Computational Competence Doesn't Guarantee Conceptual Understanding in Math

3/4/2015

Commenters on the teaching of mathematics sometimes express impatience with the idea that attention ought to be paid to conceptual understanding in math education. I get it: it sounds fuzzy and potentially wrong-headed, as though we’re ready to overlook inaccurate calculation so long as the student seems to understand the concepts—and student understanding sounds likely to be ascertained via our mere guess.

Impatience with the idea that conceptual aspects of math ought to be explicitly taught is often coupled with an assurance that, if you teach students to calculate accurately, the conceptual understanding will come. A new experiment provides evidence that this belief is not justified. People can be adept with calculation, yet have poor conceptual understanding.

Bob Siegler and Hugues Lortie-Forgues asked preservice teachers (Experiment 1) and middle school students (Experiment 2) to make quick judgments (true or false) of inequalities in this form:

N1/M1 + N2/M2> N1/M1

N and M were two digit numbers, making it hard to calculate the problem quickly in one’s head. Instead, you needed to evaluate what must be true. In this case, subjects easily recognized that the sum on the left side of the inequality had to be larger than the value on the right side. Likewise, they made few mistakes with inequalities of this form

N1/M1 - N2/M2> N1/M1

But when multiplication or division was called for, subjects made errors. Specifically, when N2/M2 amounted to less than one and was multiplied by N1/M1, subjects incorrectly thought the result would be larger than N1/M1. And division was required, subjects thought the result would be smaller than N1/M1. In fact, they answered correctly less often than chance.

Yet these same subjects were quite accurate when asked to calculate answers to problems entailed multiplication or division of fractions; middle-school students got about 80% correct. AND they showed quite good understanding of the magnitude of fractions between 0 and 1 (as shown by placing marks on a number line to represent fraction quantities).

This is a small sample, and the absolute level of performance should not be taken as representative of preservice teachers or of middle-school students. But Siegler and Lortie-Forgues suggest that the disconnect between computation and understanding is typical. That conclusion is in line with the evaluation of the National Math Panel.

So what's to be done? Teach concepts. Among other ideas, Siegler and Lortie-Forgues suggest that, once they have some competence in calculation, students might compare the results of

8/7 * 1/2

7/7 *1/2

6/7 * 1/2

There are sure to be many methods of helping with conceptual understanding, some best introduced before calculation, some concurrent with it, and some after. This latest finding points to the necessity of greater attention to understanding in instruction.

Siegler, R. S. & Lortie-Forgues, H. (in press) Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology.
http://dx.doi.org/10.1037/edu0000025

Edit, 4:11 p.m. 3/4/15: Corrected the spelling of the second author's last name. (Terribly embarrassed.)

Douglas Hainline

3/4/2015 07:04:59 am

"N1/M1 - N2/M2> N1/M1"
Don't you need to reverse the inequality here?

Daniel Willingham

3/4/2015 07:14:21 am

They get it right, meaning they correctly say "false."

Douglas Hainline

3/4/2015 05:29:42 pm

Of course! Duhh...

Not M. Name

3/5/2015 02:39:32 am

Is the quotient N2/M2 a negative number (meaning are the signs of N2 and M2 opposite)? If so, this inequality would be true.

Daniel Willingham

3/5/2015 09:50:55 pm

I think all numbers were positive.

Joe Mulvey

3/5/2015 08:48:26 am

So computational competence is not sufficient to guarantee conceptual understanding? Fair enough - intuitively, I would agree. Do you have an opinion on whether it is necessary? Lots of pedagogical guidance in recent years has de-emphasised computational practice. I struggle with teaching kids who have come from their primary education and lack facility with standard computational algorithms. I find it holds back their progress not just in further number work, but in algebra and geometry too.

Daniel Willingham

3/5/2015 09:52:03 pm

I agree, the opposite (conceptual understanding is all you need) is not true either.

Barry Garelick

3/6/2015 01:22:00 pm

"Commenters on the teaching of mathematics sometimes express impatience with the idea that attention ought to be paid to conceptual understanding in math education."

We get impatient when conceptual understanding eclipses procedural fluency, and when first graders are asked to explain "why 2 + 3 equals 5". We have always maintained that the two go hand in hand. Also, some kids pick up on the conceptual underpinning when taught, and some pick up on it later. When I learned multigit addition (without carrying and borrowing or "regrouping" as it's called today), I didn't understand the explanation of why it worked. But the next year it clicked.

Dan Willingham

3/7/2015 01:53:01 am

Great, we agree. But I would worry about teaching algorithms with the assumption that, given enough practice it will click. That sounds too much like discovery learning: set up the right conditions and assume students will get it. I'd rather rely on teaching it directly.

Barry Garelick

3/7/2015 02:17:11 am

The explanation I had been given the previous year clicked when I saw it again the next year.

Zeev Wurman

3/7/2015 09:32:18 am

I don't think one can ever guarantee that conceptual understanding will come. But if one regularly uses the procedures varied contexts, I find it highly likely that conceptual understanding will come.

That doesn't mean one should not teach or explain the reasoning behind procedures once in a while. But providing heavy duty focus on conceptual understanding (while taking away time from procedures) doesn't seem effective. Much in math is hierarchical and the processes need to be repeatedly use as scaffolding for more complex operations. If If one doesn't have them tap, one will tend to stumble.

SteveH

3/6/2015 03:00:24 pm

"Teach concepts."

Since you don't explain what this is, then you can't say that:

"People can be adept with calculation, yet have poor conceptual understanding."

There are lots of different "concepts", many of which are best learned by doing problems. One might be able to do a proof, but that doesn't mean that there is enough understanding, conceptual or otherwise.

"But Siegler and Lortie-Forgues suggest that the disconnect between computation and understanding is typical."

"Suggest?" "Typical?"

Nope. I don't buy it. Skills can't exist without some level of understanding. That can be fixed. Understanding with few skills is nowhere. The major problem in K-6 is that understanding is driven from the top-down, and surprise, skills don't happen. If you were to talk about a bottom-up, skill mastery driven approach that emphasized larger conceptual ideas, I might like it.

"There are sure to be many methods of helping with conceptual understanding, some best introduced before calculation, some concurrent with it, and some after."

Then I have no clue what you are talking about. What is the specific difference between conceptual and some other kind of understanding, and show some before, during, and after "conceptual" (as opposed to what?) understanding.

Dan Willingham

3/7/2015 01:58:28 am

The national math panel drew the opposite conclusion, namely that american students were okay (at best) on math facts, okay (at best) on algorithms, and near zero on conceptual understanding of mathematics. There may be some data showing students have conceptual understanding of a concept, but can't calculate accurately, but I can't think of one. There are, in contrast a number of studies showing the opposite.
I can't provide a full cognitive explanation between conceptual understanding and computational competence...it's usually intuitive to most that a students might be able to recognize problem types, choose the right algorithm to solve the set problem, execute it accurately, yet not understand why the algorithm provided the answer. Certainly when I taught statistics to undergraduates, I saw plenty of that.

SteveH

3/7/2015 03:35:02 am

" and near zero on conceptual understanding of mathematics"

I see no definition of conceptual (or otherwise) understanding, so much of that talk is meaningless.

"There may be some data showing students have conceptual understanding of a concept, but can't calculate accurately, but I can't think of one. "

There are many different types and levels of understanding, but you make no distinction. However, there is no such thing as mastery of skills with no understanding. The understanding might be inflexible or poorly developed, but that's a different issue. I didn't fully understand algebra until the end of algebra II. That's normal.

You need to examine your definition of "conceptual understanding." "Conceptual" is a high level term, not one that has much to do with the understanding required for low level math skills. The fact that college students use statistics as a black box is a poor comparison to what's going on in K-6 education. The "conceptual" understanding of a chi-square test needed to properly use a black box calculation is not the same as the understanding required to learn K-12 math.

K-6 educators use vague definitions of conceptual understanding, just as you do, to provide cover for full inclusion and low expectations. This is nothing new, My son started Kindergarten 13 years ago with MathLand that pushed the concepts and understanding of explaining in words why 2+2=4. That had been going on for years. Then, in first grade, it changed to Everyday Math that was all about conceptual understanding and "trust the spiral." However, in fifth grade, many bright students still had not mastered the times table. I've seen many cases where students have something that educators call conceptual understanding, but still can't do basic problems. Are you really surprised at the negative push-back on your vague promotion of conceptual understanding? Been there. Done that. It doesn't work. Top-down work on conceptual understanding has been going on for at least 20 years. Where are the results? Where are the skills?

