Not Your Mom’s Flashcards: Conceptual Understanding of Multiplication

What’s the purpose of flashcards?

There’s 2 answers I frequently hear and only one I agree with.

  • Answer 1: help kids know/memorize their facts
  • Answer 2: help students build fluency through the automatization of strategy

I know how Tracy and many others answer this question.

One thing I love about our online community is our willingness to share, grow and learn together.  Here’s 2 different sets of cards you can use with your students and share with your parents to build true mathematical fluency.

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Multiplication Subitizing Cards                                                         Multiplication Array Cards

I’m still trying to figure out who was the genius behind the subitizing cards because they’re awesome.  Here’s a post how I’ve used the cards in class and with my daughter when she was in 2nd grade.

The multiplication array cards are great because they allow students to see and use the structure in number to build fluency.  Unfortunately, I had to place my name in the pdf not for credit because that’s not how we roll.  I don’t want them to make their way on Teachers Pay Teachers.  I created them to share for free and I want them to stay that way.

Steve Wyborney is also also another teacher who is firmly committed to sharing a ton of resources that build fluency.  Oh…and they’re free!  When you visit Steve’s site just drop him a line and tell him “thanks”.  The man is awesome!

As we dive into a new school year, fluency will always be a goal.  My hope is that we do it the way it was intended. From Memory” and Memorization: There is a Difference.

If you use these cards as a parent or teacher, please report back and share how it goes. All of us are smarter than one of us.

About gfletchy

K-8 math consumer trying to listen and learn each day. Stay thirsty my friends!
This entry was posted in 3-5, Estimation, Fact Fluency, Games, Math Tools, Number Sense and tagged , , , , , . Bookmark the permalink.

30 Responses to Not Your Mom’s Flashcards: Conceptual Understanding of Multiplication

  1. Andy says:

    I’m planning to use traditional flashcards for a group of incoming 9th graders since a lack of multiplication fluency makes factoring nearly impossible. What are you thoughts on the most purposeful way to do that instead of memorization given this purpose?

    • WISE Math says:

      It never fails to shock me how many educators speak as if memory is evil and memorization should be avoided at all costs. This is antithetical to good practice and violates all that cognitive science tells us about the process of learning. Memory is the seat of learning. It is the goldmine of education, the richest seam. Solid memory work is the essential substrate upon which all understanding is built. There is no such thing as “understanding without knowing”. Understanding is the goal … but knowing is the essential priority.

      Who pushes this nonsense among educators? What great insight weighs against common sense and generational wisdom on this point?

      I support your use of traditional flashcards, Andy — they are one of many different helpful tools you could use for this, and a tried and proven one. I’m not saying there *isn’t* some other way to use them, but your first assumption should be that you will be using them for their intended purpose, more-or-less as flash-card users have always done. Why fix what isn’t broken?

      I’m also skeptical about this apparent notion many have that there’s something wrong if you aren’t reinventing the wheel each year … no, don’t abandon something that works just because it’s “old”. Thinking outside the box isn’t evil … but there’s nothing a priori to commend it over thinking inside the box first. There’s a reason it’s “the box”. There’s a reason everyone puts on socks before shoes — I’d suggest mastering that before trying some other approach. If you’ve used traditional flashcards in the conventional fashion for many years and don’t find they meet your expectations that’s one thing. But if you’ve never explored the strength of that technique, why not start by giving it a chance; if you’re unsatisfied with the results, *then* look for some way to add value.

      Dr. R. Craigen
      co-founder, WISE Math

      • gfletchy says:

        Dr. Craigen,
        Thanks so much for the thoughtful reply and it’s quite apparent that this topic strikes a chord. You use the terms memorization and memory interchangeably however I see them as different (mathematically speaking). Memorization is forced energy and when a concept is learned/built “from memory” it is built through an experience.
        Allowing students to build their understanding of facts through repetition and the repeated use of a strategy is what build’s automaticity. The is no questioning the importance of students being able to quickly recall their facts. However, I can’t ignore the disconnect happening in the majority of our classes. The issue is not in strategy development, it is in the limited number of opportunities that students have to employ, practice and automatize their strategy. This might articulate the difference more concisely.
        Flashcards are not evil, it is how we use them. We should purposefully select the cards that support the strategies being developed in class at any given time.

