- The following is a set of interview data (using diagnostic interview tasks) taken over a week in the early part of the year with a class of second grade students. The interviews went as follows:
a. “Please write the number sixty-seven.” (All can do this.) “Now write the number that is one more than that number.” (Most can do this, also.) “Now write the number that is 10 more than 67.” In the sample, all of the students counted using their fingers and only a few were successful. Not a single child was able to write the number that was ten less. All tried to do so by counting backwards.
b. The digit correspondence task was done using 53 counters exactly as described in the text on page 220. There were no students in the group that evidenced a place-value understanding of the 5 in 53.
c. Students were shown a clear plastic baggy of 57 small counters and asked, “About how many do you think are in the bag?” Students were fairly successful at making reasonable estimates. Note that this first aspect of the task requires no place value understanding. Then the students were asked to help count out the contents by putting the counters into groups of ten. That was followed with beginning the “Fill the Tens” task (shown in Figure 10.5 p 219 in the text). The teacher then filled one ten-frame card and began the second. “If we keep on filling up these cards like this, how many cards will we need before we run out of these 57 counters?” Not one child got this correct. Many said we would need 57 cards. Most students simply had no idea. In the classroom, these same students were able to write the number for a tens and ones picture of rods and cubes, could read and write numbers, and could find numbers easily on the hundreds board.
What can you learn from the students’ responses to these questions?
What would the next steps be for these students?
- Find the number given the clues below
• Less than half of 100
• The product of the digits is less than one dozen
• The sum of the digits is more than 6
• Not a multiple of 5
• Containing only digits less than 9
• Containing only odd digits
• The tens place is less than the number of sides on a pentagon
• An odd number
• A number that does not contain the smallest prime odd number as a digit
• Between one dozen and two dozen
What is the number? __
Addition & Subtraction
- Solve these without using the standard algorithm. Describe the method you used for each (e.g., empty number line, splitting, shortcut, bar diagram, partial sums, etc).
465 + 230
526 + 98
7000 – 25
342 + 153 + 481
Multiplication & Division
- Solve these without using the standard algorithm. Describe the method you used for each (e.g., complete-number strategies, partitioning, compensation, cluster, etc).
35 x 12
86 x 42
45 x 6
- Solve these without using the standard algorithm. Describe the method you used for each (e.g., missing-factor, cluster, decomposition, etc).
1224 ÷ 24
583 ÷ 4
- Would you be learning more about addition, subtraction, multiplication, division concepts if you had used the standard algorithm—or by using your approach?