Skills don't flow from conceptual understanding. There is some other understanding that has to happen, and that comes from mastery of basic skills. I can talk concepts to my tutoring students until I'm blue in the face, but they won't really understand it until they do the problems themselves. Concepts are cheap and don't get the job done.

Sarita

3/6/2015 03:37:52 pm

I completely agree with Barry. With "progressive" maths the shift away from any form of rote/memorization and learning of standard algorithms (not even mandated in the Australian National Curriculum) with a preoccupation with strategies that "supposedly" teach conceptual, deeper knowledge has essentially resulted in many students unable to do either procedural or conceptual. Although purely anecdotal I found with my daughters the lack of focus on standard algorithms and teaching times tables meant that they knew for a number of years what it meant to multiply and divide but did not posses the necessary tools to calculate these problems. It wasn't until my daughters did Kumon (ie lots of procedural practice) that they became so much more confident at mathematics and moved from the bottom of the class to the top. So, contrary to the study I noticed that in the doing they arrived at the conceptual. Also, at my daughters primary school the top maths students are all tutored, often at Kumon. Many progressives believe that introducing standard algorithms, times tables to early is damaging to a deeper understanding. This argument has managed to completely disable many students who would probably have succeeded at maths.

Dan Willingham

3/7/2015 01:59:25 am

This blog posting didn't advocate for progressive math.

sarita

3/8/2015 02:03:10 pm

I do understand that you are not arguing in favour of progressive maths. I just wanted to point out, albeit anecdotally, that my kids did seem to gain greater conceptual understanding merely by the doing. If maths students in the US are still not "getting" the conceptual part then I think a fair portion of the blame can be put on programs like New Maths. After all these were the maths programs specifically designed to teach deeper understand, yet they seemed to have failed in that endeavor. Back to the drawing board, then...

SteveH

3/7/2015 12:23:02 am

N1/M1 + N2/M2> N1/M1

Of course, my son would be the one disagreeing because N2 could be a negative number and M2 could be positive.

"Among other ideas, Siegler and Lortie-Forgues suggest that, once they have some competence in calculation, students might compare the results of "

8/7 * 1/2

7/7 *1/2

6/7 * 1/2

If they have "competence", then that would provide all of the understanding needed to answer what I guess are magnitude comparisons of these expressions. Cancel the 1/2, but you have to have the understanding/skill of knowing how to do that for an inequality equation.

I could come up with thousands of these sorts of "understanding" things, most of which are easily answered if one has proper skills. Understanding is not some sort of one-way, direct process. It is a feedback loop based on nightly homework sets. It's up to the teacher to push and manage that loop to keep all individuals up to speed. Understanding derives from skills. Understanding before skills is almost meaningless. Whenever I tutor a student, the main problems are a lack of basic skills and practice. Although I can see that they understand the "concepts", I tell them that they HAVE to do the problems themselves. They always come back with questions. Proper textbook problem sets are designed to move from simple calculations to more complex variations that require more understanding.

The fundamental problems in K-6 are low expectations and a hope that a one-direction, top-down, conceptual understanding process will work. They want to "trust the spiral" and hope that "kids will learn when they are ready". It's all about not ensuring any particular level of skills on a grade-by-grade basis. Talk of conceptual understanding is only a cover for not getting that job done. This is strictly to meet the needs of full inclusion. Fine, go ahead and teach all of the concepts you want beforehand. Does it work? What is 6*7? I think what you will find is that all you are talking about is adding more flexible understanding on TOP of skills - i.e. setting higher expectations. If you are advocating a top-down approach to math as if little understanding derives from mastery of skills, then you are just wrong. That's backwards.

Katharine Beals link

3/7/2015 01:12:39 am

"This latest finding points to the necessity of greater attention to understanding in instruction."

The place to start is the teachers, not the students: if teachers (as Experiment 1 suggests) don't understand the concepts, it's hardly surprising that students don't either.

The place *not* to start is with the type of answer we ask students to produce: verbally explained answers. As far as I know (please correct me if I'm wrong), there is no published, peer-reviewed empirical evidence that this requirement enhances conceptual understanding. And, yet, it is one of the strategies most commonly proposed by the powers that be in the education establishment (cf for example, the Common Core).

Katharine Beals

Dan Willingham

3/7/2015 02:02:46 am

Katharine
I agree, there's not much hope of teachers teaching content they don't understand.

education realist link

3/7/2015 10:32:45 am

My own experience also shows that fluency doesn't lead to conceptual understanding, as I wrote here: https://educationrealist.wordpress.com/2012/10/05/math-fluency/

I've seen many remedial level math students who can't comprehend algebra or any abstract concepts in math who know their math facts cold. I've also seen quite a few students who have strong conceptual understanding but reach for a calculator to do simple math facts--not because they weren't taught, but because they simply can't memorize facts. And of course, I've seen really weak students who can't ever memorize at all. I have personally spent hours with kids who are willing to memorize the multiples of 2 and can't manage it.

I'm not arguing against math fluency. I think prioritizing fluency is useful for mid-level students who are capable of memorizing (which is most of them). I think that we could reduce the cognitive load of algebra and other advanced topics for students who are capable of memorizing facts, and this would be a good thing. But I can understand why elementary teachers might resist insisting on memorization, if 20% of the kids memorize instantly, 60% need a month or so, and 20% never get it. Hard to spend a month or so on math fact memorization.

As a high school teacher, I think most of my students know most of their math facts. They stumble a bit on the confusing ones (6-9 multiplication and addition), but all but say 10% or so are reasonably math fluent. Figure I'm not seeing another 10% because they are so remedial they never get to algebra 2 or higher (most of what I teach these days).

So when we discuss this issue, we should recognize we're talking about moving a chunk of students from "reasonably" to "perfectly" fluent, and maybe picking up another 10% or so from "not fluent" to "reasonably fluent", while another small chunk wouldn't improve no matter what.

It's a minor emphasis change. Worth making. Not worth the fixation.

SteveH

3/7/2015 03:02:04 pm

"My own experience also shows that fluency doesn't lead to conceptual understanding, "

Define "conceptual". How does Everyday Math's "What is the One?" or pie chart conceptual understandings of fractions provide the detailed understandings necessary for manipulating rational expressions?

Skills are a required condition for proper mathematical understanding. Also, one can have mathematical understanding of the tools, but not good understanding of how and when to apply them. Many students can have Algebra I skills and understandings that fall apart with more complex rational expressions. The solution isn't more conceptual understanding. What they need is to understand how basic algebraic identities translate into more complex forms. That is much more than a "concept." I can talk till I blue in the face with the students I tutor, but they will only really understand when they actually can do the problems themselves. Unfortunately, simple ideas of really basic math skills (like the times table) gets equated to the needs of pre-algebra and above. It's not the same.

"I've seen many remedial level math students who can't comprehend algebra or any abstract concepts in math who know their math facts cold. "

Mastery of skills in basic math facts is not the same as the fluency in algebra that leads to understanding, and the level of fluency and understanding necessary for success in Algebra I is not the same as that needed in Algebra II.

A common problem area in middle school are negative signs. In algebra, it get worse. I have a great way to explain that to students. When I do, they really "understand" it. Well, not really. They get confused when they start seeing problem variations, and there are a lot of tricky variations. Full understanding only comes with practice. What good are conceptual ideas when they seem to fall apart when doing real problems. So the only way to test for true understanding is to test them with those variations.

Back when I taught math, it was easy to throw in problem variations that tested their full mathematical understanding. This was not done with words. The proof of understanding is in doing the problems. One might get away with claiming that basic arithmetic is rote, but there is no way to pass higher grade math without skill-based understanding. Understanding has many different levels and is an ongoing process, but K-6 math hurts many kids because curricula do not value mastery of basic skills. Not everything is about the times table or basic arithmetic. Too many see skills only as some sort of extrapolation of memorizing the times table. That isn't the case.

SteveH

3/7/2015 03:02:23 pm

Douglas Hainline

3/7/2015 06:24:02 pm

We seem to be talking past each other here.

