Transcript The Mathematical Knowledge Teachers Need

Capturing Growth in Teacher
Mathematical Knowledge
An Inquiry into Elementary and Middle School Teacher
Understanding of Algebraic Reasoning and Relationships
The Association of Mathematics Teacher Educators
Eleventh Annual Conference
26 January 2007
Dr. DeAnn Huinker, Lee Ann Pruske & Melissa Hedges
The Milwaukee Mathematics Partnership
University of Wisconsin - Milwaukee
www.mmp.uwm.edu
This material is based upon work supported by the National Science Foundation Grant No. EHR-0314898.
Session Goals
• Contribute to the discussions around
defining and measuring the specialized
mathematical knowledge needed for
teaching.
• Share and examine performance
assessments that look more closely at
growth in the mathematical knowledge
targeted on algebra.
What distinguishes
mathematical knowledge from the
specialized knowledge needed
for teaching mathematics?
Common vs. Specialized
Mathematical Knowledge
Encompasses
– “Common” knowledge of mathematics
that any well-educated adult should have.
– “Specialized” 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.
Mathematical Knowledge for Teaching
(MKT)
Some interesting dilemmas…
• Why do we “move the decimal point” when
we multiply decimals by ten?
• Is zero even or odd?
• For fractions, why is 0/12 = 0 and 12/0
undefined?
• How is 7 x 0 different from 0 x 7?
• 35 x 25 ≠ (30 x 20) + (5 x 5) Why?
• Is a rectangle a square or is a square a
rectangle? Why?
Capturing Growth in Teacher
Mathematical Knowledge
Setting
• Content Strand:
Algebraic Reasoning and Relationships
• Pretest: September 2005
• School Year: Monthly sessions (~20 hours)
• Posttest: June 2006
• 120 Classroom teachers:
Kindergarten - Eighth Grade
Algebraic Relationships
Expressions,
Equations,
and
Inequalities
Sub-skill Areas
Generalized
Properties
Patterns, Relations,
and Functions
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)
2
45
1
0
Gain = 0.296
-1
t = 5.584
IRT _pre
IRT _post
p = 0.000
Complete the following:
A) Draw a sketch of a rectangle to represent the
problem 46 x 37. Partition and label the
rectangle to show the four partial products.
B) Make connections from your partial product
strategy (in part A) to the traditional multiplication
algorithm, explaining how they are related.
C) Make connections from your partial products
strategy (Part A) to the problem (4x + 6) * (3x + 6),
explaining how they are related.
Reflect and Discuss
• What is the “pure” mathematical
knowledge you employed while completing
this task?
• What mathematical knowledge embedded
in this task might be accessed during the
teaching of this concept?
• Is this knowledge the same?
Performance Assessment
Gain additional insights into our teachers’ abilities to:
 Make solid connections between the area model of
multiplication and the distributive property.
 Understand and explain connections between the
standard algorithm and use of the distributive property
for multiplication.
 Generalize use of the distributive property.
Examining Teacher Work
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?
Performance Activity Results
• 16% (9/56) proficient, good explanations and
connections.
• 50% (28/56) getting there, good procedural
skills, limited explanations.
• 34% (19/56) did not accurately or completely
solve the tasks.
Next Steps
...
Next
steps…
• Do teachers’ scores predict that they teach
with mathematical skill, or that their students
learn more, or better?
• How might we connect teachers’ scores to
student achievement data?
• More open-ended items to show reasoning
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)
Knowing Mathematics for Teaching
Demands depth and detail that goes well beyond
what is needed to carry out the algorithm
 Use instructional materials wisely
 Assess student progress
 Make sound judgment about presentation, emphasis,
and sequencing often fluently and with little time
Size up a typical wrong answer
Offer clear mathematical explanations
Use mathematical symbols with care
Possess a specialized fluency with math language
Pose good problems and tasks
Introduce representations that highlight mathematical
meaning of selected tasks