Additional resources for mathematics education majors: readings, opinions, and connections with higher math


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Preservice mathematics teachers sometimes argue that the mathematics courses they take for their major (e.g proofs, linear algebra, or abstract algebra) have little bearing on their future careers: after all, they will be teaching 9th grade algebra and 10th grade geometry, not abstract algebra, and non-Euclidean geometry. They often ask the question that many students ask: "Why do I need to know this?" Of course, an answer can readily be given: these courses are part of the mathematical canon, and part of general mathematics education. However, there are other possible answers: understanding of higher-level math allows for better understanding of high school mathematics; teachers should know more math than what they will teach; topics that appear in higher-level math courses appear in a different guise in the high school curriculum as well: take for example groups, rings, and fields. The resources below elaborate on these topics. They begin to answer the question why calculus, introduction to proofs, geometry, linear algebra, statistics, and abstract algebra, are relevant to future teachers of mathematics.

The resources are especially relevant to those planning to teach in Washington State.

Contents:

Summer job options for preservice teachers:
Articles and blogs: Teaching mathematics:
Learning mathematics (standards and standardized tests):
Learning mathematics (content and careers):
Books (general):

MATH COURSES:

Pre-college mathematics: Calculus:

Apart from the fact that many high school teachers will teach it, calculus is also relevant to teaching other mathematics courses. For example, sequences and series are taught in Algebra 2. Consider this excerpt from the Washington State Standards for Algebra 2:

"...Second, they formalize their work with series as they learn to find the terms and partial sums of arithmetic series and the terms and partial and infinite sums of geometric series. This conceptual understanding of series lays an important foundation for understanding calculus. ...

A2.7.B Find the terms and partial sums of arithmetic and geometric series and the infinite sum for geometric series. Students build on the knowledge gained in Algebra 1 to find specific terms in a sequence and to express arithmetic and geometric sequences in both explicit and recursive forms."

The NCTM Principles and Standards for School Mathematics (2000) state:

"Increasingly, discussions of change are found in the popular press and news reports. Students should be able to interpret statements such as "the rate of inflation is decreasing." The study of change in grades 9–12 is intended to give students a deeper understanding of the ways in which changes in quantities can be represented mathematically and of the concept of rate of change.

Finally, according to The Mathematical Education of Teachers, calculus hones students' algebra skills. After all, calculus instructors often say that students don't fail calculus because of calculus, but because of algebra. Below is a list of resources that deal with calculus in the high school curriculum. Proofs:

Mathematics educators have long been proposing that proofs be introduced in the curriculum as early as early elementary grades. As all mathematicians know, proof is the foundation of mathematics. As one of the articles cited below (Mingus & Grassl) states,

According to Hersh (1993), "In the classroom, the purpose of proof is to explain. Enlightened use of proofs in the mathematics classroom aims to stimulate the students' understanding." It is helpful to temporarily adopt the broad definition that proof is a complete explanation, a convincing argument.... The National Council of Teachers of Mathematics Standards (NCTM, 1989) emphasized the need to teach children the basics of proof early in their education. In fact, two standards present for all grade levels are "Mathematics as Communication" and "Mathematics as Reasoning." For elementary students, the Standards suggested children be taught that justifying and explaining their reasoning as they arrive at a solution is as important as the solution. Furthermore, the Standards recommended that children be provided opportunities to practice communicating their thinking to assist them in forming links between their intuition and formal mathematical reasoning.

The NCTM Standards also state that "[r]easoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions, no matter what topic is being studied. In mathematically productive classroom environments, students should expect to explain and justify their conclusions. When questions such as, What are you doing? or Why does that make sense? are the norm in a mathematics classroom, students are able to clarify their thinking, to learn new ways to look at and think about situations, and to develop standards for high-quality mathematical reasoning (Collins et al. 1989)."

The resources below justify proofs in the high school curriculum, give examples of activities that can be used in the classroom, and discuss misconceptions university and high school students have about proof.

