*By Dick Stanley, Professional Development Program, University of California at Berkeley*

The notion of one quantity being proportional to another is certainly a very basic part of an understanding of mathematics and of its applications, from middle school through calculus and beyond. Unfortunately, the picture of proportionality that tends to emerge in school mathematics in this country is narrow and confused. Everyone learns the procedure of setting up and solving a proportion, but the connection of this to the idea of one quantity being proportional to another is tenuous.

In support of this statement, I summarize below the results of participant responses given in a workshop attended by teachers, mathematics educators, and mathematicians. The surprisingly shallow responses show a striking lack of a common, mathematically coherent understanding in this audience of the subject of proportionality.

**A. A simple problem**

In the workshop, participants first worked to solve this problem:

**Paper Stacks Problem:**

Suppose you want to know how many sheets are in a particular stack of paper, but don’t want to count the pages directly. You have the following information:

- The given stack has height 4.50 cm.
- A ream of 500 sheets has height 6.25 cm.

How many sheets of paper do you think are in the given stack?

All 18 participants found the expected result (360 sheets) by setting up and solving a proportion.

**B. What is proportional to what?**

Next, participants were asked this question:

Write down a sentence or two in response to this question:

* “In this paper stacking situation, is anything proportional to anything else?”*

The most natural response: “the number of sheets in a stack is proportional to the height of the stack” did in fact appear, but only in about a fifth of the responses. This response is in accord with a modern understanding of proportionality: a variable quantity *A* is proportional to a variable quantity *B* when there is an invariant *k* such that *A* = *kB*. In this situation the invariant is the number of sheets per centimeter.

Other responses suggested that “the height of the small stack is proportional to the height of the large stack.” But the ratio of these heights (about 0.72) is particular to these two stacks, and is not an invariant of the paper stacking situation. These two heights are not proportional in a modern sense of the term. What is getting in the way in these other responses, we feel, is a view commonly put forth in school materials: a ratio can be formed only between quantities of the *same kind*. The relationship between the number of sheets and the height of the stack cannot then be proportional, since the required “ratio” is between quantities of different kinds.

However, most disturbing is the number of responses that merely put together some scraps of remembered procedures, such as response number 4: “A proportion is the relationship of two ratios. The height of the two stacks is proportional since you are comparing one ratio to another; i.e. \(\frac{360}{4.5}=\frac{500}{6.25}\)”

**C. What does “proportional to” mean in general? **

Finally, participants were asked this question:

Write down a brief answer to this question:

* “What does it mean in general to say that one quantity is proportional to another quantity? Be as precise as you can.”*

The 18 responses are interesting enough that they are included in full:

- proportional relationship means that when one quantity in a relationship changes another will change according to some specific pattern (which won’t change in time / vary)
- “a” is prop. to “b: means that if b is altered by a factor (e.g., multiplied by t), then a is altered the same way.
- One quantity is proportional to another means the comparison is relating equal ratios.
- \[\frac{a}{b} = \frac{c}{d} \hspace{3em} ad = bc\]
- To be proportional means to have the same ratio in simplest form. The relationship between the two things is the same (in the real world like sugar:flour)
- As the numbers in the proportion change … there is a constant pattern of increase or decrease \[\frac{1\times 4}{2\times 4} \hspace{4em} \frac{4}{8} \hspace{4em} \frac{10\div 5}{5 \div 5} \hspace{4em} \frac{8 \div 2}{4\div 2} \hspace{4em} \frac{1}{2}\]

- It means that a fraction is equal to a fraction or that the two ratios are equal.
- As one part of the proportion changes the other part changes in the same relational way.
- If one quantity increases, the other quantity also increases. Or If one quantity decreases, the other quantity decreases
- It means that quantity “A” changes in a fixed or quantifiable manner as quantity “B” changes.
- The two ratios are equal. of, cross products are =
- As one quantity increases or decreases by a specified amount, the similar quantity also increases or decreases by the same amount.
- The rate of change between the two quantities is constant.
- quotients of 2 quantities are equal / constant if proportional
- The ratio of parts of each term is the same \(\frac{1\ \text{sheet}}{\text{ height}}\) is same for both

(each piece of the proportion is made up of like parts) - amount of an item will have a relation to another item
- As one quantity grows the other quantity also grows; it is a multiplicative relationship; ratio is constant; what about inversely proportional?
- When one thing is proportional to another, we can set up two fractions that are equivalent.

