[I transfer this post from my old blog.]
A paper under this name is placed on the Internet by John Baldwin. He asks a seemingly naive question: how do we justify, in school mathematics teaching, manipulations like
John Baldwin writes:
The associative law can only work on two applications of the plus sign. We generalize it to say we can regroup any sequences of additions. It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it. The second problem is that when minus signs are interspersed this gets more complicated. Since associativity fails for subtraction some further rules are required.
Indeed, notice that (3 -2) – 1 is not the same as 3 – (2-1). Subtraction is not associative!
This is a classical example of a “fuzziness” in mathematics teaching, a phenomen especially noticeable at earlier stages of school education. Very frequently, children are placed in position when they have to figure out rules which have not been made to them explicit. Of course, children learn the grammar of their mother tongue exactly that way, by absorption. Unfortunately, by the time they are taught mathematics, their natural ability to extract grammar rules from adult’s speech is already significantly suppressed.
A few days after I wrote the above text, I had to correct a student in my (university) mathematics class who was trying to apply the associativity law to conditional statements in Propositional Logic and write
p –> (q –> r) = (p –> q) –> r
–which is of course wrong, and exactly for the same reason why (3 -2) – 1 is not the same as 3 – (2-1).
This is a very delicate point indeed. How explicit should we be in formulating formal mathematical rules in (early) mathematics teaching? In teacher’s training?
I am more and more inclined to think that, in mathematics teaching, the famous quote from John von Neumann
In mathematics you don’t understand things. You just get used to them.
can be usefully reformulated:
In (early) mathematics teaching, you do not explain the rules. You just follow them, and let your students get used to that.
Admitedly, it is a contentious thesis. Peter McB in his comment to my previous post, Mathematics of Finger-Pointing, takes the position which is very different from John Baldwin’s. I quote Peter McB:
The examples of student “errors” shown on the post you linked to are, in fact, examples of the medieval-craft nature of pure mathematics, as noted 30 years ago by the computer scientist, Edsger Dikstra. The discipline still has no systematic, agreed, industrial-strength, notion of semantics: everything is still ad hoc. For instance, in some cases it is correct to cancel the symbol “n” above and below a fraction line; in other cases (when, for example, the symbol “n” is embedded inside a “sine” function), it is not. Why there is difference here is never explained, but bright students somehow master it implicitly.
As you can see, I am not in total agreement with Peter McB and will try to list some arguments in support of my thesis:
- (Young) children are exceptionally good at picking up implicit rules if adults strictly follow these rulles.
- Sometimes, formally stated rules could only confuse children.
- On the other hand, ability to explicate informal or undisclosed rules is one of the most important mathematical skills — and children should be trained in that.
- In the classroom, mathematical rigour is not children’s responsibility; it should manifest itself in the teacher’s self-discipline and consistency in following undisclosed (for a time being) rules.
- Therefore (and I differ on this point from John Baldwin) teachers, in the ideal world, should have very explicit understanding of the true nature of mathematical conventions.
I would be most happy to hear the readers’ opinions on the matter. Meanwhile, I am thinking about some further posts where I will try to give some examples of “leading by example” in teaching. The first of these posts, however, is likely to be about the first of famous “3 Rs”: Reading.
I recently was doing some volunteer tutoring and being very carefree about expressions such as the one you mentioned. Then I recalled my 8th grade teacher Betty Daniels insisting that we distribute the minus sign, apply the associative law, apply the commutative law, apply the associative law again, etc. Each time we applied the law, we had to sate the rule that was being applied. We could not abbreviate: 8th graders always want to call the associative law “the ass law.”
Gradually, as we became more acquainted with the rules, we were allowed to skip steps. For example, she allowed dropping associativity once every thing had been converted to addition and multiplication (as opposed to subtraction and division). She insisted on keeping commutativity for a while longer because she knew of algebraic systems in which commutativity did not hold.
Mrs. Daniels was an exceptional teacher. As a high school teacher she had been trained. Many (not all) future elementary school teachers resist the idea that they need a deeper understanding of mathematics.
In my tutoring lessons, I began to articulate which rules were being used when. This was difficult since my white board was less than
1.5 square meters.
