Standards or A. 8 - 6 = 2 B. 6 + 2 = 8 Curriculum? C. 8 + 4 = 12 - PDF document

Preparing Math Teachers for The mathematician's patterns, like the painter's or the poet's, must be beautiful; the Common Core: the ideas, like the colours or the words, must fit together in a harmonious way. Are they (we) ready for this? Beauty is the first test: there is no permanent place in this world for ugly mathematics. Steve Williams Husband to April and Father to Kaiden (9) and - G. H. Hardy, 1940, Bowen (3) Runner A Mathematician’s Apology Professor of Mathematics Coordinator of Secondary Mathematics Education Secondary Math Teacher at Heart Lock Haven University of PA swillia6@lhup.edu NCTM-AC 10-22-15 1 NCTM-AC 10-22-15 2 8 toys are in the chest and 6 toys are on the shelf. Which can be used to find how many toys in all? Standards or A. 8 - 6 = 2 B. 6 + 2 = 8 Curriculum? C. 8 + 4 = 12 D. 10 + 4 = 14 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., The Social Media Problems! 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). NCTM-AC 10-22-15 3 NCTM-AC 10-22-15 4 Understand addition, and understand subtraction. CCSS.MATH.CONTENT.K.OA .A.1 Represent addition and subtraction with objects, fingers, mental images, drawings 1 , sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. NCTM-AC 10-22-15 5 1

2.NBT.7 ( BEFORE standardized testing begins) Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. But don’t look now… 4.NBT.4 The Check! Fluently add and subtract multi-digit whole numbers using the STANDARD ALGORITHM!!! NCTM-AC 10-22-15 12 2

6.RP.2 Understand the concept of a unit rate a / b How carefully are we associated with a ratio a : b with b ≠ 0, and use rate language in the context of a ratio reading the standards? relationship. 3 isn't a unit rate... or is it?!?! 4 Don’t abuse fractions! They can be looked at as two integers that each mean something or as one number itself. NCTM-AC 10-22-15 13 NCTM-AC 10-22-15 14 7.G.4 Three axioms for teaching Know the formulas for the area and 1. Know the content being circumference of a circle and use them to solve presented. problems; give an informal derivation of the 2. Know more than the content relationship between circumference and being presented. area of a circle. 3. Teach from the overflow of knowledge. How would you find the area of a circle if you were given the circumference? Have you ever seen a formula that gives the area of a circle in terms of its circumference? 4. Know the standards inside and out! NCTM-AC 10-22-15 15 NCTM-AC 10-22-15 16 Ultimate Goal Goals for Presentation • To examine the Common Core State Standards (both the content standards and the standards for mathematical practice) and use them to motivate To find ways to help our participants to consider more conceptual, structured, and rigorous pedagogical strategies to future teachers become teach secondary mathematics (it isn’t always our students’ fault for not learning). competent teachers of • To motivate participants to consider other mathematics in the Common mathematical concepts where we have “short - changed” our students. Core Era. • To provide participants with some alternate ways of viewing certain math topics that are more conceptual than traditional ways of viewing them. (Consider this: We are about to ask our • To help participants begin to be able to see students to become teachers within a system mathematical concepts and structures despite in which they have not participated.) being blinded by the mindless procedures. NCTM-AC 10-22-15 17 NCTM-AC 10-22-15 18 3

