Transcript 1-8 Teacher Content

Instruction for
Mathematical Knowledge
for Teachers of
Elementary/Middle Grades
Melissa Hedges
Hank Kepner
Gary Luck
Kevin McLeod
Lee Ann Pruske
UW-Milwaukee
UWM Foundational Courses
for 1-8 Education Majors
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MATH 175: Mathematical Explorations for
Elementary Teachers, I
MATH 175: Mathematical Explorations for
Elementary Teachers, II
CURRINS 331: Teaching of Mathematics:
Grades 1-6
CURRINS 332: Teaching of Mathematics:
Middle School
UWM “Math Focus” Courses
for 1-8 Education Majors
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MATH 275: Problem-Solving and Critical
Thinking
MATH 276: Algebraic Structures
MATH 277: Geometry
MATH 278: Discrete Probability and Statistics
(Over 40% of UWM 1-8 Education Majors
choose a mathematics focus area)
Course Design Team Model
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Mathematics faculty member ensures
rigorous content
Mathematics Education faculty member
ensures strong pedagogy, and alignment with
standards
Teacher-in-Residence provides connection to
classroom practice
Topics Covered in MATH 175
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Problem-solving
Number systems
Fractions
Decimals and percent
Addition and Subtraction (meaning, and
properties)
Multiplication and Division (meaning, and
properties)
Geometry Topics Covered in
MATH 176
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Visualization (solids; nets)
Angles, circles, spheres, triangles, polygons
Constructions (patty paper; Cabri on TI-84)
Congruence and similarity
Transformations (flips, slides, turns; patty paper;
Cabri)
Measurement
Area (derivation of formulas; Pythagoras)
Probability and Statistics
Topics Covered in MATH 176
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Plots (line plots; histograms; stem-and-leaf plots;
box-and-whisker plots)
Mean, median, mode; standard deviation
Inference
Displaying outcomes (arrays; trees; sample spaces)
Probability (experimental; theoretical)
Simulation (ProbSim applications on TI-84)
Games (fair/unfair; relationship to probability)
Counting principles
Expected value
Mathematical Topics Covered
in CURRINS 331/2
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CURRINS 331: Number and operations
(number development, place value, CGI,
operation concepts); Computing devices;
Algebraic reasoning (patterns,
computational/relational thinking)
CURRINS 332: Geometry; Algebra (linear
equations); Probability; Fractions, decimals
and percents
1-8 Teacher Content
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If we spin the spinner shown below many, many
times, how many points would we average per spin?
3
8
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What is your guess? _____
1
1-8 Teacher Content
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Let’s begin with an easier example…
Perhaps it will lead us to an answer to the
previous question.
If we spin the spinner shown below many,
many times, how many points would we
average per spin?
3
8
1
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What is your guess? ____ Why?
1-8 Teacher Content
What are the similarities and the differences
in these 2 problems?
Similarities
Differences
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1-8 Teacher Content
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Suppose we would do a simulation of this
problem.
Draw a frequency histogram that you might
expect to get from spinning the spinner 100
times:
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Why did you construct the histogram as you did?
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1-8 Teacher Content
One such simulation produced the following results:
50
10
1
3
8
Region
Freq.
1
22
3
26
8
52
If you would attempt “balance” the data, where would you
locate the fulcrum?
1-8 Teacher Content
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Now, calculate the experimental
average points per spin from the data
collected:
Region
Freq.
1
22
3
26
8
52
1-8 Teacher Content
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Now, let’s calculate the theoretical number of
points per spin (or the Expected Value)
Points
1
3
8
Weighted Value
1-8 Teacher Content
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To calculate the
Expected Value, we
might consider the
following:
Points
1
3
8
Probability
1-8 Teacher Content
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What is the relationship between the 2 previous
examples?
1 x 1 + 3 x 1 + 8 x 2 = 1 + 3 + 16 = 20 = 5
4
4
4
1 x ¼ + 3 x ¼ + 8 x ½ = ¼ + ¾ +4 = 5
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Are these procedures equivalent?
Compare this to the calculation of the experimental
average.
1 x 22 + 2 x 26 + 8 x 52 = 490 = 4.9
100
100
1-8 Teacher Content
What topics in mathematics for K-8
teachers did we address in this activity?
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6.
7.
Changes to MATH 175/6
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Stabilization of instruction (hiring of Luck,
Mandell)
Improved instruction; modeling pedagogy
More hands-on activities (e.g. patty paper),
resulting in greater familiarity in CURRINS
331/2
Changes to CURRINS 331/2
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Prerequisite of C or better in MATH 176
Stronger connections to the mathematics
taught in MATH 175/6, including:
Greater emphasis on mathematical concepts
(“distributive law”, not “FOIL”; expressions vs.
equations; “opposite” vs. “inverse”)
Greater emphasis on correct notation (use of
“=” sign to indicate balance)
Use of definitions from MATH 175/6 textbook