By: Scott Carter on May 17, 2008
at 4:28 pm
In my opinion, you are all wrong. Piling up formal rules on top of formal manipulations is akin to treating students as robots, or trying to render them into robots. Just following these formal rules while demonstrating some examples and letting students guess is a bit better, at least it gives them a chance to use their intelligence.
But most of the mistakes in formal manipulations come from a disregard of the meaning of the formulas that are manipulated. The formulas are not just empty symbols, they mean something. Each time a mistake is made we should point it out to the student. It is clear, Sasha, that your logic student was either unaware of or diregarding the meaning of the logical formula that (s)he was manipulating.
Likewise, the identity (40+20) – (12+5) = (40-12) + (20-5)
becomes clear after we spell out what is in the left-hand side and what is in the right-hand side of the equality sign. Fluency in formal manipulations can help a student pass a test, but it is not a substitute for understanding. In my opinion, understanding is fundamental, and fluency just automatically comes with practice.
Just to give more rules to blindly follow to avoid mistakes doesn’t work too well, it’s much better to explain (preferably by examples), or point out the absurdity of a mistake. For example, when somebody adds fractions by adding the numerators and adding the denominators, pointing out the rule is not the best way to handle the situation. Much better is to say, “well, if you ate a half of an apple, and then you ate the other half, how much of an apple you ate? 1/2?”
By: misha on May 18, 2008
at 1:20 am
Actually, Sasha, in your second example, there is a rule at play that says “double minus is plus,” or “minus by minus is plus,” or “to subtract the negative of something, we just add it.” And in the first example the rule would be “minus by plus is minus,” or “to subtract the sum is the same as to subtract each of its terms.” My point is that these rules should not be just handed down dogmatically, they should be either derived, demonstrated by examples, illustrated by (or generalized from) the word problems, and the students should rethink the derivations of the rules, the reasons why the rules apply, every time they stumble. It’s also fruitful to see the minus sign as multiplication by -1 and then addition of the result, but it may be a problem for the students not familiar with the concept of multiplication or negative numbers.
In the previous post I meant to emphasie only one word, meaning in “… mistakes in formal manipulations come from disregard of the meaning of the formulas…,” and not the rest of the text.
By: misha on May 18, 2008
at 2:00 am
I looked again over my comments on your previous post, and I stand by them. I wonder who benefits when teachers of mathematics keep the rules implicit, instead of making them explicit? Of course, clever students may be able to guess the implicit rules eventually, and this activity may excite them. The other 95% of students, however, do not benefit from the rules being hidden. Instead, they grow up to become those adults incapable, angry and disillusioned about mathematics that we see all around us.
By: Peter on May 18, 2008
at 4:12 pm
When we played games, we followed the rules.
Sometimes, not always, the rules were in place to make the game fair. So I don’t think that children have an intrinsic distrust for following rules. Or those that do become bank robbers or politicians. I also agree with Misha, that some explanation of why the rules hold is required.
It is not enough to say that minus by minus is plus. Rather it is better to say that to negate is to reflect, and two reflections gets you back. Three reverses again. The puppet in the pupil brushes his teeth with the same hand that I do. The fellow in the mirror uses his other hand. The puppet in the pupil in the mirror has the same handedness as the fellow in the mirror.
The act of reflection is taught by gesticulations, folding the left hand over the right, demonstrating with the arms, and indeed, creating a reflection in the plane via a 180 degree rotation in space. The importance of that gesture is not fully comprehended until one is asked about the square root of -1.
BTW, most of my school mates did resent Mrs. Daniels compulsion for requiring us to articulate each and every rule. Nevertheless,
I know many of these same people 39 years later, and they all are fairly successful individuals, and those who lack success are nonetheless quite intelligent.
It is intuitive to demonstrate by examples that addition is associative and commutative by repeated examples. Properties of negation are less intuitive unless one has taken the time to contemplate the mirror person. I haven’t asked anyone else if they consider their mirror image.
By: Scott Carter on May 18, 2008
at 9:14 pm
To Scott: Mirror reflection is not the only way to explain negation, there are plenty of other ways. For example, being in debt, taking steps beckward instead of forward, to name just 2. The important thing is not to formalize too early and to emphasize the meaning.
