NCTM:Children’s Difficulty with Elapsed Time and Giving Change

Reasoning at Two Hierarchical Levels Simultaneously:

Children’s Difficulty with Elapsed Time and Giving Change
Presenter: Constance Kamii from University of Alabama

I decided to go to this session because the title caught my eye. Two of the things that I find some of my students struggle with a lot are elapsed time and giving change, so I headed to this session to learn a few strategies to help me teach these concepts.

Elapsed Time
The presentation opened with Ms. Kamii, a very entertaining presenter, explaining the difference in operational time versus intuitive time. (Most of her data came from a study done by Piaget.) She shared a scenario of a four year old student who was asked several questions about time.

Question: Are you older than your sister?
Answer: Yes because I am taller than her.
Question: Is your grandma older than your mom?
Answer: I don’t think so.
Question: When you are 10, will you be older or your sister?
Answer: I don’t know.

This four year old has a concept of intuitive time. Intuitive is based only on the observable things. She can see that she is taller than her sister, so she must be older than her. She is not able to know if she will still be older than her sister when she is 10 because she cannot see who is taller then. These things are all things that she has deduced as logical based on what she knows. She will eventually move into operational time which is based on temporal reasoning. This is a much higher level of thinking.

Piaget states that there are three types of knowledge:
1. Physical knowledge: objects in the external world (That is a cube. It is red.)
2. Logico-Mathematical Knowledge: Consists of mental relationships with the ultimate source being in your head not knowable with your eyes. Time is not something that is observable, but a series of mental relationships. This is why it requires an operational time in order to understand.

3. Social-Conventional knowledge: knowledge of things like the English language, that Christmas occurs each year on December 25, knowing when it is appropriate to say “please” and “thank you.”

Ms. Kamii stated that there is no way to really show elapsed time to children because it is something that they have to see mentally. A majority of students do not recognize operational time developmentally until third grade.

As teachers work through problems with elapsed time, they should allow students to create and build upon their own understanding in order to build their operational knowledge. Textbooks teach students “standard” ways of determining elapsed time such as counting the hours first, then the minutes, creating a sort of algorithm and subtracting, or counting only the minutes. These strategies do not allow students to construct their own understanding of elapsed time naturally, thus creating misconceptions as they go.

Students should learn elapsed time in a natural progression. How would most of your students solve this problem? Work it in as part of your Calendar Math in the morning and see what the most common answer is.

“I started reading at 8:15 and finished at 9:10. How long did I read for?”
Many second graders in Ms. Kamii’s study said 9 hours – 8 hours is one hour and 15 minutes + 10 minutes is 25 minutes, so the answer is 1 hour and 25 minutes. They do not think logically about the time and realize that their answer cannot possibly be correct because from 8:15 to 9:10 is less than one hour. They are reasoning sequentially.

One way students begin to construct knowledge is to think about the problems in ways that make sense to them. Take this problem for example: I arrived at a birthday party at 6:40. I left the party at 9:15. How long did I stay at the party? One way of making sense of the problem is to think about the 6:40 in terms of how long it will be until we reach the next hour, 7:00. See the thought representation below. Hours are represented across the top, minutes along the bottom.

To help students with elapsed time, we should:
1. Allow students to create their own understanding of elapsed time and ways to model it.
2. Give students meaningful opportunities to work with elapsed time in class. Children are not taught to think about time throughout the school day in a context that is beneficial to them creating their own understanding of it. As teachers, we tend to say “It’s time for lunch, get your things ready” or “at 9:15, you need to be ready to do Calendar Math.” Instead, we should say to our students “How much time do we have until lunch? Are we able to complete a 1 hour and 15 minute math workshop before lunch or would it be best to do a 45 minute science lesson right now?”

Making Change
The second part of the session opened with a quick video clip of a second grader playing store with her teacher. The teacher had a box of candies for 2 cents and a box of lollipops for 3 cents. The student had one dime and one penny in her money box. The teacher began with simple problems. She bought one candy for 2 cents and gave the little girl 2 pennies. The little was satisfied with it and waited for the next problem.
The teacher bought a lollipop for three cents, but gave the little girl four pennies. The little girl gave the teacher back one penny.
The teacher bought 2 lollipops and gave the little girl a dime. The little girl gave the teacher back the entire dime saying that it only cost 6 cents. When asked how much a dime was worth, the little girl said “ten cents.” She knew the value of the dime, but even with prompting from the teacher could not figure out how to use the pennies she had collected from previous transactions in her money box to make change for the dime.
To correctly solve this problem, it requires her to think about several things simultaneously: a dime is worth ten cents, subtracting 10 cents – 6 cents to make change for a dime, and giving the correct change back to her teacher.

Another similar situation, counting change by groups of tens and ones, also causes students to think about more than one concept simultaneously. Students may count this as:

As teachers, we know to transfer the counting from groups of tens to ones when we count it. We are able to think of both concepts simultaneously as we are solving the problem.

How can we correct this in our students? By using the same concepts as working on elapsed time.
1. Allow students to create their own understanding
2. Use meaningful examples.