Reconceptualizing Mathematics: Courses for Prospective and Practicing Teachers Susan D. Nickerson Michael Maxon San Diego State University

Teachers matter. From a carefully chosen sample of 2525 adults, representing a cross- section of U. S. adults, an overwhelming majority agreed that improving the quality of teaching was the most important way to improve public education.

Every Child Mathematically Proficient from the Learning First Alliance, 1998 Before It’s Too Late: A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century, 2000 What Matters Most: Teaching for America’s Future and No Dream Denied from the National Commission on Teaching and American’s Future, 1996 & 2003 The Mathematical Education of Teachers from the Conference Board of the Mathematical Sciences, 2001 Mathematical Proficiency for All Students from the RAND Mathematics Study Panel, 2003 Educating Teachers of Science, Mathematics, and Technology from the National Research Council, 2001 Undergraduate Programs and Courses in the Mathematical Sciences from MAA

Consider the following: A teacher wants to pose a question that will show him whether his students understand how to put a series of decimals in order from smallest to largest. Which of the following sets of decimal numbers will help him assess whether his pupils understand how to order decimals? Choose each that you think will be useful for his purpose and explain why Or would each of these work equally well for this purpose?

There is a growing understanding that the mathematics of the elementary and middle school is not trivial, and that teachers need more preparation and different preparation than has been common.

Looking ahead Recommendations Our focus with examples of content and instructional possibilities Guiding Questions Examples of our PD structures and delivery

In 1998, the National Research Council (NRC) appointed a committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.

The Committee’s 2001 report, Educating Teachers of Mathematics, Science, and Technology, includes recommendations about the characteristics teacher education programs in mathematics and science should exhibit.

Programs should have the following features: Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers; Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach; Teach content through the perspectives and methods of inquiry and problem- solving

Programs should have the following features: Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers; Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach; Teach content through the perspectives and methods of inquiry and problem- solving

Collaborative endeavors are a part of our: Pre-service courses Professional development: summer courses, usually at the university (3 days to 2 weeks) partnership with several districts, urban & rural classes through the university on-line hybrid courses

Integrating Learning Mathematics with Practice Director of Mathematics Math Resource Teacher Site Mathematics Administrators Mathematics Instructors (SDSU) Math Teacher Ed Instructors (SDSU) Teachers in program All Teachers at site

Programs should have the following features: Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers; Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach; Teach content through the perspectives and methods of inquiry and problem- solving

What content preparation do teachers need to have to teach middle school mathematics well? The 2001 document from CBMS, The Mathematical Education of Teachers recommends that for teaching middle school mathematics: At least 21 semester hours of mathematics, some of which should include a study of elementary mathematics, some of which should focus on the middle grades, and some of which include the study of mathematics proper.

Content recommendations in MET are made in four areas: 1. Number and Operations 2. Algebra and Functions 3. Measurement and Geometry 4. Probability and Statistics

1.Number and Operations Understand and be able to explain the mathematics that underlies the procedures used for operating on whole numbers and rational numbers.

Example 1: Multiplication of Fractions “Juanita had mowed 4/5 of the lawn, and her brother Jaime had raked 2/3 of the mowed part. What part of the lawn had been mowed and raked?”

Example 2: Why do we invert and multiply when dividing fractions? First, divide by the unit fraction: 1 ÷ 1/3 1 ÷ 1/3 is 3, or 3/1 So 2 ÷ 1/3 is twice 3, or 6, or in general k ÷ 1/n = k n Write a story problem that can be represented and solved by 2 1/2 ÷ 1/3

2. Algebra and Functions 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.

Coping strategies: 1. Just add 2. Guess at the operation to be used

Limited Strategies 3. Look at the number sizes and use those to tell you which operation to use. 4. Try all operations and choose the most reasonable answer. 5. Look for “key words.” 6. Decide whether the answer should be larger or smaller than the given numbers, then decide on the operation.

Desired strategy: 7. Choose the operations with the meaning that fits the story. Perhaps draw a picture to help understand the problem.