By: misha on May 19, 2008
at 3:12 am
Being in debt and taking steps backwards are negatives, not negations. How do you explain double-negation in terms of being in debt?
By: John Armstrong on May 19, 2008
at 4:53 am
Negation of negation: debtors owes you some money.
By: misha on May 19, 2008
at 7:41 am
…I mean, the debtor owes you some money.
By: misha on May 19, 2008
at 7:43 am
Yeah, I know what Scott and John are talking about when they are talking about negation, it’s the order 2 symmetry that switches positives and negatives, left and right, assets and debt, forward and backward.
Actually, it illustrates the other difficulty with the minus sign, it signifies sometimes a binary operation, and sometimes a unary operation, depending on the context, so we have an operation overlay here. The usage is well established, but may be confusing for an uninitiated.
The trouble is that a more pedantic notation, aside from being untraditional, is usually more cumbersome and less readable, as a result. So we have to strike a balance between precision and readability.
For example,
depends not only on what 
is and what 
is, but also on what 
is, but nobody writes 
Oh, well, life is tough…
By: misha on May 19, 2008
at 9:26 am
The multiple overlay of meanings on the negation sign may also explain the difference between how mathematicians understand which of two negative numbers is “less” than the other and how ordinary people understand this term (a topic which you have blogged about in the past). A similar situation exists with the different meanings different people give to “forwards” and “backwards” with respect to two future (or two past) time points.
By: Peter on May 19, 2008
at 11:40 am
Peter – I thinkthe benefit of implicitly-instilled rules in mathematics are the same as the benefits of teaching functional reading and writing before formal grammar (which may never be taught at all): it allows pupils to catch early sight of the wood without being confused by trying to remember each of the trees.
By: dr rick on May 19, 2008
at 2:22 pm
To dr rick: Absolutely! As for remembering trees, there is a consolation: the more you understand — the less you have to memorize.
By: misha on May 19, 2008
at 5:11 pm
dr rick — and the disbenefits, also. This is my point. For those of us who like to know what we are doing and why before we do it, grammar was a god-send.
By: Peter on May 19, 2008
at 8:28 pm
I agree, at the appropriate time – but I really doubt you were taught formal grammar before you were reading and writing reasonably fluently. Neither should you have been, in my opinion.
By: dr rick on May 19, 2008
at 10:05 pm
Many calculators distinguish between the unary operator (-) and the binary operator (-). The weaker students in the precalculus class invariably are confused.
This has been a nice discussion!
By: Scott Carter on May 19, 2008
at 11:28 pm
dr rick —
I was taught english grammar from age 7, just two years after starting to learn to read and write, through to age 18. But in math, I was only taught about associativity and commutativity from age 16, a full decade after learning the basic arithmetic operations. I learnt the rules of grammar, relatively speaking, much earlier than I learnt the rules of arithmetic, and I learnt some of the rules of grammar well before I was fluent in writing or speaking.
By: Peter on May 20, 2008
at 2:15 pm
To Peter: but you surely knew that addition and multiplication are commutative long before you were 16, you just didn’t know the lofty words to impress your friends with.
By: misha on May 20, 2008
at 2:30 pm
Misha — yes, but I am in the 5% who could infer the implicit rules, not the 95% unable to do so. (I have a university medal in a branch of mathematics.) At age 16, I still had classmates unable to always subtract correctly.
By: Peter on May 20, 2008
at 3:20 pm
But look, Peter, commutativity of addition and multiplication is totally obvious to anybody who ever bothered to think about what these operations mean, I would not even call them rules. Maybe 95% just don’t think about it in general terms because it’s not interesting to them.
By: misha on May 20, 2008
at 3:55 pm
On the contrary, Misha, these properties are certainly not “obvious” properties of arithmetic operations, since subtraction and division are not associative (the very point of this post). Why is addition associative but not subtraction? Rather than being obvious to an intelligent child, any child who actually thinks unaided about this topic is likely to become confused. Hence, the need for the rules of operation to be made explicit, at the outset, not a decade later.
