There is a significant difference between being able to apply the cross product algorithm to solve a proportion and being a proportional thinker. As outlined in the book, those who reason proportionally have a number of distinct characteristics.
Proportional thinkers -
These four characteristics are found on Transparency T-102 and are adapted from a list of 13 characteristics offered by Lamon (1999). Several of her ideas have found their way into this chapter and I recommend her books as excellent background for this topic.
- Have a sense of covariation. They understand relationships where a change in one quantity causes a corresponding change in another.
- Recognize proportional relationships as distinct from non-proportional relationships.
- Have a wide variety of informal methods for solving proportions and comparing ratios.
- Understand a ratio as an entity distinct from the quantities it compares.
Developing Proportional Reasoning
What both Lamon and the proportional reasoning project agree upon is that proportional reasoning is very important for success in mathematics and that it does not simply happen. Middle school students must engage over several years with rich activities that involve them with key ideas of ratio and proportion. These experiences should include both proportional and non-proportional situations so that students can reflect on the features of a relationship that make it proportional. These experiences should probably be both qualitative (direction of differences but without quantities) as well as quantitative. These experiences should involve a variety of modes of representation: physical models and drawings that can be measured, verbal problems with and without quantities, and numeric situations such as comparison of fractions and percentages.
Perhaps more important than the types of experiences just listed is the opportunity for reflective thinking that must accompany these experiences. Therefore, the conduct of these experiences in the classroom should include significant opportunities for students to make observations and to compare and contrast ideas among classmates - in short, there should be ample opportunity for discourse. While this does not mean that problems involving solution of proportions should not be included, the emphasis must for a long time be on the argument that produces the result and not on repetitive use of algorithms and answer getting. In short, this is an area that clearly requires practice in the sense I defined it in Chapter 5 - repeated problem-based experiences with the same big ideas.
Given what is certainly to be limited time to discuss this chapter in a methods course, the focus should probably be on problem-based examples illustrative of the four categories described below with less or even no time devoted to issues of traditional algorithms for solving proportions.
Types of Experiences for Reflective Thought
The chapter outlines five categories of activities: Identification of multiplicative relationships, selection of equal ratios, comparison of ratios, scaling activities, and construction/measurement activities.
Identification of Multiplicative Relationships. Additive relationships are easy to understand and use. They simply reflect an absolute difference between two quantities or measures. They are also restricted to like quantities or measures. Research indicates that multiplicative relationships are not as easily understood. A good place to begin this topic is with settings that allow for both additive and multiplicative relationships so that the two can be contrasted. This short section of the text encourages students to make these contrasts.
Selection of Equal Ratios. In these activities ratios are presented some of which are equal and some are not. Students are to group ratios that are equivalent and explain. Comparing Ratios. The "Lemonade" task (Activity 19.5) has proven to result in excellent discussion. The format in which it is posed (with two pitchers as shown here) tends to cause a wide variety of solution methods as well as arguments that the two pitchers will taste the same. This and the pizza task (Figure 19.7) are important ones to explore.
From a mathematical viewpoint, these activities might be viewed as the same as the comparison of two fractions. Pedagogically, even the pizza problem will prove to be different from the lemonade task, indicating the influence of context on ratio comparison activities. Note also that ratios often compare unlike quantities (as in the juice/water example) whereas fractions are strictly part/whole ratios. Also, when comparing two fractions, the size of the whole for each is always the same. This is not at all true for ratios. In fact, that is one of the things that distinguish non-fraction ratios - they can compare a ratio within a set of 12 things, for example, to a corresponding ratio within a set of 10,000 things. This fact lends itself to methods that are quite different from fraction comparison strategies.
(WATER IMAGE HERE)
Scaling Activities. These activities (finding missing entries in a table of two related quantities) if designed correctly, can involve excellent thinking in covariation. As Lamon points out, some scaling activities simply require adding like amounts along each portion of the table. While this may solve a proportion, the process does not involve proportional thinking. The book explains this difference. An activity related to scaling is to make coordinate graphs from these charts. Proportional situations are always represented by linear graphs through the origin with the common ratio being the slope of this line. It is useful to point this out whenever possible.
Measurement Activities. These activities are the most engaging because it is here that students are constructing or building, or measuring things that may or may not be proportional and then observing or predicting the ratios.
While the emphasis should not be on algorithmic solution of proportions, you should not infer that setting up proportions is not important. In problems requiring a proportion to be solved, students should be encouraged to draw a simple sketch that will enable them to see clearly what quantities in each situation are being compared. The proportions can then be set up in a variety of ways. The book distinguishes between within ratios and between ratios. I have noticed that teachers become unnecessarily concerned about this distinction. While you may find it helpful, your teachers may not.
The drawing of models to establish a proportion turns out to be especially useful for looking at percentage situations. In fact, these drawings are essentially the same as those suggested in the previous chapter. You may find it useful to look at equivalent fractions as an example of a proportion and to examine these using a small drawing (Figures19.20 and 19.21). If students can become comfortable with the fraction situations, the jump to percents is not at all difficult.