by Arthur Wiebe
(This is the third in a series of articles introducing the recently initiated AIMS program for the development of a pattern- based mathematics/science curriculum)Index of Articles | Next Article | Previous Article
Next PageThe previous article included three sets of problems for which information and answers are included in this issue. Settling the Major "T" Islands and Can You Predict? are discussed at the end of this article. The patterns in the products of two, three, four, and five consecutive integers arranged in order are discussed in detail in the activity Searching for Patterns in Products found in this issue.
We now continue examining the advantages offered by the emerging AIMS pattern-based mathematics/science curriculum.
Learning the big ideas of mathematics is the most effective way to empower students mathematically. Big ideas help to reduce student memory load, broaden understanding of and appreciation for mathematical concepts, provide connections across the discipline, and reveal the beauty of the mathematics. By contrast, efforts to learn mathematics by studying concepts in isolation subjects students to a poverty-stricken learning environment in which many are forced to take refuge in memorization rather than understanding.
Possibly the biggest idea in elementary and middle school mathematics is that of proportional reasoning. It is exciting that this big idea spans so much of mathematics with a single overarching but utterly simple pattern. Proportional reasoning generally involves four components interrelated in the pattern regardless of the context in which it occurs.
As pointed out in the Mathematics Framework for California Public Schools, "proportional relationships are at the heart of the most basic quantitative understanding of the world and are continually useful in school, in the workplace, and in everyday life. Students in the middle grades should explore a variety of situations in depth in which proportional relationships play a major role and work to develop a broad sense of the kind of proportional reasoning that applies in these situations. The central concept is the representation of one quantity as a certain proportion of another." The following are selected topics drawn from the many which are based on proportional reasoning:
| division (multiplication) | ratios |
| rate | percent |
| equivalent fractions | slope | per-unit quantities | scale relationships |
| linear functions | similarity |
| probability | proportional parts |
| circle graph computations |
Understanding these from the perspective of proportional reasoning results in a tremendous gain of mathematical power accompanied by simplicity of solution! A hallmark of the A1MS pattern-based math/science curriculum will be the development of such broadly applicable understandings.
An extended series of articles dealing with proportional reasoning appeared in Volumes III and IV of the AIMS Newsletter under the title Proportionality: A Major Concept in Mathematics. Here we will briefly review five appiications of proportional reasoning to illustrate the consistent patterns of thinking and problem solving that are involved.
It is important to understand that proportional reasoning is characterized by the facts that
We need to recognize that most children begin to use proportional reasoning at about age three when "come to understand the concept of fair shares! The concept of making fair shares, or division, is nothing more than an application of proportional reasoning!
The process by which students can make the transition from their intuitive fair share making to the creation ofd a written record in which each element accurately reflects each action at the manipulative level is discussed in detail in the above mentioned series. Here we will confine our discussion to an illustration using such a record in which 4 fair shares have been created from 12 objects. The record looks like this:
What is different here is that the 1 is written in the position shown. This creates a proportion that reflects the application of proportional reasoning. After primary students have created such a record, they are asked to write two sentences such as the following to make clear that proportional reasoning is being applied:
The relationship between the numbers in the numerical record and the two sentences is as obvious as it is consistent in meaning.
The two aspects, then, that distinguish this proportional reasoning approach from the standard paper and pencil approach are the placement of the 1 as shown and the writing of two parallel sentences to interpret and give meaning to the result. Meaning is embedded by first having the students perform the operation with real objects, then having them create a written record that traces each step at the manipulative level, and finally having them write the two sentences. (For a more extended discussion and additional illustrations, please refer to Volume III, No. 6., pp. 8-16.)
In the AIMS approach to division, the 1 is always recorded in the position indicated. As a result, if any two of the remaining three numbers are known, the unknown one can be found. There are three possible places for the unknown:
In the first, students are asked to form 4 fair shares from 12 objects. In the second, students are asked to find how many fair shares of 3 each can be made with 12 objects. In the third, students are asked to find how many objects must be used to make 4 fair shares containing 3 objects each. This last case involves the process of multiplication. That multiplication and division are inverse operations is clearly embedded in the proportional reasoning approach.
Looking at the same set of numbers without the division sign reveals another application:
These four numbers can easily be visualized as equivalent fractions that can be expressed in either of two ways:
Note that the cross-products are equal: 1 x 12 = 3 x 4. This relationship of cross products is always true in a proportion. Proportional reasoning, then, is used to create equivalent fractions. When equivalent fractions are written as (denominator, numerator) ordered pairs, their graph on the Cartesian coordinate system is a straight line! This is true for all proportions! Conversely, such ordered pairs associated with any two points along a straight line are always equivalent fractions! (See Volume III, No. 10 pages 25-27 and Volume IV, No. 1, pages 7-9.)
Students who have used and understand this standard arrangement of four components in a proportional relationship are well prepared for the study of problems involving percent. In percent problems, the objective is to find the proportional relationship between a part and the whole.
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