This chapter covers the basic concepts, skills, and concerns that apply across most of the math I'll learn. I'll introduce them here and try to highlight them where they're especially relevant throughout the rest of the project.
In his book How Humans Learn to Think Mathematically David Tall has incorporated most of these aspects of math into a framework for the development of mathematical thinking, which will probably guide a lot of my learning. I'll talk about his framework after looking at each factor.
Since I see math as primarily a way of dealing with the world, I'll start with problem solving. Problem solving is finding a way to change a situation from an undesired state to a desired one when the changes needed aren't obvious at first.
Problem solving serves a few purposes. The immediate benefit in everyday situations is that it makes a problem go away. But it also serves an educational purpose. It expands your understanding of the kinds of problems that can be solved and the kinds of solutions that are available. That is, it reveals new concepts and relationships in the problem's domain, in this case math. Thus it lets you create new knowledge structures, which is a key part of developing one's mathematical thinking.
Ideally, studying math teaches you problem solving skills. This is because mathematical activity tends to be goal oriented and because many new math concepts are hard to grasp and apply at first, so you get regular practice at thinking creatively to expand your understanding and to achieve objectives. Then you get better at both solving familiar problems and figuring out how to solve new ones.
Problem solving can also be a motivational learning tool. People like being challenged, if success feels achievable. One effective way of arranging challenges for learning is an inductive chain. Starting with simple problems, each problem to be solved teaches a concept, and each following problem uses the new concept to teach another one.
Practically every math book I find that addresses problem solving mentions George Polya's How to Solve It, a good starting point for studying problem solving, both in math and in general.
If mathematics is the science of patterns, what is a pattern? How would you know one if you saw it? If you wanted to find one, what would you look for? A good definition is harder to find than I expected, but here's one that summarizes several points I've seen made: "A Pattern constitutes a set of numbers or objects in which all the members are related with each other by a specific rule." We could also characterize a pattern as a repetition with differences. As a simple example, counting from 1 to 10 gives a pattern of whole numbers that increase by 1 each time. One procedure for recognizing a pattern might be to look for a collection of diverse items that all have something in common or that differ from each other in a repeating way. The recognition of patterns is one of the three shared human capacities that David Tall credits for our ability to develop mathematical thinking, the other two being repetition and language. See the final section for more on those.
Finding patterns is part of problem solving. If you can find the rules underlying a situation, you can often use them to learn enough about it to resolve its difficulties.
It seems logic isn't much easier to define than math, but I'll take a stab at it: Logic is the study of the rules governing implication and necessity. Roughly speaking, it deals with the truth-related relationships between statements, and those statements can be about anything. In this case, we're interested in mathematical statements.
As I said in the introduction, math seems to be about the logical properties of mathematical objects such as numbers. In Tall's terminology, math is composed of crystalline concepts. A crystalline concept is a "thinkable concept that has a necessary structure as a consequence of its context." The mathematician describes, defines, and proves the properties of these concepts and their structures.
In math one major way logic is used is in proving and disproving patterns. A mathematician finds what they think is a pattern and states it as a conjecture. Then they or others either find counterexamples to disprove it or come up with a proof for it by chaining true statements together logically, ending with the conjecture they're proving.
An algorithm is a precise, detailed procedure for achieving a result. Math operations are carried out by algorithms. Most people are familiar with the procedure for subtracting multi-digit numbers, which involves things like stacking them on top of each other, subtracting from right to left, and borrowing. People often skip carrying out the algorithms by using a calculator, but algorithms are still in play because calculators and other computers use them for every single operation.
Often there's more than one algorithm that can accomplish an operation. The subtraction algorithm I mentioned earlier is the standard one for subtraction. Schools have begun teaching children alternate algorithms for the arithmetic operations, and it has some parents frustrated and alarmed. But the new algorithms do make sense. They even reflect the way adults actually do arithmetic. For the purposes of this project, looking at multiple algorithms will help us think about the nature of the operations, and understanding that nature will let us be more flexible in solving problems that involve those operations.
We can distinguish between algorithms and relations. An algorithm is a series of actions you take on some data that takes up time and proceeds in a more-or-less causal fashion. If you add 3 to 2, you get 5 at the end. But 2 + 3 = 5 also represents a timeless relationship between the numbers that exists apart from anything you do with them. One way to visualize the relationship is that on a number line, 5 is three whole numbers to the right of 2. From this perspective, when you perform the addition of 3 to 2, you're not creating the number 5 or changing a 2 to a 5. You're simply moving your attention from one number to another in their static places on the number line.
We'll always need algorithms. But the advantages of thinking relationally are that it's simpler and more flexible. It's similar to telling someone how to get to a building. You can give them a route, or you can hand them a map and an address, or at least a pair of cross streets. A route saves them the work of finding their own route, but it might not fit their preferences (such as avoiding toll roads) or even their starting point. If they have a map, they can tailor their route to their circumstances. They can even change it mid-journey if they run across obstacles. Another advantage of thinking relationally is that if you understand the algorithms by knowing the underlying concepts, you can remember the algorithms more easily and avoid making mistakes. Edsger Dijkstra has more to say on relations vs. algorithms, which he speaks of in terms of equivalence and implication, respectively.