May I suggest that we avoid arguing over abstractions like 'conceptual understanding' vs 'rote learning', but rather talk about specific approaches -- tactics, methods, techniques -- for teaching mathematics.

Some posters here seem to believe that Dr Willingham is in the camp of the dreaded 'constructivists'. Not so. Please note that Dr Willingham, in his WHY DO KIDS HATE SCHOOL? has explained, very comprehensively, the importance of 'automaticity' in reducing cognitive load. If you have to work out what the factors of 20 are when solving a quadratic equation, you are putting an additional, unnecessary burden on your short-term memory. So you need to know a load of mathematical facts 'by heart'. I believe that there is also a very strong element of 'pattern recognition' -- related to automaticity in mathematical fluency.

But ... I also believe it DOES make sense to talk about something like 'conceptual understanding', even though we are doing this from a basis of profound ignorance with respect to what's really going on in the brain at the level of neural activity. (We don't yet have the technology is know what's really going on in the brain. fMRI scans are an extremely crude method to investigate the activities of the brain, but they're about all we have now.) So our terminology is necessarily imprecise, because our knowledge of the underlying biological reality is also imprecise.

However, I also think that there is a spectrum between pure 'comprehension', that is, a deep understanding of abstract ideas, on the one hand, and rote-learned 'facts' on the other. It's a false dichotomy to simply contrast these two ways of knowing to each other.

Thus, I have found it useful to get my students to simply know, by heart, the answer to 'How do we use the word 'ratio'? Once they have this down by heart, we can move on to looking at various interesting ratios, such as pi, the trig ratios, etc. This can then be tied to the 'ratio' numbers (misleadingly called 'rationals'). From there we can talk about non-ratio numbers, the algebraic ones like the square root of two, and the non-algebraic ones.If they don't get this sort of systematic teaching, showing how everything falls into place logically, their understanding of mathematics remains just a jumbled-up collection of half-understood words. (Thus, in the UK, for some reason, children are supposed to 'know what a surd is'. Teachers almost invariable just say, 'the square root of two is an example'. But they don't explain why, for example, pi is not a 'surd'.)

I have also found in teaching and tutoring math, that it helps if children learn about the kinds of numbers that there are, starting with the natural numbers and working up to the reals. It helps if they are shown the difference between an identity, and an equation, and between an equation and a function. It helps if they learn some basic mathematical vocabulary, such as the idea of an 'expression', a 'term',, a 'factor', an 'operator', a 'variable' (usefully called, in some parts of the world, a 'pro-number'). And they need to learn about the three laws of arithmetic, especially the distributive law. These should not be learned as dogmas, but shown graphically.

And I try to avoid using the word 'is' -- as in, 'a ratio is' -- but rather say, "We use the word 'ratio' when ..."

I believe that it helps understanding if my students realize that 'factoring' is just 'unmultiplying'. (Just as 'partitioning' is 'unadding'.)

Of course, these are just my subjective observations, based on an extremely limited and non-random sample. When I say 'I have found that', I really mean, 'I believe ...'.

Here is a concrete example: to be proficient in algebra -- or at least, to be able to do well on the sort of examinations we give kids in the UK -- my students have to recognize certain expressions as differences of two squares, perfect squares, or neither.

So you need to know the 'difference of two squares' identity by heart. You also need to practice recognizing it, from both sides, including when it is expressed as the difference of two constants squared. You also need to know that the 'X' in the X-squared term and the 'Y' in the Y-squared term, refer to any expression, not just to a single variable. You also should know that, although you will probably only see perfect squares, (and you need to be able to recognize square numbers at sight, ideally up to twenty squared), that we could also express this identify as simply x minus y, and the product of the sum and differences of the square roots of the x and the y. You need to see WHY the identity is true, and why, say, the sum of two squares, doesn't have an equivalent identity.

All this is done via talking and explaining, didactic give and take, visual presentations where possible, and doing a lot of problems. I have a worksheet which includes several dozen expressions which my students have to work through, r

SteveH

3/8/2015 12:19:36 am

"we avoid arguing over abstractions like 'conceptual understanding' vs 'rote learning'"

I assume that your comments are directed to Dr. Willingham. I was the one who pointed out the non-definition and confusion.

"but rather talk about specific approaches -- tactics, methods, techniques -- for teaching mathematics"

Then this is a new direction for this thread, but first you have to define the problems.

"Some posters here seem to believe that Dr Willingham is in the camp of the dreaded 'constructivists'."

No, most of us have been following Dr. Willingham's comments for years.

"I also believe it DOES make sense to talk about something like 'conceptual understanding',"

Concept - "an abstract idea; a general notion"

You really have to define this to make the conversation meaningful. I see it only, at best, as motivation for learning new material or understanding some overall ideas, as in a pie chart for fractions. It would not be a form of understanding that would go very far past that conceptual level of problem.

"...even though we are doing this from a basis of profound ignorance with respect to what's really going on in the brain at the level of neural activity."

I call this Brain Research Misdirection. It's not that difficult.

"So our terminology is necessarily imprecise, because our knowledge of the underlying biological reality is also imprecise."

We don't need to know what is going on in the brain to define terms well enough so that people don't talk past each other.

"However, I also think that there is a spectrum between pure 'comprehension', that is, a deep understanding of abstract ideas, on the one hand, and rote-learned 'facts' on the other. It's a false dichotomy to simply contrast these two ways of knowing to each other."

This always makes me think of a take off of the old commercial: Mentholatum, Deeeeep Thinking.

No. We can be much more precise about what understanding is required and how to get it. It is the understanding that is shown by being able to do math. What understanding is possible if you can't do the problems? If you can't define what understanding is, then you end up with only a vague sense and fall victim to the trap of thinking that math breaks into rote operations and conceptual understanding. But then you talk about "deep understanding." That sounds a lot different than "conceptual" understanding. How can you fix a problem when you really can't define what it is?

"Thus, I have found it useful to get my students to simply know, by heart, the answer to 'How do we use the word 'ratio'?"

How does this fit into the long term goals of a specific curriculum? Don't you think that proper textbooks already provide decent explanations of ratio? My son's old, and decent, Glencoe Pre-Algebra book explains it in many different ways. The homework sets start with simple problems that work their way up to ones that show more understanding.

Even if you don't know what is going on in the brain, you can define a specific curriculum that deals the vagaries of how kids think, or perhaps, not. You have to have a feedback loop and push. You may not know exactly what's going on in the brain, but we can define what success means.

So, what is/are the problems? In complex systems, you can't have vague definitions and then use some sort of top-down guess and check method for fixing it. There is no "it". This just leads to solutions that are relative because you can't separate the variables. You have to start with some specific thing that is wrong, define it well, and work backwards. Some might claim that this is an anecdotal approach, but solving specific problems rarely apply to only one student.

When my son was in fifth grade, parents (and teachers) noticed that many bright kids didn't know the times table. This could be an IQ issue, bad curricula, teacher incompetence, student laziness, or something else. However, the curriculum being used was Everyday Math that specifically tells the teachers to "trust the spiral." These kids are quite able to handle the times table (as proven by the end of that year when the teacher actually focused on the problem). This was clearly a problem with the curriculum. Some kids like my son didn't have a problem because we did the work at home. This clearly points to a fundamental flaw in K-6 math education. I would look at the EM workbooks and see decent math problems and explanations, but it was "drive-by" math. It wasn't a scaffolded spiral based on mastery of skills. It was more like circling around in a repeated partial learning approach. This is done specifically to support schools that use full inclusion and social promotion. Talk of conceptual understanding is simply Brain Research Misdirection used to hide really low expectations.

All of these "concept based" curricula are not new. My son had MathLand in first grade 12 years ago, and since then, they have used Everyday Math. Where are the results? Are you talking about new and improved concep

SteveH

3/8/2015 01:07:28 am

...

Are you talking about new and improved concepts that must be taught?

Mathematical understanding of skills versus conceptual understanding versus understandings of how to use math tools can be effectively defined to explain what is wrong and what should be fixed, but there are also more practical problems, like full inclusion and the ineffectiveness of differentiated instruction. Tracking gets hidden at home, but educators don't see (want to know about) that variable.

