Author Archives: Jason Chamberlain

Really? You’re Still Counting? (Part 3)

Counting.  Nothing in mathematics gives us more of a return on our investment.

In my first blog post in this series, we briefly looked at beginning counters, who start with rote counting and then move on to count concrete items.  These are substantial investments in their mathematical futures. For more, read my blog from May 23rd, 2018.

In the second blog, I discussed and showed examples of what it looks like as students begin to count imagined items.  This is the first payoff from all of the counting that they did with concrete items. They can imagine the concrete counting and use that to count “figuratively.”  See my blog from June 27, 2018.

Counting: Counting Movements – Counting All

With enough experiences of counting imagined finger or spatial patterns, children begin to find it unnecessary to point to specific locations of an imaginary spatial pattern or count specific finger movements.  At first, they may still move their fingers or point to a location, but they are not attending to the specific movement or location. You can see both of these evidenced by Lucas in the following video.

Students at this stage of their development may begin to tap their finger, nod their head, swing their leg, or perform some other movement in sync with the counts.  In additive situations (YES, THEY’RE STILL COUNTING!), such as Lucas’, they tend to count movements for the first addend (in his case, 18) and then find a way to keep track of the second addend (5) with a specific finger or spatial pattern developed in the previous stage.  They will usually use fingers to keep track or maybe a spatial pattern if it is convenient, such as the girl in this next video:

Here is another example of that student not paying attention at all to the specific finger taps she is making on the dice:

Counting: Counting Number Words – Counting All

With enough experiences of counting movements, students will begin to count only their number words.  In the following video, watch closely to see Ally’s lips move as she counts to 19 for the first addend and then uses her fingers to keep track of the second addend (3).  She goes on to explain her solution

Again, we most commonly refer to this behavior as adding, SHE IS STILL COUNTING!  This usually only happens for a brief time, because students soon figure out they don’t need to count all of the numbers of the first addend.  When they realize this, we can say they can “count on” with conceptual understanding.

Next time, we will look at what happens as students get a significant return on their investments of counting all from:

  1. Just saying a number word sequence
  2. Counting concrete items
  3. Counting in the imagination
  4. Counting movements
  5. Counting number words.  

The payoff for all of this is “counting on.”

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4×4=1?

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The Power of Imagination in Mathematics

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