What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game.

— Keith Devlin

Should Children Learn Math by Starting with Counting?

It seems obvious that our children must have a wide range of experience with real world objects before counting, addition, or subtraction mean anything to them. But are other topics, such as calculus, better learned as abstract rules — as a game that we play with symbols? And what about the topics in the middle? For instance, how best can we break our algebra students of common errors such as distributing the square or canceling out addition terms?

To teach effectively, I need to understand how students learn. Do different approaches work best with different concepts? Or at different ages or stages of development? I can think of at least 3 ways that I have learned math — what about you? How do you and your children learn?

## I Learn by Discovery

- Collect lots of experiences.
- Notice patterns or trends.
- Develop mental metaphors to explain the patterns, and postulate rules from the metaphors.
- Test the rules against future experiences. Do the rules always work? Do we need to modify them for different situations?

**Advantages of this method:** Discovery learning can be applied in many situations, and it does not require a teacher. When I am stuck on a math puzzle, “guess and check” can get me going again.

**Potential problems:** Discovery learning is slower than other approaches, making it an inefficient use of limited class time. It’s great for occasional projects, but impractical as the main course. Also, too heavy an emphasis on the discovery approach can lead teachers to discount the importance of practice — but much practice is needed to help students master and retain math concepts (and avoid those common errors!)

Also, while I can discover quite a bit about math by exploring numbers, shapes, and patterns on my own, if I do not get a teacher eventually — even if that “teacher” is simply a good library book — I can only go as far as the limits of my own imagination will take me. And experience alone does not offer a reliable foundation for future mathematics.

The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.

— Leonhard Euler

quoted in the Mathematical and Educational Quotation Server

## I Learn by Constructing a Logical Framework

- Start with some basic intuitions (axioms) and definitions.
- Build logical connections between these foundational concepts.
- Deduce further relationships, applications, or extensions of the concepts.
- Apply the mathematical concepts to solving problems.

**Advantages of this method:** This is a time-tested approach to teaching and learning math. The directed, highly-structured lessons, when led by a capable teacher, help to limit students’ confusion or misunderstanding.

**Potential problems:** Intuitions and connections do not always stick in the student’s mind, no matter how logically presented, and distinctions between various concepts may not make sense until later, as students practice applying the math. Also, many students are impatient with the slow process of building a framework and want to skip or skim through the initial stages.

The study of mathematics is apt to commence in disappointment….We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this greatest science eludes the efforts of our mental weapons to grasp it.

## I Learn by Mastering the Rules of the Game

This is the approach described by Devlin in How Do We Learn Math?

- State the rules. Like the rules of chess or any other game, the rules of math do not have to “make sense” at first.
- Practice the rules.
- Apply the rules in a variety of situations.
- Internalize the rules, developing a personal sense of why and how they work.

**Advantages of this method:** Teaching math as an abstract game can give us the fastest results. It is also much easier to track a student’s progress (by testing the application of specific rules) than with either of the other approaches mentioned above.

**Potential problems:** Memorizing too many rules that don’t make sense builds up into a confusing mental jumble. Also, not everyone proceeds to step 4 at the same rate, if at all — but if a student stalls out at step 3, he or she has no foundation on which to build the next mathematical concept. And if the teacher has not progressed to step 4, then the students are truly in trouble! (See Knowing and Teaching Elementary Mathematics by Liping Ma.)

Free math (Available here Monday through Friday). But you must bring your own container, and you must fill it with much or little according to its capacity and the amount of work that you are willing to do. The learning assistant (sometimes euphemistically called a “teacher”) will provide expertise, advice, guidance, and will set an example. But in the final analysis it is you who must do the work needed for your learning… Here it is — this wonderful stuff called math. If you want it, come and get it. If you don’t want it, kindly step out of the way — as not to impede the progress of those who do. The choice is yours.

— L. M. Christolphe, Jr.

quoted in the Mathematical and Educational Quotation Server

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.

It occurred to me a while ago that the brain is designed counter to math and science in the first place.

The animal brain has evolved to be a pattern-recognition engine. This seems to be good for math and science–developing theories first requires the ability to recognize a pattern where one hasn’t been seen before.

But the next step involves strict proof, or rigorous scientific testing, which is not natural to the animal brain at all. Instead, we’re wired to generalize broad truths from specific anecdotes. This is why so many people are superstitious about silly things, why first-grade teachers spend months telling kids that “taked” isn’t a word, and why prejudice and post-traumatic stress disorder exist at all.