Douglas Hainline

3/7/2015 06:28:30 pm

I have a worksheet which includes several dozen expressions which my students have to work through, recognizing each expression as the difference of two squares, perfect squares, or neither. The ability to quickly recognize an expression as a perfect square is critical, including knowing that a variable raised to an even power is the square of that variable raised to half that power. (X^12 is X^6, squared.)

There is plenty of room for 'tricks' to help rote memorization of routines, such as recognizing when a three-term quadratic expression is a perfect square. Is 4X^2 -12X + 9 a perfect square? Is X^2 - 16X + 36 a perfect square? Yes in the first case, no in the second, and a quick way to decide is to apply the jingle, "rooty-toot-toot, twice the square roots" to the coefficient of the middle term.

They also have to appreciate WHY we solve most quadratics differently to the way we solve linear equations. This means understanding the special nature of zero. So I start off my explanation of how to solve quadratics by first showing how the solution methods that worked for linear equations, do not work for most quadratics. (Of course if there is only a quadratic unknown, as in 3X^2 +1 = 49, we can use the methods that we used for linear equations.)

I do this by giving my students an arithmetic problem to solve: 675 x 445 x 86 x 0 x 6453 x 231 = ? When they 'get it', I then ask them to solve this algebra problem: 765 x 45 x 213 x 'X' x 43 x 3456 = 0 ... what does 'X' HAVE to be? Finally, if the above equation becomes 765 x 45 x 213 x (X - 3) x 43 x 3456 = 0, again, what does 'X' HAVE to be? THEN we can understand why we try to factor a quadratic equation if we can, to find a solution. (And if we cannot factor it, how by completing the square, we can get a non-integer solution, if one exists.)

In summary: don't argue about 'Direct Instruction' vs 'Consructivism'. Argue about what day to day practices are best for learning mathematics.

Douglas Hainline

3/8/2015 03:49:52 am

I'm an American, but I live in England. I can testify that mathematics and science education here, at the school level, is .... far from perfect.

Kids -- at least the ones I deal with -- are taught isolated techniques, well or poorly, but very little of the mathematical background to them. They don't know the difference among identities, equations, and functions. They don't know the way in which a sine is like pi, or even, many of them, what pi is. ("Uh ... about 3.14?") They don't know WHY you add exponents when multiplying different powers of the same base, or why X to the zero power is 1. They might memorize the latter fact, but they don't know why.

Until the last Labour government made some changes in the direction of common sense, many of them didn't know their times tables. This includes students from private schools. This has improved somewhat, but not nearly enough.

My students learn a lot of things by heart, including the cubes to 12, definitions, tricks for factorizing, whether exponentiation distributes over addition, and so on. We do lots of flashcard rapid-response drills.

But at the end of the day, I want them to be able to attack non-bookwork problems, like "How many zeroes are there in the numbers from 1 to 100 000?" ."How many times can I divide 6 evenly into 69!"? To be confident in working on problems like this, they need to have a good store of rapid-recall math facts. But something else is also needed, whatever we call it.

I find most of the arguments about mathematics education vacuous. We need to talk about specifics, concrete methods, what you do on Tuesday afternoon.

J.D. Salinger

3/8/2015 12:45:21 pm

The study does not provide details of how each group solved the A/B * C/D > A/B problem. The objective for those questions is apparently to test whether or not subjects "Understand" that the truth or falsity of the statement A/B * C/D > A/B (for positive quantities) depends solely on the magnitude of C/D, relative to 1.

Were the problem given as 5/6 * 2/3 > 5/6, the fact that multiplication by a fraction results in a value less than 5/6 might have been obvious without calculating, but as the report states, using such simple fractions would result in people actually calculating the answer. As an adult who majored in math and who has gotten away from much arithmetic thinking I took an algebraic approach and divided both sides by 31/56 resulting in 17/42 < 1 which is clearly false.

I got through much of the paper without observing that the point was to test the arithmetic understanding of directionality of products. The testers wanted candidates to see that A/B appears on both sides, and use an arithmetic understanding to draw a conclusion.

There are too many problems with this as a measure of understanding, isolated from skill. For one, there will be students, particularly in the math/science group who DO see the common factor right away, and simply cancel it as I did Is that "understanding" or "procedure"? Yes, "understanding" is involved in deciding if it's true or false. What is really being measured?

The premise is that subjects will pick up on the common factor on the two sides of the inequalities and invoke arithmetic thinking. But the evidence on hand is that well-informed, skilled and "understanding" adults--particularly those versed in algebraic thinking--can easily miss that and make a leap to other attacks on the problem instead.

J.D. Salinger

3/8/2015 12:46:54 pm

Oops, typo in my second paragraph. I meant to say "17/42 > 1 which is clearly false" in last sentence.

Douglas Hainline

3/8/2015 02:41:11 pm

I think Mr Salinger has raised the roofbeams here, if I understand him right.

I teach elementary algebra. One of the things I teach is the 'difference of two squares' identity. But it's perfectly possible for a student to 'know' this identity, and to be able to use it to solve a problem where it is presented explicitly, but yet fail to recognize it when it appears in slightly altered form ... say, with the negative term first.

The classic case is to fail to see that you need to use this identity to solve the following problem:

What is 1000000000^2 - 999999999^2 ?

Even students who 'know' the difference of two squares identity are likely to reach for their calculators when they see this problem for the first time.

So you have to give students practice in looking for this underlying structure (and others too). That means showing them idenity in several forms, and also showing them near-examples which are not really examples.

Are they then getting a 'deeper' understanding of algebra? Or just practice in exam technique? Or just becoming better at pattern-recognition? I don't know, and unless we can really define such concepts explicitly, I think we risk getting tangled up in semantic knots.

I think what I am teaching them is useful, but I know that there is a whole aspect of mathematical thinking that we never touch, perhaps because it is not examinable: so I would like to get them thinking, spontaneously, about things like, if it's useful to define ratios of the sides of a right-angle triangle on a circle, might it be useful to do the same on some other figure, such as a hyperbola?

Put another way, I would like to go some way in addressing the concerns raised in Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

SteveH

3/8/2015 03:36:39 pm

"Put another way, I would like to go some way in addressing the concerns raised in Lockhart's Lament: "

Ugh, not Lockhart's superficial analysis again. I've seen this link for years. If you talk about basic competence (teacher, student, curriculum?) problems with things like the times table or basic skills that are actually covered in their textbooks, then trying to fix them from the top-down using concepts or engagement (a big thing around here) is not directly tackling the problem. It won't work.

"What is 1000000000^2 - 999999999^2 ?"

I could come up with hundreds or thousands of these things. They are the bread and butter of math competitions, but that's not how you learn math. Many of these are tricks you have to "see." That is neither a necessary or sufficient condition for "understanding" math. My son did the AMC competitions each year so I'm quite familiar with these questions. They are also common on the SAT test, which some claim is really not a math test, but one that checks to see if you can spot the key shortcut or hidden radius. A classic one is to give an expression and ask whether the result is odd or even. Any number of these questions is not enough to show proper development in math. They do not define a curriculum. However, they do make fun or interesting after-school math team questions.

However, those who are very good on the AMC test all start with the basic traditional math curriculum sequence that is based on mastery of nightly homework sets. The major problem I saw in math was that my students had very poor curricula in K-8 that did not value the importance of nightly homework. They would be allowed to go through the motions and not properly graded or tested.

