your school board a copy now!

-- BarryGarelick - 11 Dec 2006

"It didn't seem to occur to them that long division might be used elsewhere in maths - in polynomial division. "

I would love to be able to raise this point in the context of the curriculum discussions we're having in our school district, but I don't have a good grasp on what polynomial division is, and at what stage a student would encounter it. I would really appreciate it if someone could fill me in!

-- DaleA - 11 Dec 2006

For multiplication, ask them to prove why every algorithm requires you to multiply each digit of one number by each digit of the other number. If they want to talk about understanding, let's talk about real understanding. Have them use that information to design a new algorithm.

Exactly my point. The fuzzies mistakenly call traditional math teaching (i.e., the one that is familiar to most of us and how most of us here learned) rote learning and claim it leaves students not understanding "why" it works. That's why I recommended putting those who are pushing for EM to explain why/how the student invented and traditional algorithms work. Have them explain why the lattice method preserves place value, for example. If they want kids to understand, are they really teaching understanding? Do they really understand it themselves for all their talk? If the "invert and multiply" rule for dividing fractions seems like rote to them, can they then explain why one DOES invert and multiply? Saxon provides a means to exlain this. Singapore does not, but starting in fourth grade, it paves the way for the pattern, showing that dividing 10 by 2 is the same as multiplying 1/2 x 10, and going on with similar patterns until sixth grade when the students are then given the last piece of the jigsaw and asked to make the leap that dividing by a fraction requires inversion of the divisor. Based on patterns, yes. I thought the fuzzies were all about "patterns". Aren't they constantly saying math is about patterns? Throw it back to them.

-- BarryGarelick - 11 Dec 2006

"So we need to discriminate between reliable and unreliable research."

The problem with research is that it says nothing about assumptions. You have to define and agree upon where you are going. What are the grade-by-grade expectations?

-- SteveH - 11 Dec 2006

"When you have disagreement between two camps of experts, and you must make a decision, you have to decide which camp you're going to listen to."

Experts in what? Journalists can't quite figure it out. Many people don't bother figuring it out and use the simple method of assuming that the truth is somewhere in the middle. That's why people grab onto "balance" so quickly. They don't have to think or research anymore. They don't want to admit that they really know very little about math.

When someone talks about balance, just say OK, let the math professors define the content and year-to-year expectations, and then let the teachers figure out how to get the job done. That's an appropriate balance.

Another technique, when told about "understanding", is to ask them very nicely and persistently to define exactly what it means for manipulating fractions. I'm sure their idea of understanding for 1/2 + 2/3 won't hold up to an understanding of 1/(x+2) + 2/(x+3).

For multiplication, ask them to prove why every algorithm requires you to multiply each digit of one number by each digit of the other number. If they want to talk about understanding, let's talk about real understanding. Have them use that information to design a new algorithm.

Another test of understanding is to give them M < N, M = N, and M > N, and have them explain what kinds of problems fall into each group and explain various techniques for their solution. Understanding and balance are all they can grab on to, and they don't know what they're talking about. Experts in what? Their own opinion.

-- SteveH - 11 Dec 2006

Hirsch addresses CA Board of Education

When I talked to Mr. Lucia about the context in which I would be speaking, he mentioned that California law requires education policy to be research-based. That's the theme I shall focus on.

The enormous problem faced in basing policy on research is that it is almost impossible to make educational policy that is not based on research. Almost every educational practice that has ever been pursued has been supported with data by somebody. I don't know a single failed policy, ranging from the naturalistic teaching of reading, to the open classroom, to the teaching of abstract set-theory in third-grade math that hasn't been research-based. Experts have advocated almost every conceivable practice short of inflicting permanent bodily harm.

So we need to discriminate between reliable and unreliable research. And of course my recommendation is going to be that only reliable research should guide policy. Now it is possible to give some rules of thumb for determining scientific reliability, but there is no formula adequate to all situations. The distinguished sociologist of science, Stephen Cole in his Harvard Press book, called Making Science has found a continuous spectrum of reliability in most of the natural and social sciences. At the core of each discipline, there develops a consensus of the learned, and this consensus is highly dependable. It is close enough to being right that you can bet your life and your children's lives on that core. But out at the edge, on the frontier of the discipline, there is a lot of disagreement, and we can't tell for sure which rival theory is right. When lawmakers say that education policy should be based on research, the spirit of the law implies reliable, consensus research. Any other interpretation would mean, and has meant, carrying out unwarranted human experimentation on our own children.

