Hope you are having a wonderful Wednesday! We continue our book study of Teaching Student-Centered Mathematics PreK-2 by Van de Walle, Lovin, Karp, and Bay-Williams with chapters 14 & 15 about fractions and measurement. To read past posts, you can visit our book study archive.
Chapter 14 – Exploring Early Fraction Concepts
As PreK-2 teachers, it is our job to develop students’ “informal” understanding of fractions before they are immersed in formal instruction of fraction in later grades. Therefore, the authors present three different meanings of fractions that are important for students to understand at this level.
This meaning of fractions is the most widely developed. It often involves shading equal-sized parts of a whole. “Although the part-whole meaning of fractions is is important, too often instruction based only on the part-whole relationship leaves children with little sense that fractions are numbers.”
Equal Sharing (Division)
The authors remind us that equal sharing is something that children understand from a young age because of their experiences wanting to share things fairly. Continuous quantities (those that can be divided into equal-sized pieces/portions–example, 3 whole candy bars to be shared) and discrete quantities (those made up of individual items that comprise a whole–example, 12 cookies to be shared) can be shared. The authors suggest having students explore the sharing of discrete quantities using word problems that will yield no remainder. Pages 256-263 provided teachers with a progression for using story problems to teach equal shares as well as a number of examples and activities to help students develop a strong understanding of equal sharing.
Connections between fractions and measurement naturally arise when measuring length and telling time. “Measuring situations by their very nature consist of measuring a quantity that we could cut into as many equal-sized pieces as we need and so involve continuous quantities.”
The authors go on to address the importance of making students understand that fractions are numbers. When students are engaged in solving word problems that require equal sharing (partitioning), and are asked to determine how much each child will get, the idea that fractions are numbers is reinforced. In this way, just like when solving word problems with whole numbers, students are asked “how many” and this serves as a reminder that fractions have value. As we know, children also learn to count whole numbers before learning to add and subtract. This idea can be applied to fractions as well when students are taught to count fractional parts of a whole and tell how many equal-sized parts (example, three thirds) comprise a whole (iterating). As a former fifth grade teacher, many students did not have the foundational understanding of fractions as numbers, and it was essential to help students develop this understanding in order to move any further. The fact is, this should not be happening in fifth grade. It is our job as PreK-2 teachers to develop this understanding early. For such a big idea, it seems relatively easy.
A variety of models for fractions should also be used with students. They overview and give examples of area, length, and set models. Area models include pie pieces, folded paper, pattern blocks, geoboard models, grid/dot paper drawings, and rectangles. Length models include folded paper strips, fraction bars/strips, Cuisenaire rods, and line segment drawings. Set models can be any set of objects. One model that I have NOT been using with students is a line segment drawing. I plan to show my students how to use line segment drawing this year. This will help build a foundation for their future work with fractions on the number line. A line segment drawing is a length or measurement model for fractions that involves the drawing of a line segment and dividing the line into equal parts with hashes. Then parts can be shaded to represent a given fraction. Below is an example of a line segment drawing for 1/3.
I am thinking this type of model could also be drawn and divided into a different number of equal parts so students can compare the size of parts, as in the following task I have had my students do in the past. Draw three rectangles (line segments) of the same size (length). Divide one in half, on into thirds, and one into fourths. Compare the size of the shares. What do you notice? Line segment drawings also lend to an understanding of units when measuring.
Equally as important as being able to model fractions in different ways is the use of fraction language. Students should use “whole, half/halves, thirds, and fourths/quarters” when talking about fractions. They need to be able to describe a whole that is divided/cut into equal-sized parts. Fraction language should also be used to describe the number of parts and that the parts are equal shares of the whole. The authors recommend waiting until the end of second grade to introduce students to the use of symbols to represent the fractions they have thus far been describing using words. They also caution the use of a slash to represent fractions. Using a horizontal line makes it easier for student to determine which number is on the top and bottom of the fraction bar. When teaching students to write fractions using symbols, much time must be spent making sure students understand what the top and bottom number mean. The authors provide some great suggestions for doing this.
Chapter 15 – Building Measurement Concept
What is measurement? Measurement is a comparison of an attribute of an item (or situation) with a unit that has the same attribute. Hugh? What does that mean? To understand this definition, you need an understanding of attribute and unit.
Let’s use a table as an example:
- We could select one of the following attributes to measure–the table’s length, height, area, or weight.
- We would then decide an appropriate unit of measure to compare–paperclips, paper strips, straws, square tiles (nonstandard units), inches, feet, pounds (standard units).
The key to understanding is that a comparison is being made. Three steps must be taken each time something is measured:
- Decide what attribute to measure.
- Select a unit that has that same attribute.
- Compare the units (by filling, covering, matching, or using some other method) with the attribute being measured.
The authors do well to stress the importance of developing students’ abilities to compare early even though no measurement is required. An example would be a kindergartener being able to tell if an object is heavier or lighter than another or if something appears longer or shorter. These kinds of opportunities will lead into the use of nonstandard units to measure a specific attribute of an object in first grade. Then, in second grade, students need to develop a strong understanding of standard units.
I find the the three goals for understanding measurement at the PreK-2 level suggested by the authors to be especially helpful:
- Familiarity with the unit. Students should have an idea of the size of commonly used units such as inches, feet, yards, minutes, and hours. This knowledge should be used to make estimation of length given a particular object. This is just as important as being able to successfully measure an object using an appropriate tool.
- Ability to select the appropriate unit. Students need to be able to select a reasonable standard unit. Equally important is knowing how precise to be when measuring. For example, when playing a game you may need to be about 10 feet from your partner, but when you are buying a carpet for a specific space an exact measure is be needed.
- Knowledge of relationship between units. Students should know the relationship between minutes and hours as well as between inches, feet, and yards.
As always, the authors provide us with some fabulous activities for measuring length, reading clocks, and understanding money. While length, time, and money are the main focuses of measurement at the PreK-2 level, the authors suggest we provide students with some initial experiences the the attributes of area, volume/capacity, and weight/mass. Helpful suggestions and activities for doing so are given at the end of the chapter.
Well, there are only two chapters left! We will be back again on Sunday with chapter 16 which focuses on geometric reasoning and concepts. AND, as always, we welcome you to chime in with your thoughts and comments about chapters 14 and 15.
AND, before you go, we are excited to let you know that you can sign up to receive our brand-new bi-monthly newsletter! It’s simple—just enter your email on our sidebar and click Subscribe.
All the best–