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The number 10 is dominant in our mathematics because the numeration system used is base 10. The understanding and using of tens is essential when working with place value, addition, subtraction, multiplication, and many other areas. How do children think about and use 10 in their mathematics?
Baron, a 1st-grader, was asked to show three fingers. He slowly put up one finger at a time while whispering “1, 2, 3,” then tentatively extended his hand to show the teacher his three fingers. “Good job! Can you show me six fingers?” Once again he extended his fingers one at a time while quietly counting. When he said six, he looked at his fingers and then touched each one while counting again, “1, 2, 3, 4, 5, 6.” Finally, he extended his hands toward the teacher to show his fingers. “Nice! Can you show me ten fingers?” Baron quickly threw both hands in front of him with all his fingers flailing out saying, “10!.”
“1-2-3-4-5-6-7-8-9-10-20-30-40. What?”. AJ stopped and stared down at his blocks with a puzzled look on his face. He started counting again, “1-2-3-4-5-6-7-8-9-10-20-30.” He stopped a second time and said, “Wait.” Now he looked confused. Counting is something he has done a lot. He has counted beyond 100 multiple times and this time he had just 13 blocks sitting on the ground in front of him. It has been a few months since we have done anything like this, but this was unexpected. He has never counted by tens in previous attempts, so it surprised me, and it also seemed to surprise him. My guess is that his kindergarten class has been practicing counting by tens.
Reno, a 2nd-grader, excitedly asked me if I knew what 10 and 10 are, then quickly said, “It’s 20!” He did the same for 20 and 20 along with 50 and 50, eagerly letting me know the answer for each. His questions led right into a line of questioning I had planned for him. I place a card in front of him with the numeral 14 on it and then a second card with the numeral 10 and asked for the total. He sat for a while staring at the cards before beginning to count while extending fingers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30…” He continued counting until he reached 52 and realized he was lost in his counting.
Another time while working in a 2nd-grade classroom, I asked a student to count out 23 snap cubes. Once he had done so, he was asked to connect cubes to makes stacks of 10. He made two stacks of 10 and had three loose cubes remaining that he pushed to the side. These were covered, and then he was told to count out ten more cubes and make another stack of 10 from them. Finally, he was asked how many cubes he has all together. He begins counting “23”, then extending his fingers sequentially counted “24-25-26-27-28-29-30-31-32-33” until he had ten fingers sticking up. “There’s 33!” Given the opportunity to check the total number of cubes, he lifted the cloth and pointed to each stack saying “10-20-30.” Then pointed to each of the individual cubes and said, “31-32-33.”
Keifer is a 2nd-grade student with whom I was continuing to explore ideas of tens in math situations. On one occasion I placed two cards in front of him with the numerals 33 and 10. I then asked for the total. Keifer quickly responded, “43.” When asked how he knew, he told me that “there’s 30 here”, pointing to the card with 33 printed on it, and then he added “and a ten here, which makes 40.” He then counted three more saying, “41-42-43” while extending his fingers. I gave him 24 and 21 more. He began counting “24”, then sequentially extended his fingers and said, “25-26-27-28-29-30-31-32-33-34-35…45” until he had extended 21 fingers while counting.
In each of the stories, what is the child’s understanding of 10, and what does that mean for their mathematical understanding? How could we as teachers use their understanding to help guide their learning?
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