Building Multiplication Fluency in Middle School

  • Building Multiplication Fluency in Middle School

    By Michael Wiernicki, posted November 23, 2015 –

    I recently had the opportunity to sit with a sixth-grade teacher who had some deep concerns about some of her students who were struggling with their basic facts, mainly multiplication. Her worries were written all over her face: “How can I get these students to know their facts and get them through the sixth-grade curriculum by May?”

    Her worries were valid, and she is certainly not alone. I’ve had similar conversations with teachers in high school as well as elementary school. To teach mathematics, the problems we give to students must be memorable. Although this may not be easy (it’s much easier to hand students a deck of flash cards or give them a 5-minute speed test), it is worth the effort. If we want to fill mathematical gaps, we need to help students develop understanding. With understanding, connections are made, and the mathematics becomes memorable.

    To be clear, there’s nothing wrong with practice. Even drill can have a place. But what are we drilling? If it’s memory, forget it (no pun intended). Memorization will fail unless connections are made through building understanding. If we’re drilling strategy, then we’re on to something. But before we get to drill, we’ve got to have some investigation and some purposeful, memorable practice.

    Through discussion, we came up with a plan to help these students develop their understanding of multiplication, make the facts memorable, and not lose sight of the sixth-grade content (and even preview some seventh-, eighth-, and ninth-grade content . . . shhh).

    We began by showing students this picture:

    2015-11-23 art1 

    We asked the students what they noticed. Before sharing their thoughts with the class, we asked them to first talk with some group members about their ideas. When it was time to share, all students were engaged. A sample of what they noticed will appear in the comments section one week after this post. In the meantime, please share in the comments. What do you notice?

    The students noticed more and more as the discussion progressed. Each time students shared, we had them explain what they were noticing so that everyone could understand. We thought it was very important for these students to realize that they were capable of not only noticing mathematics but also explaining the mathematics they saw. The teacher was impressed as she heard these often-struggling math students use mathematical language, communicate with a high degree of precision, notice patterns, and explain number relationships.

    This sharing took a lot more time than we planned, but it was worth it for the engagement and mathematical discussion we got in return.

    As we wrapped up the discussion, some students asked questions; this was perfect because we were going to ask them what they were wondering about. There were not as many “I wondered about” as “I noticed,” but students were still a bit curious about the picture itself. I will discuss these student wonderings in the comments next week. In the meantime, please share in the comments. What do you wonder?

    The focus of this first lesson was to get students talking and get them to realize that they really are capable of making sense of mathematics, using mathematical descriptions of the arrays in the picture (not just color), and asking mathematical questions. We did not have time to allow students to answer any of the questions they had, but we did continue it the next day. The next post will detail that.

    2015-11-23 Wiernicki

    Mike Wiernicki,, is a math specialist in Georgia. He tweets at @mikewiernicki and blogs about teaching mathematics at


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    Michael Wiernicki - 2/13/2016 10:54:34 AM
    Excellent observations everyone! The math that students see and connect to the diagonals and Jane Rebecca noted above helps them make sense of facts and builds a nice little coat hanger for them to hang the knowledge of squares, square roots, etc. on when they come to it later! Better that they hang it somewhere where it's easily accessible than it clutter up the floor of their brains!

    Jane Porath - 12/4/2015 6:29:06 PM
    As a middle school teacher, I often see students who do not know their foundational facts. Actually this problem has gotten worse as I have gotten older. Students often struggled to do 7 x 8, but now 4 x 8 is a challenge for many. I think our students are developing a conceptual understanding, but they are not becoming fluent. This is where the importance of the practice comes into play.

    Jane Porath - 12/4/2015 6:25:07 PM
    I notice that there are perfect squares moving diagonally from top left to bottom right. I notice that each new color, inclusive of the previous colors, also creates a perfect square on number of rectangles.

    Rebecca Damas - 12/3/2015 10:38:01 AM
    The high school teachers I work with lament on occasion that their high schoolers often lack fluency with mathematical facts. This image would be an interesting opportunity for high schoolers to connect what they are studying with some of those facts that keep slipping their minds. I'm going to forward this to them pronto! Meanwhile, I'm loving the diagonal line of symmetry, and I wonder how many students would connect this image to algebra tiles, representations of completing the square, or representations of FOILing.

    Michael Wiernicki - 11/30/2015 2:21:36 PM
    Chris, Nice noticings. The students we worked with didn't dive into the perimeter ideas, but may have if we had done this investigation later in the year. The total dots is a fascinating investigation as well, especially if the pattern of colors is used in determining the total. Thanks for the comment! I'll be posting the student notices and wonders below, later this afternoon. Mike

    Harold Greist - 11/26/2015 8:04:15 AM
    I notice that the longest side of each rectangle determines the color of the rectangle.

    Chris Bolognese - 11/25/2015 7:36:27 PM
    I wonder how many total dots/pegs there are collectively. I wonder what the total perimeter is of all the individual rectangles.