Dan Willingham

3/9/2015 04:16:43 am

This has been a lively, wide-ranging discussion, much of which I can't contribute to, although I've been following it. I'll add just a few observations on the core point the article made, and which prompted me to blog about it.
This is one more (not the first) article that highlights a dissociation in student ability. Students are fairly competent in using a mathematical construct in one context, but are stumped when confronted by using it in another context. Sometimes the other context is a different problem type, sometimes it's a request for an explanation of why one would carry out a procedure.
Inflexibility is characteristic of skills in the early stages. What's odd about math is that many kids develop flexibility only slowly, if at all. For example, many late elementary children can't give a reasonable answer to the query "what does this "=" symbol mean?" even though they’ve been using it correctly for years. Researchers use the term "conceptual" for knowledge supporting this behavior not only in math but other knowledge domains. Conceptual knowledge is sometimes differentiated by content (in contrast to procedures), sometimes by the richness of interconnections, sometimes both.
SteveH you seem to think that the idea of "conceptual knowledge" is not useful. (You're in good company...B. F. Skinner used to get quite irritated by talk of "understanding" that was separated from “solving problems.") I really have no argument with that. I think the goals of nearly everyone in math is the same (which puts math in sharp contrast to ELA, history, et al.) Striving for "conceptual understanding" is, in my book, a shorthand for flexibility and competence and it’s a shorthand most people are comfortable with. So if by "conceptual," you're satisfied with "can deploy skills flexibly," we're on the same page. (With my cognitive psychologist hat on, I’m interested in the mental apparatus underlying such competence and couldn’t do without specification of different types of knowledge.)
A separate question is "how to get kids there?" Siegler and Lortie-Forgues just touch on this toward the end of their article and I gave one of their examples. I think their larger point was this: the fact that children can be competent in executing standard problems, yet show inflexibilty indicates that simply doing a lot of problems doesn’t produce flexibility for some (perhaps most) kids. In the comments some seem to suggest that practice is the key to flexibility. If the suggestion is that “if students REALLY knew the basics, they’d have the flexibility,” I’d say “some will get to spontaneously, some won’t.” As I said earlier in this thread, this suggestion smells suspiciously like discovery learning. In general, you get what you teach. Teaching one thing really well in the hopes that it will yield something else doesn’t work.
I think most of the commenters in this thread meant that practice will yield flexibility if it is varied and if carefully selected problems used. (This suggestion is contrasted with the objection that much of what has been proffered as boosting conceptual understanding has been useless and has soaked up time that could have been better spent. I agree.) So here’s the real challenge: if you believe that the Siegler & Lortie-Forgues results are typical, and that it’s a consequence of a LACK of this sort of effective practice, what can be done to make it more common?

Sarita

3/9/2015 02:43:08 pm

I'd wouldn't like to think my post was advocating for discovery maths. The problem in Aust., as in the US, is that the emphasis is on conceptual while procedural is somehow seen as the "poor cousin". I believe that concepts should be explicitly taught, however, in early - mid primary do we really need to teach conceptual for several years before we equip our students with the skills they need to execute? At this level the concepts are fairly basic and, in my view, should not be given the enormous amounts of time that they are. In addition, the concepts are taught using different strategies that are convoluted and confusing for many students hinder their ability to "understand" sending them backwards rather than propelling them forward. In Aust. students are not expected to know their 10 * 10 tables until the end of year 4! This means they will be working on the concepts, using "strange" strategies for years before they are actually able to to complete a sum. So in my view strategies don't work very well. And I agree with Steve, I dislike the spiral. Far preferable would be a mastery program. Harping on with conceptual understanding can impede progress. It must be taught but it has taken over a large chunk of the Aust. class room. In the mean time standard algorithms are not required by the Aust. National curriculum. When I explained that my kids learnt by "doing" the procedural" I meant that the constant practice was more beneficial than spending another year talking about what multiplication was, which was retarding their growth. With so much to learn to much class room time has be taken up with strategies that don't seem to be successful in teaching concepts anyway.

SteveH

3/9/2015 07:47:18 am

"you seem to think that the idea of "conceptual knowledge" is not useful. "

I didn't say or imply that in any way.

"Researchers use the term "conceptual" for knowledge supporting this behavior not only in math but other knowledge domains."

Your "this behavior" example could be for many reasons. It's not a clear definition of "conceptual."

"(You're in good company...B. F. Skinner used to get quite irritated by talk of "understanding" that was separated from “solving problems.") "

I almost agree with this. What understanding could possibly be useful if you can't actually use it to solve problems?

There are different types of understanding and they can be more clearly defined for math. Conceptual is a general idea that might be useful at a high level, like the number line or pie chart for simple fractions. However, this kind of understanding, or lack thereof, does not define what is wrong in math, especially K-6 math. That kind of understanding will leave you high and dry until you work your way through many problem sets to really understand what is going on.

"I think the goals of nearly everyone in math is the same.."

No they aren't. CCSS specifically does not expect a STEM level teaching of math in K-6, no matter how much they talk about conceptual understanding and problem solving. This is the environment our kids live in. We parents ensure mastery (and understanding) at home. Schools just "trust the spiral" and talk about concepts and critical thinking.

"Striving for "conceptual understanding" is, in my book, a shorthand for flexibility and competence and it’s a shorthand most people are comfortable with."

Few agree with what this means in terms of the specific type and level of mastery of math skills. Most in the K-6 world use talk of conceptual understanding and critical thinking to promote student-centered, mixed ability, top down, development of skills. It hasn't gotten the job done and these curricula have been in use for at least 20 years. It's a cover for low expectations.

"So if by "conceptual," you're satisfied with "can deploy skills flexibly," we're on the same page."

You can't use the word "conceptual" that way. It does not apply to how K-6 pedagogues use the word. A key to all of this is whether the understanding, whatever it's called, is driven from the top down or the bottom up. Students taught from the bottom up via skills might show inflexibility and limited (or age appropriate) understandings (we don't expect kids to multiply in octal), and those can be fixed, but a top down approach via "conceptual" understandings assumes that skills derive from general ideas of what is going on. They do not. Skills are never rote once you get past the times table and basic arithmetic, although those are the classic pedantic justifications for discovery.

"...yet show inflexibilty indicates that simply doing a lot of problems doesn’t produce flexibility for some (perhaps most) kids."

I can't tell you how wrong this is. Proper math development consists of a feedback loop based on doing lots of nightly homework sets and review in class. If it isn't this, then high schools and colleges are doing it all wrong. They aren't. It's K-6 math that's screwed up.

"If the suggestion is that “if students REALLY knew the basics, they’d have the flexibility,” I’d say “some will get to spontaneously, some won’t.” As I said earlier in this thread, this suggestion smells suspiciously like discovery learning."

This is not even wrong. It's backwards. Those valuing discovery care little for skills, and skills are not either "spontaneous" or not. I've never seen that sort of effect.

"if you believe that the Siegler & Lortie-Forgues results are typical,..."

No. The results of this sort of limited test would not tell me anything.

N1/M1 + N2/M2> N1/M1

Competition math is full of these things, some more tricky than others. Math is not about "seeing" things. It's about skills that you can apply consistently and knowing when to apply them. There is the understanding necessary to master the tools and there is the understanding of how to mix and match the tools to solve many problems. If you add in "conceptual", that can only refer to basic, high level ideas, not the detailed understanding that only comes from mastering problem sets day after day and year after year.

K-6 math has abdicated this role and no amount of talk of concepts and discovery will fix it.

David Wees link

3/9/2015 08:17:48 am

Hey Steve H,

A link to some evidence justifying your position would be great, since Dan has at least attempted to provide that. Otherwise your claims are just words.

David

SteveH

3/9/2015 02:20:37 pm

I always love it when I get "Prove it" arguments after I question ("at least attempted") grand leaps of argument and redefinition. Go ahead and redefine conceptual understanding to mean something that is good by definition, whatever that is. Then go ahead and tell others to meet some much larger level of proof. K-8 educational pedagogues do that to kids and parents all of the time.

"...yet show inflexibilty indicates that simply doing a lot of problems doesn’t produce flexibility for some (perhaps most) kids."

"Perhaps most"? Really? No more problem sets. Whoopee!

How about the argument of common sense.

SteveH

3/10/2015 01:55:47 am

When you get past the times table and basic arithmetic, which is never just rote (divide 23 into 163), things change. In algebra, everything is justified with basic identities. These are very simple conceptually, like a/1 = a, a/a = 1, and a+b = b+a. Few have difficulty (conceptually) understanding these identities. However, one's understanding gets challenged when the equations get more complicated than 2X+3=5. I distinctly remember in 8th grade algebra (which I got into with absolutely no help at home) that we had to do only one step for each change in an equation and we had to justify it with an identity. We students would get annoyed and want to do multiple steps at once, but the goal was not only to ensure understanding (at more than a conceptual level), but to enforce a methodical and detailed thinking process.

Then comes Algebra II, which many consider to be the most difficult high school math class. It's not so much that students didn't understand what these identites meant, but they didn't have enough practice expanding their use with more difficult rational equations. One might be able to do a proof, but not have the understanbding to use it as a tool to solve many problems.

