“algebra” in elementary schools: it’s not about x’s and...

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“Algebra” in Elementary Schools:It’s Not About x’s and y’s

2007 NCTM Conference

Atlanta, Georgia

Tad Watanabe

Kennesaw State [email protected]

http://science.kennesaw.edu/~twatanab

• What should elementary school algebra looklike?

• What are the fundamental ideas of algebrathat are appropriate for investigation byelementary school students?

• What ideas related to algebra are discussedin the Japanese elementary schoolcurriculum?

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Materials Studied

Content Strands in the JapaneseCourse of Study (COS)

QuantitativeRelations


Numbers & Shiki

Geometric Figures

Numbers & Calculations

Quantities & Measurements

Geometric Figures

Gr. 7 - 9Gr. 3 - 6Gr. 1 - 2

Lower SecondaryElementary

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(shiki)

• 3 + 5 = 8

• 8 + 4

• 11 > 8 + 5

• 3x + 4 = 16

• 2x + 3y

• etc.

Content Strands in the JapaneseCourse of Study (COS)



Numbers & Shiki

Geometric Figures

Numbers & Calculations

Quantities & Measurements

Geometric Figures

Gr. 7 - 9Gr. 3 - 6Gr. 1 - 2

Lower SecondaryElementary

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Quantitative Relations

The objectives and contents of thisdomain cover a wide range, but can bedivided into three categories: idea offunction, writing and interpretingmathematical expressions, andstatistical manipulation.

(Teaching Guide, p. 36)

Ideas of Functions

• How a product changes when the multiplierincreases or decreases by 1

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• Comparing and ordering numbers

• To view a number as a product of other numbers

• How a product increases when the multiplierincreases by 1

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• One-to-one correspondence

• To view a number as the sum or difference ofother numbers

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TopicsGrade

5

Ideas of Functions (cont.)

• Ratio and the value of ratio

• Proportion and its graph

• Inverse proportion

• Proportional relationships

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• Examination of quantitative relationshipsrepresented by formulas such as A x B = C

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• Dependency relation of two numbers and itsgraph

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TopicsGrade

Grade 1: Composition &Decomposition of Numbers

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Teachers’ ManualOne of the main objectives of this unit is to helpstudents see a number from multiple perspectives.This is a foundation for functional (algebraic)thinking because, in order to see a number inrelationship to others, we must pay attention toany dependency relationship or rule forcorrespondence. Thus, with decomposition of ten,once you pick “1”, then the other number, “9,” isdetermined. Furthermore, if the first numberincreases 1, 2, 3…, the other number will decrease9, 8, 7,…

Teachers’ ManualFirst graders cannot develop such aperspective automatically, and teachers maywant to order the (written) combinations ordisplay the blocks so that a pattern might benoticed visually.

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Grade 1: Addition

Grade 1: Subtraction

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Grade 2: Multiplication• What number do you need to add to 3x4 to

get the answer for 3x5. Find the answer for

3x6 by looking at the answer for 3x5.

Grade 2: MultiplicationIf you increase the multiplier of 4x3 by 1, how

much larger will the answer be? What if you

increase the multiplier of 4x4 by 1?

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Grade 2: Multiplication

• Constructing 6’s facts

“In the 6’s table, when themultiplier increases by 1,the product increases by6. So, …”

Grade 4: Covariation

• Fill a 1-liter measuring cup half-way.

• What happens if you tilt the measuring cup?– When the number on the left increases, the

number on the right ________.

– When the number on the right increases, thenumber on the left _______.

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Grade 5: Formulas

• Area of a parallelogram

• Area of a triangle

• Circumference of a circle

• Area of a circle

Grade 5: Examination offormulas

There is a parallelogram

with the height of 8 cm.

Let’s investigate what

happens to its area if its

base changes from 1 cm,

2 cm, …, 5 cm, without

changing the height.

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Grade 5: Examination offormulas

Let’s investigate how

the circumferences of

circles change when

their diameters

change.

Grade 6: Proportions

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Mathematical Expressions

• Writing and interpreting mathematical expressionsinvolving addition, subtraction, multiplication anddivision.

• Comparing numbers and quantities, and expressingtheir relationships.

• Writing and interpreting mathematical expressionsinvolving ( ).

• Using symbols such as and in mathematicalexpressions, and evaluating such expressions bysubstituting specific values.

Grade 1: Addition

Write shiki.