I grew up in Australia, where almost everyone learns to play Cricket in the summer and either Rugby League or Australian Rules football in the winter. Most children have no difficulty learning the rules of these games (even though the rules of cricket are truly arcane). But, here’s the difference: the rules of these games are taught explicitly, at school and in vacation play groups. No one expects children to master these games without being told the rules. That would be absurd. It’s absurd in mathematics, too, as Edsger Dijkstra argued.
Sadly, I think your comment reflects the educational attitude I deplored in a comment above — the desire to teach for the benefit of the clever 5% of children while ignoring the needs of the other 95%. Not only children suffer from this attitude, but mathematics and related disciplines also suffer, since so few students go on to study these disciplines after completing school.
By: Peter on May 20, 2008
at 4:49 pm
I have to agree with you on number 4: “In the classroom, mathematical rigour is not children’s responsibility; it should manifest itself in the teacher’s self-discipline and consistency in following undisclosed (for a time being) rules.”
I don’t think I would have understood induction, had it not been for Dr. Coleman.
Teachers are students mini-maths role models and they should take this role seriously, even if they do not feel that they are so. You lead by example…
By: beans on May 20, 2008
at 5:34 pm
I was talking about commutativity of addition and multiplication in response to Peter’s remark. If you tell any sane person what addition is (like taking 3 apples and addind 2 more), it will be clear to him that it’s commutative. Likewise, if you tell that multiplication is like counting the number of soldiers in a rectangular formation, say, of 6 ranks and 5 files, it will be clear that it’s commutative. Non-commutativity is such a surprising thing because people have the inpression that all the algebraic operations are commutative, based on their experience with numbers. Most people even don’t know the word “commutative,” but they will tell you that a+b=b+a and ab=ba if you ask, and they will also tell you that a+(b+c)=(a+b)+c and a(bc)=(ab)c without ever hearing of associativity. It’s just common sense.
By: misha on May 20, 2008
at 11:00 pm
If we admit that at least a part of mathematics is a language, we can find some support for dr rick’s position in Steven Pinker’s “Language Instinct,” where he gives a lot of evidence that we are preprogrammed to pick up patterns and construct grammars, and that’s how we learn how to speak and how to understand speech.
Anyway, I don’t think there is any one-size-fits-all approach to teaching mathematics (or any subject for that matter), because what works for one student may only annoy or confuse the other. I advocate emphasizing reasoning and nontrivial problem solving because I see them as the most essential, and also the most valuable not only in mathematics, but in general.
By: misha on May 21, 2008
at 3:13 am
Peter – certainly we teach the fact that the operations are commutative, and indeed we expect our students to know that before they come to us at eleven. We also teach that subtraction and division are anticommutative, but that they commute when thought of as inverses.
We would rarely use the word “commutative” to talk about it, of course, because most youngsters have enough difficulty with necessary-for-exams five-dollar words like “integer” and “perpendicular” without bothering them too often with others. Associativity is deeper waters, but certainly students are very explicitly told that they can’t just throw in brackets without thinking about signs – which is what they need to know.
I would assert that the fact that you, like me, are in that talented few percent that win maths prizes at university disqualifies us completely from talking about how mathematics ought to be taught to less abnormal children unless and until we have actually done it; our intuition on the subject is utterly worthless. Based on my experience, I believe that asking children to work from the fully-stated rules of arithmetic would be a very profound example of the teaching only to the talented 5% that you rightly condemn (and I think the experience of the attempts at “new math” over the last forty years lend some support to that opinion). It is in fact exactly childrens’ attempts to learn mathematics as a series of abstract rules of manipulation that causes them to fail, and in particular to fail in ludicrous ways; inevitably they get “the wrong rule”.
Misha – I have nothing to say about your last post except that I agree with it very strongly.
By: dr rick on May 21, 2008
at 9:07 pm
The “new math” reform also didn’t work out because of the idiotic attempt to introduce it at all grades simultaneously, instead of phasing it in gradually, as I heard from somebody who was in school at the time. Paraphrasing an old dictum, those who can not do, teach and those who have no idea about teaching, manage education.