When I taught college algebra students, I would have them start by circling all of the terms of an equation. Next, I would have them convert each term into a rational expression by dividing it by 1 unless it was already a rational expression. Then I would have them circle all of the factors. Then I would have them give each factor an exponent - one, if it didn't already have an exponent. I would tell them that these are the "a's" and "b's" of the basic identities. I would tell them that they can move any factor up or down to or from the numerator and denominator just by changing the sign of the exponent. [1/a = a^(-1)] I would explain how the basic identies work for more complicated equations.

Too many students think that they have to get rid of negative exponents. Too many students think that a simplified form of an equation is the only right one. Order of operations is drilled into kids' heads these days, but that really is wrong. Math equations are 2D and graphical, not linear text. It's not about any particular sequence of solving, but what is allowed by the identities and what is not.

Call this conceptual understanding if you want, but still, students won't really understand until they practice, practice, practice. You can try, but there is no way to teach or discover enough understanding to be able to solve all of the problems in a homework set. As I always say, I discovered so many things on my own doing homework. However, modern pedagogues seem to think that discovery is only an in-class, group thing. That's because it's used as justification for having the teacher as the guide on the side. They really don't care about discovery because they never check to see if it worked. There is no feedback loop. Group discovery is not the same as individual discovery.

Forget the methodical skills you learn by doing lots of problems. Practice is NOT just about speed or accuracy. It's about really, truly understanding the implications of those concepts and even proofs.

SteveH

3/10/2015 03:52:23 am

Then there are word problems which require different skills and understandings. This is not some sort of Polya-like process of thinking, dawing pictures, or working backwards. There are specific skills that have to be learned - ones that will grow with students to help them in the future, say for solving problems with Bernoulli's equation. Incredibly, some K-8 pedagogues seem to think that the best way to learn basic skills is with word problems, as if that provides more understanding. No. That just proves the need for parental school choice.

You can start with simple equations like this before algebra:

? = 2*50
? = 50*2
100 = ?*50
100 = 2*?

THEN, you can add words like "If a car is traveling at 50 miles per hour, how far will it go it two hours." Then, "If you traveled 100 miles and the car is going at 50 miles per hour, how long did it take. You have to have word problem statements that cover each variation. Still, students won't fully understand until they do the problems. How do you validate understandings without problems? With words? No. All of the students I tutored could show me understanding with words. Then when they did the problems themselves, they would get confused. So much understanding is buried in the details. Words fail. Only results show understanding. Understanding without results is meaningless. Skills with inflexible understandings can be fixed, but not understanding with few skills.

Beyond K-6, many word problems center around a governing equations like D=RT, work, and mixture. A student has to develop the skills to identify the governing equation and learn how to translate the words into the values that fit that equation. Not all problems are like this, but this is the bread and butter of math. Students need the skill to be able to identify when a governing equation, like D=RT, needs to be applied. If they had the sort of preparation I showed above, they are then ready to deal with DRT word problems where each of the variables is missing.

But it's more than that. There are so many variations and the more complex ones form the basis of many math competition problems. One does not get good at these competitions via conceptual understandings. One gets good by doing lots of different examples. A classic DRT variation is one where you travel to and from a location 100 miles away. On the trip out, you travel at 40 miles per hour, and on the way back, you travel at 60 miles per hour. How long did the trip take? This is the classic average rate trap. One cannot pre-understand all problem variations.

Then there is the understanding provided by units and dimensional analysis. Let the math skills give you the understanding, as I like to say. Many times you can create a proper equation just by making sure that the units work out. Students have to understand that units behave just like factors, but they have to do the problems to really understand what that means.

There are so many things to learn and understand, and a direct understanding --> skills process does not exist. It has been and will always be a feedback loop, and hopefully, the teacher is more than a potted plant-on-the-side hoping that engagement and grit will get the job done. There have been many times when I looked into the faces of my students and realized that my explanation didn't work, but I was able to change tacks immediately. I was not a guide on the side where some groups are wasting their time or cooling their heels waiting to ask me questions. Struggle is neither a necessary or sufficient condition. Students struggle enough as it is without specifically including it in the curriculum. Struggle is not a sign of better understanding.

There are different types and levels of understanding specifically for math. There are the conceptual understandings that motivate and provide a framework, the skill-based detailed understandings, and there are the understandings of how to mix and match skills to solve word or other problems. One does not need a proof understanding in K-6. One only needs to be able to do all variations of age-appropriate (and not below STEM-level) problems.

Zeev Wurman

3/10/2015 06:10:23 pm

I would like to go back to Dan's attempt to summarize.

"So if by "conceptual," you're satisfied with "can deploy skills flexibly," we're on the same page."

And then:

"children can be competent in executing standard problems, yet show inflexibilty indicates that simply doing a lot of problems doesn’t produce flexibility for some (perhaps most) kids."

In K-12 mathematics, like in most anything K-12, we do not attempt to cultivate mathematicians, scientists, or even engineers. We attempt to cultivate young people that have good basic understanding of what a professional would simply call "routine problems." Those, in turn, can serve as a stepping stone to cultivating professionals in college, some of whom -- rather few, in fact -- will actually ever try to solve a "novel" or "original" problem.

To me, the implication is obvious. Flexibility with application is easily developed by systematic demonstrations and practice with of solutions to many/most generic types of problems. Steve H mentioned a few. They include D=V x t (or V = R x I for electricity), mixture problems, work problems, exponential growth and decay, distance under constant acceleration, solution through decomposition to partial fractions, solutions through (integer?) factorization, etc. While the list is not tiny, it is still quite limited and can be easily learned over 4-6 years of middle and high school.

Now, suddenly, kids "can deploy skills flexibly." Certainly to all practical purposes -- they will solve anything you are bound to throw at them at this level. And we know how to develop this "flexibility" (I'd simply call it "knowledge") without any need for discovery learning.

Ahh, some will say. That's not a TRUE flexibility. Perhaps. But those kids will find getting a bachelor's degree in anything, from sociology to STEM, rather easy. And some of them will develop -- now that they do have a broad and solid basis -- a true flexibility in college allowing them to attack truly novel problems. But these are -- and will be, perhaps forever -- only 0.1% of the students. And meanwhile we've solved the 99.9% problem too.

education realist link

3/18/2015 05:41:51 pm

I know I'm late to this, but:

" Flexibility with application is easily developed by systematic demonstrations and practice with of solutions to many/most generic types of problems. Steve H mentioned a few. They include D=V x t (or V = R x I for electricity), mixture problems, work problems, exponential growth and decay, distance under constant acceleration, solution through decomposition to partial fractions, solutions through (integer?) factorization, etc. While the list is not tiny, it is still quite limited and can be easily learned over 4-6 years of middle and high school."

Is absolutely untrue. It's not even close to true.

I would be surprised if anyone posting here has a broader and more varied experience base in teaching math than I do, and I'm saying categorically that we can't teach anywhere near half the kids the "generic types of problems" in algebra. It may possibly be true if applied to a small fraction of the overall high school population. 10%. Maybe 20%. The argument is how small the group is.

But we are tasked to teach *everyone* algebra, and as I recall, Ze'ev promotes this idea. And nowhere near 50% of the population can learn what he proposes. I would guess 20% is optimistic.

Of course, he's just at the other end of the real problem, which is we persist in pretending that our educational goals are realistic.

Zeev Wurman

3/18/2015 07:05:51 pm

You are asking us to believe your -- anonymous -- expertise and agree with you that only a small fraction of high school students -- no higher than 20% -- are capable of being taught to competently solve those classes of problems.

Here is a text by Kunihiko Kodaira, a rather famous Japanese mathematician and mathematics educator, from the 1992 translation of his Japanese grades 7-12 mathematics curriculum series.
------------------
"The first nine grades are compulsory, and enrollment is now 99.99%. According to 1990 statistics, 95.1% of age-group children are enrolled in upper secondary school, and the dropout rate is 2.2%. In terms of achievement, a typical Japanese student graduates from secondary school with roughly four more years of education than an average American high school graduate. The level of mathematics training achieved by Japanese students can be inferred from the following data:

Japanese Grade 7 Mathematics (New Mathematics 1) explores integers, positive and negative numbers, letters and expressions, equations, functions and proportions, plane figures, and figures in space. Chapter headings in Japanese Grade 8 Mathematics include calculating expressions, inequalities, systems of equations, linear functions, parallel lines and congruent figures, parallelograms, similar figures, and organizing data. Japanese Grade 9 Mathematics covers square roots, polynomials, quadratic equations, functions, circles, figures and measurement, and probability and statistics. The material in these three grades (lower secondary school) is compulsory for all students.

