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.
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.
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