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Here are scenarios from two different teachers’ classrooms, each involving students’ developing ideas about number and operations. Snippets of students’ discussions are given for each scenario. For each student, consider the following:

*Understanding or Misunderstanding*: What does the statement reveal about the student’s understanding or misunderstanding of number and operations ideas? Which ideas are embedded in the student’s observations?*Next Instructional Moves*: If you were the teacher, how would you respond to this student? What questions might you ask so that the student would ground his or her comments in the context? What further tasks and situations might you present for the student to investigate? Note 3

Ming Hui and Kenneth were working to translate the base five number 1234 to a base ten number. The teacher has asked them not to use manipulatives. Below is a snippet of their conversation:

**Ming Hui:**We can’t use the tiles this time, so let’s try to remember what tiles are put under each place.

**Kenneth:** Okay, put the 1 tiles under the 4. That’s 4.

**Ming Hui:**And then the 5s are under the 3. That’s 15 more.

**Kenneth:** And then the 25s are under the 2. That makes 50 more.

**Ming Hui:**So far we have 4 plus 15 plus 50. That’s 69.

**Kenneth:** And then we have one more. That must be the 100s. We’ve got 169 in all.

**Ming Hui:**Yes, the base ten number is 169.

**a.**What methods did the students use to solve the problem? What do these methods tell you about how the students are thinking about the problem? What mistake did they make in their conversion process? Why do you think they made this common error?

**b.**How would you help the students deal with any misconceptions they have?

**a. **These two students imagine using the tiles to solve the problem. They correctly imagine choosing four 1 tiles for the digit on the right, three 5 tiles for the second digit, and two 25 tiles for the third digit. At this point, Ming Hui correctly adds what they have so far. Then Kenneth seems to stop thinking about the tiles and chooses 100, rather than 125, as the value of the next power of 5. Ming Hui appears to be happy with this answer. The students most likely made this error because they inadvertently switched to thinking in base ten, and therefore derived 100 as the next place value instead of 125.

**b. **These students need a reminder of the values of each power of 5. You could suggest that they always make a place-value chart before doing conversions in either direction. For this problem, their chart would look like this:

You could then suggest that they work from left to right when converting. This number would be (1 • 125) + (2 • 25) + (3 • 5) + (4 • 1) = 125 + 50 + 15 + 4 = 194.

Brad and Kent were working to translate the base ten number 342 to a base five number. Below is a snippet of their conversation:

**Brad:** That’s three 25s and four 5s and two more.

**Kent:** No, we’re going the other way.

**Brad:** Oh, you’re right. Then the three 100s are twelve 25s. That’s two 125s and two 25s.

**Kent:** And the four 10s will make eight 5s, or one 25 and three 5s.

**Brad:** Okay. So far we’ve got two 125s, three 25s, and three 5s. All we need is two more. So the number is 2332.

**a.**What methods did the students use to solve the problem? What do these methods tell you about how the students are thinking about the problem?

**b.**How would you help the students deal with any misconceptions they have?

**a.**These students clearly understand place-value systems but are using an unconventional method to translate the numbers. Brad first assumes 342 is a base five number and correctly translates it to base ten. When corrected by Kent, he translates from each base ten value to its base five equivalent. So instead of working with the number 342 from the beginning, he starts with 300. He correctly states that 300 is 12 • 25, which in turn is (2 • 125) + (2 • 25).

**b.**

These students clearly understand the system. However, you might suggest that translating each place of a base ten number separately could become messy and lead to mistakes. These students might also benefit from a base five chart. They could use a chart like the one below to remind them of the powers of 5 that are the place values and then record and subtract the number of each power of 5 as they keep a running total, as shown:

This tells us that the number is 2332 in base five.