Creating Mathematically

Powerful Students:

Computation Strategies

that Help Math

Make Sense

Math Matters

The single strongest predictor of completion of a bachelor’s degree is the highest level of mathematics completed in high school. Completing a course beyond advanced algebra, such as pre-calculus or statistics, more than doubles the chance that a student entering college will complete a degree.

The number of college and university mathematics courses taken is the single greatest predictor of lifetime earning potential, cutting across gender and other demographic groups.

Adelman, Cliffford, 1999. “Answers in the Toolbox: Academic Intensity, Attendance Patterns, and Bachelor’s Degree Attainment.” U. S. Dept. of Education. http://www.ed,gov/pubs/Toolbox/toolbox.html. Accessed 11/6/2006.

Who will get the best jobs

in a flat world?

Great collaborators Great synthesizers & connectors Great leveragers Great explainers Passionate personalizers Great adapters Great localizers Anything green Math lovers

- think algorithmically

Thomas Friedman, March 21, 2007, Opening Address, National Council of Teachers of Mathematics Annual Meeting, Atlanta, GA

Specialized Jobs

New Middle Jobs Localized Jobs done face-to-face

Computation Strategies &

AlgorithmsInstead of learning a prescribed (and limited)

set of algorithms, many curricula now encourage students to be flexible in their thinking about numbers and arithmetic. Students begin to realize that problems can be solved in more than one way. They also improve their understanding of place value and sharpen their estimation and mental-computation skills.

The following slides are offered as an extension to the parent communication from your child’s teacher. We encourage you to value the thinking that is evident when children use such algorithms—there really is more than one way to solve a problem!

Mentally think about this problem:

1004 – 697

What was your thinking?

Share at your table.

An algorithm consists of a precisely specified sequence of steps that will lead to a complete solution for a certain class of problems.

Five Important Qualities of Algorithms

Accuracy (or reliability)– Does it always lead to a right answer if you do it right?

Generality– For what kinds of numbers does this work?

Efficiency (or complexity)– Is it quick enough? Do students persist?

– Does it lead to easier calculations in the end?

Ease of accurate use (vs. error proneness)– Does it minimize errors?

Transparency (versus opacity)– Can you SEE the mathematical ideas behind the algorithm?

Hyman Bass. “Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective.” Teaching Children Mathematics. February, 2003

Some Advice for

Working with Algorithms

• Watch your language

• Don’t lie

• If you don’t like the numbers, change them

James C. Brickwedde

Project for Elementary Mathematics

Hamline University, St. Paul. MN

Focus AlgorithmsPartial Sums

Partial Products

Partial Differences

Partial Quotients

Area Model for Multiplication

Trade First or Ready-Set-Go

735+ 246

900Add the hundreds (700 + 200)

Add the tens (30 + 40) 70Add the ones (5 + 6)

Add the partial sums(900 + 70 + 11)

+11

981

356+ 247

500Add the hundreds (300 + 200)

90Add the tens (50 + 40)

Add the ones (6 + 7)

Add the partial sums(500 + 90 + 13)

+13

603

429+ 989

1300100

+ 18

1418

Generalizability

What about decimals?

In what ways would Partial Sums help

students be able to do AND understand

addition? What mathematics do they

take into further study of mathematics?

What do you notice about:

• Efficiency/Complexity

• Ease of Accurate Use

• Transparency

5682

4,00048010012+

4,592

80 X 50

80 X 6

2 X 50

2 X 6

Add the partial products

5276

3,500140300

12+

70 X 50

70 X 2

6 X 50

6 X 2

3,952Add the partial products

50 2

40

6

2000 80

12300

5246

2,000

30080

12

2,392

A Geometrical Representation of Partial Products

(Area Model)

Generalizability

What about decimals?

Do You Remember Quadratic

Equations?

(x + 2) (x + 3) = y

X2 + 3x + 2x + 6 = y

X2 + 5x + 6 = yFOIL?

Think about the formula for the

area of a rectangle

(x + 2) (x + 3) = y

l x w = A

x + 2

x

3

l x w = (x + 2) (x + 3) = A

+

X

3

X + 2

x2 2x

3x 6

A = (x + 2) (x + 3)

= X2 + 3x + 2x + 6

= X2 + 5x + 6

Combining

Algebra and

Geometry+

In what ways would Partial Products or the

Area Model for Multiplication help

students be able to do AND understand

multiplication? What mathematics do they

take into further study of mathematics?

What do you notice about:

• Efficiency/Complexity

• Ease of Accurate Use

• Transparency

12

723459

611

2

13

64

Students complete all regrouping before doing the subtraction. This can be done from left to right (or right to left). In this case, we need to regroup a 100 into 10 tens. The 7 hundreds is now 6 hundreds and the 2 tens is now 12 tens.