**D. What has gone wrong? **

The confusing jumble of responses here is disturbing. At the very least it points to a lack of a common understanding within the school mathematics community of this very basic and important subject. It would certainly be wrong to blame teachers. Rather, I believe the culprit is a general lack of mathematically sound grade-level appropriate presentations of proportionality that have been available to teachers. In addressing this lack, mathematicians must certainly play a major role.

**E. Comments**

The subject of proportionality in school has a long, complex, and fascinating history. Here, I will simply suggest the range of relevant issues.

**Euclid**

All school approaches to proportionality have their origins in Euclid’s treatment in Book V of *Elements*. This is where the brilliant treatment of ratio by Eudoxus appears. However, Euclid’s treatment of proportionality is essentially that of *discrete* quantities: four magnitudes that have the same ratio are called proportional. (See Definition 6.) Today, proportional relationships are understood as being between two *variable* quantities. In my view, the inadequate understanding of proportionality shown by many responses in the workshop is due to the failure of school mathematics materials to sufficiently stress the role of variable quantities in a modern understanding of proportionality. We elaborate on this idea in the next section.

**Proportions and missing the crucial invariant**

Finding the numerical solution to a problem such as the paper stacks problem by setting up and solving a proportion is fully reasonable, and we all do it. However, the mathematically interesting point in a situation such as this is that there is an ** invariant,** namely the number of sheets per centimeter (80 sheets per cm).

In an approach that focuses only on setting up and solving a proportion, this invariant never needs to be found. All that is found is an unknown (360 sheets) in one particular situation. This means that the crucial relationship between the variable quantities *n* = number of sheets and *h* = height of a stack is never seen: \[n = 80h.\] And in fact, seeing proportionality as involving a relationship between *variable* quantities was the key point missing from most responses in the workshop. Repeating this point from Part B above:

A variable quantity

Ais proportional to a variable quantityBwhen there exists an invariantksuch thatA=kB.

This statement includes two hard but very important mathematical ideas, the idea of an invariant, and the idea of a variable quantity. It is my feeling that work toward bringing out these ideas should begin as soon as the language of proportionality is introduced in middle school.

**Analogy: the Law of Sines**

To make an analogy, consider the Law of Sines for triangles: \(a/\sin\alpha = b/\sin\beta = c/\sin\gamma\). These three ratios are not only equal, but their common value is an important invariant of a triangle: the diameter of the circumcircle. Bringing out the invariant and its meaning is an essential part of a fully mathematical treatment of the Law of Sines. A focus on the invariant as the common value of a set of ratios should be an essential part of a mathematical treatment of proportionality as well.

**The Common Core State Standards in Mathematics**

The approach to proportionality suggested in the Common Core State Standards in Mathematics promises to be of real help, since the emphasis is directly on proportional relationships and the constant of proportionality. In fact, the approach is remarkable in that the term “ratio and proportion” does not appear at all, nor does the idea of “setting up and solving a proportion.” Instead, the central concept is proportional relationships themselves.

However, my observation is that old habits are hard to break. It will not be easy to overcome tradition in developing and implementing this far more reasonable approach.

**F. Conclusion**

I think we would all agree that a reasonable treatment of proportionality should lead to students being able to understand a statement such as this:

The gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance separating them.

This requires a rather flexible understanding of the idea of “proportional to.” We have argued that a traditional approach to proportionality that focuses on setting up and solving a proportion is not adequate. Instead, what is needed is an approach that emphasizes the role of variable quantities and their invariant ratio. The responses in the workshop seen in Section C above would have been rather different if these ideas had been more prominent in school materials.

This entry outlined an important issue and increased my understanding of an obstacle to student understanding. Concise and helpful.

Thank you. For me, the main obstacle to understanding in many school materials is that so much of the treatment of proportionality has been reduced to performing a memorized procedure (setting up and solving a proportion), with little opportunity to look at the logic underlying the procedure. This logic is itself concise (two variable quantities with a constant ratio), but students need lots of time and opportunity to think about it.

The paper Stacks problem can be easily done in fifth grade, as follows. We should find out how thick one sheet is: let us say h cm. We are given (h+h+…+h) (500 times) is equal to 6.25 cm. In 5th grade, students learn that for any fraction h,

500 h = (h+h+…+h) (500 times).