An abstraction is an idealized generalization of some state of affairs. It embodies certain common attributes of the state and not other attributes that vary from one circumstance to another. This process wraps up a complex concept or procedure inside a simpler one that represents it and hides its details and variations. For example, the male and female restroom icons are abstracted representations of men and women. For mathematical examples, multiplication is an abstraction of repeated addition, and exponentiation is an abstraction of repeated multiplication.
Abstraction is one place patterns show up in math. The commonalities in the relationships between the elements of the pattern get abstracted into a rule that describes the pattern.
Abstraction plays a major role in Tall's model, where it also goes by the phrase compression of knowledge. He divides it into three tracks: structural abstraction for working with objects, operational abstraction for actions, and formal abstraction for axioms. I'll return to these tracks in the final section.
The abstractness of math and its technical use of symbols can obscure one of its important features. Math is expressed through a language, or at least a register, which is a subset of a language meant to fulfill a certain function. The language of math has both written and spoken forms. It has a vocabulary (number names, shape names, operators, etc.), a syntax, and a set of symbols. And like any language, it's used to communicate.
In Tall's model, language is one of the shared human capacities that enables mathematical thinking. It allows us to compress experiences and procedures into concepts and specify their properties. In other words, language aids abstraction.
People sometimes say that math is a language, but I like to take a broad view of math and say that it has a language, though it is more than that language. We can distinguish between the words of a language and the objects or concepts the language is referring to. So the numeral 1 is a word, and the corresponding concept is the idea of one itself.
This is similar to the relation-algorithm distinction I covered earlier. Relations and concepts both refer to the mathematical patterns that exist on their own, and algorithms and words refer to methods humans use to deal with those realities. The distinction is important to keep in mind because in the end, the word we pick to represent a concept is arbitrary, which is why there are so many languages. Math symbols and the ways we string them together into formulas had to be invented. We should always leave the door open to finding new ways to think about and represent the underlying mathematical concepts when the old ways become less helpful.
One part of speech that math makes special use of is a hybrid of verb and noun that Tall calls a procept. For example, the expression 2 + 3 acts as both an instruction to add 3 to 2 (a verb) and an object on its own that can be manipulated and reasoned about (a noun). The addends can be flipped to create the equivalent expression 3 + 2, for instance. Or 2 + 3 can be viewed as a stand-in for the result of the calculation. There are other contexts in which people treat verbs and even whole clauses as nouns, but it's an especially prominent feature of the mathematical language. And since procepts make it possible to reason about and to build conceptually on every mathematical action we perform, they're an especially important part of developing mathematical thinking.
The linguistic issues surrounding math extend beyond the symbols we use to represent its concepts and operations. We use language to talk about math for several purposes with both ourselves and other people. We use it to clarify math concepts or problems in our own minds, to teach concepts to others, to make mathematical requests ("Measure this length," for example), and so on.
All these ways of encoding and communicating about math have to be learned, and mathematical language is both similar enough to and different enough from everyday ways of speaking and thinking that in the beginning learners will be confused. So it's important to pay careful attention to language while learning and teaching the subject.
As they're learning the basic concepts of math, children work with small collections of physical objects. These unsuspecting children are actually learning the basic properties of and operations on sets. Math has an area that defines these properties and operations known as set theory, and in fact, mathematicians have apparently determined that they can form a logical foundation for most of mathematics partly on a particular version of set theory. It'll be a while before I know enough to understand this, so here I'll just mention it.
But we can make use of the main idea, that set theory has something to say about the fundamental concepts of math. It's another potentially helpful angle from which to view math concepts. As I see it the main advantage of drawing from set theory at this point is that it directly addresses some of the ways we begin to learn about math and lets us be precise about them. So even though we're starting our math self-re-education with numbers, I'm also going to bring in a different mathematical object, the set.
How mathematical thinking develops
David Tall organizes his theory of math education around three mathematical worlds, several mechanisms of thought, and three stages of development.
Worlds of mathematics
Mathematics occupies three realms: objects, actions, and axioms. Thinking in each realm involves different mental capacities and different methods of development.
The world of objects deals primarily with space and shapes. We interact with it through processes of mental imagery based on experiences with physical objects, processes that Tall calls conceptual embodiment. We grow our understanding of objects through our sensory capacity for recognition, which allows us to see patterns, similarities, and differences. We use our capacity for language to categorize objects based on these observations. Structural abstraction, this grouping of concepts through categorization, is one way we form new concepts within the world of objects. The first stage of structural abstraction is empirical abstraction, in which children play with objects to learn their properties. The second stage is Platonic abstraction, in which physical objects become idealized mental objects, such as sizeless points and widthless lines. Tall calls the methods of development in the world of objects conceptual embodiment.