As an evolved response, this is a good thing–if you see a tiger eat your cousin, it’s no good for your long-term survival to sit around and think, “Well, just because THAT particular tiger was hungry, that’s no good reason to generalize about ALL tigers.”

But it’s a habit that’s hard to break. And it’s why, when I teach math, it’s easier for me to get kids to learn algorithms for solving particular types of algebra problems, than to get them to complete abstract proofs of the same algorithms.

Thank you for addressing this subject. Math frankly intimidates me and my dear husband has suggested that maybe it’s because I never had the experience of learning it in a way that fit me. I hope to pass a different experience on to my kiddos. Thanks again!

Great topic, Denise.

Reading your post reminded me of a wonderful phrase in Devlin’s article “Should Children Learn Math by Starting with Counting”–‘linguistic creation’:

“If indeed there are these two, essentially different kinds of mathematical thinking, that must be (or at least are best) learned in very different ways, then a natural question is where, in the traditional K-university curriculum, the one ends and the other starts. And make no mistake about it, the two forms of learning I am talking about are very different. In the first, meaning gives rise to rules; in the second, rules eventually yield meaning. Somewhere between the acquisition of the (whole) number concept and calculus, the process of learning changes from one of abstraction to linguistic creation.”

“Linguistic creation.”

And then this got me thinking about how people (especially my own children) learn language. It’s a meaningless game, at first. It really is. To put it another way: at no point during any of my three children’s lives did I have to ‘build up to’ language. They learn language FIRST; they get all the different “meanings” and shades of meanings SECOND.

Marty’s example of “taked” is a good one. I would bet a thousand dollars that not one of those kids that Marty is referencing EVER heard the word “taked” from their parents or any other caregivers. Yet their little Broca’s areas do their magic to produce this non-normative vocalization. Why?

Because for most verbs, the past tense (in English) is formed by adding “d” or “ed.”

They figure out the rules FIRST.

Some have said that math is just another language . . .

I think this is a great article. I know I was personally very bad at discovering patterns before someone pointed a bunch out to me. I had to learn a bunch of rules before I was able to discover anything interesting. Then it took a long time before I was able to understand why a proof was necessary. (I was definitely a believer of the human-survival-tiger method.) But as my experiences in learning math grew I appreciate the discovery process more and more. I wonder if there is one ordering of teaching/learning styles that would be more productive than another?

This article also makes me think of the discussion between Montessori schools vs how public schools are usually taught. Certainly, methods of learning is not a math-only question.

Thanks, everyone, for the comments! I’m glad I’m not the only one who finds these ideas interesting.

I am planning an elementary class at our homeschool co-op next fall, which is part of what started me thinking about how students learn. In the past, I have done a Rules of the Game class on algebra. This time, for the younger students, I want to structure the lessons on a directed-discovery approach, but with a strong framework of logical connections behind each activity.

I sure wish I could find a copy of the Davydov math program that Keith Devlin described here. It sounds

exactlylike what I want to do, and I’m sure it would be much better than whatever I cobble together on my own.Very interesting post — lots to think about here. Thanks for laying it out like this.

good stuff for sure. sorry i’m so late to the party.

my take: an awful lot of damage is done by acting like

everybody should just choose some method or point of view

and then *commit* to it and hold all other methods

in suspicion at best or treat them as *opposing*

points of view… when “really” everybody has

to get at least a little bit good at *everything*.

the extremes aren’t all that far apart:

if the students can do the calculations

but not talk about meanings,

you’ve got nothing.

if the students can talk about the meanings

but can’t do the calculations

you’ve still got nothing.

every lectures-only teacher wants good student feedback.

every guide-on-the-side sometimes needs

for everybody to just shut up and listen for a while.

there are no two opposing points of view here

but only various (constantly changing) positions

along some continuum… “arrows” on the number line

are *not* points of the set but only *directions*…

so too for “direct” versus “constructivist” methods.

and the whole thing’s a distraction anyway

to *my* working life. the *real* math wars aren’t about

how i should deal with my *students* at all…

but how i should deal with the *boss*.

they’ve got everybody squabbling about

what the *right way* to push everybody around

might be… when the *real* issue is

what are schools *for*?

if it’s about getting people to accept being pushed around

then it’s working pretty well as designed. if not…

what then are we to do?

homeschoolers have essentially *solved* this problem.

the rest of us are in a hell of a fix.

I’m not sure we homeschoolers have solved anything, but we definitely have more flexibility in some areas. We don’t have to deal with the administrative hassles I read about on many teacher blogs, so we can focus more on trying to discover a “right” way to teach this particular topic to this particular student at this particular time.

