Contibutions of Curricula

A Historical Look at How We Teach Addition

Purpose of Project

• Down with reform – Back to the basics! Wait! Didn’t we try that before?

• Learn more and start earlier!Focus on key concepts each year.

• How much has curricula really changed over the years? In what ways? In what ways has it stayed the same throughout the reforms?

Methods

• Choose Focus

• Chart

• Categorize

1897 1918 1929 1937

1958 1964 1969 1974 2004

• Collect Textbooks

Textbook Philosophies

• Problem Solving "The problem material is drawn from the life in which children and their parents are living to-day. It is within their knowledge or experience, and presents real, rather than imaginary situations" (1918, p. iii).

1897 1958 1969 19741918 1929 1937 20041964


• Problem Solving

1897 1958 1969 19741918 1929 1937 20041964

• Drill and Memorization

“Rote counting and the learning of facts from flash cards, for example, are not acceptable procedures” (Stokes et al., 1958, p. 2).

Drill can “reinforce concepts that have been approached as part of the over-all structure of mathematics” (Eicholz et al., 1969, p. 5).

When once a child has sensed a number fact, he should be made to memorize it, and use it until its use becomes as automatic and unconscious as walking or talking” (1918, p. iv).


• Problem Solving


• Fun/Engaging vs. Social

1897 1958 1969 19741918 1929 1937 20041964

“No pupil should have to moan, “Aw! It’s the same old stuff,” when he flips through his math book. The pages…are varied. They’re lively. They look like fun” (1974, p. vi).

“The teaching program must revolve around social living. All plans, whether they pertain to study materials or to teaching methods, must have a social approach” (1958, p. 1).


• Problem Solving


• Role of Teacher

1897 1958 1969 19741918 1929 1937 20041964


"Do not allow the notes to stifle your own effective teaching methods and creative efforts" (1969, p. 17).

"The teacher's position is unique in that she alone has sufficient insight into the backgrounds of her pupils to know how to make the subject meaningful to them” (1964, p. 1).

"The teacher should recognize differences in children, and give the stronger ones an opportunity to test their power” (1918, p. 3).


• Problem Solving


• Role of Teacher

• Role of Parent

1897 1958 1969 19741918 1929 1937 20041964


“Because the parent is the first teacher, the text is so constructed that the parent can readily gain insight into the concepts presented on each page” (1964, p.1).

Addition Philosophies

• Scope

1897 1958 1969 19741918 1929 1937 20041964

OLD

"In many schools, particularly those that accept children six years of age in the first grade, it may be inadvisable to attempt any formal work with numbers; while other first-grade classes whose members are seven, eight, or more years of age may accomplish more work than is here indicated" (1918, p. 1).

NEW

“Everyday Mathematics begins with the premise that students can, and must, learn more mathematics than has been expected from them in the past” (2004, p. ii).


• Scope

• Symbols

1897 1958 1969 19741918 1929 1937 20041964

• Meaning and words before symbols

+ means “and”, say “plus”

+ means “and”, say “and”

“The children use the terms and symbols plus (+) and minus (-) from the beginning” (1964, p. 18).


• Scope

• Symbols

1897 1958 1969 19741918 1929 1937 20041964

• Physical Manipulation

"Every wide-awake primary teacher knows the value of dramatization, playing store, the game element, etc. and is quick to invent and adopt the proper device as needed" (1918, p. 1).

“You should allow them to use these concrete objects as long as necessary. However, to discourage the children’s dependence on these crutches, put the materials someplace where they can be readily used if needed, but do not pass them out to each child” (1969, p. 95).


• Scope

• Symbols

• Creating Problems

1897 1958 1969 19741918 1929 1937 20041964


• Make own problems

• Describe methods

• Interesting gap


• Scope

• Symbols

• Creating Problems

• Proof

1897 1958 1969 19741918 1929 1937 20041964


"Beginning at the top of the column, add the first two numbers, and to this sum add the third number…Now, to be sure you are right, begin at the bottom and add upward" (1929,p. 68-9).