Geometry:

Nobody can argue that future math teachers need to take a geometry class. The links below are to Washington State and NCTM standards, to provide comparison between high school geometry curriculum and the content of a college geometry course. The other readings are related to teaching high school geometry.
Linear algebra:

"It is not surprising to see the linear and abstract algebra requirements for middle school and high school mathematics teachers. Beyond simple arithmetic, linear algebra is one of the most widely used mathematical subjects today. It has much in common with high school algebra and forms a bridge between high school algebra and abstract algebra. In terms of abstract algebra, classroom textbooks even at the elementary level reinforce the axioms (properties) of the whole numbers and the real number system. It is important for middle school and high school mathematics teachers to have an understanding and working knowledge of the mathematical structures that encompass these properties." (from the description of a linear and abstract algebra course for teachers at CSU, Stanislaus)

The NCTM Standards state:

"High school students can use their understanding of numbers to explore new systems, such as vectors and matrices. By working with examples that include forces or velocities, students can learn to appreciate vectors as a means of simultaneously representing magnitude and direction. Using matrices, students can also see connections among major strands of mathematics: they can use matrices to solve systems of linear equations, to represent geometric transformations (some of which can involve creating computer graphics), and to represent and analyze vertex-edge graphs.

Properties that hold in some systems may not hold in others. So teachers and students should explicitly discuss the associative, commutative, and distributive properties, and students should learn to examine whether those properties hold in the systems they study. The exploration of the properties of matrices may be particularly interesting, since the system of matrices is often the first that students encounter in which multiplication is not commutative.

Students should also extend their understanding of operations to number systems that are new to them. They should learn to represent two-dimensional vectors in the coordinate plane and determine vector sums. Dynamic geometry software can be used to illustrate the properties of vector addition. As students learn to represent systems of equations using matrices, they should recognize how operations on the matrices correspond to manipulations of such systems."

Below are some other interesting resources: Probability and statistics:

Calculus courses in high schools are increasingly replaced with statistics courses, while the middle school and high school curriculum abound with topics in probability and statistics.

The NCTM Standards (2000) have a lot to say about statistics. Here is a small sample:

"Students' experiences with surveys and experiments in lower grades should prepare them to consider issues of design. In high school, students should design surveys, observational studies, and experiments that take into consideration questions such as the following: Are the issues and questions clear and unambiguous? What is the population? How should the sample be selected? Is a stratified sample called for? What size should the sample be? Students should understand the concept of bias in surveys and ways to reduce bias, such as using randomization in selecting samples. Similarly, when students design experiments, they should begin to learn how to take into account the nature of the treatments, the selection of the experimental units, and the randomization used to assign treatments to units.

...Describing center, spread, and shape is essential to the analysis of both univariate and bivariate data. Students should be able to use a variety of summary statistics and graphical displays to analyze these characteristics.

The shape of a distribution of a single measurement variable can be analyzed using graphical displays such as histograms, dotplots, stem-and-leaf plots, or box plots. Students should be able to construct these graphs and select from among them to assist in understanding the data. They should comment on the overall shape of the plot and on points that do not fit the general shape. By examining these characteristics of the plots, students should be better able to explain differences in measures of center (such as mean or median) and spread (such as standard deviation or interquartile range). For example, students should recognize that the statement "the mean score on a test was 50 percent" may cover several situations, including the following: all scores are 50 percent; half the scores are 40 percent and half the scores are 60 percent; half the scores are 0 percent and half the scores are 100 percent; one score is 100 percent and 50 scores are 49 percent. Students should also recognize that the sample mean and median can differ greatly for a skewed distribution. They should understand that for data that are identified by categories—for example, gender, favorite color, or ethnic origin—bar graphs, pie charts, and summary tables often display information about the relative frequency or percent in each category.

A parameter is a single number that describes some aspect of an entire population, and a statistic is an estimate of that value computed from some sample of the population. To understand terms such as margin of error in opinion polls, it is necessary to understand how statistics, such as sample proportions, vary when different random samples are chosen from a population. Similarly, sample means computed from measurement data vary according to the random sample chosen, so it is important to understand the distribution of sample means in order to assess how well a specific sample mean estimates the population mean.

In high school, students can apply the concepts of probability to predict the likelihood of an event by constructing probability distributions for simple sample spaces. Students should be able to describe sample spaces such as the set of possible outcomes when four coins are tossed and the set of possibilities for the sum of the values on the faces that are down when two tetrahedral dice are rolled.

High school students should learn to identify mutually exclusive, joint, and conditional events by drawing on their knowledge of combinations, permutations, and counting to compute the probabilities associated with such events. They can use their understandings to address questions such as those in following series of examples.

Abstract algebra: