1 NCTM 2008 Dr. Eric Milou Rowan University Department of Mathematics x3876 Basic Skill and Conceptual Understanding: Not Dichotomous at All

2 Overview  National Math Panel Recommendations  Conceptual vs. Procedural Debate  Number Sense & Computation Proficiency

3 National Math Panel (NMP)  A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.

4 NMP  Instructional practice should be informed by high-quality research, when available, and by the best professional judgment and experience of accomplished classroom teachers. High-quality research does not support the contention that instruction should be either entirely “student centered” or “teacher directed.”

5 NMP  A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).

6 Response (Gary Stager)  The Report of the National Mathematics Advisory Panel does not dispute that teachers spend lots of time teaching fractions. The report merely urges that teachers do even more of the same while hoping for a different result. A definition of insanity comes to mind.  It would be bad enough if wasted time was the only consequence of the fanatical fraction focus, but too many students get the idea that they can’t do math. This damages their inclination towards learning other forms of mathematics. Given the importance of mathematics and the widespread mathphobia sweeping the land, students can ill afford to a diminution in their self-image as capable mathematicians.

7 Response (Peanuts)

8 Education Week 11/1/06  We cannot afford to waste time on polarization. What is important is that we pragmatically address critical target areas to improve mathematics education. We cannot be distracted from our primary mission—to match tactical initiatives in other, newly technological societies that are snatching our competitive advantage in innovation—while we bicker over modest differences in approach. (Jere Confrey)

9 Compute the following: 4 x 9 x 25 How many ounces are in a gallon? 4 x 4 ÷ 4 x 4 30 ÷ 3/4

10 What’s “Typical?” in US

11 Third International Math & Science Study (TIMSS) Procedures vs. Concepts

12 Stated vs Developed

13 We need a BALANCE  Balance  Direct Instruction  Constructivism  Balance  Conceptual Understanding  Algorithmic Proficiency  These are NOT Dichotomous

14 Conceptual Understanding  24 ÷ 4 = 6  24 ÷ 3 = 8  24 ÷ 2 =12  24 ÷ 1 = 24  24 ÷ 1/2 = ??

15 Fractions - Conceptually More than 1 or Less than 1 Explain your reasoning The F word

16 Which is larger?  (2/3 + 3/4 + 4/5 + 5/6) OR 4  12.5 x 45 OR 4.5 x 125  (1/3 + 2/4 + 2/4 + 5/11) OR 2

17 Conceptual Fraction task  Kim’s teacher asked her class to design a flag using four colors, dividing a square into parts, and to color the parts as follows:  1/2 is colored red  1/4 is colored blue  1/8 is colored green  Any other part is to be left white

18 Flags

19 Harder Task  A chocolate bar is separated into several equal pieces.  If Laura eats 1/4 of the pieces; and  Paul eats 1/2 of the remaining pieces;  There are six pieces left over  Into how many pieces was the original bar divided?

20 Chocolate Bar LAURA P A U L 16 pieces

21 Decimals  1000 ÷ 1.49 = = Torture!  Big Macs Sell for $1.49, how many Big Macs can I buy for $10.00?  1 is $1.50  2 are $3  4 are $6  6 are $9 Mental Mathematics is a vital skill

22 Computation is Important  Engaging & Active  Less passive worksheets  More thinking & reasoning

23 Numbers Are Everywhere

24 Computational Practice Target #:

25 Active Computation  Fifty (1, 2, 3, 4, 5, 6 and addition)  Buzz (3)  Product Game

26 Conceptual & Contextual  = ?  How do we teach this? xxx x xx x x x x x x x x x x x

= ? = 15

= / / > --> 10 --> --> --> --> --> --> --> 17

29 How Many Circles? 50

= ? = = = 1000

= ? =

32 Multiplication  13 x 17 = ? 1 3 x

33 Conceptual approach leads to ? x 7 x3x3 x2x2 3x 7x 21  Algebra: (x + 3) (x + 7) =

34 Fact #1 A

35 Fact #2 B

36 Fact #3 C

37 Fact #4 D

38 Fact #5 E

39 Fact #6 F

40 Fact #7 G

41 Fact #8 H

42 Fact #9 I

43 What is this?

44 What is this? F A C E

45 What If? AB C DE F GH I

46 Try Again

47 Try Again D E C A D E

48 What’s the Point?  Isolated Facts  Less likely to retain information  Connected Facts, Patterns, Facts in Context  More likely to retain information

49 Characteristics of a good mathematics program  CONCEPTUAL  CONTEXTUAL  CONSTRUCTIVISM  COMPUTATION

50 Thank You Dr. Eric Milou Rowan University