Textual Expression of Knowledge in Curricula: Illuminating Opportunities to Learn Area MeasurementLorraine Males & Funda Gönülateş*

Michigan State University

0 1 2 3 40%

20%

40%

60%

80%

100%

Percentage of Content Expressed in Student Materials

SFAWEMSaxon

Grade

Research Questions

How is knowledge presented to students through textual elements in curricula for area measurement in grades K-4?

Framework & Analysis

Results

Implications

Rationale

• Students were asked to do, rather than know (little conceptual content)

• Students did not have direct access to conceptual content (mostly in Teacher materials)

• Stephan & Clements (2003) Covering and Counting recommendation expressed more than their other three recommendations (mostly expressed by procedural Queries)

• Poor performance in measurement including area (Blume, Galindo, & Walcott, 2007; Lehrer, 2003; Stephan & Clement, 2003; Kamii & Kysh, 2006).

• Instructional approaches become crucial in developing a strong understanding of area measurement

• With the reliance on curriculum materials by many teachers (Remillard, 2005) it is important to know:

• what area measurement content is provided in materials

• how area measurement is treated in these materials

Important Points

Curriculum materials provide:

• A Procedural focus, Mostly Queries (Q + P)

• As grades increase more content appears in Student materials

• Conceptual Knowledge is provided mostly by Statements in Teacher materials

Data Sources

• Michigan Mathematics (Foresman-Addison Wesley, 2008)

• Everyday Mathematics (The University of Chicago School Mathematics Project, 2007)

• Saxon Math (Larson, 2004)

Percentage of Knowledge Types in Each Curricula

Conceptual Procedural Conventional Total

SFAW

K 2.1 97.9 0.0 2921 0.2 99.8 0.0 492 2 3.4 96.6 0.0 2983 10.3 88.1 1.7 4774 4.6 94.1 1.3 1039

Saxon

K 0.0 100.0 0.0 2331 1.3 98.7 0.0 2402 1.0 98.5 0.5 3913 2.5 93.6 3.9 5144 1.7 89.9 8.6 888

EM

K 0.0 100.0 0.0 201 2.8 97.2 0.0 1442 6.5 87.6 5.9 1853 6.0 91.8 2.2 7304 8.1 89.2 2.7 732

ConceptualStatements

SStatements

TQueries

SQueries

T

SFAW

K 0 0 0 41 0 1 0 02 2 4 2 23 4 7 12 204 6 14 15 13

EM

K 0 0 0 01 0 3 1 02 0 9 3 03 8 16 6 124 10 11 24 8

Saxon

K 0 0 0 01 0 0 0 32 0 2 0 23 0 6 6 14 0 5 0 6

Taking a closer look at Conceptual Knowledge

K 1 2 3 4 K 1 2 3 4 K 1 2 3 4SFAW EM Saxon

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

100%

67% 64% 70%

100% 100%

67%52%

100% 100% 100%

33% 36% 30% 33%48%

Conceptual Statements in Student and Teacher Materials

Student Teacher

K 1 2 3 4 K 1 2 3 4 K 1 2 3 4Total 292

Total 492

Total 298

Total 477

Total 1034

Total 20

Total 144

Total 185

Total 727

Total 731

Total 185

Total 240

Total 391

Total 514

Total 861

SFAW EM Saxon

7%3% 2% 2% 2% 3% 4% 2% 3%

11%7%

13%18%3%

3% 6% 7% 6% 2%3% 6% 2%

2%

33%

19%5%

30%

3%

10%

15%2%

21%

2%2%

16% 24%

39%

18%

58%

74%

83%

55%

86%90%

78%83%

63%

80% 83%

70%66%

44%

62%

4% 5%3%

4%

11% 8% 9%11% 4%

3%3%

Percentage of Textual Elements in Each CurriculaDemonstrations Worked Examples Statements Questions Problems

Area Coding Scheme ExamplesStatements Questions Problems Demos WEs

S T S T S T T S

Conceptua

l Knowledge

Unit Measure Compensation 1(C)

Definition of Area 1(D)

Meaning of Area Measure 1(D)

Order/Equality 1(S)

ProceduralKnowledge

Measure Area on a Grid, counting whole 2(F)

1(L)

Measure Area on a Grid, counting whole and fractional 1

(J)

Measure Area of a Rectangle on Grid, row and columns 1

(M)

Measure General with support 11(N, O, P, R)

Measure by Covering with Sufficient Non-Standard Units 2

(B, C)

Visual Estimation of Area 2(G, I)

Read an array 2(A)

Draw region X standard units with Support

1(H)

Conventional

Knowledge

Definition of non-standard area units

4(D,

E, K)

Stephan & Clements(2003) provide the following recommendations:

a) construct the idea of measurement units (including measurement sense for standard units)

b) have many experiences covering quantities with appropriate measurement units and counting those units

c) structure spatially the object they are to measure

d) construct the inverse relationship between the size of a unit and the number of units used in the measurement (p.13-14)

Building an Understanding of Area

Michigan Mathematics, Scott Foresman Addison Wesley, Grade 3, p. 468

* This work is based on the work of the STEM Project, Principal Investigator: Dr. Jack Smith, Michigan State University

Project Team: Rachel Ayieko, Kuo-Liang Chang, Yi-Ling Cheng, Dan Clark, Leslie Dietiker, Hanna Figueras, Funda Gönülateş, Pat Greeley, Jia He, Danny Johns, KoSze Lee, Lorraine Males, Aaron Mosier, Matt Pahl, Gulcin Tan Sisman (Middle East Technical University), Ashley Taglauer, Alexandria Theakston