If this distinction between core and non-core research is rightly understood, and if its implications are followed in California, then I think the days of faddism, guruism, partisanship, and unwarranted experimentation may be numbered. I'm not saying that research can decide the aims of education. In a democracy, those are decided by the people. But core science can determine how best to achieve them. Take reading. As a people we have decided that we want all our children to read well. Mainstream research has been saying for some years that a naturalistic approach cannot achieve that goal for all children. The reasons why that core research was not heeded is a subject for intellectual and social history, some of which I traced in my recent book, The Schools We Need & Why We Don't Have Them.

I was forced to conclude that in the field of psychology, which is the key field for education research, much of what is accepted within the educational community has been required to conform to a so-called "constructivist" ideology that does not represent the consensus in mainstream psychology, and is almost certainly incorrect. One distinguished psychologist who receives grants from the education division of the National Science Foundation (NSF) expressed dismay at the ideological, anti-empirical sermons, as he called them, which he hears at the education division of NSF meetings in psychology.

[snip]

Scientific consensus is not just a matter of counting heads. If you counted all the experts who have gotten on board the performance-test bandwagon, they would outnumber by far the toilers in the psychometric vineyards who publish meticulous articles in the best journals. Counting heads is not the way to determine a scientific consensus. The number of people who believe in flying saucers is greater than the total number of astrophysicists in the world.

[snip]

It is an uncomfortable thing to say, but the average quality and reliability of science in the best educational journals is below the quality and reliability of science in the best mainstream journals. We laypersons cannot judge the quality of research. Figures don't lie, but how do we know which figures are accurate, complete, and rightly interpreted? Our only recourse is to depend on the reputations of the most highly-regarded journals and scientists. Sensible persons would not quickly challenge Lee Cronbach any more than they would challenge a Nobelist like Herb Simon. Such highly-regarded sources are not always right, but they are far more likely to be right. The consensus of the learned in first-rate scientific work is one of the closest connections we have with the reality principle.

Let me turn to math education. I read a recent report in Education Week which stated that there were two rival math groups in California vying for your approval. On the one side there is what Education Week called the "reform" group who want to put in place the standards of the National Council of Teachers of Mathematics (NCTM), and on the other, the so-called "anti-reform" group that calls those standards variously "fuzzy math" and "whole math." I thought that the tone of the Ed Week report was typical of current educational reporting in that the NCTM approach, which reflects the dominant view among educators, was labeled "reform" while the dissident group that is trying to effect change was labeled "anti-reform." That kind of ideological bias in reporting is characteristic of the education world, and it well illustrates the need for constant vigilance.

To this Board I hardly need to restate the details of the math debate. The NCTM group stresses conceptual understanding over mindless drill and practice, while the dissident group stresses the need for drill and practice leading to mastery. To resolve the issue, which researchers should you listen to? Here are three suggestions: John Anderson, David Geary, and Robert Siegler -- three highly distinguished scientists in the psychology of math education. What are they likely to tell you? I believe you will get strong agreement from them on the following points: that varied and repeated practice leading to rapid recall and automaticity is necessary to higher-order problem-solving skills in both mathematics and the sciences.

They would probably explain to you that lack of automaticity places limits on the mind's channel capacity for higher-order problem-solving skills. They would tell you that only intelligently directed and repeated practice, leading to fast, automatic recall of math facts, and facility in computation and algebraic manipulation can one lead to effective real-world problem solving. Anderson, Geary, and Siegler would provide you with reliable facts, figures, and documentation to support their position, and these data would come not just from isolated lab experiments, but also from large-scale classroom results. If these top scientists agreed on all these points, that is the consensus you should trust, no matter how many pronouncements to the contrary might be made by national educational bodies.

-- CatherineJohnson - 10 Dec 2006

It didn't seem to occur to them that long division might be used elsewhere in maths - in polynomial division.

Now that I've learned polynomial division, I'm wondering how kids who haven't learned long division fare at this skill...

I may be ready to read the Milgram-Klein paper on long division, which was over my head a few months ago.

I had never been taught polynomial division!

Not in two years of high school algebra.

So I was seeing it for the first time in Saxon, and I was teaching it to myself.