The textbook Japanese Grade 10 Mathematics covers material that is compulsory. This course, which is completed by over 97%of all Japanese students, is taught four hours per week and comprises algebra (including quadratic functions, equations, and inequalities), trigonometric functions, and coordinate geometry.

Japanese Grade 11 General Mathematics is intended for the easier of the electives offered in that grade and is taken by about 40% of the students. It covers probability and statistics, vectors, exponential, logarithmic, and trigonometric functions, and differentiation and integration in an informal presentation.

The other 60% of students in grade 11 concurrently take two more extensive courses using the texts Japanese Grade 11 Algebra and Geometry and Japanese Grade 11 Basic Analysis. The first consists of fuller treatments of plane and solid coordinate geometry, vectors, and matrices. The second includes a more thorough treatment of trigonometry and an informal but quite extensive introduction to differential and integral calculus."
----------------

The content described by Kodaira easily covers most or all of the problem classes I describe. Using his numbers, about 93% of the Japanese cohort masters this content reasonably well. Indeed, their TIMSS results attest to that.

So unless you believe that the Japanese are genetically superior to us, it is much more than 20% that can master this content with proper instruction.

Douglas Hainline

3/18/2015 11:29:09 pm

The problem with trying to analyze and improve education is that there are so many variables.

It may be that there is an uneven distribution of innate cognitive ability among the various tribes of mankind. Maybe the Japanese and Chinese are just naturally smarter than Europeans or other tribes. We'll probably know for sure in another couple of decades, despite the efforts of the Thought Police to make this question impossible to ask.

But ... even if this is the case, there are probably other variables which affect how well young people can learn mathematics, variables which we can do something about. Note that in Japan, Shanghai, SIngapore, their culture, and their methods of instruction, the way their teachers are trained, the way instruction is carried out in the schools, differ a lot from the United States.

Which doesn't mean that if we identify these factors, we can translate them to the US. If in one culture, high academic achievement is admired, and in another, the high achievers are ridiculed as 'geeks' or put down for 'acting white', there may be little we can do about it.

However, there are probably other things that can be transferred.

Or perhaps we already have some home-grown examples of successful practice. Please look at the following half-hour video (warning: poor sound quality at the beginning -- just persevere.).:
https://www.youtube.com/watch?v=j9SjFsimywA

Of course, as the speaker says, 'experimental programs' are not proof of anything. (A private side note here: even a certain 'data-cooker' is probably, in person, an inspiring teacher, and could get good results from her students, in spite of and not because of her wrong ideas about how to teach math.)

But surely Mr Engelmann's methods should have been thoroughly investigated by educational researchers? And yet they haven't been. When a large-scale experiment in the early 70s ('Project Follow-Through) indicated that his approach, as opposed to others, was successful ... the results were ignored. (More on Project Follow Through here: https://en.wikipedia.org/wiki/Project_Follow_Through and here:
http://www.zigsite.com/for_readers_not-familar_with_project_follow_through.html

And what makes me uneasy about Educational Realist's views on the uneducability (in serious math) of 80% of American children is that they dovetail with and reinforce the faddish nonsense of the PC educational establishiment, when we may have, decades ago, worked out instructional methods that could make a real difference for the 80%.

SteveH

3/19/2015 12:45:38 am

"I would be surprised if anyone posting here has a broader and more varied experience base in teaching math than I do, and I'm saying categorically that we can't teach anywhere near half the kids the "generic types of problems" in algebra."

You've been using this "argument from authority" for ages with no proof of your expertise and only a vague calibration based on the fact that you've tried really, really hard and it doesn't work. You even once claimed that teachers don't matter. You dismiss clear examples of systemic problems in K-6 because you are a one-variable person - IQ. You add nothing to these discussions except for a soapbox.

There is an easy solution. More parental and school choice. I would then absolve you of your real complaint - "But we are tasked to teach *everyone* algebra", which is really what you want - and leave you to the reap the results of your teaching skills.

"10%. Maybe 20%."

That can be the motto of the school where you teach.

SteveH

3/19/2015 01:14:59 am

"The problem with trying to analyze and improve education is that there are so many variables."

In this thread, I've given a process for separating the variables. It is done from the bottom up, not with top-down studies. I've also given very specific examples. I've also seen it with my "math brain" son who needed help with math at home in K-6 with basic skills. I saw nightly low expectations with Everyday Math. His schools sent home notes telling us parents to work on "math facts." I saw bright kids who didn't know the times table in fifth grade get fixed in short order once the school actually paid attention to the problem rather than just "trust the spiral" which is what EM tells the teachers. This is not necessarily a "brain" issue, but one of pedagogy, competence, and low expectations. There is the problem that curricula based on the PARCC test sets as their highest level ("distinguished") the goal of passing a course in college algebra, and this low expectation starts in Kindergarten. This is what ALL kids get no matter what their IQ.

Schools could ask exactly what we parents of successful students did at home. Kumon is not doing well because parents are stupid. Even the research pointed to in this thread was uninterested in finding out why their good math student sample ended up that way. I find that amazing.

Douglas Hainline

3/11/2015 10:02:53 pm

I don't know about anyone else here, but I find this thread to be (1) very interesting, and (2) very frustrating.

We don't seem to have two counterposed approaches to mathematics education being debated, where the result of argument could at least be more clarity if not consensus.

To put it concretely: if we had videos of one-hour algebra classes taught by each of the posters here, could we link classes to posters?

SteveH

3/12/2015 02:39:05 am

This is not about how any one particular class works. This is about the whole curriculum process and how it works over years.

Everyone wants understanding. Everyone has always tried to teach it in spite of the fact that modern ed school pedagogues like to claim otherwise. I wasn't taught two+ digit subtraction with borrowing out of the blue. I was taught to count up and to look at what was a reasonable answer or not. We knew exactly why borrowing worked. Really. Long division requires number sense. How many 23s go into 163? We had to do that in our heads and virtually everyone learned to start from the left: "How many 20s go into 160? Then you worked up or down to find the number that didn't go over. That was done as n*20 + n*3. It is extraordinarily frustrating for many to hear talk that these skills are mostly rote. Besides, when you get to algebra and word problems, any sniff of roteness disappears. Solving equations is not an algorithm.

So what is the main or general problem?

Results on tests are bad. In other words, kids can't DO the problems (rote or otherwise and with or without conceptual understanding), and many of the problems are very, very simple. When I look at NAEP fourth grade results, I'm astounded. This is almost a competence issue, not one of conceptual understanding or critical thinking. Is it the competence of the child, the teacher, the curriculum, or a combination of the three? You have to separate the variables and look at specific cases. This can't be a top-down guess and check relative analysis. You have to find individual, clearly-defined problems you can analyze. You have to do this over and over and over. This is like debugging a program. You can't use guess and check. You have to find a specific wrong number, understand it, and then work backwards to the source. It probably won't fix all problems, but more often than not, it solves many or it points to issues with educational assumptions.

In my son's fifth grade Everyday Math class, many bright kids didn't know the times table and some still used their fingers for adding 7+8. That was a specific and concrete problem. Are these kids just not that bright? No. When the fifth grade teacher began to focus specifically on and ensure skills, they got done quickly. Why wasn't this done before? The main reason is that Everyday Math didn't require it. It's a spiral curriculum where repeated exposure to the material is supposed to get the job done naturally. The purpose of this is to allow for full inclusion classrooms in K-6. My "math brain" son went through this curriculum for five years and I saw exactly how it worked (or didn't). It's what I call a "go through the motions" curriculum. It wasn't a spiral curriculum based on mastery of scaffolded skills and learning, it was more like repeated partial learning. There were maybe 6 simple homework problems each night before the curriculum moved on to another topic. I call it a curriculum for nervous chipmunks. Clearly, its main purpose is to allow for full inclusion where everyone is at some different level of skills and understanding. It's supposed to work naturally. The teachers tested, but results were graded with rubrics, not percent correct scores. Clearly, this natural assumption was not working. We even had to ensure mastery of basic skills with our son. Amazingly, the school would send home notes telling us parents to work on basic "math facts." (That is a huge problem in and of itself.) EM is a curriculum that emphasizes understanding and discovery - select your own algorithm, but there it is in black and white; bright kids did not know the times table. The curriculum didn't reuquire it. CCSS might set slightly higher standards for assurance of basic skills, but EM is still used in our schools and their assumptions have not changed.