By: misha on May 22, 2008
at 3:57 am
The “new math” or “Kolmogorov” reform in Russia is a fascinating subject. One reason for its failure was insufficient attention to retraining of teachers, and one interesting consequence (rarely discussed) — a dramatic (perhaps temporary) lowering of level of mathematical thinking in participants of mathematical competitions. At the time, my explanation was that teachers were too busy coping with new syllabus to pay attention to, and work individually with, mathematically able children. And these were, almost by definition, best teachers! Teacher training is the key for programme development; it does not matter what is printed in the new textbook, what matters is how the new material transformed in teachers’ heads.
Regarding the minus sign — a mathematician colleague who was a pupil in a French school (but not in France) in 80-s, told me that minus as as unary operation was not used, instead there was function opp(x).
By: Alexandre Borovik on May 22, 2008
at 6:43 am
It may be a better idea to use the minus only as a unary opearation, and write a+(-b) instead of a-b. But both ideas look rather silly to me.
As for Kolmogorov, he was just too smart to be a good teacher, to most people he appeared incoherent and incomprehensible, only extremely bright students like Arnold could follow him. Let’s face it: to be a good teacher you have to be stupid enough to relate to an average pupil.
By: misha on May 22, 2008
at 9:26 am
dr rick —
I agree with you that the school mathematics syllabus should not be oriented to the top 5%, but to the vast majority. I said as much in my comments above. But the school math education I received WAS oriented to the very top 5%, since little was presented explicitly. Despite that fact that I was in this top 5%, I still found it difficult, unintuitive and hard to follow. Asking that students somewhere guess or infer the rules, as commentators above have supported, is not orienting the education system to the majority, but only to a tiny minority — not just a clever minority, but a clever minority who think in a certain way.
I believe there is a divide here between people who learn best in a top-down fashion, asking for the rules to be made explicit upfront — and people who learn best in a bottom-up, examples-first fashion, with the rules only being mentioned (if at all) after the techniques are mastered. I learn best top-down, as perhaps do most computer scientists.
School mathematics education should NOT be aiming to educate the next Fermat or the next Goedel, but instead the entire generation of skilled people our society needs across a range of mathematical disciplines – mathematics, statistics, computer science, physics, engineering, economics, even marketing. This means helping many more than the top 5%. I believe my views are as valid as those of anyone else in this debate, regardless of where I fall in the spectrum of abilities.
By: Peter on May 22, 2008
at 4:37 pm
Peter – You believe that teaching children from explicit rules at a very early stage would help them; implicit in your argument seems to be the opinion that it would help the weaker ones more. I disagree with this, in fact I think it is *exactly* wrong. There we will, I think, have to agree to differ.
It may of course be that we are talking at cross purposes. If you are simply asking that children be told explicitly that something they might think will work in another situation does not, and why, then I absolutely agree, as would any competent practitioner. It did not seem to me that that was what you were asserting; I think you are asking for something more profoundly different, in which very explicit rules of abstract manipulation are laid down at the very outset. My own belief is that this would be, at the very best, confusing to most students.
I agree that many people, once they have crossed the disjunction between primarily abstract and primarily concrete thinking, learn best in a top-down rules-based fashion. However, I assert that virtually nobody does so before they have passed this point, and that that includes the vast majority of pre-pubescent children.
I also believe that without a well-formed framework of acquired numerical intuition to aid them, it is impossible for normal children to retain and correctly apply all the different but confusingly similar-looking rules they would otherwise need. Anybody who watches children who are out of recent practice trying to calculate with fractions will recognise this. Here is a situation where they have certainly been taught clear, explicit rules of how to proceed, and it has in most cases been little better than useless.
You are of course as entitled to your opinion as I am mine, though I would argue and have argued that perhaps the fact that I spend most of my time actually teaching children does put me in a somewhat stronger position here.
By: dr rick on May 22, 2008
at 7:14 pm
To Misha 28:
I think it is very important to emphasise a+ (-b) = a-b, it is essentially the definition of a-b. Subtraction is not really something fundamental, it is the product and the inverse for the group operation that matter.
Gentlemen,
has anyone thought what would happen if one teaches vector algebra
before arithmetic? Maybe it would make life much easier…
By: Another Peter on May 23, 2008
at 4:27 am