I found it somewhat "natural" and intuitive, because it really did make sense as a form of long division.

Moreover, my mistakes - and I suspect people make many mistakes doing polynomial division simply because there are so many "working parts" - weren't mystifying or crippling.

I could recover from them quickly; often my mistakes made sense and were even helpful.

I'm pretty sure this is all because having relearning elementary math I not only had procedural fluency in long division, I had a fairly sound conceptual understanding of it.

-- CatherineJohnson - 10 Dec 2006

• The Phase 4 course as it stands is “a mile wide and an inch deep.” It appears to be a classic spiral curriculum, with the majority of the topics taught in each one of the 3 years of middle school. This means that the 1st year in the cycle is profoundly difficult for a number of students because they’re seeing dozens of topics in a very short period of time. “If this is Tuesday it must be negative exponents.”

• This is where the tutors and the parent reteachers come in. When time on topic is so brief the student hasn’t understood the material, the tutor or the parent reteaches the concept.

• Even when a student has understood the material, he has not had enough practice to master it, so he quickly forgets what he’s seen in class. He thinks he can do the problems because he could do them at school, but he can’t. So the tutor or parent provides worksheets, writes new problems himself, etc.

• In the end, the parent becomes responsible for his child’s math education. Parents won’t let their children fail if they can help it. The parent responds to the first grades of C, D, and F by going to the teacher and sending his child for extra help. Then, when neither of these options solves the problem, he reteaches the material himself and/or hires a tutor. Tutoring and parent reteaching don't work well, either, but they make the difference between just getting by and outright failure.

• The issue of fairness arises from the fact that parents are unacknowledged educators. Obviously, IUFSD does not intend to create math tracks and courses that only the most privileged students – privileged in terms of family income, family education, or both – can take. But when a significant number of students in a course need extensive parental reteaching and/or tutoring, the result is that the course is effectively “closed” to less privileged students.
  

Hirsch has a nice presentation to the CA Board of Ed.

He says that when you have dueling experts you have to make a choice.

Constructivists have managed to put across the argument that since disagreement exists, we should listen to them.

But that's not the case.

When you have disagreement between two camps of experts, and you must make a decision, you have to decide which camp you're going to listen to.

-- CatherineJohnson - 10 Dec 2006

Of course, the problem is that things are a bit murkier, since EVERYDAY MATH actually does contain math! (Thanks, Anne!)

The probably for a lot of the math warrior mathematicians is that (I think) they frequently find themselves in the position of having to argue pedagogy...

-- CatherineJohnson - 10 Dec 2006

"I hope that was sarcasm."

Yet, it was sarcasm. I had trouble with math in high school. So the idea that someone could actually get a Ph.D in math and then teach it to Stanford students impresses me. And I'm more inclined to listen to that person than to the administrators in my local school district.

-- RobynW - 10 Dec 2006

Are the professors right? I don't have enough of a math background to evaluate their comments independently.

The fact is, no one has enough of any kind of background to evaluate anyone's opinions outside his or her field.

The entire nature of expertise is that IT'S EXPERTISE.

The rest of us respect expertise when we see it - and non-experts have a general ability to recognize expertise.

A Stanford mathematician is an expert in math. Period. There's nothing further to discuss.

If Jim Milgram says EVERYDAY MATH is an inferior mathematics textbook, and a math teacher says it's a great mathematics textbook, I'm going to listen to Jim Milgram on the math and listen to the math teacher on teaching - and I'm going to go with Jim Milgram.

Absolutely, keep repeating: why have we selected a math curriculum opposed by Nobel prizewinners?

Would parents have signed off on this if they knew?

-- CatherineJohnson - 10 Dec 2006

It points to a larger problem of education because this happens (more or less) in all subjects. It's up to parents to ensure learning. This doesn't just involve making sure that the homework gets done.

I had the most fascinating conversation with a Tarrytown parent today - amazing.

I'll get it up front later.

-- CatherineJohnson - 10 Dec 2006

I think experts in a content area are qualified to speak about the content students need to learn and master in order to succeed in higher math and science courses.

All parents should insist on this at all times.

Subject matter content specialists should be part of curriculum decisions.

If you don't have parents in the community with degrees or careers in the subject, you should recruit or hire specialists from local firms or colleges.

-- CatherineJohnson - 10 Dec 2006