Summary of Chapter 4: The Mathematical Education of Teachers (CBMS)
Recommendations for Middle Grades Teacher Preparation

"The more sophisticated content of middle grades mathematics necessitates that mathematics specialists teach in these grades and that these specialists have a well-developed understanding of the mathematics they teach."

Keypoints:

• at least 21 semester-hours of mathematics for prospective middle grades mathematics teachers


• courses must be designed that will lead prospective teachers to develop a deep understanding of the mathematics they will be teaching. 


• coursework should be carefully selected from the options offered by the department, and would require a precalculus or college algebra background. 


• Number theory and discrete mathematics can offer teachers an opportunity to explore in depth many of the topics they will teach. 


• A history of mathematics course can provide middle grades teachers with an understanding of the background and historical development of many topics in the middle grades curriculum. 


• A mathematical modeling course, depending on the level and substance of the course, can provide prospective teachers with understanding of the ways in which mathematics can be applied. 


• If the prospective teachers are likely to teach algebra, coursework in linear algebra and modern algebra would be appropriate. 


• If, in addition, the teachers might be expected to teach a full-year course in geometry, then they should have the same geometry coursework as prospective secondary teachers. (These options would most likely require more than 21 semester-hours.)
 

"Making sense of mathematics should be a cross-cutting theme throughout K-12 mathematics instruction and in courses for prospective teachers. For many prospective teachers learning mathematics has meant only learning its procedures and, they may, in fact, have been rewarded with high grades in mathematics for their fluency in using procedures. Although procedural fluency is necessary, it is not an adequate foundation for teaching mathematics. An orientation towards making sense of mathematics must be considered fundamental both to learning and to teaching mathematics."

Number and Operations

Coursework for prospective middle grades teachers should lead them to:

• Understand and be able to explain the mathematics that underlies the procedures used for operating on whole numbers and rational numbers.
  
• Understand and be able to explain the distinctions among whole numbers, integers, rational numbers, and real numbers, how they are placed on the number line, and how field axioms hold or do not hold depending on the types of numbers being used.
  
• Convert easily among fractions, decimals, and percents.
  
• Demonstrate facility in using number and operation properties, including mental computation and computational estimation.
  
• Understand and be able to explain fundamental ideas of number theory as they apply to middle school mathematics.
  
• Make sense of large and small numbers and use scientific notation.
  
• Apply proportions appropriately and provide explanations.
  

Algebra and Functions

Prospective middle grades teachers should:

• Understand and be able to work with algebra as a symbolic language, as a problem solving tool, as generalized arithmetic, as generalized quantitative reasoning, as a study of functions, relations, and variation, and as a way of modeling physical situations.
  
• Develop an understanding of variables and functions, especially of different equivalent relationships between variables.
  
• Understand linearity and how linear functions can illustrate proportional relationships.
  
• Recognize change patterns associated with linear, quadratic, and exponential functions.
  
• Demonstrate algebraic skills and be able to give a rationale for common algebraic procedures.

Measurement and Geometry

Prospective middle grades teachers should be able to:

• Identify common two- and three-dimensional shapes and list their basic characteristics and properties.
  
• Make conjectures about geometric shapes and then prove or disprove them.
  
• Demonstrate how rigid motions in the plane result in congruent figures.
  
• Demonstrate understanding of how similar figures result from a dilation, and the role of proportional relationships in determining similarity.
  
• Demonstrate ability to visualize and solve problems involving two- and three-dimensional objects.
  
• Connect geometry to other mathematical topics, and to nature and art.
  
• Understand the common forms of measurement and choose appropriate tools and units for measuring.
  
• Understand, derive, and use measurement techniques and formulas.

Data Analysis, Statistics, and Probability

Coursework should provide prospective teachers avenues to:

• Design simple investigations and collect data (through random sampling or random assignment to treatments) to answer specific questions.
  
• Understand and use a variety of ways to display data.
  
• Explore and interpret data by observing patterns and departures from patterns in data displays, particularly patterns related to spread and variability.
  
• Anticipate patterns by studying, through theory and simulation, those produced by simple probability models.
  
• Draw conclusions with measures of uncertainty by applying basic concepts of probability.
  
• Know something about current uses of statistics and probability in many fields.