As you may have read from a previous post, I have spent quite a bit of time and thought on how to assist my daughter in developing schema for dividing decimals. Even knowing the importance of prior knowledge on understanding new information, I have still been amazed at how this "building schema" work has paid off. In about a month we will be working on a unit on adding and subtracting fractions with unlike denominators. Kylie's schema for fractions is actually pretty good, but we have been working on schema for this unit for some time. I thought I would share with you the approach that I have used. First of all, I need to cite the following book as being very influential on the process that I will be describing.
Minilessons for Operations with Fractions, Decimals, and Percents: A Yearlong Resource
by Kara Louise Imm, Catherine Twomey Fosnot, and Willem Uittenbogaard
ISBN-13: 978-0-325-01029-8 OR ISBN-10:0-325-01029-3
http://www.firsthand.heinemann.com/
For those of you familiar with the Fosnot/Dolk books we have read in EDU 355, you will recognize these as the 'minilessons for efficient computation.' Minilessons are fairly short (appx. 10 minute), highly focused lesson starters which address strategy development. It is important to realize that all of the minilessons discussed below are given to the students WITHOUT first telling them a procedure. The idea is that they use their schema to make sense of situations, and that doing so extends their schema. By carefully sequencing the problems and by suggesting models for thinking, students are able to be successful without an enforced step-by-step procedure.
At the beginning of 5th grade, I gave Kylie daily minilessons with the prompt to "think about money as you attempt to solve these". This prompt suggested the use of money as the model. The first minilesson string of problems I gave her was:
1/2 + 1/5
2/5 + 1/2
1/2 + 1/10
2/5 + 3/10
2 1/2 + 1 2/5
10 1/2 + 5 3/5
99 3/5 + 1 1/2
In each case the denominator was easily transferable to a money context. Kylie was able to solve all of these with a bit of think time and there were some other desirable consequences from this exercise. She wrote answers in both decimal and fraction form, which helped her think about connections between fractions and decimals. She also developed her relational thinking. For example, "Well, 1/5 is .20 so 2/5 must be .40." This focus on developing relational thinking continued over the next few weeks as the strings became more complex, continually building schema. They involved both addition and subtraction, and ended with explorations regarding how changing the denominator by a power of ten influenced the value of the fraction. For example, thinking about the difference between 5/10 and 5/100 (both of which are easily represented with the money model) to 5/1000 (which isn't as easily represented, but which can be deduced by careful thought about the patterns present in the first two fraction/decimal equivalents. Using what we learned from this discussion, Kylie was able to project this thinking onto other, less obvious fractions, such as relating 3/4 to 3/40. The very last string I gave her with the money model prompt was:
2 7/100 - 1 7/50
2 18/200 - 1 9/100
4 12/25 - 3 2/5
1/5 - 1/10 - 1/20
1/5 - 1/25 - 1/100
1/4 - 1/25 - 1/50
The next day we had a discussion about certain denominators not working very well with the money model. We looked at a clock face and talked about what kinds of fractions might we be able to represent using the face of a clock. For each fraction, we talked about how that would be represented, sometimes with a variety of denominators. For example, 1/12 can also be 5/60. After time spent exploring the clock we began with the following string:
1/2 + 1/3
1/6 + 1/2
1/6 + 3/12 + 1/4
1/6 + 1/4 + 7/12
3/4 + 2/12
We continued to do strings daily for the next 2-3 weeks extending this thinking. Sometimes she had a fraction that didn't seem to fit well with the clock but it was one that could be reduced (ex. 6/18 = 1/3) and used appropriately. Eventually, subtraction was also included. It is important to realize that at no point has the idea of common denominators been introduced. It is not necessary to use this concept with all addition/subtraction of fractions with unlike denominators. After the opportunity to explore and use these two models, there were a few minilessons in which Kylie had to choose which model made sense "for those particular numbers." This gave her the opportunity to exercise metacognitive decision-making.
For most of late October, November, and December our minilessons explored aspects of multiplication and division to build schema for those operations with decimals. Recently we have returned to fractions. In "Phase 3" of strategy development, I wanted to introduce a model that will be particularly useful in helping her think about common denominators without implanting this idea in some kind of formulaic way. In addition, she still will be exercising decision-making by choosing which common denominator is most user-friendly for her schema.
The latest model is the "double number line." In this approach, students look at the two or more numbers they are adding or subtracting and they decide what they would like to use as the common whole. For example, the first problem I gave Kylie was 1/4 + 1/5. In the formulaic way we teach students, they are supposed to choose the least common denominator of 20. In reality, this is only one of an infinite number of possible denominators to choose from. Kylie chose to make the whole 100. She labeled a line with 0 on one end and 100 on the other. These whole numbers were labeled on the under side of the line. On the top of the line she drew lines showing the approximate position of these fractions (this is also an opportunity to build conceptual understanding of fractions--"where should you draw the line for 1/5? To the right or left of the line for 1/4?) (The blog format is not allowing the model below to be viewed appropriately. The 20 should appear directly underneath the 1/5. The 25 should appear under the 1/4. The 100 should appear under the far right end of the line.)
_______1/5__1/4___________________________________
0 20 25 100
As a result of this visual model, she could see that her final answer would be 45/100. A different student might have represented the same problem as follows: (Again, the 4 should appear under the 1/5, the 5 should appear under the 1/4, and the 20 should appear under the far right end of the line.)
________1/5___1/4_________________________________
0 4 5 20
This representation results in a visual model that shows 1/5 + 1/4 = 9/20. Equivalence can become a natural part of the dialogue here. Did these students get answers that were equivalent or not?
The first string in the sequence for double number line was:
1/4 + 1/5
2/4 + 1/5
1/4 + 45/100
1/2 + 1/4 + 1/5
55/100 + 25
Just to give you an idea where this goes, the last of these 12 strings includes 5/7 - 30/70 and 75/150 - 1/4. As you can see, her preparation for a unit on adding and subtracting fractions with unlike denominators will be much more extensive than is typically developed in most commercial curriculum.
This last model is by far the most flexible, but the earlier models built important ideas about adding and subtracting fractions. When Kylie has a bit more experience with the double number line in adding and subtracting fractions, then she will be prompted to "choose your model" for subsequent problems. In other words, the focus is not on having her adopt a particular model, but rather on her developing her ability to think about what makes sense for a particular set of numbers and operation. This type of decision-making is essential in fostering mathematical thinking that will transfer to the future.
Sunday, February 1, 2009
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