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In our program, we made a direct link from percents—which by now the students thought of in proportional terms—to decimals. In this way of thinking, a decimal represents an intermediate distance between two numbers e. To begin the lessons in decimals, the students were given LCD stopwatches with screens that displayed seconds and hundredths of seconds the latter indicated by two small digits to the right of the numbers; see Figure After experimenting with the stopwatches, the children noted that there were of these small time units in 1 second.

We continued our work with decimals and stopwatches, with a focus on ordering numbers. After discussion, they learned to record their times as decimals. So, for example, 20 centiseconds was written as. Next, the students compared their personal quickest reaction time with that of their classmates, then ordered the times from quickest to slowest. In this exercise, the students could learn from their experience of trying to get the quickest time that, for example, 0.

In this second level of the instructional program, the students were introduced to decimals for the first time. Students worked on many activities that helped them first understand how decimals and percents are related and then learn how to represent decimals symbolically. As the decimal lessons proceeded, we moved on to activities designed to help students to consider and reflect on magnitude. Thus the final activities included situations in which students engaged in comparing and ordering decimals. As noted earlier, although the curriculum began with percents as the initial representation of rational numbers, we found that the students made many references to fractions.

Now, at this final level of the program, our goal was to give students a chance to work with fractions more formally and then provide them with opportunities to translate flexibly among fractions, decimals, and percents. In a first series of activities, students worked on tasks in which they were asked to represent a fraction in as many ways as they could.

Number of Rational Terms in Binomial Expansion - Class-11 CBSE/JEE Maths

While students initially used fractions in these equations, they soon incorporated the other representations in challenges they composed. How much more to make one whole? The students carried out further work on conversions with the LCD stopwatches used earlier in the program. For example, given a relatively simple secret code, e. This allowed them to increase their understanding of the possibility of fluid movement between representations.

In one set of lessons, I gave the students a set of specially designed cards depicting various representations of fractions, decimals, and percents e. The students used the cards to design games that challenged their classmates to make comparisons among and between representations.

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In the first game, the leader dealt the cards to the students, who in turn placed one card from their hand face up on the classroom floor. The challenge was to place the cards in order of increasing quantity. Students who disagreed with the placement of a particular card challenged the student who had gone before. This led to a great deal of debate. This was a fraction that the students had not previously encountered in their lessons, and Sarah was not sure where to place it. The cards took up the entire length of the classroom by the time every student had placed his or her cards on the floor!

A second card game employing the same deck of cards, invented by a pair of students, had as its goal not only the comparison of decimals, fractions, and percents in mixed representations, but also the addition and subtraction of the differences between these numbers. This game again used the LCD stopwatches introduced earlier in the lessons. The two students who invented the game, Claire and Maggie, based it on the popular card game War.

What happened next is transcribed from the videotape of their play:.

The Rational Number Project

So mine is more. Hum …. At this point in the lessons, most of the students were comfortable thinking about percents, decimals, and fractions together. In fact, they assumed a shorthand way of speaking about quantities as they translated from fraction to percent. To illustrate this, I present a short excerpt from a conversation held by a visiting teacher who had watched the game the two girls had started and asked them to explain their reasoning.

I was interested to know how you figured out which of the numbers is more,.

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Ok, it is like this. One-eighth is half of one-fourth, and one-fourth is 25 percent. Well, you certainly know percents very well. But what about decimals?

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  • We began with some formal activities with fractions and equivalencies, including tasks in which students had to work with and devise lengthy equations. We also had the students make up their own games and challenges to help them gain more practice in this kind of flexible movement from one operation to another. One of our primary goals here was to provide students with habits of mind regarding multiple representations that will be with them throughout their learning and lay the foundations for their ability to solve mathematical problems. To date, variations of our curriculum have been implemented and assessed in four experimental classrooms.

    From the very first lessons, students demonstrated and used their everyday knowledge of percents and worked successfully with percents in situations that called on their understanding of proportion. Our particular format also allowed students to express their informal knowledge of other concepts and meanings that are central to rational number understanding. Recall that when working with the pipes and tubes and the beakers of water, students successfully incorporated ideas of the rational-number subconstructs of measure, operator, and ratio.

    What was also evident was that they had a strong understanding of the unit whole and its transformations. Similarly, when decimals were introduced in the context of stopwatches, the students readily made sense of this new representation and were able to perform a variety of computations. Finally, by the end of the experimental sessions, the students had learned a flexible approach to translating among the representations of rational numbers using familiar benchmarks and halving and doubling as a vehicle of movement.

    While the class as a whole appeared to be engaged and motivated by the lessons, we needed to look at the improvement made by individual students at the end of the experimental intervention. We were also interested to see how the performance of students in the experimental group compared with that of students who had traditional classroom instruction. To these ends, we assessed the experimental students on a variety of tasks before and after the course of instruction and administered these same tasks.

    Briefly, we found that students in the experimental group had improved significantly. Specifically, the experimental group made more frequent reference to proportional concepts in justifying their answers than did the students in the nonexperimental group. Let us return to the question posed to Wyatt at the start of the program and excerpted at the beginning of this chapter and look at the responses of two students, Julie and Andy, whose reasoning was typical of that of the other students at the end of the program.

    Can I use this paper to try it out? As can be seen, both Andy and Julie correctly ordered the numbers using their knowledge of percent as a basis for their reasoning. Andy, a high-achieving student, simply converted these quantities to percents. Julie, identified as a lower achiever, used paper folding as a way of finding the bigger fraction.

    Both used multiplicative solutions, one concrete and one abstract.


    Can any fractions fit between one-fourth and two-fourths? And if so, can you name one? In a final example, students were asked to compute a percent of a given quantity—65 percent of Although this type of computation was performed regularly in our classrooms, 65 percent of was a significantly more difficult calculation than those the students had typically encountered in their lessons.

    Despite these differences, students found ways to solve this difficult problem. Okay, 50 percent of is Half of 80 is 40, so that is 25 percent. So if you add 80 and 40 you get So, take away 16 is Then I did 10 percent of , which is Then I did 5 percent, which was 8. For anyone who has seen a colleague pause when asked to compute a percentage, as one must, say, to calculate a tip, the ease with which these students worked through these problems is striking. These are only a few examples from the posttest interviews that illustrate the kinds of new understandings and interconnections students had been able to develop through their participation in the curriculum.

    Our research is still in an early stage. We will continue to pursue many questions, including the potential limitations of successive halving as a way of operating with rational numbers, downplaying of the important understandings associated with the quotient subconstruct, as well as a limited view of fractions. Furthermore, we need to learn more about how students who have been introduced to rational numbers in this way will proceed with their ongoing learning of mathematics. First, our program began with percents, thus permitting children to take advantage of their qualitative understanding of proportions and combine that understanding with their knowledge of the numbers from 1 to , while avoiding or at least postponing the problems presented by fractions. Second, we used linear measurement as a way of promoting the multiplicative ideas of relative quantities and fullness.

    Finally, our program emphasized benchmark values—of halves, quarters, eighths, etc. One well-established insight is that rational-number teaching focused on pie charts and part—whole understandings reinforces the primary problem students confront in learning rational number: the dominance of whole-number reasoning. One response is to place the multiplicative ideas of relative quantity, ratio, and proportion at the center of instruction.

    However, our curriculum also builds on our theory and research findings pointing to the knowledge students typically bring to the study of rational number that can serve as a foundation for conceptual change.