Next, we need to regroup a 10 into 10 ones. The 12 tens is now 11 tens and the 3 ones is now 13 ones.

Now, we complete the subtraction. We have 6 hundreds minus 4 hundreds, 11 tens minus 5 tens, and 13 ones minus 9 ones.

10

802274

79

5

12

28

14

946568

813

3

16

78

Subtract the hundreds(700 – 200)

Subtract the tens(30 – 40)

Subtract the ones(6 – 5)

Add the partial differences(500 + (-10) + 1)

500– 245

1491

10

736

Subtract the hundreds(400 – 300)

Subtract the tens(10 – 30)

Subtract the ones(2 – 5)

Add the partial differences(100 + (-20) + (-3))

100– 335

77

20

412

3

Generalizability

What about decimals?

How might Trade First or Partial

Differences change students ability to do

AND understand subtraction? What

mathematics do they take into further

study of mathematics?

What do you notice about:

• Efficiency/Complexity

• Ease of Accurate Use

• Transparency

4

11110

5

19 R3

120

60

23112

5148

3 19

Students begin choosing partial quotients that they recognize!

Add the partial quotients, and record the

quotient along with the

remainder.

I know 10 x 12 will

work…

10

112650

25

85 R6

800

272632

326320

6 85

Compare the partial quotients used here to the ones that you

chose!

1600

Generalizabiity

What about decimals?

How might Partial Quotients help students

do AND understand division? What

mathematics do they take into further

study of mathematics?

What do you notice about:

• Efficiency/Complexity

• Ease of Accurate Use

• Transparency

Connections with Rational

Numbers

. . . many students view fractions, decimals, and percents as three isolated topics, connected only because they are asked at some point to convert among them.

If they learn about these concepts as different but equivalent representations of rational numbers, they have a better chance of understanding how to use any representation of a rational number more effectively.

“Given a pile of jigsaw puzzle pieces and told to put them together, no doubt we would ask to see the picture they make . . . Without the picture, we probably wouldn’t want to bother with the puzzle. Ironically, this situation is very much like what we ask young people to do all the time in school.

To students, the typical curriculum presents an endless array of facts and skills that are unconnected, fragmented, and disjointed…”

Beane, 1991

Make Connections Transparent

Rational Numbers

Decimals

Fractions Percent

Measurement

expre

ss a

s part

s to

pow

ers

of

ten

use

d in

SciMathMN Minnesota K-12 Mathematics Framework, 1998

The most basic idea in mathematics is that

math makes sense to ALL students.

John Van De Walle

Algorithms

Helping children become comfortable with algorithmic and

procedural thinking is essential to their growth and

development in mathematics and as everyday problem

solvers.

Extensive research shows the main problem with teaching

standard algorithms too early is that children then use

the algorithms as substitutes for thinking and common

sense and they don’t watch their place value language.

Other algorithms might be used for extended periods of

time to make mathematics more transparent and

accessible to more students. This also gives a firmer

foundation for exploring mathematics in later years.

Focus algorithms are powerful, relatively efficient, and easy to understand and learn.

Students may use various algorithms or strategies in solving problems.

When students have not settled on a particular algorithm or are making errors in computation teachers may encourage the use of the focus algorithm for each operation.

The aim of this approach is to promote flexibility while ensuring that all students know at least one reliable method for each operation.

Everyday Mathematics Operations Handbook. SRA/McGraw Hill ©2002, p. 5

Focus Algorithms

A Special Thank You

to Sue Wygant, Teacher on Special

Assignment, Burnsville-Eagan-Savage

Schools, for developing the algorithms for

staff and parent use as part of a

Mathematics Partnership Grant provided

by the Minnesota Department of Education

Adapted and used with permission.

Alternative algorithms have been shown to increase student understanding and accuracy. The following research and professional literature resources are among those that support the use of alternative algorithms:

Bass, Hyman. “Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective.” Teaching Children Mathematics. February 2003. NCTM. www.nctm.org

Heibert, James. Making Sense: Teaching and Learning Mathematics with Understanding. 1997. Heinemann. www.heinemann.com

Ma, Liping. Knowing and Teaching Elementary Mathematics. 1999. Lawrence Erlbaum Associates. www.erlbaum.com

National Research Council. Adding It Up: Helping Children Learn Mathematics. 2001.National Academy Press. www.nap.edu

National Research Council. Helping Children Learn Mathematics. 2002. National Academy Press. www.nap.edu

National Council of Teachers of Mathematics (NCTM). Curriculum Focal Points for PK-8: A Quest for Coherence. 2006. NCTM www.nctm.org

_____. Teaching and Learning Algorithms in School Mathematics. NCTM 1998 Yearbook. NCTM www.nctm.org

_____. Teaching Children Mathematics – Focus Issue on Computational Fluency. February 2003. NCTM www.nctm.org