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Title: Doing Mathematics as a Vehicle for Developing Secondary Preservice Math Teachers


1
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

2
How much do all 3 chickens weigh? Each chicken?
Kindt et al, 2006
3
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

4
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)

5
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

6
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

7
Methods Course
  • First semester
  • - Observing teaching
  • - Curriculum
  • - Designing lessons
  • Second semester
  • - Equity
  • - Assessment
  • - Designing lessons

8
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
9
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
10
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

11
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
12
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

13
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?

14
Different mathematical approaches

15
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
16
Strategies
  • Calculus
  • Trigonometry
  • Pythagorean Theorem
  • Coordinate geometry
  • Triangle theorems

17
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.

18
Surface mathematical connections

19
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?

20
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.

21
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.

22
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

23
Manage discussions

24
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
25
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
26
Student designed problems
27
What is Changing?
  • A problem from Japan
  • In the figure, as the step changes, also
    changes.

Step 1 2 3
Peterson, 2006
28
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
  • .

29
What is the rule and why?
  • Number of squares

Step 1 2 3
30
Instruction Managing solutions

31
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)

32
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)

33
Burrill, 2004
34
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
35
Burrill, 2004
36
Burrill, 2004
37
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
38
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)

39
A disconnect that needs explaining
40
Knowledge for Teaching
  • Unpacking the mathematical story
  • Making connections
  • Curricular knowledge
  • Making assumptions explicit

41
Knowledge for teaching?
42
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
43
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
44
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
45
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

46
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
47
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
48
habits of mind

Not all math is equal underlying concepts should
drive instruction Not all solutions are equal
49
habits of mind

Math makes sense Chickens Ratio problem Farmer
Jack
50
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
51
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)

52
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)

53
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.

54
  • 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.

55
  • 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.
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