Doing Mathematics as a Vehicle for Developing

Secondary Preservice Math Teachers Knowledge of

Mathematics for Teaching

- Gail Burrill
- Michigan State University
- Burrill_at_msu.edu

How much do all 3 chickens weigh? Each chicken?

Kindt et al, 2006

Pedagogical Content Knowledge Lee Shulman, 1986,

pp. 9-10

- For the most regularly taught topics in ones

subject area - The most useful representations of ideas
- The most powerful analogies, illustrations,

examples and demonstrations - Ways of representing and formulating the subject

that make it comprehensible to others - A veritable armamentarium of alternative forms of

representation - Understanding of why certain concepts are easy or

difficult to learn

Mathematical Knowledge for TeachingDeborah Ball

Hyman Bass, 2000

- a kind of understanding ..not something a

mathematician would have, but neither would be

part of a high school social studies teachers

knowledge - teaching is a form of mathematical work

involves a steady stream of mathematical problems

that teachers must solve - Features include unpacked knowledge,

connectedness across mathematical domains and

over time (seeing mathematical horizons)

Mathematical Knowledge for teaching

- Trimming- making mathematics available yet

retaining mathematical integrity - Unpacking-making the math explicit
- Making connections visible- within and across

mathematical domains - Using visualization to scaffold learning
- Considering curricular trajectories
- Flexibly moving among strategies/ approaches
- adapted from Ferrini-Mundy et al, 2004

Secondary Preservice Program at MSU

- Three precursor general ed courses
- Year-long methods course (4 hours a week) as a

senior blended with 4 hours per week in the field

and 2 hours a week of teaching lab, special ed

and minor - Mathematics Majors
- Post graduate fifth year-long internship program
- General secondary program goals- no specific

guidelines for math

Methods Course

- First semester
- - Observing teaching
- - Curriculum
- - Designing lessons
- Second semester
- - Equity
- - Assessment
- - Designing lessons

Goals

Course Goals

- Deepen and connect mathematical content knowledge

with student mathematical understanding. - Analyze from a new perspective what mathematics

is and what it means to learn, do and teach

mathematics. - Learn to listen to and look at students work as

a way to inform teaching, using evidence from

these to make decisions.

Adapted from Roneau Taylor, 2007

Goals

Course Goals

- Learn to design and implement lessons to engage

students in learning (tasks, sequence, discourse,

questioning, use of technology) - Learn to reflect on practice both from a

perspective as a teacher, a researcher, a

learner, and from the perspective of what you see

students learning - Recognize what is meant by equity and access to

quality mathematics for students, parents and

communities (including attention to policy)

Adapted from Roneau Taylor, 2007

Weekly math problems

- Quarterly problem sets
- Algebra
- Geometry
- Number
- Data and statistics
- Chosen to reflect the scope and depth of the area
- Assigned as homework,discussion managed by a pair

of randomly assigned students who meet with

instructors to discuss problem, solutions and

misconceptions

Algebra Geometry

- Beams
- Chickens
- Manatees
- Men/Women Salaries
- What is Changing
- Farmer Jack
- Jawbreakers

Construct rhombi Minimize distance Minimize area

triangle Paper folding Isosceles Triangle Car and

Boat

Problem characteristics

- Accessible by different approaches at the same

level - Accessible by different mathematical approaches
- Surface mathematical connections
- Usually involve a connection between symbols and

some other representation - Provide opportunities to surface misconceptions
- Lend themselves to exploiting different ways to

manage student mathematical discussions - Different types or nature of problems

Different tasks

Sum is more than the parts - confidence

interval Multiple interpretations that lead to

thinking hard about the mathematics Patterns

emerge across different problems - simulations

Make concept explicit -construct

rhombi Constructing own problems -What is

changing?

Different mathematical approaches

A rope is attached from a car on a pier or wharf

to a boat that is in the water. If the car drives

forward a distance d, will the boat be pulled

through a distance that is greater than d,

less than d or equal to d?

Source unknown

Strategies

- Calculus
- Trigonometry
- Pythagorean Theorem
- Coordinate geometry
- Triangle theorems

A B

d

C-d

C

A

A

B

- A2B2 C2 A2B2 (C-d)2
- If B B-d, then boat would have moved

horizontally exactly d. If BgtB-d, the boat

would move less than d if B lt B-d, then the

boat would move a horizontal distance greater

than d.

Surface mathematical connections

Making connections

- How many handshakes are possible between 2

people? What about 3, 4, 5, 6, and 7 people?

