Really? You’re Still Counting (Part 5)

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 types of counting are huge investments in their mathematical futures.

In the second post, 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 are able to imagine the concrete counting and use that to count “figuratively”.

In the third post, there was some talk and video showing how students begin to just count their movements and then begin to count by just saying number words.

Last time, we discussed how students begin to count on but only count by ones to keep track of the second addend in an additive situation.

Counting: Counting Their Counts – Counting On

If you read the previous blog post, you may have noticed in the video that the student used her fingers to keep track of her counts for the second addend.  The payoff for doing this repeatedly happens when students begin to be able to count their counts. One place this behavior shows up is in missing addend situations.

Students like this one (without enough experience counting on by ones) may do something that looks like it might provide a solution, but they get lost because they don’t count their counts.  Although this student initially was able to provide the correct solution, he wasn’t sure of what he did and really got confused.

With more practice counting on by ones, solving missing addend problems becomes possible.

If students are able to count backward, subtractive (counting off) situations are also possible.  In the following video, the students had 24 blocks under their cloth and they took out 5. The question was, “How many are left?”  This student, without giving away her solution, was explaining how she solved it:

In all of these situations, students at this level know they need to count how many times they count.  These students also begin to use easy composites (usually 10s, because they are emphasized in our base 10 system) and then count/track the ones separately.

Counting: Counting With Composites – Counting On

With enough experiences of counting on (or off) by counting their counts, students begin to count with composites.  The student in this video liked basketball, so he was presented with a situation where there were 14 players in the gym, some more showed up, and then there were 51 (missing addend problem).  You may notice that he first counts by 10s to get to 44, and then counts by 1s to get the rest of the way to 51. This is made possible by being able to count with composites of 10. He also began to count by composites other than 10 as he moved into this stage of counting.

This next student counted with various composites in a situation where he was given 63 – ? = 37:

Very advanced students at this level can go either way fluently, like this student:

After all that counting, in the words of Dr. Leslie Steffe, children’s mathematics guru from the University of Georgia, now students “can really do some math.”

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

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