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

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? 38 1

1-8 Teacher Content What are the similarities and the differences in these 2 problems? SimilaritiesDifferences

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?

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

1-8 Teacher Content Now, calculate the experimental average points per spin from the data collected: RegionFreq

1-8 Teacher Content Now, let’s calculate the theoretical number of points per spin (or the Expected Value) PointsWeighted Value 1 3 8

1-8 Teacher Content To calculate the Expected Value, we might consider the following: PointsProbability 1 3 8

1-8 Teacher Content What is the relationship between the 2 previous examples? 1 x x x 2 = = 20 = 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 x x 52 = 490 =

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

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

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