Private: Learning Math: Number and Operations
Classroom Case Studies, 6-8 Part B (continued): Reasoning About Number and Operations – Examining Students’ Reasoning
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?
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).
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.
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.