So 500 h = 6.25 and therefore h=6.25/500 = (2 x 6.25)/(2 x 500) = 12.5/1000 = 0.0125 cm (by the definition of a finite decimal). How many of these would add up to 4.5 cm? Let us say y sheets. Thus 0.0125 y = 4.5 and

y = 4.5/0.0125

= (8 x 4.5)/(8 x 0.0125)

= 36/0.1 = 360 sheets.

For DISCRETE proportional reasoning problems (such as the Paper Stacks), there is never any need to worry about proportions. It is in the continuous case that proportions must come in, BUT WHAT IS IN THE LITERATURE MESSES UP THIS POINT.

That is another story.

This is certainly a nice way to solve the problem, and it uses good reasoning at each step.

But students also need to know something more general, namely, what a proportional relationship is and what is meant by a constant of proportionality. Knowing this serves as a key to solving many problems. It is especially important, given that proportional relationships appear so frequently in mathematics and its applications.

My concern is that the workshop responses show that many people can solve this problem but be totally at sea when it comes to understanding the more general idea of a proportional relationship. A further concern is that materials available to teachers tend to give a very poor picture of the mathematics of ratio and proportional relationships. A mathematically more sound picture of this part of mathematics needs to find its way into schools.

I realize it’s not likely that you will see this comment 5 years later, but I’ll my question anyway. Can you suggest any of your work (or the work of others) on the difference in the discrete and continuous cases?

Please see Section 7.2 of my book, TEACHING SCHOOL MATHEMATICS: ALGEBRA http://tinyurl.com/y6ulr5kq (pp.144-154) for a discussion of both the discrete case and the continuous case. More importantly, this section provides a critical examination of the fictitious concept of “proportion reasoning”, and explains why it has no mathematical basis. A “proportion” appears naturally if we have a “linear function without constant term” (see p. 139, loc. cit.), so one’s effort should be spent on understanding such functions and not on understanding “proportions”. Such misplaced emphasis is in fact rather common in the existing school curriculum. For example, there is in fact no mathematical concept called a “variable” and, instead of straining to understand what a “variable” is, one’s effort should be directed to understanding what a function is (please see pp. 1-3, 28-29, 38-39, loc. cit.).

Incidentally, the afore-mentioned book was published without an index, but the index can be found at this link: http://tinyurl.com/haho2v6

Thanks so much for this help. I have read most of your work and felt sure I had seen this discussed. I have the book you cited and will look it up (and thanks so much for the index). This issue seems to me to be a really important one for teaching students in middle school, and your explanation, as always, is truly enlightening for those of us in education. (As you can probably tell, I’m a HUGE fan of your work. Thanks for sharing your invaluable insights with us non-mathematicians!)

Thank you for the comments. If you have further comments on this issue, it may be easer to discuss them in private. Please write me at wu@berkeley.edu

Thank you for your question about discrete and continuous. Let me try to give an answer. I apologize that it is a little long. But it illustrates the basic point being made in my blog.

In my blog I did not use the term “continuous”, and I used the term “discrete” just once: “Euclid’s treatment of proportionality is essentially that of discrete quantities: four magnitudes that have the same ratio are called proportional. (See Definition 6.)”

I realize that “discrete” is not an ideal term here. “Specific” is perhaps better. In any case, the distinction I was making is really quite natural. It is between two ways of expressing a quantity: using a specific instance of the quantity, or using a variable. In today’s school mathematics, a specific instance is generally a number. In Euclid it was a geometric quantity, such as a specific line segment in a figure.

The importance of the distinction is that it is at the core of two different approaches to problem situations: A traditional middle school approach uses specific instances of quantities (numbers) in cases where variables would be more natural. In more up to date approaches, variables are used.

We illustrate with a typical example problem: Suppose we are told that in an enlargement of a photo, a 6 inch line segment becomes 15 inches. We are asked what a 8 inch line segment becomes.

A traditional approach sets up a proportion 15/6 = x/8. The solution, x = 20 comes from solving this proportion equation. (The reasoning behind this approach is often left vague.)

A more modern approach reasons that, in an enlargement, each line segment is enlarged by the same dimensionless factor, call it e. If we let any length before the enlargement be “b” and the corresponding length after the enlargement be “a”, then we know that a = e b. The given information gives us a numerical instance of this relationship: 15 = 6 e. Solving this equation gives the numerical value e = 2.5. This allows us to express the general relationship: a = 2.5 b. The solution to the problem is then found by substituting b = 8 into this relationship: a = 2.5 x 6 = 20.

