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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