Doing Mathematics as a Vehicle for Developing Secondary Preservice Math Teachers’

Knowledge of Mathematics for Teaching

Gail Burrill Michigan State University

Burrill@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 one’s 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 Teaching

Deborah 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’ teacher’s 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 coursesYear-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 MajorsPost 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

•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.

Course Goals

Adapted from Roneau & Taylor, 2007

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)

Course Goals

Adapted from Roneau & Taylor, 2007

Weekly math problems

Quarterly problem sets• Algebra • Geometry• Number• Data and statisticsChosen 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

BeamsChickensManateesMen/Women SalariesWhat is ChangingFarmer JackJawbreakers

Construct rhombiMinimize distanceMinimize area trianglePaper foldingIsosceles TriangleCar and Boat

Problem characteristics

Accessible by different approaches at the same levelAccessible by different mathematical approachesSurface mathematical connections Usually involve a connection between symbols and

some other representationProvide opportunities to surface misconceptionsLend themselves to exploiting different ways to

manage student mathematical discussionsDifferent types or nature of problems

“Different” tasks

Sum is more than the parts- confidence interval

Multiple interpretations that lead to thinking hard about the mathematicsPatterns emerge across different problems

- simulations Make concept explicit

-construct rhombiConstructing 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

A2+B2 = C2 A2+B’2 = (C-d)2

If B’ = B-d, then boat would have moved horizontally exactly d. If B’>B-d, the boat would move less than d; if B’ < B-d, then the boat would move a horizontal distance greater than d.

A A

dC-dC

B’

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

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

A

B

C D

E

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 JapanIn 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 Jack’s table for the ten year period. (Burrill, 2004)

Solution I: ‘Mis-reading the Situation’0 01 3000 +30002 6000 +30003 9000 +30004 12000 +30005 15000 +30006 18000 +30007 21000 +30008 24000 +30009 27000 +300010 30000 +3000 Burrill, 2004

Solution II: ‘Dividing Into Equal Parts’Year Bushels

per yearTotalBushelsof corn

1 3000 30002 3000 60003 3000 90004 3000 120005 3000 150006 3000 180007 3000 210008 3000 240009 3000 2700010 3000 30000

year Bushes per year

Total

1 x

2 x+x

3 x+x+x

4 4x

5 5x

6 6x

7 7x

8 8x

9 9x

10 10x

Total

Using Variables

Burrill, 2004

1211109876543210

1000

1500

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3000

3500

4000

4500

5000

5500

6000

Farmer Jack's Corn Production

Year

bushels

Burrill, 2004

1 2 3 4 5 6 7 8 9 10

0

1000

2000

3000

4000

5000

6000

Column 3

Farmer Jack's Corn Production

Years

Bushels

Year Bushelsper year

1 21002 23003 25004 27005 29006 31007 33008 35009 370010 3900

Burrill, 2004

“Let d be the yearly increase and an be the amount harvested in year n. Then an+1 = an+d 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)(a1+an), so S10 = (10/2)(a1+a10) = 5(a1 + a1+ 9d) = 30000. So 2a1+9d = 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-y=y1+ m(x-x1) (Confrey & Smith,

1994)

109876543210

500

1000

1500

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3000

3500

4000

4500

5000

5500

6000

Farmer Jack's Corn Production

Year

Production (bu)

400035003000250020001500100050000

50

100

150

200

250

300

350

400

450

500

550

600

Farmer Jack's 10-Year Corn Production

Starting amount

Increase

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 wasn’t just them who had the mistake. Before this course I couldn’t 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 didn’t get directly to the pair who wrote the proof. The teacher asked questions of them, too.”

Preservice Student

“habits of mind”

Need for precisionVocabularyexpression/equationconstruct/draw“lines are similar”

Trimmingdivision never makes biggera1 in recursive definitions

“habits of mind”

The nature and role of proof: mix converse/statementassume what provingprove by exampleprove by pattern

DefinitionsAssumptions and their consequences

“habits of mind”

Doing math is a way of thinkingMore than routine proceduresProblems out of context of unitTakes time Errors can be productive

“habits of mind”

Not all math is equalunderlying 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

Polya’s Ten Commandments

Read faces of students

Give students “know how”, attitudes of mind, habit of methodical work

Let students guess before you tell themSuggest it; do not force it down their throats (Polya, 1965, p. 116)

Polya’s Ten Commandments

Be interested in the subject

Know the subject

Know about ways of learning

Let students learn guessing

Let students learn provingLook 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.