    • gfletchy says:

      I completely agree on the importance of multiplication fluency, especially at the high school level. Sadly, the majority of students who are entering 9th grade have only been asked to memorize facts to this point in their life. So it’s not so much a question as to whether it’s new math or old math, rote memorization is skill that many of them are unable to perform.
      When we (you and I) learned our facts we begin with 1 and worked our way right up through 12. There were no connections made to previous facts or what we had already learned. I’d suggest purposefully selecting the order in which introduce the cards in order to achieve mastery. Here is a helpful resource that suggests a progressional sequence to obtain fact mastery.
      Here’s a game to make practice a little more enjoyable:

      Hope it helps Andy and let me know how it goes.

      • Andy says:

        Thank you. I will need to see where they are at. It is likely an issue for some students, so it will be small group support. The larger issue for some is that nobody tried to teach them multiplication facts since 4th grade, even though some did not know it then. If there are large gaps, this progression will be very helpful.

    • Math Grab says:

      Hi Andy – I would you suggest you check out Math Grab ( to help your students learn multiplication. Full disclosure it was created by my husband and I but we have been getting some great feedback from kids & parents that it’s really helping kids learn multiplication! Let us know what you think!

  2. Outstanding stuff. Thanks!! This will really help multiplication fluency in the 4th grade!

  3. howardat58 says:

    Two things:
    1. Given the complications of using x for multiply and x as a variable (sooner or later) I think it would be better to use a plain sans serif x for multiply on the cards. (I am a pain in the a…..)
    2. You might find the following post of mine fun and useful

    • gfletchy says:

      Love the illustration you’ve shared Howard. It will be a nice little exploration to throw in front of kids. What do you notice? What do you wonder?
      Thanks for the suggestion on the cards. Once I make the change I’ll update the file!

  4. Pingback: Quick Images Number String | monkeyinaroundwithmath

  5. I used the subitizing cards to create a number string using a PowerPoint presentation. It went really well. Wrote about it here *, and linked the powerpoint in case anyone is interested. I will try to get some pictures from the number talk to add if I can. I can’t wait to try some more!

  6. Pingback: Multiplicative reasoning – Part 1: Times tables | This is my classroom

  7. Pingback: Multiplicative reasoning – Part 1: Times tables | Penn Wood CPD

  8. Math Grab says:

    Hi All – My husband and I recently created a fun Multiplication Card Game called Math Grab designed to help kids memorize their multiplication times tables in a fun family setting! We have been getting excellent feedback from teachers, parents, and students. I wanted to share this with everyone!

  9. Thanks for the links to the subitizing and array cards. Here is another resource that I think would complement these for teachers and students. They are flashcards posted on Youcubed from Jo Boaler:

  10. Pingback: Multiplication and Division: From Memory and Memorization | Musings Of A Math Teacher Blog

  11. Nate Rosin says:

    Love the multiplication subitizing cards. I shared them with a 3rd grade teacher in my district. I am looking forward to hearing what she thought of them.

    I do have one question: why do the cards only go through 7?

  12. Pingback: The Multiplication Sundae and the Bad Taste of Incentives | Questioning My Metacognition

  13. Emily says:

    Can you please give more info about how to use the array cards (why are some blocked shaded – what is the strategy?)
    Thank you !

    • gfletchy says:

      Hi Emily,
      The shaded boxes are introduced to move students away from one-to-one counting. Once students have a conceptual understanding of multiplication, the cards are introduced to move students towards imaging the numbers instead of having to see and physically count them.

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    I’ll bookmark your blog and check once more right here
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    right right here! Best of luck for the following!

  15. Pingback: Fact Fluency- What is it? | SMathSmarts

  16. Pingback: Math Fact Fluency Ideas - Sparkling in Second Grade

  17. Mary Lee Lee Newman says:

    Thank you for the flashcards and resources, especially for providing them for free! I refine my thoughts about multiplication fact fluency every year. This year I am going to add the array cards for each student and a parent meeting to get parents involved in practicing math facts.

  18. Sarah Johnson says:

    I just found these cards this weekend and have already used them with 3rd-grade intervention groups this morning. I’m curious about one thing with the multiplication subitizing cards. Why is 7 groups of 7 the highest multiple on the cards? What do you recommend for a visual/strategy for the higher multiples?

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