On teaching math
Suggested standards for math teaching*
- Model sense-making. Practice and model the understanding and conviction that mathematics makes sense.
- Model math learning. Practice and model autonomous as well as collaborative mathematical learning.
- Teach origins. Teach the rationales from which mathematical concepts, procedures or problems arise, e.g., pursuit of efficiency and elegance, historical development of a particular concept or problem, concrete problems that stimulated mathematical work.
- Teach mathematical reasoning. Teach the accepted techniques used to derive or justify knowledge in a mathematical system, e.g., use of specific examples or counterexamples, proof by induction, or the role of definitions, conjectures and proofs within mathematics
- Teach representations. Employ a range of mathematical representations--images used to picture or describe an object or process--to make sense of problems. Relate representations to each other, and to results.
- Teach applications. Represent and model that fundamental mathematics developed to solve a particular problem can often be applied to a range of seemingly unrelated problems.
- Teach specific subject matter knowledge: numbers and operations, number theory, algebra, geometry and measurement, data analysis and probability and statistics, calculus and analysis.
Commentary on the standards*
Mathematics often provides insight into difficult problems and yields concise, well-defined processes with which to analyze those problems effectively. Mathematical pursuits also include the seeking of greater efficiency and elegance in solving existing problems. Nowadays, specific mathematical areas and concepts also result from advances in technology that allow mathematicians to begin to address new problems or to look at old ones in new ways.
In general, a mathematical approach involves defining a problem through conjecturing in an established mathematical area. Conjecturing may be supported by technology, by compelling ideas based on past work, computation or pattern exploration. The precise explanation of a mathematical solution, regardless of the specific problem, starts with definitions and axioms and builds from there to basic properties, which are then used to obtain desired results. Further understanding of the mathematical ideas so developed can then result from new theorems proven within the existing system, or from exploration---driven by analysis of the system's limitations---of its underlying assumptions, which can lead to new axioms, definitions and theorems.
These aspects of mathematical ways of thinking are interrelated and build upon each other. A new problem gives rise to new definitions that give rise to new operations and properties that give us new theorems and results, and those results themselves may lead to further definitions and results. The process never ends.
Knowledge of mathematics for prospective teachers consists of five aspects that are involved in this mathematical approach: origins, mathematical reasoning, specific knowledge, representations and applications. Below we elaborate on each type of knowledge:
The word origins refers to the rationale from which a mathematical concept, procedure or problem arises. This rationale may include the seeking of greater efficiency and elegance in solving existing problems, and it may include historical background on a particular concept or problem. Origins may also be psychological or pedagogical. Mathematics often originates in concrete problems. The resulting mathematical ideas may evolve further in order to address a broader range of problems perhaps not tied to the original one.
One of the simplest examples prospective teachers should understand is the evolution of number systems. The natural numbers are sufficient for counting but not for measuring length or for solving the simplest linear equations. For these tasks we need to extend the system of natural numbers to include the rational numbers. However, the rational numbers are inadequate for solving problems about area or for solving quadratic equations, which require construction of the real and complex numbers.
Mathematical reasoning refers to the accepted techniques used to derive or justify knowledge in a mathematical system. Thus, it includes different forms of mathematical argument (e.g., use of specific examples or counterexamples, proof by induction or proof by contradiction). Mathematical reasoning also includes aspects of logic (e.g,. the equivalence of an if-then statement and its contrapositive; the role of definitions, conjectures and proofs within mathematics; and consequences of changes in assumptions or definitions).
One way to understand a mathematical concept more fully is to represent it. Mathematical representations consist of the images used to picture or describe an object or process. Many mathematical objects or processes can be represented in many ways. One of the beauties of beginning to see mathematics as a whole is to see the interplay among various areas of the subject. A particular way of representing a problem or concept may lead to an especially efficient or enlightening result. We are able to use mathematics to make sense of problems by employing appropriate representations. Examples:
- We can use area models to represent multiplication of multi-digit numbers or binomial expressions or to illustrate probabilities.
- We can view the Binomial Theorem as a formal result about raising a polynomial to a power and employ the laws of algebra to give an inductive proof. Equally well, we can think of the Binomial Theorem as telling us how many subsets there are with a given number of elements, or as a result about probability.
- We can view matrices as rectangular arrays of numbers on which are defined certain algebraic operations, or we can view them as linear transformations. Matrices can represent stochastic processes or differential equations. Viewing linear transformations as matrices can advance further understanding through computational experimentation and visualization enhanced by computer technology.
Applications of mathematics abound. Fundamental mathematics developed to solve a particular problem can often be applied to seemingly unrelated problems. Consider these examples:
- Calculus was invented to solve problems involving motion, but now it is useful for many other situations.
- The notion of Fourier series, originally intended to solve problems in mathematical physics, plays a fundamental role in the analysis of signals.
Alternatively, one might use the opposite process of going from a particular problem, context or data set, to a generalization. This could involve existing mathematics or could require developing new mathematics (which connects to the theme of origins). For example:
- Problems about mixtures or population growth lead to abstractions about rates or proportional reasoning.
Experiences with both types of applications help students learn that mathematics both makes sense and can be used to make sense out of the real world.
Specific subject matter knowledge
Specific subject matter knowledge consists of the key assumptions, definitions, algorithms and theorems of the specific mathematical field being studied. At the K- 12 level the mathematical fields typically include numbers and operations, algebra, geometry and measurement, data analysis and probability. At MSU specific mathematics subject matter for teachers includes numbers and number theory, calculus and analysis, geometry, algebra and statistics.
*Excerpt from Teachers for a New Era. (2004). Teacher knowledge standards. East Lansing: Michigan State University.