“…particular topic, particular student…”

that’s great. I would essentially expect a teacher to use a hybrid of the latter two, with a bit of discovery from time to time. But with the flexibility you imply.

Jonathan

To chip in on the subject of counting vs measurement, Gardner suggests in his article a kind of false dichotomy, that you can either “start with counting” (which leads to the counting numbers 0,1,2,3,…), or “start with measuring” (which leads to the real numbers). Now this is a false dichotomy. It is impossible to get anywhere in measuring with no understanding of counting, whereas discrete counting can get along without measurement. You cannot choose to start without counting, because it is fundamental to measurement (you have to choose a physical item as 1, and compare it to the item which is to be measured, by counting how many are required to make the sizes match up).

What does not depend particularly on counting is our geometric intuition. Our experience of counting and our experience of geometric observation are the two things that do not depend on each other, and both lead to mathematics. However there is no opposition between the two, no duality. It is not light vs dark, or good vs evil, but two different things and nothing more. Therefore it is wise to avoid pitting the two against each other.

But this is what Gardner gets dangerously close to doing in his article. In mathematics you think about your counting experience and your geometric experience, and using logic, as well as imagination, you advance. What I am trying to advise people against is to have some sort of struggle between the two types of experience for supremacy over the other. It is true that there is something less reliable about geometric experience than counting experience, as revealed by the saga of non-Euclidean geometry, and the realisation that our own physical universe may not behave as our geometric intuition tells us it should. However the drawbacks of geometric intuition need not concern schoolchildren, only scientists and mathematicians.

Hi, Dan! Do you mean Devlin’s article, or is there one by Gardner that I’m not aware of?

The Davydov approach that Devlin wrote about intrigues me because I am preparing for a homeschool co-op class of elementary students, all of whom are studying some other math program at home. I want to help them think more deeply about math, but I can’t approach it in the “same old way” as their textbooks, or they will just shrug and assume they know everything already.

In addition, I have seen how well the bar model diagrams work in Singapore math. These are, in a way, a geometric approach to solving algebra word problems, and they make it very easy for students to extend concepts from whole numbers to the reals.

this is AWESUM ! 🙂

thx hu ever created it !

Who is Gardner? Obviously I meant Keith Devlin. Wow, what a cock-up. But we can come back from this. Just focus. Anyway…

Hi Denise, I am talking about the same article, although I was more talking about the title rather than what the previous posts had been talking about. Math teachers tend to have a poor understanding of the nature of mathematics, but mathematicians are worse.

To your second paragraph, I will get to that in a minute. To your third paragraph, I am only a math graduate, not a teacher, and I don’t know what bar model diagrams are, and I don’t want to tread on anybody’s toes.

I was impressed that Devlin managed to identify counting experience and geometric experience as the two driving factors behind mathematical developement, which puts him 998475^28385 factorial steps ahead of axiomatic set theorists, who are simply far too busy with (and there’s no nice way to say this) axiomatic set theory. However, he then gets slightly confused. This isn’t Tekken. It’s not “counting experience” vs “geometric experience”. But he seems to think it is! The truth is, they both complement the study and enhancement of understanding of the other, particularly the ability of the former to enhance the latter. You just have to develop both as you see fit, no radical “championing” of one over the other will ever give a good math education.

From what I can tell, there’s a lot more to this Davydov syllabus than a fatuous emphasis on geometry over counting. It seems it involves teaching algebraic concepts early, in fact I would say that is by far the main idea of the syllabus.

But a quick post on axiomatic set theory. I am a graduate student in math (graph theory, combinatorics, set systems) and it just mystifies me how this subject even came to be. I literally believe that this subject is a pointless as flicking noodles at a wall. In fact I would rather do that than study this subject.

I make no attempt at explaining how axiomatic set theory came to have the status it does today. It’s a mystery. The foundation of mathematics. If anyone is to decide about that, it should be the teachers and the students, because mathematics is a human discipline and they have the easiest access to it. Not that a mathematician cannot access it by meditation, only that he probably will not. Mathematics starts when you count things, and experience that, and when you experience geometric thoughts, it’s as simple as that. You then apply your logic and your imagination to these experiences and understandings.

For me, the fact that the “new math” approach using axiomatic set theory to teach mathematics to schoolchildren failed miserably is not only proof that it is inappropriate to teach children axiomatic set theory, but that axiomatic set theory itself is simply a load of bollocks.

People champion Russell and Whiteheads’ 1000 page proof that 2+2=4, as a kind of mathematical eccentricity, but the truth is that their proof is simply bollocks. 2+2=4 is one of the simplest theorems with the simplest proofs (even a 2-line formal proof!). An 8-year old will tell you this, and he is right, and anyone who says otherwise is wrong.

