My colleague EHK told me today about a difficulty she experienced in her first encounter with arithmetic, aged 6. She could easily solve “put a number in the box” problems of the type
buy counting how many 1’s she had to add to 7 in order to get 12
but struggled with
because she did not know where to start. Worse, she felt that she could not communicate her difficulty to adults.
A brief look at Peano axioms for formal arithmetic provides some insight in EHK’s difficulties. I quote Wikipedia, with slight changes:
The Peano axioms define the properties of natural numbers, usually represented as a set N or
[I skip axioms for equality relation — AB.]
The […] axioms define the properties of the natural numbers. The constant 1 is assumed to be a natural number, and the naturals are assumed to be closed under a “successor” function S.
- 1 is a natural number.
- For every natural number n, S(n) is a natural number.
Axioms 1 and 2 define a unary representation of the natural numbers: the number 2 is S(1), and, in general, any natural number n is Sn-1(1). The next two axioms define the properties of this representation.
- For every natural number n other than 1, S(n) ≠ 1. That is, there is no natural number whose successor is 1.
- For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.
The final axiom, sometimes called the axiom of induction, is a method of reasoning about all natural numbers; it is the only second order axiom.
- If K is a set such that:
- 1 is in K, and
- for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
Thus, Peano arithmetic is a formalisation of that very counting by one that little EHK did, and addition is defined in a precisely the same way as EHK learned to do it: by a recursion
Commutativity of addition is a non-trivial (although still accessible to a beginner) theorem. Try to prove it — you will be forced to feel some sympathy to poor little EHK. If it is a trivial task for you, write a recursive rule for multiplication and prove, from Peano axioms, commutativity of multiplication. Then write a recursion for exponentiation and try to explain, why this time commutativity fails, even if the recursive scheme appears to be the same.
In an ideal world (I emphasise, in an IDEAL world), primary school teachers should be taught Peano arithmetic — of course not because they have to teach it to their pupils, but because they have to appreciate intellectual challenges that their pupils have to overcome.
It is a strange concept of British model of teachers training that teachers need to know only the stuff that they pass to pupils. For a successful teaching, at least in mathematics, a teacher has to know much, much, more.
I have to emphasise: I do not propose to introduce Peano arithmetic into teacher training courses. I talk about an IDEAL world.
Mathematics under the Microscope 
That might not be an IDEAL way of making student thinking numbers like that. I believe students in their early days should learn math though intuition.
But since 4th grade calculus is possible… that won’t be such a bad idea…
By: Mgccl on May 14, 2008
at 11:27 pm
Mgccl: I repeat again, formal arithmetic should be taught to TEACHERS, not pupils — because at least some of the pupils are likely to reinvent it is intuitive level. I emphasise , Peano arithmetic is just simple principles of counting.
By: Alexandre Borovik on May 15, 2008
at 4:05 am
I slightly changed Axioms, bringing them closer to original form proposed by Peano: in most modern books natural number start with 0, not with 1. Children start their arithmetic with 1.
By: Alexandre Borovik on May 15, 2008
at 4:13 am
The poor girl hasn’t figured out yet that
. I’m sure a careful teacher would have figured out what the difficulty had been and would have explaiend. As for the Peano axioms for teachers, I’m not sure that it is such a great idea, it’s a bit dry. Maybe they need more practical discussions about the difficulties that the students have with arithmetics and how to help them.
To Mgccl: 4th grade calculus? What are they going to do with it? Isn’t it better to develop some problem solving skills?
By: misha on May 15, 2008
at 7:27 am
I don’t think it is correct to say that in the “British model of teachers training that teachers need to know only the stuff that they pass to pupils”. To be a primary school teacher you need to have a higher level pass in GCSE Maths (a much higher level than taught in primary school). To be a secondary school teacher you need to have a maths (or heavily mathematical) degree, which is again a much higher level than taught in secondary school.
The situation you describe does crop up at university level though: I recall when I was an undergraduate going to my tutor for help on an algebra problem. He replied: “look, I’ve not done any algebra since I was a undergrad! you know much more about it than me”. It was even worse for a course on knot theory, most of which he was (understandably for someone who was an undergraduate in the 70s) completely unaware of.
By: matthew on May 15, 2008
at 7:54 am
Misha: yes, EHK confirms that she did not understand precisely commutativity of addition, because addition done via counting by ones did not appear to be commutative. Of course, the teacher had to help. The whole issue is HOW the teacher helps a child in this situation: in case of EHK the best solution would be to give her sufficient number of examples like 2+3, 3+2, etc. However, in most cases, the teacher uses a simplest form of proof: by intimidation. EHK told me that that was the last time she was trying to think for herself, from that unfortunate episode on she just blindly followed what teachers told her.
At one time of my life I was involved in running maths competitions for children and the selection procedure for FMSh, Physics and Matheatics School at Novosibirsk University (a preparatory boarding school). There was a whole genre of interview problems, designed and selected to pick particular traits of mathematical thinking in the interviewees. There was an interesting subgenre of problems on “hidden counting”.
For example: a rectangle 19 by 99 is divided, by lines parallel to its sides, into
equal squares. How many of these squares does a diagonal of the rectangle intersect?
To solve this problem, it could be useful remember that “to count” means, in the initial meaning of this world, “to count by ones”.
