Math Games to Build Skills and Thinking Claran Einfeldt, Cathy Carter,

What is “Computational Fluency”? “connection between conceptual understanding and computational proficiency” (NCTM 2000, p. 35)

Conceptual Computational Understanding 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 mental- computation skills.

Before selecting an algorithm, consider how you would solve the following problem 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 and then What was your thinking? = = = 847

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 Partial Quotients Lattice Multiplication Click on the algorithm you’d like to see! Trade First

Add the hundreds ( ) Add the tens ( ) 70 Add the ones (5 + 6) Add the partial sums ( ) Click to proceed at your own speed!

Add the hundreds ( ) 90 Add the tens ( ) Add the ones (6 + 7) Add the partial sums ( )

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56 × 82 4, , X X 6 2 X 50 2 X 6 Add the partial products Click to proceed at your own speed!

52 ×76 3, X X 2 6 X 50 6 X 2 3,952 Add the partial products

× 46 2, ,392 Click here to go back to the menu. A Geometrical Representation of Partial Products (Area Model)

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

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Subtract the hundreds (700 – 200) Subtract the tens (30 – 40) Subtract the ones (6 – 5) Add the partial differences (500 + (-10) + 1) – 2 4 5–

Subtract the hundreds (400 – 300) Subtract the tens (10 – 30) Subtract the ones (2 – 5) Add the partial differences (100 + (-20) + (-3)) – 3 3 5– Click here to go back to the menu.

R Click to proceed at your own speed! Students begin by 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…

Click here to go back to the menu R Compare the partial quotients used here to the ones that you chose!

× Compare to partial products! 3 × 7 3 × 2 5 × 7 5 × 2 Add the numbers on the diagonals. Click to proceed at your own speed!

× 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 mathgames/ m htm ematics/Mathematics.html