Transcript Games to Build Mathematical Skills and Thinking

Math Games to Build
Skills and Thinking
Claran Einfeldt, [email protected]
Cathy Carter, [email protected]
http://www.cmath2.com
What is
“Computational Fluency”?
“connection between conceptual
understanding and
computational proficiency”
(NCTM 2000, p. 35)
Conceptual
Understanding
Computational
Proficiency
Place value
Operational
properties
Number
relationships
Accurate,
efficient, flexible
use of
computation for
multiple purposes
Computation
Algorithms:
Seeing the Math
Computation Algorithms in
Instead of learning a prescribed (and
limited) set of algorithms, we should
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 mentalcomputation skills.
Before selecting an algorithm, consider how
you would solve the following problem.
48 + 799
We are trying to develop flexible thinkers who recognize that
this problem can be readily computed in their heads!
One way to approach it is to notice that 48 can be renamed
as 1 + 47 and then
48 + 799 = 47 + 1 + 799 = 47 + 800 = 847
What was your thinking?
Important Qualities of Algorithms
Accuracy
 Does it always lead to a right answer if you do it right?
Generality
 For what kinds of numbers does this work? (The larger the set
of numbers the better.)
Efficiency
 How quick is it? Do students persist?
Ease of correct use
 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.
Table of Contents
Partial Sums
Partial Products
Partial Differences
Trade First
Partial Quotients
Lattice Multiplication
Click on the algorithm you’d like to see!
Click to proceed
at your own
speed!
Add the hundreds (700 + 200)
Add the tens
(30 + 40)
Add the ones (5 + 6)
Add the partial sums
(900 + 70 + 11)
735
+ 246
900
70
+11
981
Add the hundreds (300 + 200)
Add the tens
(50 + 40)
Add the ones (6 + 7)
Add the partial sums
(500 + 90 + 13)
356
+ 247
500
90
+13
603
Click here to go
back to the menu.
429
+ 989
1300
100
+ 18
1418
Click to proceed
at your own
speed!
80 X 50
80 X 6
2 X 50
2X6
Add the partial
products
56
× 82
4,000
480
100
12
+
4,592
52
× 76
70 X 50
70 X 2
6 X 50
6X2
Add the partial
products
3,500
140
300
12
+
3,952
A Geometrical Representation
of Partial Products
(Area Model)
50
2
40
2000
80
6
300
12
Click here to go
back to the menu.
52
× 46
2,000
80
300
12
2,392
Students complete all regrouping
before doing the subtraction. This
can be done from left to right. 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.
11 13
6 12
723
459
2 64
Now, we complete the subtraction. We have 6
hundreds minus 4 hundreds, 11 tens minus 5 tens,
and 13 ones minus 9 ones.
9 12
7 10
802
274
5 28
Click here to go
back to the menu.
13 16
8 14
946
568
3 78
Subtract the hundreds
(700 – 200)
Subtract the tens
(30 – 40)
Subtract the ones
(6 – 5)
Add the partial differences
(500 + (-10) + 1)
736
– 245
500
10
1
491
Subtract the hundreds
(400 – 300)
Subtract the tens
(10 – 30)
Subtract the ones
(2 – 5)
Add the partial differences
(100 + (-20) + (-3))
Click here to go
back to the menu.
412
– 335
100
20
3
77
19 R3
12 2 31
1 20 10
1
11
Add the partial
Students begin
60
5
quotients, and
by choosing
record the
partial quotients
51
quotient along
that they
with the
48
4
recognize!
remainder.
3 19
Click to proceed
at your own
speed!
I know
10 x 12 will
work…
85 R6
32 2726
1 60 0 50
1126
Compare the
partial quotients
800
25
used here to the
ones that you
326
chose!
3 20 10
6 85
Click here to go
back to the menu.
Click to proceed
at your own
speed!
5
3
35 × 7 23 × 7 7
3 5
1
15 × 2 03 × 2 2
8 0
6
Add 1the numbers
6
on the diagonals.
53
× 72
3500
Compare
210
to partial
100
products!
+
6
3816
16
1
6
× 23
1 2
0
200
2
2
120
0
1 3
30
3 3
8
+ 18
8
6
368
Click here to go
back to the menu.
Algorithms
“If children understand the mathematics behind the
problem, they may very well be able to come up with a
unique working algorithm that proves they “get it.”
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.”
Importance of Games
Provides . . .
. . .regular experience with meaningful
procedures so students develop and draw
on mathematical understanding even as
they cultivate computational proficiency.
Balance and connection of understanding
and proficiency are essential, particularly
for computation to be useful in
“comprehending” problem-solving
situations.
Benefits
Should be central part of mathematics
curriculum
Engaging opportunities for practice
Encourages strategic mathematical
thinking
Encourages efficiency in computation
Develops familiarity with number system
and compatible numbers (landmark)
Provides home school connection
Where’s the Math?
What mathematical ideas or understanding
does this game promote?
What mathematics is involved in effective
strategies for playing this game?
What numerical understanding is involved
in scoring this game?
How much of the game is luck or
mathematical skill?
Games Require Reflection
Games need to be seen as a
learning experience
Where’s the Math?
What is the goal of the game? Post this
for students.
Ask mathematical questions and have
students write responses.
Model the game first, along with
mathematical thinking
Encourage cooperation, not competition
Share the game and mathematical goals
with parents
Extensions
Have students create rules or
different versions of the games
Require students to test out the
games, explain and justify revisions
based on fairness, mathematical
reasoning
Games websites
www.mathwire.com
http://childparenting.about.com/od/makeathome
mathgames/
http://www.netrover.com/~kingskid/Math/math.ht
m
http://www.multiplication.com/classroom_games.
htm
http://www.awesomelibrary.org/Classroom/Mathe
matics/Mathematics.html
http://www.primarygames.co.uk/
http://www.pbs.org/teachers/math/