That’s why treatises like Principia Mathematica are a load of bollocks, because they are made by people who thought about philosophy, and very abstract mathematics and what people were doing in very abstract mathematics, and how to formalize with axioms what these people were doing, but they didn’t think about how they learned mathematics as a child, and thus failed, completely.

I just ate 40 king prawns.

Foo Fighters:

Dave Grohl

Nate Mendel

Taylor Hawkins

Chris Shiflett

Girls Aloud:

Cheryl Cole

Nadine Coyle

Sarah Harding

Nicola Roberts

Kimberley Walsh

Join the two bands together:

Foo Fighters Aloud:

Dave Grohl

Chris Shiflett

Nadine Coyle

Cheryl Cole

Sarah Harding

Nicola Roberts

Nate Mendel

Kimberley Walsh

Taylor Hawkins

But Denise, I didn’t reply to your 2nd paragraph. The previous post was more of a rant than anything, but I didn’t want to leave hanging what I said in the first.

To your home students, I can only suggest something that would at least alleviate the impression (maybe you have done that already) that mathematics is finished, done, nothing more to do. Because, as you are no doubt aware, it isn’t. I can only suggest an exercise. An exercise just to think about, that’s all.

Ah, wait, elementary students, never mind. But as a riddle for any more advanced students who might be able to appreciate it:

[In (1) and (2) the answer is meant to be “yes, definitely” or “no, not necessarily”].

(1) Let d be at least 3. There is a town where the most sociable person knows d other people, and in this town there is no clique of people (a group of people who all know each other) with more than d people in it. A tee-shirt salesman comes to town selling tee-shirts in d different colors. He wants to sell one tee-shirt to each person in such a way that no person knows someone who has the same color tee-shirt as them. Can the merchant sell his tee-shirts in this way?

(2) The students at a college have formed bands. Moreover, they have done so in such a way that for any two bands that they have formed, they have also formed a band by joining those two bands together (see above). Is there a student who is in most of the bands?

(3) One of the previous problems is an unsolved problem in mathematics, the other is not. Can you tell which is which?

This is my idea of a way to introduce the idea to students (of an appropriate ability and inclination) to the idea that some problems in mathematics have actually not yet been solved, even ones that aren’t too difficult for them to understand.

Just to clarify, I am a masters student in mathematics, I am not a proper postgrad at this point in time!

Gardner would be Martin Gardner.

As for the counting vs. measurement topic, it is more a matter of emphasis than of complete dichotomy. Typical American math books emphasize whole numbers and counting for the first 3-4 years, and then they gradually move toward working with real numbers over the next 3-4 years. The Davydov program starts with comparing and measuring real-world lengths, weights, and volumes, which means they are dealing with real numbers from the beginning.

This, plus the use of algebra and geometric diagrams to help children discuss and reason about underlying concepts, is what makes the Davydov program so interesting to me. I think it would be different enough from what the kids are doing at home that it would draw their attention.

It is an interesting article – UK schools believe in a practical approach – which seems fine – but still we have slid in the world rankings – and being poor at maths in the UK seems acceptable.

In Romania, they are very academic and little time is spent on practical skills – oddly the employers don’t like this and they are moving to a more practical curriculum.

I think you are right that we need to work on a balance.

I also was bad at spotting patterns, but this article makes it clear that people do in fact learn by patterns and methods. I think this is more human nature and something to do with survival instinct. When looking at how children learn math, there are are a few effective ways to do it. One of them being add colourful images and make it more suited to them.

The debate over the best method for the delivery of math instruction is not likely to be resolved any time soon. I’m not sure we should ever attempt to declare a winner. By the time we all agree upon the best practice the state of education will have changed again and we will have to re-think our approach. There are benefits to the process however and it is important to work toward improving the system.

As a high school math teacher who works predominately with ESL students my focus is on trying to convince my students that their education has value. For many of my students the cultural divide poses a significant barrier to embracing the ideals of public education. It is far more important for these students to find a compassionate and encouraging environment than to be overly concerned with their performance on standardized tests.

That being said I firmly believe that good teachers know their students. They mold their instruction to meet their student’s needs. If something is not working they change their approach. Education is not a static enterprise. Teachers for the most part want their students to learn and they are willing to do whatever is necessary to accomplish that goal. It is the function of professional development to keep teachers abreast of the latest information that is emerging from educational research. Teachers will take what they can use from the current research and discard what they can’t use. There is no perfect method nor is there a perfect teacher. We are all works in progress.