A central feature of the more modern approach is that it uses simple linear functions such as a = 2.5 b, where the quantities a and b are often called variables. A relationship expressed in a simple linear function such as a = 2.5 b is called a proportional relationship. The advantage of the more modern approach is that it makes explicit the general relationship underlying the problem situation. In the more traditional approach this underlying relational relationship is never found.

The blog illustrates the unfortunate consequences of the traditional approach, and urges the adoption of a more modern approach.

You say that the idea of an invariant ratio and variable quantities should be introduced as soon as the language of proportions are introduced in middle school.

Consider introducing them as soon as multiplication is introduced in elementary school. They are not too advanced for elementary school children. A typical multiplication word problem uses an invariant ratio. We just don’t point that out to students. We instead use additive thinking of equal groups (additive comparison of group size) and repeated addition.

Nunes and Bryant, in Children Doing Mathematics, emphasize the invariant ratio as the essential element of multiplication. This relationship can be thought of as a many-to-one correspondence, for example, 12 inches per foot, or $3 for every pound. Michael Goldenberg goes into more detail on the Nunes and Bryant arguments on his blog.

http://rationalmathed.blogspot.com/2010/03/terezinha-nunes-and-peter-bryant-dole.html

Park and Nunes did a study testing repeated addition versus correspondence as a basis for understanding multiplication. In multiplication problems, the students who were taught multiplication through correspondence significantly outperformed the children taught multiplication through repeated addition. The 6-year olds in the study were able to understand the correspondence and the replication of that correspondence—i.e., that invariant ratio y number of times. (I see a flaw in that study, but it doesn’t affect the ability of the students to understand the replicated correspondence.)

Park, J-H. & Nunes, T. (2001). The development of the concept of multiplication. Cognitive Development, 16, 763-773.

Your suggestion is very attractive: the idea of an invariant ratio and variable quantities should be introduced as soon as multiplication is introduced in elementary school. I agree. Thank you also for the reference to the work of Nunes and Bryant, which seems quite useful.

The whole idea of connecting ratio to multiplication is one that is not used nearly as often as it might be. The CCSSM lays the groundwork by emphasizing the “times as much” interpretation of multiplication in grades 4 and 5. When ratio is introduced in grade 6, it would be helpful to build on this. For example, we could work with a situation where we are told that one book costs 3 times as much as another. A natural question to ask here is how someone might know this. The answer is that we divide one cost by the other and find the quotient 3. In ratio language, there is a 3 to 1 ratio of the costs of the two books.

So understanding ratio involves appreciating the close relationship of multiplication and division. Unfortunately, many school treatments of ratio do not even connect ratio to division. Instead they focus on equivalent ratios. In this they follow Euclid, but such an approach makes little sense in a curriculum where students have seen division for a few years before ratio appears.

Getting back to your point about connecting ratio to multiplication, I certainly agree that getting past multiplication as repeated addition is very important. I also appreciate using the term “correspondence” to refer to invariant ratios such as a price of 3 dollars per pound, as is the idea of explicitly seeing a product such as “pounds • dollars per pound” as involving an invariant ratio “dollars per pound”.

I suggest that we could learn much by studying Davydov’s curriculum. It says much about what students are capable of learning and how we might improve our teaching. Here is a problem for Grade 3 that appears in the Schmittau and Morris article at http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_1/Schmittau.pdf :

“At 8:00 in the morning a motor vessel (ship) left a pier. At the same time from another pier a motor boat moved toward the vessel. The ship traveled at a speed of 35 km/hr, and the motor boat at 40 km/hr. The distance between the piers is 150 km. When did they meet? How many kilometers farther than the ship did the boat travel?” (Davydov et al., 2001). [a student’s solution is explained]

They write, “Davydov and his colleagues identified foundational concepts, such as quantity, direct and indirect measurement, units, and part-whole relationships, that underlie many mathematical ideas. The children’s understanding of these conceptual antecedents is thoroughly developed. As new topics are introduced, these foundational concepts are the constituent understandings that are needed to understand the new topic. Students are never confronted with a complex topic such as proportional reasoning unless all the component ideas are carefully developed well before its introduction.”