Consider this problem: Dieter A: I lost 1/8 of my weight. I lost 19 pounds. Dieter B: I lost 1/6 of my weight, and now you weigh 2 pounds more than I do. How much weight did Dieter B lose?

First, make a list of relevant quantities Dieter A’s weight before the diet Dieter A’s weight after the diet Fraction of weight lost by A Amount of weight lost by A Difference in weight of A and B before diet Difference in weight after diet Etc. Dieter A’s weight before the diet Dieter A’s weight after the diet Fraction of weight lost by A Amount of weight lost by A Difference in weight of A and B before diet Difference in weight after diet Etc.

Now we consider the values of the quantities Dieter A’s weight before the diet: ? Dieter A’s weight after the diet: ? Fraction of weight lost by A: 1/8 Amount of weight lost by A: 19 pounds Difference in weight of A and B before diets: ? Difference in weights after diets: 2 pounds (A is 2 pounds less than B Etc. Dieter A’s weight before the diet: ? Dieter A’s weight after the diet: ? Fraction of weight lost by A: 1/8 Amount of weight lost by A: 19 pounds Difference in weight of A and B before diets: ? Difference in weights after diets: 2 pounds (A is 2 pounds less than B Etc.

Dieter A’s original weight (before diet) Before diet Shows weight loss of 1/8 of original diet. Weight lost was 19 lb. After diet

A’s weight before diet: 19 x 8 = 152 A’s weight after the diet 152–19=133 B’s weight after the diet: = is 5/6 of B’s weight before the diet so B weighed 162 pounds B lost 162 – 135 = 27 pounds

2. Algebra and Functions Recognize change patterns associated with linear, quadratic, and exponential functions. Example: Growing DotsGrowing Dots

3. Measurement and Geometry Identify common two-and three dimensional shapes and list their basic characteristics and properties Example here GSP quadrilaterals vennquadrilaterals venn

4. Data Analysis, Statistics, and Probability Draw conclusions with measurements of uncertainty by applying basic concepts of probability Ex: three card pokerthree card poker

Programs should have the following features: Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers; Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach; Teach content through the perspectives and methods of inquiry and problem- solving

Cynthia has 1 cup of sugar and each recipe requires: a) 1/2, b) 1/3, c)2/3, d) 3/4, e) 4/5 of a cup of sugar. How many recipes can she make? Model and draw each case. What number sentence describes each? Later generalize for more or less than one cup of sugar.

In a report by the MAA Committee on the Undergraduate Preparation in Mathematics (2004): “..does not mean the same thing as preparation for the further study of college mathematics. For example, while prospective teachers need knowledge of algebra..the traditional college algebra course with primary emphasis in developing algebra skills does not meet the needs of elementary [and middle school] teachers.”

Looking ahead Recommendations Our focus with examples of content and instructional possibilities Guiding Questions Examples of our PD structures and delivery

Decisions about what to include rely on answers to guiding questions Pre-service courses Professional development: summer courses, usually at the university (3 days to 2 weeks) partnership with districts classes through the university on-line hybrid courses

Guiding Questions: 1)What is the content at their grade level? With what have they had only superficial exposure? What would be difficult to learn from the curriculum? 2)How does this content need to be extended? 3)How does the research literature characterize the difficulties & student misconceptions? 4)Does this group of teachers have particular needs?

1)What is the content at their grade level? With what have they had only superficial exposure? What would be difficult to learn from the curriculum? Ratio and proportional reasoning Minimal and sufficient definitions of quadrilaterals

Procedures can sometimes be learned from the curriculum. When we talk about content we are talking about conceptual understanding and procedural fluency.

2) How does this content need to be extended?

3) How does the research literature characterize the difficulties & K-12 student misconceptions? Specialized knowledge for teaching

Difficulties in algebra include Misunderstanding the equals sign Comprehending use of literal symbols as generalized numbers or variables Expressing relationships in a variety of ways such as tables, graphs, and equations Understanding the role of the unit Ratio and Proportional reasoning Misunderstanding the equals sign Comprehending use of literal symbols as generalized numbers or variables Expressing relationships in a variety of ways such as tables, graphs, and equations Understanding the role of the unit Ratio and Proportional reasoning

Misunderstanding the equals sign Students tend to misunderstand the equal sign as a signal for “doing something” rather than a relational symbol of equivalence.

