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The Problem

This article highlights the problem of weak mathematics skills among elementary school teachers and proposes solutions to provide them with a deeper understanding of the math they teach.

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The Problem

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  1. The Problem Many (perhaps most) elementary school teachers are weak in mathematics. They lack a deep understanding of the math they teach. * When I refer to elementary teachers, I mean all teachers of math at the elementary grades regardless of whether they are certified as “elementary” or “special education” teachers. Richard Bisk - Mathematics Department - Worcester State College

  2. If our third grade teachers read at the sixth grade level, there would be cries for action. • However, many elementary teachers can’t “do” sixth grade math. Richard Bisk - Mathematics Department - Worcester State College

  3. How do you teach a mathematical subject when you aren’t proficient in it? You focus on rules, procedures and memorization; or on manipulatives, games and activities that you can’t readily connect to concepts. Richard Bisk - Mathematics Department - Worcester State College

  4. Why Has This Happened? Few teachers were asked to learn much math in their preparation programs. Compare this to the number of courses they take that emphasize reading and writing. Often the math courses they take are not relevant to the task of teaching elementary school mathematics. Richard Bisk - Mathematics Department - Worcester State College

  5. We think K-6 math is easy. Elementary Math is not elementary Richard Bisk - Mathematics Department - Worcester State College

  6. Easy?? • Reading a first grade book is easy • So teaching reading to first graders is easy • First grade math is easy • So teaching first grade math is easy Richard Bisk - Mathematics Department - Worcester State College

  7. Goal Provide elementary school teachers with a deep understanding of the math they teach. Richard Bisk - Mathematics Department - Worcester State College

  8. Mathematical Habits Math must be taught at all levels with a focus on understanding. Memorization of number facts is essential; but it’s easier with understanding. 8 x 6 = ? It’s double 4 x 6 Richard Bisk - Mathematics Department - Worcester State College

  9. The habit of memorizing rules and algorithms without understanding is counterproductive. Its negative effects become particularly apparent as students move into middle school. Teachers who don’t understand math well can’t teach for understanding. Richard Bisk - Mathematics Department - Worcester State College

  10. What are the Changes? Most preservice elementary teachers will need at least three relevant college level math courses. (They also need a math methods course to be taken after the math content courses.) MTEL will include a separate math section that must be passed. Richard Bisk - Mathematics Department - Worcester State College

  11. Barriers to Change at the Colleges • Our students are passing MTEL now. • With Dual Majors – no room for more requirements. • Math Fear – students won’t take more math voluntarily. Richard Bisk - Mathematics Department - Worcester State College

  12. More Barriers • Many select elementary education to limit math demands in their coursework. Elementary math is simple. • Many fail Accuplacer and need remedial work before taking college math courses. • Pressure to improve retention rates. Richard Bisk - Mathematics Department - Worcester State College

  13. What Courses? Here’s what we’re working on at WSC: • Number and Operations (aka arithmetic) • Geometry, Measurement, Probability and Statistics • Patterns, Functions and Algebra (aka algebra for teachers) Richard Bisk - Mathematics Department - Worcester State College

  14. Number and Operations • Place Value • Definitions of the four operations and models for explaining them • Problem Solving • Development of algorithms, Why they work • Basic number theory – primes, divisibility, gcf, lcm • Fractions • Ratios, Percents, Rates • Negative Numbers • Decimal Fractions Richard Bisk - Mathematics Department - Worcester State College

  15. This is the foundation course. Arithmetic is fundamental to the rest of mathematics and much of science in the same way that reading is fundamental to all of education. Richard Bisk - Mathematics Department - Worcester State College

  16. Course Goals • Present arithmetic as a coherent topic based upon definitions and fundamental axioms • Shift in perspective from how to why • Arithmetic is not just a bunch of rules to memorize Richard Bisk - Mathematics Department - Worcester State College

  17. Example - Fractions Many college students don’t know how to add and multiply fractions. It’s important for teachers to understand that operations on fractions are just extensions of the same operations on whole numbers. Richard Bisk - Mathematics Department - Worcester State College

  18. Example - Multiplication 3 x 5 Richard Bisk - Mathematics Department - Worcester State College

  19. 3 ×12 = (3 × 10) + (3 × 2) 3 10 2 Richard Bisk - Mathematics Department - Worcester State College

  20. 24 × 32 30 2 20 600 40 120 8 4 Richard Bisk - Mathematics Department - Worcester State College

  21. a b a a2 ab ab b2 b Richard Bisk - Mathematics Department - Worcester State College

  22. A Few Final Exam Questions • How many millions are there in 23 billion? • Convert 2546 to a base ten number. • Explain why 9 ÷ 0 is undefined • Use the distributive property and the fact that 999 = 1000-1 to find 999 x 213 Richard Bisk - Mathematics Department - Worcester State College

  23. Make up a short word problem to illustrate measurement division for 15 ÷ ¾ • If a car goes 4/5 mile in 2/3 minute, what is its speed in a. miles per minute b. miles per hour. • To multiply 2.37 x 1.256 we first multiply 237 x 1256. We then count 5 places from the right and insert a decimal point. Explain why this is correct. (Please don’t tell me it’s the rule!) Richard Bisk - Mathematics Department - Worcester State College

  24. Geometry, Measurement, Probability and Statistics Richard Bisk - Mathematics Department - Worcester State College

  25. Geometry and Measurement • Attributes of geometric figures • Properties of lines and angles • Similarity and congruence • Geometric construction • Basic proofs/deductive geometry • Dimensional analysis • Development of standard formulas for area and volume • Pythagorean Theorem and distance formula Richard Bisk - Mathematics Department - Worcester State College

  26. Probability and Statistics • Descriptive statistics Graphing to display data Central tendency and variation • Basic probability definitions and concepts • Counting; multistage experiments • Misuse of statistics Richard Bisk - Mathematics Department - Worcester State College

  27. Patterns, Functions and Algebra taught for first time last Fall To get enough students enrolled to be allowed to run the course, I had to offer it as a joint graduate/undergraduate course. We only met 12 times. Next fall, two sections will be offered Richard Bisk - Mathematics Department - Worcester State College

  28. Patterns, Functions and AlgebraApproach • Began with a college algebra course and asked: • How do I connect this to arithmetic? I want them to recognize algebra as a natural extension of arithmetic. • How do I help teachers connect algebra to activities in the elementary classroom? Richard Bisk - Mathematics Department - Worcester State College

  29. Main Topics Connections between algebra and arithmetic • “What’s my rule?” problems • Number sequences • Handshake problem • Triangular and square numbers • Singapore math model drawing • Geometric and algebraic representations of odd and even numbers Richard Bisk - Mathematics Department - Worcester State College

  30. Main Topics • Definitions of functions and relations • Linear, quadratic and exponential functions and their applications • Rate of change • Proportionality Richard Bisk - Mathematics Department - Worcester State College

  31. Main Topics • Connecting graphs and functions – use of graphing calculator for discovery activities. • Transformations of functions – vertical and horizontal shifts, reflections • Symmetry – odd and even functions • Composition of functions and inverse functions Richard Bisk - Mathematics Department - Worcester State College

  32. Materials • Bisk College Algebra Manuscript • Graphing Calculator • Activities and Papers from Sybilla Beckmann Tom Fortmann Herb Gross Bill McCallum Jim Milgram Richard Bisk - Mathematics Department - Worcester State College

  33. Final Comments • Teaching these courses is professionally satisfying. • There is surprising depth to K-8 mathematics. • Teaching elementary teachers can break the cycle of math failure that we see in our classrooms every day. Richard Bisk - Mathematics Department - Worcester State College

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