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Grade 1: Addition with a sum > 10

Takeshi picked 9 acorns. Miki picked 4acorns. How many acorns did they pickaltogether?

(1)Write an expression

(2)Let’s think about the method ofcalculation.


How many chocolatesare left in the box? Let’sfigure out different waysto find the answer.

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Treatment in the lower grades(Teaching Guide, p. 39)

Teaching about writing and interpretingmathematical expressions already started at thestage of learning addition in first grade… But,since concrete numbers are used in lower gradesand calculation immediately leads to one number,children rarely become aware of the fact that 3+4represents a concrete phenomenon. Therefore, itis important to teach children focus on themeaning of mathematical expressions instead ofpaying attention solely to getting results.

Grade 4: How many dots?Think about different ways to calculate the numberof dots.

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Grade 4: Writing as one shiki

• Makoto bought a hamburger for 130 yen anda cup of soup for 210 yen. He gave the clerk a500 yen coin. How much change will Makotoreceive?

Makoto: 130 + 210 = 340, 500 - 340 = 160.

Ritsuko: Can we represent this in one shiki?

Shiki with words

• [Amount You Paid] - [Cost] = [Change]

500 - (130 + 210) = 160

• [Area of Parallelogram] = [Base] x [Height]

• [Circumference] = [Diam.] x [Ratio ofCircum.]

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Shiki with words

Shiki with , , and Grade 2

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A sheet of construction paper costs 30 yen.Write a shiki that can be used to calculate thetotal cost of construction paper, yen, whenyou buy sheets of construction paper.

If is 40, what number does represent?

Before x’s and y’s:• Represent relationships among quantities usingshiki with words.

• Use or to stand for unknown quantities towrite a shiki to represent relationships amongquantities.

• Substituting different values in in shiki, ordetermine the value of by reversing theoperations.

• When a relationship between two quantities andare represented in a shiki, the value of is

determined when we know the value of

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Representations to model problemsGrade 4

For the school sports festival, two fourthgrade teams are making 40 postersaltogether. Team B will make 8 moreposters than Team A. How many posterswill each team make?


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Concluding Thoughts

• Algebra in elementary schools goesmuch beyond the study of patterns.

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Algebra Standards (PSSM)

Instructional programs from prekindergarten

through grade 12 should enable all students

to—

– understand patterns, relations, and functions;

– represent and analyze mathematical situations

and structures using algebraic symbols;

– use mathematical models to represent and

understand quantitative relationships;

– analyze change in various contexts.

Principles and Standards forSchool Mathematics

When students notice that operations seem to have

particular properties, they are beginning to think

algebraically. For example, they realize that changing the

order in which two numbers are added does not change

the result or that adding zero to a number leaves that

number unchanged. Students' observations and

discussions of how quantities relate to one another lead

to initial experiences with function relationships, and

their representations of mathematical situations using

concrete objects, pictures, and symbols are the

beginnings of mathematical modeling.

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Concluding Thoughts

• Algebra in elementary schools goesmuch beyond the study of patterns.– Teachers need a deep understanding of

elementary school mathematics inrelationship to algebra.

– Curriculum materials need to articulatealgebraic thinking in elementary schoolmathematics more explicitly.

Grade 1Teachers Manual

•Algebra - Patterns:

children develop algebraic

thinking by finding all the

parts for a given total.

•After children discuss the

question on the page, ask:

how many ways can you

make a train for 4?”

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• Look at the facts with 3 as a

factor. What patterns do you

notice? (Problem for Ss)

• Students identify patterns,

which is important in

developing algebra sense.

Have students make a list of

multiplication facts with 3 as a

factor to see that they can

always add 3 to a fact to find

the next fact. (Manual)

Quantitative Relations

The contents of this domain include… itemswhich are useful in examining ormanipulating contents in other domains. …An important aim of this domain is tounderstand the contents of other domainsusing the ideas and methods discussed in thisdomain.

(Teaching Guide, p. 36)

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Concluding Thoughts

• Algebra in elementary schools goes muchbeyond the study of patterns.– Teachers need a deep understanding of

elementary school mathematics in relationshipto algebra.

– Curriculum materials need to articulatealgebraic thinking in elementary schoolmathematics more explicitly.

• Algebra in elementary schools is as much aprocess standard as a content standard.

“algebra” in elementary schools: it’s not about x’s and...

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