The world of actions starts with arithmetic and algebra. We interact with it by mentally moving the positions of symbols, a process Tall calls functional embodiment. We grow our understanding in this realm through our motor capacity for repetition, which allows us to practice action sequences until we can perform them unconsciously. Language enables us to encapsulate these processes. Grouping actions through encapsulation is called operational abstraction. The first stage in this track is pseudo-empirical abstraction, in which children learn the properties of actions on objects. For example, the operation of counting leads to the concept of number. The operation of sharing leads to the concept of fraction. The second stage is reflective abstraction, in which actions become objects to be reasoned about, or procepts. At this level, the action of addition becomes the concept of sum, and repeated addition becomes the concept of product. Tall calls the methods of development in the world of actions operational symbolism.
The world of axioms is the subject of formal mathematics at the university level and deals largely with sets. We grow our understanding of axiomatic math through our capacity for language, which allows us to define thinkable concepts that we assemble into increasingly sophisticated knowledge structures. This labeling and defining of concepts is also how language compresses knowledge. Language then allows us to deduce the properties of these concepts through logical proofs, the process of formal abstraction. Tall calls the methods of development in the world of axioms axiomatic formalism.
You can imagine these worlds of mathematical development as three circles in a Venn diagram that overlap in four areas. Embodied symbolism is the area leading from conceptual embodiment to operational symbolism. Embodied formalism is concerned with Euclidean proof. Symbolic formalism is the area of algebraic proof. And the area occupied by all three worlds covers proofs that combine embodiment and symbolism. I'll call each of these seven areas a region, the four areas of overlap plus the three that purely concern a particular world.
Mechanisms of thought
In addition to all the processes mentioned above, Tall discusses a couple more that act across all three worlds.
The first is met-befores, a play on metaphor. These are concepts the learner has encountered earlier that they use to understand a new concept. Met-befores can be either supportive or problematic for understanding the new concept. When a learner proposes a solution or explanation, met-befores enable the teacher to ask the question, "What have you met before that makes you think that?"
The second mechanism is blending. The mind creates new knowledge structures and thinkable concepts by compressing knowledge through abstraction, by connecting the thinkable concepts into knowledge structures, and by blending earlier knowledge structures into new ones. Blending involves linking different modes of thought and experience together, such as vision, touch, and abstract concepts. For example, real numbers are a blend of the physical number line (embodiment), the idea of decimal numbers (symbolism), and a particular set definition (formalism).
Stages of development
Mathematical reasoning and proof develop in three stages, each of which involves multiple worlds. The first stage is practical mathematics, which explores geometry using physical objects and explores arithmetic using calculation. The second is theoretical mathematics, which covers Euclidean and algebraic proofs. The third is formal mathematics, which involves proving theorems from set-theoretic axioms.
Putting all the stages, capacities, and processes together, we arrive at an outline that looks like this:
- Stage: practical mathematics
- Region: embodiment
- Concepts: space and shape
- Capacity: recognition (perception)
- Method: conceptual embodiment
- Embodiment: conceptual
- Abstraction: empirical
- Region: embodied symbolism
- Capacity: repetition (action)
- Region: symbolism
- Concepts: number, arithmetic, generalized arithmetic
- Method: operational symbolism
- Embodiment: functional
- Abstraction: pseudo-empirical
- Region: embodiment
- Stage: theoretical mathematics
- Region: embodied formalism
- Abstraction: Platonic
- Region: symbolic formalism
- Concepts: algebra, algebraic proof
- Abstraction: reflective
- Region: proof combining embodiment and symbolism
- Region: embodied formalism
- Stage: formal mathematics
- Region: formalism
- Method: axiomatic formalism
- Abstraction: formal
- Region: formalism
Tall summarizes, "The whole development of mathematical thinking is presented as a combination of compression and blending of knowledge structures to produce crystalline concepts that can lead to imaginative new ways of thinking mathematically in new contexts."
- What gave us the idea that numbers are ruled by logic?
- What is the relationship between numbers and sets?
- Billstein et al. 2007, 2
- Tall 2013, 176
- Somers 2011
- Krypton Inc n.d.
- Hatfield et al. 2008, 124-125
- Billstein et al. 2007, 22-23
- Tall 2013, 21
- Hofweber 2011
- Tall 2013, 27
- Billstein et al. 2007, 23-24
- Mehta 2014
- Dijkstra 1985
- Dijkstra 1986
- Tall 2013, 10, 16
- Pimm 1987, xiii, 7-20, 75
- Tall 2013, 12, 21, 24
- Zheng 2015
- Tall 2013, 12-14
- Wikipedia, s.v. "Hilbert's Program," last modified January 3, 2014, http://en.wikipedia.org/wiki/Hilbert's_program
- Tall 2013, 6-8
- Tall 2013, 12
- Tall 2013, 21
- Tall 2013, 10, 15
- Tall 2013, 9
- Tall 2013, 16
- Tall 2013, 11
- Tall 2013, 21
- Tall 2013, 9-10
- Tall 2013, 16-17
- Tall 2013, 21
- Tall 2013, 16
- Tall 2013, 17
- Tall 2013, 18-19
- Tall 2013, 22-23
- Tall 2013, 24-25
- Tall 2013, 18-20
- Tall 2013, 17, 19
- Tall 2013, 28