Layout

1897 1958 1969 19741918 1929 1937 20041964

ToolsMessyNarrativeBlended Topics

Order: 1897

Order: 1918

Order: 1929

Order: 1937

Order: 1958

Order: 1964

Order: 1969

Order: 1974

Order: 2004

Order: Composite

Order: End LevelTotals to 9

Totals to 10

Totals to 19

2-Digit

Application to Reading

“It would be interesting to compare the order and duration of concepts taught over the year among the countries. I wonder if either of these factors influences how well children learn the various concepts.”

What about assessments? Have they changed as much as the curricula?

Articles on TIMSS and International Studies

Examples: Missing Addend1897

1918

1937

1958

1969

2004

1964

Examples: Combinations1897

1969

1974

1964

Examples: Combinations1958

1937

2004

Examples: Vertical

1897 1918

1929 1937

• Early

• No symbol

Examples: Vertical1958

• Both forms

Examples: Vertical

1964

“Once the children thoroughly understand addition and the idea of solving equations, vertical notation for addition problems is introduced” (1969, p. 86).

1969

1974

2004

Examples: Commutative1897

19581969

1974

1929

2004

Examples: Associative

1969

1974

1964

Examples: Identity

1969 1974

1929

1964

2004

Examples: Expanded Form

1897

1969

1964

Examples: Other Operations1958

1897

Strategies For Adding Two, Single-digit Numbers1897 1918 1929 1937 1958 1964 1969 1974 2004

Manipulatives X X X X X X X X

Counting Up X X

Number Line X XPractice (Dot Cards/Dominoes) X X X X

Ten Trays XBreak Apart/ Combinations X

1969

1897 1918 1929 1937 1958 1964 1969 1974 2004

Ovals/Parentheses X X XCross Out and Write Sum X

Add Downward X X

Pictures X

Dot Cards X X

Strategies For Adding Several One-Digit Numbers

1969

1897

1897 1918 1929 1937 1958 1964 1969 1974 2004

Ten Frames XBundles and Single Sticks

XBeads on Grid XPlace Value Chart XAddition Columns XBreak Apart/ Expanded Form

X

Strategies For Adding a Single-digit and Two-Digit Number

1964 19741929

1897 1918 1929 1937 1958 1964 1969 1974 2004

Place Value Chart XAddition Columns: + Ones Then Tens

XBase-10 Blocks X

Tens Frames XBreak Apart/ Expanded Form

X

Strategies For Adding Two, Two-Digit Numbers

1964

1929

2004

1974

Other Thoughts Related to Readings

• A few of the textbooks advocate explaining your answers or assert that math should be “social”, yet they do not express a philosophy toward discourse.

• How closely did teachers follow these curricula?

• If the pages for each topic from these books were combined, which would teachers use?

Resources

1897 1958 1969 19741918 1929 1937 20041964

Baird, S.W. (1897). Graded Work in Arithmetic: First Year Numbers From 1 to 20. New York: American Book Company.

Buswell, G.T., Brownell, W.A., & Dolch, M.P. (1937). Jolly Number Tales: Book One. Boston: Ginn and Company.

DeVault, M.V., Greenberg, H.J, Frehmeyer, H., & Bezuszka, S.J. (1974). SRA: Mathematics Learning System Text: Level 1. Chicago: Science Research Associates, Inc.

Eicholz, R.E. et al. (1969). Elementary School Mathematics (2nd Edition): Book 1. Ontario: Addison-Wesley (Canada) Limited.

Elwell, C.E., Stanislas, S.M., & Fitzgerald, J.F. (1964). Teacher's Edition of New Ways in Numbers: Book 1. Boston: D.C. Heath and Company.

Fowlkes, J.G. & Goff, T.T. (1929). The Modern Life Arithmetics: Six-Book Series, Book One. New York: The Macmillan Company.

Stokes, C.N., Adams, B., & Bauer, M.B. (1958). Arithmetic in My World: 1. Boston: Allyn and Bacon, Inc.

The University of Chicago School Mathematics Project. (2004). First Grade Everyday Mathematics: Teacher's Lesson Guide Volume 1 & 2. Chicago: Wright Group/McGraw-Hill.

Watson, B.M. (1918, 1922, 1924). Simplified Primary Arithmetic. Boston: D.C. Heath and Company.

Contibutions of Curricula

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