Transcript Mathematics Tasks as a Vehicle to Help Teachers Become

Doing Mathematics as a Vehicle for
Developing Secondary Preservice
Math Teachers’ Knowledge of
Mathematics for Teaching
Gail Burrill
Michigan State University
[email protected]
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 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
GoalsGoals
Course
•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
Course
GoalsGoals
•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
Beams
Chickens
Manatees
Men/Women Salaries
What is Changing
Farmer Jack
Jawbreakers
Geometry
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
d
A
A
B
C
C-d
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
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
B
E
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
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
1
2
3
4
5
6
7
8
9
10
0
3000
6000
9000
12000
15000
18000
21000
24000
27000
30000
+3000
+3000
+3000
+3000
+3000
+3000
+3000
+3000
+3000
+3000
Burrill, 2004
Solution II: ‘Dividing Into Equal Parts’
Year
1
2
3
4
5
6
7
8
9
10
Using Variables
Bushels Total
per year Bushels
of corn
3000
3000
3000
6000
3000
9000
3000
12000
3000
15000
3000
18000
3000
21000
3000
24000
3000
27000
3000
30000
year
Bushes
per year
1
x
2
x+x
3
x+x+x
4
4x
5
5x
6
6x
7
7x
8
8x
9
9x
10
10x
Burrill, 2004
Total
Total
Farmer Jack's Corn Production
6000
5500
5000
bushels
4500
4000
3500
3000
2500
2000
1500
1000
0
1
2
3
4
5
6
Year
7
8
9
10 11 12
Burrill, 2004
1
2
3
4
5
6
7
8
9
10
Bushels
per year
2100
2300
2500
2700
2900
3100
3300
3500
3700
3900
Farmer Jack's Corn Production
6000
5000
Bus he ls
Year
4000
3000
Column 3
2000
1000
0
1
2
3
4
5
6
7
8
9
10
Years
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)
Farmer Jack's 10-Year Corn Production
6000
550
5500
500
5000
450
4500
Production (bu)
600
400
Increase
Farmer Jack's Corn Production
350
300
250
200
150
100
4000
3500
3000
2500
2000
1500
1000
500
50
0
0
0
500
1000 1500 2000 2500 3000 3500 4000
Starting amount
1
2
3
4
5
6
Year
A disconnect that needs explaining
7
8
9
10
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.”
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 precision
Vocabulary
expression/equation
construct/draw
“lines are similar”
Trimming
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
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
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)
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: 135164.
•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.