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1 NCSM 2008 Dr. Eric Milou Rowan University Department of Mathematics 856-256-4500 x3876 Conceptual Understanding and Basic Skills: Discussion.

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Presentation on theme: "1 NCSM 2008 Dr. Eric Milou Rowan University Department of Mathematics 856-256-4500 x3876 Conceptual Understanding and Basic Skills: Discussion."— Presentation transcript:

1 1 NCSM 2008 Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876 Conceptual Understanding and Basic Skills: Discussion Points for Teachers and Parents

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

3 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 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 5 NMP  A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).

6 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 7 Response (Peanuts)

8 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 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 10 What’s “Typical?” in US

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

12 12 Stated vs Developed

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

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

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

16 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 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 18 Flags

19 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 20 Chocolate Bar LAURA P A U L 16 pieces

21 21 Decimals  1000 ÷ 1.49 = 671.1409396 = 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 1.491000

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

23 23 Numbers Are Everywhere

24 24 Computational Practice Target #: 6 3 8 17 1 3

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

26 26 Conceptual & Contextual  8 + 7 = ?  How do we teach this? xxx x xx x x x x x x x x x x x

27 27 8 + 7 = ? 2 5 10 + 5 = 15

28 28 17 - 8 = 1 7 - 8 / / 0 17 2 7 8 --> --> 10 --> --> --> --> --> --> --> 17

29 29 How Many Circles? 50

30 30 1000 - 279 = ? 279+1 = 280+ 20 = 300+700 = 1000

31 31 1000 - 279 = ? 1000 - 1 = 999 999 -278 721

32 32 Multiplication  13 x 17 = ? 1 3 x 1 7 1 2 9 031 2 2 1 ------- 10 7 10 3 1 0 0 3 0 7 0 2 1 221

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

34 34 Contextual Problem Solving ≠ More Word Problems

35 35 Example 1: Sneakers SECOND purchase Nike ConverseReebok Nike Converse Reebok FIRSTFIRST

36 36 100 Students  50 1st time Nike buyers  30 1st time Converse buyers  20 1st time Reebok buyers  How many would buy Nike the second time?  50 x.4 + 30 x.2 + 20 x.1 = 28 NC R N C R

37 37 Example 2: Drug Steady State  10 mg Zrytec daily  50% of Zrytec is eliminated daily  What % is in the body  after 7 days  10 days  n days?

38 38 10mg Daily of Zrytec DayMg. of Zyrtec 1 2 3 4 5 6 10 15 17.5 18.75 19.375 19.6875

39 39 Fact #1 A

40 40 Fact #2 B

41 41 Fact #3 C

42 42 Fact #4 D

43 43 Fact #5 E

44 44 Fact #6 F

45 45 Fact #7 G

46 46 Fact #8 H

47 47 Fact #9 I

48 48 What is this?

49 49 What is this? F A C E

50 50 What If? AB C DE F GH I

51 51 Try Again

52 52 Try Again D E C A D E

53 53 What’s the Point?  Isolated Facts  Less likely to retain information  Connected Facts, Patterns, Facts in Context  More likely to retain information

54 54 Characteristics of a good mathematics program  CONCEPTUAL  CONTEXTUAL  CONSTRUCTIVISM  COMPUTATION

55 55 Thank You Dr. Eric Milou Rowan University milou@rowan.edu http://www.rowan.edu/colleges/las/departments/math/facultystaff/milou/eric.html


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