Try to come up with an equation for n number of

people. Make a list or table of the number of

possible handshakes for each amount of people.

Do you know what these numbers are?

Making connections

- Study the table of Pythagorean triples.
- Make a conjecture about all of the Pythagorean

triples that have two consecutive integers as a

leg and the hypotenuse that is not true for all

Pythagorean triples.

Making connections

- Suppose you have a bag with two different colors

of chips in it, red and blue. If you draw two

chips from the bag without replacement, how many

of each color chip do you need to have in the bag

in order for the probability of getting two chips

of the same color to equal the probability of

getting two chips, one of each color.

Making connections

- Find the pattern if the sequence continues. Find

an equation for the number of dots in the nth

figure. Make a list of the number of dots for

the first 6 figures. Do you know what these

numbers are? - Figure 1 Figure 2

Figure 3

Manage discussions

Isosceles Triangle

Given the isosceles triangle ABC where AB BC

12. AC is 13. BD is the altitude to AC, and D

is on AC. AE is the altitude to BC, and E is on

BC. Find DE

Isosceles Triangle

- students check the papers of their peers. a

great way to increase the understanding.Three

indicators of understanding communicate a

concept to another person, reflect on a concept

meaningfully, or apply a concept to a new

situation, When a student is asking questions

of the original paper owner the two are

communicating about math, conveying some

understanding. The grader is reflecting about the

method the first student used to solve the

problem and the original student reflects about

the comments and questions posed by the grader.

If the methods of solving are different they have

to look in detail at how someone else did the

problem.

Preservice student

Student designed problems

What is Changing?

- A problem from Japan
- In the figure, as the step changes, also

changes.

Step 1 2 3

Peterson, 2006

What is changing?

- Area
- Perimeter
- Length of longest side
- Number of intersections
- Number of right angles
- Sum of interior angles
- Number of parallel line segments
- Number of squares
- .

What is the rule and why?

- Number of squares

Step 1 2 3

Instruction Managing solutions

Patterns/Reasoning Proof

- What constitutes valid justification?
- Lack of connection to a geometric scheme that

established a relation between the rule and the

context. - Focus on particular values rather than making

generalizations - Inability to generalize across contexts (Lanin,

2005) - Algebraic notation often confusing and not used

(Zazskis Liljedah, 2002)

Farmer Jack

- Farmer Jack harvested 30,000 bushels of corn over

a ten-year period. He wanted to make a table

showing that he was a good farmer and that his

harvest had increased by the same amount each

year. Create Farmer Jacks table for the ten

year period. (Burrill, 2004)

Burrill, 2004

Using Variables

year Bushes per year Total

1 x

2 xx

3 xxx

4 4x

5 5x

6 6x

7 7x

8 8x

9 9x

10 10x

Total

Burrill, 2004

Burrill, 2004

Burrill, 2004

Let d be the yearly increase and an be the

amount harvested in year n. Then an1 and and

an a1 (n-1)d. The condition is that the 10

year total harvest is 30000 bushels, thus, S10

?an 30000 where S10 is the total number of

bushels after 10 years. Now, Sn (n/2)(a1an),

so S10 (10/2)(a1a10) 5(a1 a1 9d) 30000.

So 2a19d 6000. Any pair (a,d) where a and d

are both greater than 0 will produce a suitable

table. There are an infinite number of tables if

you do not restrict the values to be positive

integers.

Burrill, 2004

Research on Functions

- Teaching issues
- Students accept different answers to same problem

rather than reject a procedure they feel is

correct or explore why the difference (Sfard

Linchveski, 1994) - Form has consequences for learning
- (y mx b vs y b x(m)
- point slope form-yy1 m(x-x1) (Confrey Smith,

1994)

A disconnect that needs explaining

Knowledge for Teaching

- Unpacking the mathematical story
- Making connections
- Curricular knowledge
- Making assumptions explicit

Knowledge for teaching?

Misconceptions

- They chose solutions that built off of one

another, and the first solution was actually a

misconception and the last was a general solution

to the problem. JJ presented his misconception

first and admitted that he did it wrong. He

went through his thought process and then

explained how he figured out it was a

misconception. After the solutions had been

presented the class talked about how the

misconception helps other students who also had

this misconception feel justified that it wasnt

just them who had the mistake. Before this

course I couldnt think of why you would want to

show a misconception to the class, but I now

understand that talking about a misconception can

be used to help students understand. If a

student can explain what they have done wrong in

a problem, it means that they have learned

something.