Where did this stuff come from? Three of my favorite • Seven years of teaching all levels of secondary mathematics • Seventeen additional years of working problems to get us thinking with preservice teachers and talking to them about potential conceptual outside of just procedures deficiencies or misunderstandings • Twenty-four years of trying to closely examine as many mathematical concepts in as much detail as possible • A sincere interest to have my students (and myself) develop that Profound Understanding of Fundamental Mathematics by looking at concepts in different or non-traditional ways NCTM-AC 10-22-15 19 NCTM-AC 10-22-15 20 My First “Favorite” Problem An Algebraic Perspective    7 ax b 9   7 b 9 b    2 and 6 a a Commentary It’s not about teaching your students to solve this     2 a b 7 type of problem. It’s about teaching them to think         1 15 a 1 ; b 15 7 x 9 about structure and use known tools to solve it.    4 2 4 2 6 a b 9 NCTM-AC 10-22-15 21 NCTM-AC 10-22-15 22 A Transformation Perspective, 1 A Transformation Perspective, 2 The inequality interval has been shrunk and moved. 1 The inequality interval has been shrunk and moved. It has been shrunk by (from 8 to 2). 1 4 It has been shrunk by (from 8 to 2).   1   3     4 1 1 So: 2 and 6 4 4     2 2        1 1 move first: 2 m 7 and 6 m 9 m 30 4 4        Now move: 1 m 7 and 3 m 9 m 7.5 2 2 1         Thus, 7 4 x 30 9 will simplify to 2 x 6.       Therefore, 7 1 x 15 9 will simplify to 2 x 6. 4 2 NCTM-AC 10-22-15 23 NCTM-AC 10-22-15 24 4

What one student saw this semester: -2 needed to get mapped to 7 and 6 needed to get mapped to 9. So he found the equation of the line between these two points: 1 15   y x 4 2 A Combination Perspective Thinking like this allowed us to write The inequality interval has been shrunk and moved. any degree function that would fit  between the horizontal lines y 7 It has been shrunk by and moved by . a b      and y 9, transforming the form 2 a ab 7            move first: a 2 b 7 and a 6 b 9     n     y a x p q 6 a ab 9 1           1 Thus, b and b 30, so 7 x 30 9 will simplify to 2 x 6. 4 4 The pedagogical implications here are huge and should fundamentally change the way that most teachers teach the concept of linear functions. NCTM-AC 10-22-15 25 NCTM-AC 10-22-15 26 CCSS Content The point here is to give deep thought to the underlying concepts rather than only considering procedures to common problem types. NCTM-AC 10-22-15 27 NCTM-AC 10-22-15 28 CCSS Mathematical Practices CCSS Mathematical Practices NCTM-AC 10-22-15 29 NCTM-AC 10-22-15 30 5

My Second “Favorite” Problem The Developmental Progression of Circle the fractions from the following: Fractions in the Curriculum      2 5 2 25 3, 2, , 3, 4.6, 5.3, , , , , 0 Fractions 3 2 3 5 Counting #s Whole #s Rational #s Integers Concentrate on these! A “first” definition of a fraction: A part of a whole To a high school teacher or a college professor, the diagram above makes no sense. The CCSS definition of a fraction: There is no set called “fractions” NCTM-AC 10-22-15 31 NCTM-AC 10-22-15 32 My Third “Favorite” Problem Why is this important? Write an equation that has the following solutions:    x 8 or x 2     8 0 2 0 (what are the factors?) x or x       x 8 x 2 0     2 x 8 x 2 x 16 0    2 x 6 x 16 0 x   But what about 3 5 ?      x 3 5 or x 3 5    x 8 or x 2 NCTM-AC 10-22-15 33 NCTM-AC 10-22-15 34 What does this have to do with… How (and why) do we think about this problem? Write an equation that has the following solutions:    CCSS 7.NS.1b/c x 8 or x 2 If we talked like this, maybe we could take x = 8 and x = -2 and create an absolute value equation instead of only a quadratic one. Questions to ask: What number is the same distance away from 8 and -2; and what is this distance? Answers: Half way between, or (8 + (-2))/2 = 3 and the    distance is (8 – (-2))/2 = 5. 3 5 8 means that 8 is located 5 5 units from 3 in the right direction.          3 5 2 means that 2 is located 5 5 units from 3 in the left direction. x   Therefore, 3 5          3 5 8 means that 8 is 5 units from 3 in the left direction.     What numbers are 5 units away from 3? 3 5 2 means that 2 is 5 units from 3 in the right direction. NCTM-AC 10-22-15 35 NCTM-AC 10-22-15 36 6