Reconceptualizing Mathematics: Courses for Prospective and Practicing Teachers Susan D. Nickerson...

Reconceptualizing Mathematics: Courses for Prospective and

Practicing Teachers

Susan D. NickersonMichael Maxon

San Diego State University

Reconceptualizing Mathematics: Courses for Prospective and

Practicing Teachers

Susan D. NickersonMichael 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.

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

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.

.123 1.5 2 .56

.60 2.53 3.14 .45

.6 4.25 .565 2.5

Or would each of these work equally well for this purpose?

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.

.123 1.5 2 .56

.60 2.53 3.14 .45

.6 4.25 .565 2.5

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.

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 aheadRecommendations

Our focus with examples of content and instructional possibilities

Guiding QuestionsExamples of our PD structures

and delivery

Looking aheadRecommendations

Our focus with examples of content and instructional possibilities

Guiding QuestionsExamples 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.

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.

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;

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

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

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

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.

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 Operations2. Algebra and Functions3. Measurement and Geometry4. Probability and Statistics

Content recommendations in MET are made in four areas:

1. Number and Operations2. Algebra and Functions3. Measurement and Geometry4. 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.

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 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?”

23×45=2×43×5=815 45 refers to th eentire lawn 815 refers to th eentire lawn 23 refers to the partof the lawn that has been mowed

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

Continuing in this fashion, find 2 1/2 ÷1/3. How many thirds are in 2 1/2? 1 2 3 4 5 6 7 1/2 2 1 /2 ÷ 1/3 = 2 1/2 x 3/1 = 7 1/2

Generalizing, ab÷1c=ab×c1=acb.

Next, consider 1 ÷ 2/3 ask:

How ma 2/ny 3 are in 1 ?whole unit

There are 1 1/2, or 3/2.

2 ÷ 2/3 mus t hav e twice a s man y 2/3 a s doe s 1,

S o, 2 x 3/2 is 6/2 or 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.

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.

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.

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

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.

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.

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?

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 quantitiesFirst, 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 quantitiesNow 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)

Dieter A’s original weight (before diet)

Before dietShows 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: 133 + 2 = 135

135 is 5/6 of B’s weight before the diet so B weighed 162 pounds

B lost 162 – 135 = 27 pounds

A’s weight before diet: 19 x 8 = 152

A’s weight after the diet 152–19=133

B’s weight after the diet: 133 + 2 = 135

135 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 Dots

2. Algebra and Functions• Recognize change patterns associated with linear, quadratic, and exponential functions.

Example: Growing Dots

3. Measurement and Geometry• Identify common two-and three dimensional shapes and list their basic characteristics and properties

Example here GSP quadrilaterals venn

3. Measurement and Geometry• Identify common two-and three dimensional shapes and list their basic characteristics and properties

Example here GSP quadrilaterals venn

4. Data Analysis, Statistics, and Probability• Draw conclusions with measurements of uncertainty by applying basic concepts of probability

Ex: three card poker

4. Data Analysis, Statistics, and Probability• Draw conclusions with measurements of uncertainty by applying basic concepts of probability

Ex: three 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

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.

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.”

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 aheadRecommendations

Our focus with examples of content and instructional possibilities

Guiding QuestionsExamples of our PD structures

and delivery

Looking aheadRecommendations

Our focus with examples of content and instructional possibilities

Guiding QuestionsExamples of our PD structures

and delivery

Decisions about what to include rely on answers to guiding questionsDecisions 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 includeDifficulties 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

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

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 equationsExpressing relationships in a tables, graphs, and equations

Three Burning Candles

Group 1 Group 2 Group 3

a. Which set of candles is represented by the graph?

b. What was the starting height of each candle?

c. At what rate did each candle burn? How do you know?

d. What is the slope of each line?

e. Write an equation to represent each candle’s burning.

f. Write an equation for a fourth candle, Candle D,

whose graph is parallel to the graphs of Candles A, B, and C.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Dashed line gives Rabbit's

average speed.

Turtle: 6 m/s for 100 m

Rabbit: 8 m/s over (50 m) and 4 m/s back (50 m).

Average speed is 100 ÷ 18 3/4 = 5 1/3 m/s.

T

R

R

2018161412108642

time (seconds)

distance

(meters)

110

100

90

80

70

60

50

40

30

20

10

Understanding the unitUnderstanding the unita. Can you see 3/5 of something in this

picture? Where? Be explicit. (3/5 of what?)

b. Ca n you se e 5/3 of some thing in this pictu re? Whe re ?

How did you cha nge the way you looke d at the picture

in order to see 5/ 3?

c. Can you see 2/3 of so meth ing in this pictu re? Whe re ?

d. Ca n you see 5/ 3 of 3/5 in this pict ure ? How did you have to cha nge the way you looked at the pict ure in order to s ee

5/3 of 3/5

e. Can you see 1 ÷ 3/5?

Ratio and Proportional Reasoning

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?

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 curriculumSpanning 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 aheadRecommendations

Our focus with examples of content and instructional possibilities

Guiding QuestionsExamples of our PD structures

and delivery

Looking aheadRecommendations

Our focus with examples of content and instructional possibilities

Guiding QuestionsExamples 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 7-12 (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.

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)

“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)