Preservice student

Managing discussions

- As the students were writing up their solutions,

the rest of the class was supposed to figure out

the different solutions presented. This was

discussed in class as a way to keep all the

students engaged in the lesson. Watching the

video, it seems this might not be the best way to

keep students engaged because most of the class

was no longer looking at the solutions instead

they were having side conversations with one

another.

Preservice student

Defending thinking- evidence of understanding

- students were asked to do a think, pair, share

discussion. The students thought individually

about the problem as homework, came to class with

their completed proofs, paired off and each pair

discussed how they did the problem. The pairs

picked one proof to put up on the board, and

students walked around the room and took notes

about the other proofs. - After the gallery walk the students were brought

back together, and asked questions about what

they didnt get directly to the pair who wrote

the proof. The teacher asked questions of them,

too.

Preservice Student

habits of mind

Need for precision Vocabulary expression/equatio

n construct/draw lines are similar Trimming d

ivision never makes bigger a1 in recursive

definitions

habits of mind

The nature and role of proof mix

converse/statement assume what proving prove by

example prove by pattern Definitions Assumptions

and their consequences

habits of mind

Doing math is a way of thinking More than

routine procedures Problems out of context of

unit Takes time Errors can be productive

habits of mind

Not all math is equal underlying concepts should

drive instruction Not all solutions are equal

habits of mind

Math makes sense Chickens Ratio problem Farmer

Jack

Making connections

- Solve each problem using at least two different

approaches students might use. - 1. Which is the best buy for barbecue sauce
- 18 oz at 79 cents or 14 oz at 81 cents?

NRC, 2001

Polyas Ten Commandments

- Read faces of students
- Give students know how, attitudes of mind,

habit of methodical work - Let students guess before you tell them
- Suggest it do not force it down their throats

(Polya, 1965, p. 116)

Polyas Ten Commandments

- Be interested in the subject
- Know the subject
- Know about ways of learning
- Let students learn guessing
- Let students learn proving
- Look at features of problems that suggest

solution methods (Polya, 1965,p. 116)

References

- Roneau, R. Taylor, T. (2007). Presession

working grouop at Association of Mathematics

Teacher Educators Annual meeting. - Ball, D.L. Bass, H. (2000). Interweaving

content and pedagogy in teaching and learning to

teach Knowing and using mathematics. In J. - Burrill, G. (2004). Mathematical Tasks that

Promote Thinking and Reasoning The Case of

Farmer Jack in Mathematik lehren - Confery, J. Smith, E. (1994). Exponential

functions, rates of change, and the

multiplicative unit. Educational Studies in

Mathematics. 26 135-164. - Ferrini-Mundy, J., Floden, R., McCrory, Burrill,

G., Sandhow, D. (2004). Knowledge for

teaching school algebra challenges in developing

in analytic framework. unpublished paper - Kazemi, E. Franke, Megan L. (2004). Teacher

learning in mathematics using student work to

promote collective inquiry. Journal of

Mathematics Teacher Education, 7, 203-235.

- Kindt, M., Abels, M., Meyer, M., Pligge, M.

(2006). Comparing Quantities. In Wisconsin Center

for Education Research Freudenthal Institute

(Eds.), Mathematics in context. Chicago

Encyclopedia Britannica - Lannin, John K. (2005). Generalization and

justification the challenge of introducing

algebraic reasoning through patterning

activities. Mathematical Thinking and Learning,

73(7), 231-258. - National Research Council. (1999). How People

Learn Bain, mind, experience,and school.

Bransford, J. D., Brown, A. L., Cocking, R. R.

(Eds.). Washington, DC National Academy Press. - Polya, G. (1965). Mathematical discovery On

understanding, learning, and teaching problem

solving. - Peterson, B. (2006) Linear and Quadratic Change

A problem from Japan. The Mathematics Teacher,

Vol 100, No. 3. PP. 206-212. - Sfard, A., Linchevski, L. (1994). Between

Arithmetic and Algebra In the search of a

missing link. The case of equations and

inequalities. Rendicondi del Seminario

Matematico, 52 (3), 279-307.

- Shulman, L.S. (1986). Those who understand

Knowledge growth in teaching. Educational

Researcher. 15 (2) 4 - 14. - Zazkis, R. Liljedahl, P. (2002). Generalization

of patterns the tension between algebraic

thinking and algebraic notation. Educational

Studies in Mathematics 49, 379 402.