This points to a very basic philosophical issue of education. Is it natural or not? Educators don't like tracking so it ends up at home. (They tell parents to do it at home!) If they dump full inclusion and offer different levels, then the upper levels will be filled with parents who set higher expectations and push. This isn't saying much since schools don't teach to a STEM level in K-6. Schools have to set high standards and push. I got to calculus in high school with no help from my parents. This was not possible with our math brain son. Schools used to keep kids back a grade or require summer school. This doesn't happen anymore. They increase the range of ability, willingness, and disruptiveness in the classroom but somehow think (or at least tell the parents) that they can do a better job academically for individual students. They do this with talk of differentiated instruction, conceptual understanding, and fairy dust. If understanding is king and the process is supposed to work naturally, then if things go south, you blame the kids, parents, society, SES, and anything else. No. education is a feedback loop and schools have to push and en

SteveH

3/12/2015 04:32:57 am

No. education is a feedback loop and schools have to push and ensure mastery. This is not some magical and automatic conceptual-->skills process. It's not top down and natural.

This evaluation is NOT an analysis of understanding except for the fact that these "understanding" curricula have been around for 20+ years and show no improvement in scores. If, however, students did better on the basic skills tests (rote or otherwise), then nobody would be talking about understanding or critical thinking. Only after we separate the noise and obfuscation about understanding generated by K-6 ed school pedagogues can we have a proper discussion of the different types and levels of mathematical understanding. When my son was in pre-school I thought about an education that provided more mathematical understanding than when I was growing up. Then I saw that our schools used MathLand and realized that they were going in the wrong direction, but incredibly, they were basing it on better understanding. I've been fighting against wrong ideas of understanding ever since.

SteveH

3/12/2015 04:20:10 am

Let's look at the "error" defined in journal article in this thread:

"Impatience with the idea that conceptual aspects of math ought to be explicitly taught is often coupled with an assurance that, if you teach students to calculate accurately, the conceptual understanding will come. A new experiment provides evidence that this belief is not justified. People can be adept with calculation, yet have poor conceptual understanding."

Being able to "calculate accurately" is NEVER, I repeat, NEVER done without teaching some type and level of understanding.

This isn't about examining an error they happened upon. This is posed in a way that implies that the experiment will find what the authors are looking for. Confirmation bias.

Students always have problems with how fractions (above and below 1) affect an equation. They have understanding problems with all sorts of things. I remember that I had some really weird ideas about algebra. That didn't mean that I was being taught with little understanding. It's never a one-way street. It's a feedback loop process where the teacher has to push and ensure. While understanding how the brain may work for some kids might be helpful, it's not a lynchpin in the process of providing good math education, especially at the stinking bad current basic skill test result level.

"So what's to be done? Teach concepts. Among other ideas, Siegler and Lortie-Forgues suggest that, once they have some competence in calculation,..."

"Teach concepts"

That isn't being done???

"... once they have some competence"?

What do they think is going on in math class? Each new unit is introduced and explained. Concepts are taught. Then students are given a homework set that carefully moves from simple uses of the material to ones that require much more understanding. My son's old Glencoe Pre-Algebra book unit on "Multiplying Rational Numbers" starts out with graphic explanations and then moves on to what it calls the "Key Concept":

"To multiply fractions, multiply the numerators and multiply the denominators."

a/b * c/d = (a*c)/((b*d), where b,d are not zero

and then they give an example. They don't offer a proof, but even if a student could do one, it wouldn't necessarily help them when they got to multiplying 2+1/3 * 3/8 or 1/(x+3) * 3/8.

In this unit, they include an introduction to "dimensional analysis" and units which is a hugely important conceptual idea. There is NO competence first going on here. It starts with concepts. Then how does the homework set go?

The first section has problems like 1/4 * 3/5. Question 13 is a word problem using geography. Problem 14 introduces variables, like

a/b * 5b/c

and you are told to write them in simplest form.

Question 17 is a word problem based on D=RT and uses units.

Questions 18-35 include all sorts of variations using mixed numbers and positive and negative fractions. Eighteen problems. there is nothing like doing many of these to make you absolutely sure of your skills AND understandings.

I could go on and on. There are 67 problems in the homework set, including ones that require you to convert units, say from inches to cm or ft^2 to m^2. Isn't it clear that the squared units problem forces the student to better understand what is going on? The final question is a "Writing in Math" question. Then there is a section in the book called "Spiral Review" that inlcudes a link-back to where each problem was explained.

Most teachers assign a collection of odd and even problems (the odd questions have answers in the back of the book) and the homework is gone over the next class period. Repeat and ensure problem set success class after class and year after year, and you will have success in math at every level. I liked to use weekly quizzes to "push" students. There is nothing natural about this process.

It's all there in proper textbooks that only appeared starting with my son's Pre-Algebra textbook. This process didn't exist with Everyday Math. However, by the time many students get to the tracking split in math in seventh grade, they have so many gaps in skills and understandings that it's all a downhill struggle from this point on. And, as kids get older, it's so much easier to blame them, their parents, their peers, society, lack of engagement and grit.

The problem is not a lack of concepts, but proving the concepts and full understandings that are taught by successfully doing problems.

Angel C. de Dios link

3/13/2015 02:52:11 am

We may be missing some important points from this study. First, the participants were clearly adept in arithmetic, and if were allowed to calculate during the test would have gotten the correct responses. The test, however, specifically instructs not to calculate, but guess. Second, these are "true or false" questions so randomly guessing gives 50%. However, the scores are lower than 50%. Therefore, it is not a mere lack of understanding, but a presence of misconceptions. Misconceptions are not taught during procedural teaching. When a child learns to add, subtract, multiply and divide, these are all procedures. Misconceptions are learned when concepts and understanding of mathematical procedures are taught like when a teacher incorrectly points out that multiplication always leads to a bigger result and division always leads to a smaller result.

SteveH

3/15/2015 04:02:48 am

Specifically about the study:

“These predictions were examined with three populations: pre-service teachers attending a school of education, students attending a middle school, and math and science majors at a highly selective university.”

“In summary, pre-service teachers’ conceptual understanding of fraction multiplication and division, as indicated by performance on the direction of effects task, was weak. This was true even for participants who flawlessly executed the fraction multiplication and division algorithms.”

“Thus, with children as with adults, successful execution of fraction arithmetic computations was no guarantee of understanding the procedures.”

“Thus, results of Study 3 showed that students highly proficient in math exhibited strong conceptual understanding of fraction arithmetic. This ruled out the interpretation that the task precluded accurate judgments.”

Those math/science students were:

"highly proficient in math exhibited strong conceptual understanding.."

"Highly proficient."

And of course I looked and looked and looked for any curiosity or examination of how those math and science students got that way and found nothing. All that the authors do is guess on how to fix the problem by just looking at poor results, not successful results. It really is quite incredible. I have my hand raised. I have one of those kids. Please ask us parents what we had to do at home. Many of us have been though this process and have gotten there ourselves. I can tell you about many of the misunderstandings I had, why, and how they got fixed.

The authors don't even show that this is something special about fractions. What happens to a number greater or less than one when you square it or take the cube root? What about the common problem of being able to "understand" problems with nice numbers, but not with ugly numbers? Did they test them with 5*1/2 < 5?

What about explaining the shape and domain of a curve, like y=1/x or y=log x without plotting points? Can students identify linear or quadratic equations? Is there ever any direct path between teaching (skills, concepts or whatever) and obtaining proper understanding? What is proper grade level understanding? When did those math/science students figure out how to do their problems? Math is a feedback loop no matter how well you use words or examples to teach concepts or understanding.

What about different levels of understanding? When should a student understand the implications of parametric analysis, which is what this is all about? Many of these things are not normal and provide fodder for math competitions. What do you think the best math students do to get better at those understandings? They don't start with words. They do sample problems. Those drive questions and then explanations and then understanding. It's about practice, not words. It's about feedback, not a one-way process.

Comments are closed.

Computational Competence Doesn't Guarantee Conceptual Understanding in Math

Purpose

Archives

Categories