Comprehending the many uses of literal symbols A=L x W 40=5x 2a + 2b = 2(a + b) Y = 3x + 5 A=L x W 40=5x 2a + 2b = 2(a + b) Y = 3x + 5

Expressing relationships in a tables, graphs, and equations

Understanding the unit

Ratio and Proportional Reasoning Proportional reasoning has been called a watershed concept, the “capstone of elementary arithmetic and the cornerstone of all that is to follow” Situations that call for understanding the close relationship between fractions and ratios, both of which are represented by the same notation, cause particular problems. Proportional reasoning has been called a watershed concept, the “capstone of elementary arithmetic and the cornerstone of all that is to follow” Situations that call for understanding the close relationship between fractions and ratios, both of which are represented by the same notation, cause particular problems.

A donut machine produces 60 donuts every 5 minutes. How many donuts does it produce in an hour? a. Identify the rate in this problem. What unit rate is associated with this rate? b. Write a proportional statement that could be used to find the number of donuts produced in an hour. c. Made a graph of the donuts produced every 5 minutes and a graph of the donuts produced every minute. Are the graphs different? Why or why not? d. What is the slope of each line?

Spanning the curriculum Students must have an understanding of how to write a useable mathematical definition and establish equivalence among definitions. Students develop a disposition for problem solving. Students must learn to use valid reasoning to reach and justify conclusions. Students must have an understanding of how to write a useable mathematical definition and establish equivalence among definitions. Students develop a disposition for problem solving. Students must learn to use valid reasoning to reach and justify conclusions.

4) Does this group of teachers have particular needs?

Looking ahead Recommendations Our focus with examples of content and instructional possibilities Guiding Questions Examples of our PD structures and delivery

Pre-service courses for middle school math teachers: 5 courses for elementary: Number & Op (3) Algebra (2 or 3) Geo & Meas (3) Prob & Stats (2 or 3) Children’s mathematical thinking (1.5) 2 courses for middle school: Math for Middle School Teach (3) Transition to Higher Math or History of Math (3) Other courses: Calculus I Transition to Higher Math or History of Math (3) Statistics (3) Number theory (3) Min 32 units; 3 in each content area

Math for Middle School Teachers Pre-service course Guided by NCTM Curriculum Focal Points (Gr 6-8) Division of Fractions Operations with Integers Ratio & Proportion Functions and relations Children’s thinking in these same content areas

Professional development courses Immediacy. Teachers back in classes tomorrow, bring questions about their own experiences and students. Collaborative team encompasses school administrators and educators with a focus on scaffolding of ideas Usually more flexibility to design course particular to group/team of teachers. Connections tight among concepts.

Designing courses for professional development Summer courses Partner with school district 1 to offer 4 courses to qualify teachers for NCLB Partner with School District 2 to provide PD to 7th through Alg 1 in pull out days (only 30 hours) each of 2 years

Designing courses for professional development Partner with School District 3 to provide PD to middle school (either pull-out or after school) Partner with School District 4 to provide PD to (summer and after school) On-line planned for middle school

Teach content so they know well, understand deeply, and use effectively the fundamental content they teach Teach content through the perspectives and methods of inquiry and problem- solving It’s not just what we teach teachers, but how we teach teachers.

Discussion

That students do not regard algebra as a sense-making and useful subject is often due to the way that algebra is often taught and the way that students are prepared for algebra. Middle school teachers themselves must make sense of algebra and its underpinnings.

“Algebra has been experienced as an unpleasant, even alienating event, mostly about manipulating symbols that do not stand for anything. (But) algebraic reasoning in its many forms, and the use of algebraic representations such as graphs, tables, spreadsheets and traditional formulas, are among the most powerful intellectual tools that our civilization has developed.” (Kaput 1999)