by Christy Plummer
When children gather around a game board – smiling and talking and sharing ideas – it's not always clear to the casual observer whether the math game they're playing is "just for fun" or a legitimate opportunity for learning. As teachers, we know the difference usually depends on us. Did we select the game for the sake of selecting a game, quickly inserted as a transition between "the lesson" and whatever comes next? Or did we intentionally introduce it – to the right learners at the right time – as a teaching tool? It sounds simple enough, but in the complex arena of real-world instruction, it is more challenging to do than anyone at our school anticipated. So let me tell you what we've learned from a couple of years of concentrated work on it.
A few years ago, questions about fact fluency began to bubble up in conversations among teachers. In our classrooms, we had a sense of which students were stronger with regard to fact knowledge, who seemed to be progressing at an appropriate pace, and who needed support. What we didn't have was a common tool to gather specific information that could help us understand students' growth in fact knowledge over time. In the spring of 2016, I attended an Indiana Partnership for Young Writers workshop with Courtney and Ryan Flessner and learned about Fact Fluency Interviews (Madison Metropolitan School District [MMSD], 2007, 2006). We were so excited to have found a tool to help us investigate our questions! We experimented with the interviews during the last months of school, and by May had made a decision to go all-in for the coming school year.
Fact Fluency Interviews at Our School:
A couple of times each year, each K-4 student sits beside a teacher with a list of decontextualized problems (e.g., 3 + 7 = ___). As a student responds verbally to each problem, the teacher records the answers, takes note of mathematical strategies, asks well-timed questions to gain insights into the child's thinking, and keeps a silent count as one measure of the child's automaticity with a given number relationship. These one-to-one exchanges have the feel of conversations and, in our experience, minimize anxiety associated with timed fact tests such as "mad math minutes" (Boaler, 2014).
A few months later, reflecting on our hard-won first round of schoolwide data, we had a shared be-careful-what-you-wish-for moment. The fact fluency profiles – created from the interviews for individuals, classrooms, and grade levels – revealed needs and opportunities that got our attention. Lots of us began to wonder, If we take care in introducing young readers to just-right books, shouldn't we be equally intentional about introducing young mathematicians to just-right games? The answer – Well, of course we should! – was obvious. We reasoned that identifying a just-right game was as simple as thinking carefully about the strategies and understandings exhibited by a group of learners and searching through our repertoire of games with an eye toward possible next steps. When children demonstrate automaticity with doubles relationships, for instance, a next step might be the introduction of game that involves "doubles plus one" or "doubles minus one" strategies (e.g., using 8+8 to solve 8+7 or 9+8). Sounds easy enough, right? We thought so, too.
But it took perseverance and a willingness to work together across grade levels to make real progress. Teachers are learning together through conversations about "what's next" for a given group of students, and we're doing a better job of proactively bridging students' experiences across grades. We've also developed a culture of sharing quality games that might otherwise have lived and died within a single classroom or grade level.
As we've worked through the hard parts, we've come to appreciate that matching students with just-right games depends equally upon three interrelated aspects of practice: current information about children's strategies, awareness of mathematics learning trajectories, and ready access to a bank of worthwhile games.
In our case, information about children's strategies was perhaps the easiest of the three to come by. We've studied Cognitively Guided Instruction (Carpenter, Fennema, Franke, Levi, & Empson, 2015) and used flexible problem-solving interviews to examine shifts in children's strategies across a range of problem types. Teachers in the early grades draw on the research of Robert Wright and his Math Recovery colleagues (Wright, Ellemor-Collins, & Tabor, 2012; Wright, Martland, & Stafford, 2006; Wright, Stanger, Stafford, & Martland, 2015) to understand and support foundational knowledge such as children's facility with number word sequences, numerals, and the structure of our number system. The information gleaned from Fact Fluency Interviews provides another lens through which to understand our mathematicians' strategies. Teachers' ongoing attention to strategy use between assessment periods keeps this information current.
