Content Knowledge
for Leadership
District Model to Strengthen and Assess
Math Teacher Leaders’ Understanding of the
Big Ideas of Algebra
Connie Laughlin, Melissa Hedges, & DeAnn Huinker
Milwaukee Mathematics Partnership
University of Wisconsin-Milwaukee
National Council of Supervisors of Mathematics
Annual Conference, March 2007
www.mmp.uwm.edu
This material is based upon work supported by the National Science Foundation under Grant No. 0314898. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).
Session Goals
◊ Examine a sequence of activities used to deepen math teacher leaders’ knowledge of the big ideas of algebra.
◊ Share the impact on teachers’ mathematical knowledge for teaching algebraic reasoning
and relationships.
What distinguishes
mathematical knowledge from the specialized knowledge needed
for teaching mathematics?
Common vs. Specialized Mathematical Knowledge
“Common” knowledge of mathematics that any well-educated adult should know.
“Specialized” knowledge to the work of teaching and that only teachers need to know.
Source: Ball, D.L. & Bass, H. (2005). Who knows mathematics well enough to teach third grade? American Educator.
Algebraic Relationships
Professional Development
Goals
· To develop the MTLs’ content knowledge in algebraic reasoning by actively engaging them in learning mathematics.
· To increase their ability to recognize and develop algebraic thinking in their students.
· To provide them with resources that they could take back to their schools in order to facilitate similar types of learning opportunities with their staff.
Sequence
Aug / Patterns, Variables, and Equivalent Expressions
Sept / Describing Change
Oct & Nov / Equality, Equivalence, and the Meaning of the Equals Sign
Jan & Feb / Generalized Properties: Distributive Property
Mar / Expressing Relationships
Apr / Moving Among Representations
June / The Big Picture
Sample Tasks
· Equality & Relational Thinking
· Representations:
From Symbols to Stories
· Representations:
From Graphs to Stories
Representations
· Context (situation)
· Verbal (sentences)
· Numerical (tables of values)
· Visual (graphs, diagrams)
· Symbolical (algebraic equations)
8 + 4 = £ + 5
· Solve.
· Explain your reasoning to a partner.
Share with a colleague at your table:
· What does the equal sign mean?
· How would your students answer the same question? Why do you think they would answer that way?
· As you talk through these questions, begin to think about a working definition of the equal sign that might make sense to both you and your students.
8 + 4 = £ + 5
Percent of children offering solutions
Grade / 7 / 12 / 17 / 12 & 17 / Other / n1 / 0 / 79 / 7 / 0 / 14 / 42
1 & 2 / 0 / 54 / 20 / 0 / 20 / 84
2 / 6 / 55 / 10 / 14 / 15 / 174
3 / 10 / 60 / 20 / 5 / 5 / 208
4 / 7 / 9 / 44 / 30 / 11 / 57
5 / 7 / 48 / 45 / 0 / 0 / 42
6 / 0 / 84 / 14 / 2 / 0 / 145
Falkner, K., Levi, L., & Carpenter, T. P. (1999). Children’s Understanding of Equality: A Foundation for Algebra. Teaching Children Mathematics.
8 + 4 = £ + 5
¨ Computational approach
Compute the “answer” of 12 and reason from there.
¨ Relational thinking approach
Reason using relationships and properties and do not need to find the “answer” of 12.
“5 is 1 more than 4, so the number in the box must be 1 less than 8.”
True or False ??
13 Î 9 = 90 + 27
13 Î 9 = 130 – 13
Use each approach:
¨ computational strategy
¨ relational thinking strategy
Big Idea
Relational thinking with numbers and operation needs to become “explicit” for it to lead to algebraic reasoning.
“One of the central purposes of algebra is to model real situations mathematically, and a key to this is the capacity to translate from natural language to algebraic expression.”
Driscoll (1999, p. 119)
Fostering Algebraic Thinking
Describe a story or context
that could be modeled by
13x + 5 = y
Then Group Discussion . . . .
Session Goals
◊ Continue developing the algebraic habit of mind of “expressing relationships.”
◊ Explore translations among representations for algebraic situations including equations, pictures, words, tables, and graphs.
Examine the work samples for
13x + 5 = y
Examine the work as pairs or small groups…
· What might account for the variety of answers/responses to the prompt?
· What does this work show us about their understanding?
Big Idea of Algebra
Variable
Numbers or other mathematical objects can be represented abstractly using variables.
Relationships between mathematical objects can often be represented abstractly by combining variables in expressions, equations or functions.
Big Idea of Algebra
Representation
Mathematical relationships can be
represented in equivalent ways:
· Verbally (carefully worded sentences)
· Numerically (tables of values)
· Visually (diagrams, graphs)
· Symbolically (algebraic equations)
Describe a story or context
that could appropriately be modeled by this graph.
·
Examine the work samples for
telling the story from the graph
As you reflect on teacher work samples consider the following:
· Is the mathematics correct? Are mathematical symbols used with care?
· Are the connections between representations clear?
· Are explanations mathematically correct and understandable?
Is the “change” . . .
· Increasing or decreasing or both?
· Steady (constant) or does it vary?
· Occurring quickly or slowly?
Big Idea of Algebra
Linearity
A relationship between two quantities is linear if the
“rate of change” between the two variables is constant.
Setting
· Content Strand:
Algebraic Reasoning & Relationships
· Pretest: September 2005
· School Year: Monthly sessions
(~20 hours)
· Posttest: June 2006
· 120 Classroom teachers:
Kindergarten - Eighth Grade
Items
· Measure mathematics that teachers use in teaching, not just what they teach.
· Orient the items around problems or tasks that all teachers might face in teaching math.
· MMP performance assessments to give insight into depth of teacher knowledge developed around monthly seminars.
Teacher Growth in Mathematical Knowledge for Teaching (MKT)
Gain = 0.296
t = 5.584, p = 0.000
Knowing mathematics for teaching includes knowing and being able to do the mathematics that we would want any competent adult to know.
But knowing mathematics for teaching also requires more, and this “more” is not merely skill in teaching the material.
Ball, D.L. (2003). What mathematical knowledge is needed for teaching mathematics? Secretary’s Summit on Mathematics, U.S. Department of Education, February 6, 2003; Washington, D.C. Available at http://www.ed.gov/inits/mathscience.
Mathematical knowledge for teaching must be serviceable for the mathematical work that teaching entails, for offering clear explanations, to posing good problems to students, to mapping across alternative models, to examining instructional materials with a keen and critical mathematical eye, to modifying or correcting inaccurate or incorrect expositions.
Ball, D.L. (2003). What mathematical knowledge is needed for teaching mathematics? prepared for the Secretary’s Summit on Mathematics, U.S. Department of Education, February 6, 2003; Washington, D.C. Available at http://www.ed.gov/inits/mathscience. (p. 8)