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Instruction for Mathematical Knowledge for Teachers of Elementary/Middle Grades

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. MATH 175: Mathematical Explorations for Elementary Teachers, I

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Instruction for Mathematical Knowledge for Teachers of Elementary/Middle Grades

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  1. Instruction for Mathematical Knowledge for Teachers of Elementary/Middle Grades Melissa Hedges Hank Kepner Gary Luck Kevin McLeod Lee Ann Pruske UW-Milwaukee

  2. UWM Foundational Courses for 1-8 Education Majors • 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

  3. UWM “Math Focus” Courses for 1-8 Education Majors • 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)

  4. Course Design Team Model • 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

  5. Topics Covered in MATH 175 • Problem-solving • Number systems • Fractions • Decimals and percent • Addition and Subtraction (meaning, and properties) • Multiplication and Division (meaning, and properties)

  6. Geometry Topics Covered in MATH 176 • 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)

  7. Probability and Statistics Topics Covered in MATH 176 • 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

  8. Mathematical Topics Covered in CURRINS 331/2 • 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

  9. 1-8 Teacher Content • If we spin the spinner shown below many, many times, how many points would we average per spin? • What is your guess? _____ 3 8 1

  10. 1-8 Teacher Content • 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? • What is your guess? ____ Why? 3 8 1

  11. 1-8 Teacher Content • What are the similarities and the differences in these 2 problems? Similarities Differences

  12. 1-8 Teacher Content • 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: • Why did you construct the histogram as you did?

  13. 1-8 Teacher Content One such simulation produced the following results: If you would attempt “balance” the data, where would you locate the fulcrum? 50 10 1 3 8

  14. 1-8 Teacher Content • Now, calculate the experimental averagepoints per spin from the data collected:

  15. 1-8 Teacher Content • Now, let’s calculate the theoretical number of points per spin (or the Expected Value)

  16. 1-8 Teacher Content • To calculate the Expected Value, we might consider the following:

  17. 1-8 Teacher Content • 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 • 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

  18. 1-8 Teacher Content • What topics in mathematics for K-8 teachers did we address in this activity?

  19. Changes to MATH 175/6 • 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

  20. Changes to CURRINS 331/2 • 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

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