Subitizing to Foster Multiplicative Thinking

Learning basic facts through tricks or a mnemonic song lead students down a path of memorization, not understanding.  I previously discussed the idea of conceptual development here.  When students practice their multiplication facts, they’re expected to move from concrete to abstract too quickly.  This takes time!!!

Moving too quickly forces memorization and avoids any possibility of multiplicative thinking.  First students are asked to model 3 rows of 6 using colored tiles…

Screen Shot 2015-02-10 at 9.06.33 AM

Then after a day or two of exploring arrays, students are magically expected to remember that…

Screen Shot 2015-02-24 at 8.52.05 AM

Intensions are good but strategy development is the key underpinning of automaticity.  In the past I’ve asked myself the following questions:

  • When do students practice multiplication that’s not written as AxB? (for the purpose of automaticity) 
  • When/how is multiplicative reasoning fostered?
  • When do multiplication strategies become automatized?
  • When do number strategies become number knowledge?

My answer to all 4 questions….I use multiplication subitizing cards with my daughter whose in 2nd grade so I figured I’d videotape and share.

My takeaways:

  • It’s easy to identify which facts she’s comfortable with and has automatized.
  • At 1:15– she demonstrates how she used multiplicative thinking for 6×6.
  • At 2:00- she could use her understanding of the commutative property for multiplication to build fluency.  She was quickly is able to recognize that 7×5=35 at 1 minute, but it took some time to figure out 5×7.  She owns this strategy for addition so I’m waiting for it to click with multiplication….no rush:-)
  • At 2:20– I missed a fact when she was wrong (doh!)
  • At 2:48 and 3:55 she explains her multiplicative reasoning.

5 minutes spent purposefully building fluency and I gained so much information about my daughter. The great thing is that it works for every 8 year old…even the ones in your school.

Disclaimer: I didn’t create the cards and don’t know who did but they deserve a virtual math hug!  If you know who did please let me know so I can give proper attribution!


About gfletchy

K-8 math consumer trying to listen and learn each day. Stay thirsty my friends!
This entry was posted in 3-5, 6-8, Estimation, Fact Fluency, Games, Math Tools, Number Sense, Strategy Development. Bookmark the permalink.

13 Responses to Subitizing to Foster Multiplicative Thinking

  1. steven waters says:

    Subitizing activities are an integral part of the development of all facets of number sense. Some kids do not develop this mental capacity as automatically as some others and these activities are such a wonderful method to get those kids caught up, or in your case to begin the process of developing multiplicative reasoning. Thanks for sharing.

    • gfletchy says:

      Absolutely!!! I think as teachers we all get the “big ideas” but it’s identifying the ways to bridge the learning in-between that we sometimes find difficult!
      Hopefully this piece of the puzzle will help pick up a couple more students.
      Thanks for checking in Steven and for the comment.

  2. Andrew says:

    Can’t wait to use these cards with my students working on developing multiplicative thinking! Thanks Graham!

  3. Marian Dingle says:

    I tried using the cards with a fifth grader last week, hoping that they would help her develop her multiplication fluency. Well, what I discovered was that she had a hard time subitizing even 1 group of numbers, let alone multiple groups in the circles. She had to keep counting them on each card.
    When asked, she said she had never used such cards, so I took this to mean she had never been exposed to subitizing with addition or subtraction. It really never occurred to me that she wouldn’t recognize 7 dots. So next week, we will be working on just the lower numbers and then build up from there. This is a great reminder to me to continually search for gaps in understanding. Thanks so much for these cards!

  4. Love this! I plan to try them with my 5th graders who are struggling with math fact memorization. The picture in the middle of the page says 3 X 6 = 30. What am I missing? Is that supposed to be a mistake or…

  5. Pingback: Not Your Mom’s Flashcards: Conceptual Understanding of Multiplication | Questioning My Metacognition

  6. Can anyone explain to me why the cards use a different pattern for the smaller pattern of 7 dots inside the circle vs the 7 circles of dots ? To me it does not make sense to have 2 different patterns at micro/macro level.. This is really cool stuff – I will try on my kids (and myself)

    • gfletchy says:

      Good question Fredrik and to be honest I’ve never really thought about it, but I have noticed it.

      I am in the process of creating some similar cards since I am not the one who created these in the first place. I wish I knew who was the creator. I’ll be sure to keep the pattern consistent throughout to help build automaticity. Thanks for pointing that out again.

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