Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Content Objective

Fractions have always represented a considerable challenge for students, even in the middle grades. Results of NAEP testing have consistently shown that students have a very weak understanding of fraction concepts (Werne &Kouba, 2000) This lack of understanding is then translated into many difficulties with fraction computation, decimal and percent concepts, the use of fractions in measurement, and ratio and proportion concepts. One reason for the weak performance is that students have limited exposure to fractions. They are introduced at the end of second grade, where according to the Common Core State Standards students will partition shapes and move on to learning about equal shares. In third grade, students learn to develop an understanding of fractions by the following standards. They learn that a fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts and that a fraction a/b is the quantity formed by a parts of sizes 1/b. They move on to understand a fraction as a number on the number line and to represent fractions on a number line diagram.  In the last third-grade fraction standard the student is asked to explain the equivalence of fractions in special cases and to compare fractions by reasoning about their size.  In Van De Walle’s book Elementary and Middle School Mathematics: Teaching Developmentally he found that “Few, if any programs provide students with adequate time or experiences to help them with this complex area of the curriculum”. He states that “We must explore a conceptual development of fraction concepts that help students at any level construct a firm foundation, preparing themes for the skills that are later built on these ideas.”⁵

He further states that “When teaching fractions, there are several ways to help students make connections for deeper understanding:”

1. Connect the informal knowledge that students bring to fraction instruction by relating new concepts, procedures, and symbols to real-world contexts and situations outside of school that are meaningful to students, and then build on them.

2. Connect the different models used in instruction so students learn to seamlessly progress from one to another and back again, gradually coming to appreciate their similarities and differences.

3. Connect the models used in instruction to the words articulated (new vocabulary, math talk), actions modeled (movements, gestures), and symbols manipulated (of the form a/b). Students who experience a variety of ways to think, talk, and act about fractions, and who are expected to move back and forth between them, develop more flexibility in their fraction understanding.⁶ In the Van De Walle book, he states that we should start teaching fractions by developing concepts.  What should students learn about fractions? What are the big ideas? The things that I will teach in second grade are as follows. These concepts are adopted from Van De Walle’s big ideas in his book.

1. Fractional parts are equal shares or equal-sized portions of a whole or unit. A unit can be an object or a collection of things. More abstractly, the unit is counted as 1. On the number line, the distance from 0 to 1 is the unit.
2. Fractional parts have special names that tell how many parts of that size are needed to make the whole. For example, thirds require three parts to make a whole.
3. The more fractional parts used to make a whole, the smaller the parts. For example, eighths are smaller than fifths. The denominator of a fraction indicates by what number the whole has been divided to produce the type of part under consideration.
4. Thus, the denominator is a divisor. In practical terms, the denominator names the kind of fractional part that is under consideration. The numerator of a fraction counts or tells how many of the fractional parts (of the type indicated by the denominator) are under consideration. Therefore, the numerator is a multiplier- it indicates a multiple of the given fractional part.

In the Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra seminar, we learned models for illustrating the key ideas of fractions that go beyond the traditional "pieces of pie "pictures and that promote thinking of fractions as ratios, or as making comparisons between different quantities of the same type, are important for advancing student thinking. We looked at the linear model(length) and the area models. We saw that the length model can be valuable for comparing fractions and for understanding addition and subtraction. Area models can help give insight into the processes of multiplication and division. During our seminar, we had a chance to work on word problems and discover multiple ways to solve problems. The Rational Number project states that “There is no single manipulative aid that is “best” for children and for all rational number situations. A concrete model that is meaningful for one child in one situation may not be meaningful to another child in the same situation nor to the same child in a different situation. The goal is to identify manipulative activities using concrete materials whose structure fits the structure of the particular rational number concept being taught.

According to Van De Walle, “Sharing tasks are generally posed in the form of a simple story problem. Suppose there are four square brownies to be shared among three children so that each gets the same amount. How much (or show how much) will each child get? Task difficulty changes with the numbers involved, the types of things to be shared (regions such as brownies, discrete objects such as pieces of chewing gum), and the presence or use of a model”. What better way to teach equal shares than to have students share items. Most of the time when students share things, they share the objects one by one.⁷

When this process leaves leftover pieces, it is much easier to think of sharing them fairly if the items can be subdivided. Typical “regions” to share are brownies(rectangles), sandwiches, pizzas, crackers, cakes, candy bars and so on. The problems and variations that follow are adapted from Empson (2002) When I present these or similar problems to my class, I will suggest that students draw models to help them solve the problem.  I would also use this time to get my students to stop and reflect by asking the following questions. Which do you think is most difficult? Which of these represent essentially the same degree of difficulty? What other tasks involving two, four, or eight shares would you consider as similar, easier, or more difficult than these?