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

            Gail Burrill
      Michigan State University
    How much
    do all 3

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
• 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
• 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
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
            Course Goals
•Deepen and connect mathematical content
knowledge with student mathematical
•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
             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
  Algebra            Geometry
                     Construct rhombi
                     Minimize distance
Men/Women Salaries   Minimize area triangle
What is Changing     Paper folding
Farmer Jack          Isosceles Triangle
Jawbreakers          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
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

Pythagorean Theorem
Coordinate geometry
Triangle theorems
      A                  d
      A       A
          B                   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.
Surface mathematical
       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
– 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

                        B                        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                   D                   C
             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
      Patterns/Reasoning & Proof
• What constitutes valid justification?
• Lack of connection to a geometric scheme that
  established a relation between the rule and the
• Focus on particular values rather than making
• Inability to generalize across contexts (Lanin,
• 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      0
1      3000         +3000
2      6000         +3000
3      9000         +3000
4      12000        +3000
5      15000        +3000
6      18000        +3000
7      21000        +3000
8      24000        +3000
9      27000        +3000
10     30000        +3000                 Burrill, 2004
Solution II: ‘Dividing Into Equal Parts’       Using Variables
Year    Bushels Total                      year     Bushes     Total
                                                    per year
        per year Bushels
                                           1        x
                 of corn
1       3000     3000                      2        x+x

2       3000     6000                      3        x+x+x
3       3000     9000                      4        4x
4       3000     12000                     5        5x
5       3000     15000
                                           6        6x
6       3000     18000
7       3000     21000                     7        7x
8       3000     24000                     8        8x
9       3000     27000                     9        9x
10      3000     30000
                                           10       10x
          Burrill, 2004                    Total
                     Farmer Jack's Corn Production





                 0     1   2   3   4   5     6    7   8   9   10 11 12

                                           Year                          Burrill, 2004
                                    Farmer Jack's Corn Production
Year   Bushels
       per year              6000

1      2100                  5000

2      2300                  4000
3      2500

                  Bushel s
4      2700                  3000                                              Column 3

5      2900                  2000

6      3100                  1000
7      3300
8      3500                     0
                                      1   2   3   4   5   6   7   8   9   10
9      3700                                           Years

10     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
                                    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,
           Farmer Jack's 10-Year Corn Production                                                Farmer Jack's Corn Production
            600                                                                      6000
            550                                                                      5500
            500                                                                      5000
            450                                                                      4500

                                                                   Production (bu)
            400                                                                      4000

            350                                                                      3500
            300                                                                      3000
            250                                                                      2500
            200                                                                      2000
            150                                                                      1500
            100                                                                      1000

             50                                                                       500

              0                                                                             0
                  0   500   1000 1500 2000   2500 3000 3500 4000                                1   2   3   4   5   6   7   8   9   10

                             Starting amount                                                                    Year

                       A disconnect that needs explaining
    Knowledge for Teaching
• Unpacking the mathematical story
• Making connections
• Curricular knowledge
• Making assumptions explicit
Knowledge for teaching?
• “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.”
          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
• …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 precision
    “lines are similar”
    division 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
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
    Ratio problem
    Farmer Jack
       Making connections

Solve each problem using at least two
 different approaches students might

1.Which is the best buy for barbecue
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 them
Suggest 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 proving
Look at features of problems that suggest
solution methods (Polya, 1965,p. 116)
•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-
•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
•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.