Our effort to understand strategies led to new questions about children's mathematical development in the broader context of learning trajectories. A few years into our work with contextually based problems in Cognitively Guided Instruction, for example, we had questions as to why some of our problem-solvers seemed to get "stuck." The Math Recovery learning framework helped us appreciate the interplay between children's understanding of number and their skillfulness in problem solving (Wright et al., 2006; Wright et al., 2015). Most recently, a group of K-5 teachers read and talked our way across each domain in the Common Core math standards to better understand the vision for students' mathematical growth across time. We've drawn additional insight from the grade-band overviews presented in NCTM's Principles and Standards for School Mathematics (2000) and the domain-specific Progressions Documents that support the Common Core Math Standards (ime.math.arizona.edu/progressions). Familiarizing ourselves with learning trajectories helps us to avoid pitfalls of grade-level thinking, and better positions us to see and respond to the actual mathematicians who inhabit our classrooms.
Developing a bank of worthwhile math games has proven to be an ongoing process of gathering and culling. Our starter game collection came from two instructional resources published years ago by teachers and researchers in Madison, Wisconsin: Learning Mathematics in the Primary Grades (MMSD, 2006) and Learning Mathematics in the Intermediate Grades (MMSD, 2007). We're adding tried-and-true favorites from curricula we've taught through the years (e.g., Investigations, Everyday Math), as well as promising games from newer-to-us resources (e.g., Flessners' workshops, Math Recovery, Contexts for Learning Mathematics, M2, Stenhouse's Well Played series). One thing I've learned through playing games with the Flessners and others in the Partnership's math community is the importance of distinguishing between a game that happens to involve math and a truly worthwhile math game. In my experience, worthwhile games are those that:
It is a rare and treasured game that satisfies all the criteria (think Yahtzee), but holding this list in mind is helpful whether you're evaluating the potential of an existing game, adapting a game to better meet the needs of your students, or inventing a game in response to an unmet need.
Inventing a game, you say? Absolutely! Check out our original game, Collect the Tens. If it meets your criteria for a worthwhile game, we'd be honored for you to add it to your game bank. As you discover ways to adapt it for your students, we'd love to learn from your innovations.
By now I expect you realize that matching students with just-right games is challenging because it requires us to answer the same essential questions that arise whenever we set out to differentiate for young learners: Who is in front of me? Where do we want to go from here? and What tools are available?
Yes, it's big work, but we find that the investment in time and energy is yielding equally big returns for teachers and students. Are you game?
Christy Plummer is the Lower School Mathematics Coordinator for University School of Nashville in Nashville, TN. She transitioned to this role two years ago, following 13 years of classroom teaching. She earned her Master's Degree from Vanderbilt's Peabody College and joyfully drives to Indianapolis each month to participate in the Partnership's Math Teacher Research Group.
Boaler, J. (2014). Research suggests that timed tests cause math anxiety. Teaching children mathematics, 20(8), 469-474.
Carpenter, T., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2015). Children's mathematics: Cognitively guided instruction (2nd ed.). Portsmouth, NH: Heinemann.
Dacey, L., Gartland, K., & Lynch, J. B. (2016). Well played: Building mathematical thinking though number games and puzzles. Portland, ME: Stenhouse.
Madison Metropolitan School District (MMSD). Teaching and Learning—Math Division. (2007). Learning mathematics in the intermediate grades. Madison, WI: MMSD.
Madison Metropolitan School District (MMSD). Teaching and Learning—Math Division. (2006). Learning mathematics in the primary grades. Madison, WI: MMSD.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Wright, R. J., Ellemor-Collins, D., & Tabor, P. D. (2012). Developing number knowledge: Assessment, teaching, and intervention with 7—11-year-olds. London: Sage Publications.
Wright, R. J., Martland, J., & Stafford, A. K. (2006). Early numeracy: Assessment for teaching and intervention. London: Sage Publications.
Wright, R. J., Stanger, G., Stafford, A. K., & Martland, J. (2015). Teaching number in the classroom with 4—8 year olds (2nd ed.). London: Sage Publications.