Elementary mathematics contains a number of hidden structures; usually they sit in ambush exactly in dangerous points where lower leve concepts and techniques are integrated into a higher level context, like counting by one into addition. Children normally jump over such stfethes of white water as salmons over a waterfal, by process of reification (you can find more on that in my book). For a teacher, it is not enough to understand psychological difficulties experienced by a child in reification, the teacher should also be able to understand mathematical difficulties involved and hidden dangers awaiting his/her pupils.
By: Alexandre Borovik on May 15, 2008
at 10:56 am
This is fascinating! Just one thing… I think it was probably the last time I tried to think for myself at primary school. I had a wonderful teacher from age 11 who gave us problems to investigate rather than exercises from text books. She certainly encouraged us to think for ourselves and (something which is absolutely not emphasized enough in school maths) write down our arguments.
By: EHK on May 15, 2008
at 1:35 pm
Sasha: Giving more examples, as you suggested, would certainly help, but I’m not sure it’s the best way to explain. With your suggestion, here we go again, theorem-proof-example or fact-example approach again, formalizing too early, not giving enough explanations of where the formalism came from, not making enough with something outside of the formalism. After all, what does it mean to explain? Doesn’t it mean to point out a connection to something that the student already knows? Whay does it mean to understand something? Doesn’t it mean to make a connection to something else that we are already familiar with? What does it mean to master a topic? Doesn’t it mean to make enough connections it with the other topics, how to use the developed tools to solve problems, to freely go from one formulation to the other?
I suspect that EHK had not been exposed enough to the word problems that would connect the formal addition to something familiar to a child, would give her some informal ways of thinking about addition, rather than in terms of counting to 7 and then continue counting 5 more times to get to 7+5. Such a sipmle problem as “there are 2 baskets of apples, 5 apples in one and 7 apples in the other, how many apples are in two baskets together?” would make clear that addition is commutative. Thinking about addition in terms of adding lengths (like putting one stool on top of the other, how high the top of the stool on top will be?) would also help.
My diagnosis: too many topics and factoids taught in isolation, too much mechaniccal drilling, too few informal explanations, too few connections with other topics and subjects, too few word problems. After all, the bulk of our brain is made of connections, that’s what makes us smart. Teaching should be easy on facts and heavy on connections between the facts, on explanations of these facts, on letting students think about what they learn and letting them find their own explanations and connections, on figuring things out. The most interesting thing about something new is usually not the fact itself, but some innovative way to get to it, some novel way to figure it out. Unfortunately it’s not what is discussed in most of the articles and textbooks, especially in mathematics, the true springs of discovery and understanding are obscured by dry formalism and heavy terminology.
And now we are wondering why education, the way it is, doesn’t work so well. It’s like cutting out a big part of a frog’s brain and then wondering why it doesn’t jump. Calculus, by the way, is a prime example of this phenomenon.
To EHK: I see I guessed it right.
By: misha on May 16, 2008
at 3:35 am
In my previous comment: “…not making enough with something outside of the formalism” should be “…not making enough connections with something outside of the formalism.” sorry for sloppy editing
By: misha on May 16, 2008
at 3:41 am
I agree with Misha. My chiild started to count at 6.5 (in Europe we start much later that in Russia for exalmple). The first rule I explained him was X+Y=Y+X. It’s a basis of arithmetics.
Now, 8 months later, he counts billons, multiplies, knows decimals, fractions, his IQ is 143 and my youngest is 150. We live in Europe and constate that education is much poorere that in Russia. I have to follow Russian normal program in math, because the local one is too easy for him. In Russia they have a great math program by Peterson. Arythmetics is teached by using groups theory. Child understand how to groups objects by diferent characteristics: by color or by shape or size – same group, and N° of objects in each group would differ, but som is the same, because they are still same objects. My son does it easily, because it’s anyway 5+3 and he could count 365.48+154.65, etc, but logic points inclusion, repetition, creative presentation and challenge makes this program fun for him and he asks for “math work”. The prof of Gifted kids school liked Russian materials I’ve shown her. The copybooks are in Russian and, normally, the concept should be teached in a class before done at home, but I just give him several pages with saying nothing. he could not read Russian and just started to read Dutch, but he understand everything himself.
If you need to make child progress very quickly in math, to understand concepts and have more fun – use Peterson’s books!
By: Inna on January 9, 2009
at 8:15 pm
Sorry – I was in a hurry and made many mistakes. Just to add that due to my experience, simply following Peterson’s copybooks would solve problem with simple repetition, because there is a repetition in a book, but in different forms and that’s why is not boring. And also 2+3 is explained on a level of apples, but it could be 5 apples – some of them are green, other yellow, then some big, other – small, some with leaves, some – without. And then you create from the whole set 2 groups of big-smal, which would represent 2+3 for ex, or group of red-green (1+4), etc.
Then you know WHY do you need to count 1+4 and 2+3, etc. It’s better than apples. You could think it’s more complicated, but there are different levels and in the beginning children compare groups of shapes, then count them and then have same example using numbers and same example using expressions (X+Y=Z, then Y=Z-X). So, form the beginning they have the whole picture. I’d say Russian level for 5-6 y.o. kids should be equalent to 7-8 y.o. european program taking into the account that parents in Russian work with their kids at home on school subjects and that Russian kids should read 25 words per minute when they go to first grade, count and write NICELY. I do not think kids living in Europe are stupid – they simply need to have a proper program, motivation, challenge and some effort from parents.
By: Inna on January 9, 2009
at 8:30 pm
Inna — thanks for your comments. Best luck to your children!
By: Alexandre Borovik on January 10, 2009
at 2:06 pm
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at 12:58 am