Monday / March 27

# 4 Great Classroom Games to Get at Real Mathematical Strategy and Fluency

A mathematics strategy game typically refers to a game that is won by out-maneuvering one’s opponent. Such games abound in apps. Some more common to the mathematics classroom include Nim and Mancala. These games include quantitative and deductive reasoning, and they are fun! But we would like to propose a new definition of strategy game — a game that helps students become proficient at using reasoning strategies. There are really two versions of such strategy games: 1) games that help students become proficient at using a strategy and 2) games that help students become proficient at choosing a strategy. Incorporating a combination of such games provides meaningful, enjoyable practice that supports the development of mathematical fluency. As defined by the National Research Council (2001) and illustrated in Figuring out Fluency in Mathematics Teaching and Learning (Bay-Williams & SanGiovanni, 2021), Fluency includes three components, each having observable actions.

Figure 1. Procedural fluency components and actions.

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The label “fluency game” is often given to speed games that focus getting faster at recalling basic facts. These are anything but fluency games. Students cannot think about which strategy is efficient when forced to hurry. Speed games like “Around the World” have a tragic impact on far too many students, causing anxiety and leading them to think they are not good at mathematics.

Here we share some real fluency games. No speed component; students don’t compete to get an answer faster than their opponent. Instead, these games provide students the opportunity to think and talk through strategies and to learn with their opponents.

### Strategy Games: Games that focus on a reasoning strategy

A Winning Streak. Make Tens and Compensation are two very useful reasoning strategies. For example, to add 39 + 17, a person might ‘move one over’ to rethink the problem as 40 + 16 (i.e., Make Tens) or they might just pretend the 39 is a 40, add, then subtract one from the 17 (i.e., Compensate). These useful strategies also apply to decimals and fractions where Make Tens becomes Make a Whole. In  A Winning Streak, students roll a 10-sided die (or draw a card) and add it to the given number on the game board (See Figure 2). This game provides students an opportunity to practice Make Tens or Make a Whole (and/or Compensation). New game boards are easy to make to provide more practice. Did you notice that students can also be strategic in which spots they claim on the game board in order to score more points. Can we call this a Strategy-Strategy Game?

Figure 2. Two versions of A Winning Streak.

You can download the Winning Streak +39 gameboard here and the +1.9 gameboard here.

Highs or Lows. How might a student fluent in multiplying fractions solve this problem: ? What they won’t do is change  into a fraction (. Yet this method is commonly used for problems, like this one, that are more efficiently solved by the Break Apart to Multiply strategy (i.e., using the distributive property). In Highs or Lows, players draw three (or four) cards,* make a multiplication problem, and solve it. Templates are provided to focus on a number type (e.g., one-digit times two-digit or one-digit times mixed number). They flip a coin to determine the goal (highest or lowest) – flip before arranging cards to put a focus on strategy or after to incorporate luck. The player with the product that fits the goal wins the round. Sample templates are shown in Figure 3.

*You can create access digit cards, 0-9, or use playing cards (queen = 0, ace = 1, remove 10s, jacks, and kings)

### Fluency Games: Games that focus on choosing and using different strategies

These real fluency games give students an opportunity to think about when they want to use a strategy, moving beyond a focus on accuracy to also focus on efficiency and flexibility.

Strategy Spin: Grab your scissors and turn a dull worksheet into this fluency game! But, first make sure the problems on the page lend to a variety of strategies.  Players take turns spinning the strategy spinner, and then look for a problem that is best solved using that strategy. If they find a problem and solve it correctly using that strategy, they keep the card. Play until the cards are gone. The player with the most cards wins. Beyond the four operations, this game can also be played with ratios and solving equations for unknowns (see spinners in Figure 4).

Figure 4. Two versions of Strategy Spin spinners.

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You can download the Strategy Spinner templates for subtraction here and for division here.

Strategories: This is not a misspelling, but an adaptation of a popular game. Students are given a recording card with strategies listed and a blank cell to write in an example problem. For addition, the cards look like one of these (whole number version and decimal/fraction version), as shown in Figure 5.

Figure 5. Strategories game boards for addition (one for whole numbers and one for fractions and/or decimals).

Place students in groups of three. Player 1 asks one of the other players to solve a problem on player 1’s card for _______ strategy. If the player explains the problem using that strategy, they score 5 points. If not, the third player gets a chance to “steal” by explaining the problem using that strategy. If the third player cannot, the author of the problem must explain using that strategy. If they cannot, they lose 10 points. Continue taking turns till all problems are solved (or play 3 rounds). High score wins.

### Strategic Use of Games

As teachers, we select games to help our students become competent and confident. The games shared here are designed to help students first become competent and confident in using strategies and then become competent and confident in choosing strategies. This type of practice is so much more useful, interesting, and cognitively demanding than solving a page of problems all the same way. That is not fluency practice – at best it is accuracy practice. We close this blog with a little checklist to help your strategic selection of games.

Game Characteristics to Seek Game Characteristics to Avoid
• Students have think time
• Students explain their reasoning aloud (and we can listen and assess)
• Involves the enactment of a strategy students need to learn
• Involves choosing a reasoning strategy
• Is fun!
• Can be played over and over again (either in its given form or by adapting it).
• Involves speed
• Students compete to solve the same problem
• Winner of the game is based on who is faster/stronger at using a strategy.
• Involves player elimination

Sherri Martinie is an associate professor of curriculum and instruction at Kansas State University, where she teaches undergraduate and graduate courses in mathematics education. Prior to taking her position at Kansas State, she taught elementary, middle, and high school mathematics for a combined 20 years. Sherri is an author, professional development leader, grant writer, and conference speaker.

Jennifer M. Bay-Williams is a professor of mathematics education at the University of Louisville, where she teaches preservice teachers, emerging elementary mathematics specialists, and doctoral students in mathematics education. She has authored numerous books as well as many journal articles, many of which focus on procedural fluency (and other aspects of effective mathematics teaching and learning).

Sherri and Jennifer are co-authors on the Figuring Out Fluency in Mathematics series. You can find many  more games, routines, centers, and worked examples like these in the series newest classroom companions: Figuring Out Fluency: Addition and Subtraction with Fractions and Decimals and Figuring Out Fluency: Multiplication and Division with Fractions and Decimals.