The authors list the concepts that children need to understand before they can solve problems like this one. The article goes into detail about how some of the earlier concepts are developed. Proportional reasoning is taught in the third grade. What makes it possible to teach proportions or any concept at any grade is an understanding of prerequisite concepts. I suggest that our main problem is we don’t teach true understanding of concepts. We teach how to compute answers. Even as we strive for rigorous instruction, we are still attached to the idea that if students can compute the answers to problems then they understand the concepts involved. We may say otherwise, but that idea is deeply embedded in our way of thinking.

To overcome this limitation, we need to study and discuss examples of truly rigorous teaching, like in this article. We need to collaborate and make our teaching public so that we can share our insights. We need the input of others to grow beyond our own interpretations of what we read and see. Otherwise, we interpret reform recommendations according to our current thinking and our current teaching practices and don’t change the core of our practice.

Your reference to Davydov is helpful and relevant, since understanding such foundational concepts as quantity, direct and indirect measurement, units, and part-whole relationships is crucial for understanding proportional relationships. This builds on the points in your earlier post, stressing the importance of understanding the interaction of multiplication and ratio. And I agree fully that a serious problem is the tendency to teach how to compute answers rather than teaching true understanding of concepts.

Your remark “… we interpret reform recommendations according to our current thinking and our current teaching practices and don’t change the core of our practice” rang especially true for me. A case in point is the approach to the mathematics of proportionality mapped out in the CCSSM, which emphasized proportional relationships and did not even mention setting up and solving a proportion. In spite of this, some recent discussions of proportionality I have heard go right back to the old, familiar emphasis on proportions.

Well, I fully agree with Burt Furuta. The problem seemes to be that we are unable to picture what we never have experienced. It is hard to believe that students can learn more math on a third och fourth of the time used today. The foundation for this is a thorough plan for how to make the students understand any concept, not overlooking anything. This is what Davydovs model have achieved in most parts.

When it comes to proportionality, it is not an issue that can wait until third grade. The mere number system (with place values) is founded on proportionality. This is where Davydovs concept has a drawback: the model of part-whole relationships is more adapted for additive comparisons than for multiplicative ones. That problem can be solved by making a rigorous definition of what is ment by a comparison. And it must be concrete – not founded on some formula. Alas, this is not an issue that can be fully explained in a blog comment. So I leave that for now.

But to give you hope: it is possible to make students (at least from grade 4 to my experience) understand such things as fractions or algebra within hours. And of course: this is achieved with a kind of teaching that is quite close to Davydovs way of doing it, but light-years away from traditional teaching. It leans heavily on developing general mathematical ideas and models by the use of hands-on materials. Just like Davydov do.

Thank you for your comments. Picking up on one thing you said, I have also long felt that it is important to make a rigorous definition of what is meant by a comparison. And I like the idea that it must be concrete. With this basis, students will understand the formulas better when they are introduced.

Well, as you too feel the need for a rigorous definition of “comparison”, maybe I have better break the news: It’s all there in Category Theory. Professor emeritus Ronald Brown stresses (in a paper about “analogy and comparison, on his home page) that comparison is one of the basic elements in describing structures. In Category Theory there are ‘objects’ and ‘arrows’ (also called morphisms) between objects forming so called ‘categories’. The arrows are similar to “functions”. What Ronald states is that the arrows describes the comparison between two objects. He have some interesting points on that, you can read on his page. An attribute of the arrows, that may be pointed out, is that they are “dynamic” and operational.

The hard “trick” is to make Category Theory digestible for pupils in compulsory school. (Partly renaming of concepts are helpful in that.) The main observation to do is that when we tell the comparison of two values, we mention a number (and an unit, if present), but moreover we mention an operation (!), that often comes in disguise. For example we may say that 5 m are 2 m more than 3 m! The operation becomes visible when we translate “more” to Latin: it is “plus”. As can be expected “less” is “minus” in Latin. Ronald points out that comparisons, like arrows, have a direction. In a comparison, one of the objects we compare, works as a reference object. This object I name “root”, for reasons that will show. Now to the definition (confined in its form for use with numbers in school):

A comparison is a way to tell the size of a value by making reference to a root value and tell what “to do” (operation+value) with the root to get the value to be described.

All this can readily be depicted in a kind of diagram. I use the example above to outline the diagram:

The value 5 m, that we want to describe is placed to the left, and the root 3 m to the right, with a left pointing arrow in between. Above the arrow we write +2 m.

(You may construct this diagram in your blog.)

The left pointing arrow can be read: “compared to”, and the comparison result is expressed above the arrow.

I don’t know what conclusions you may draw from this. I can only tell it took me years to realise all its implications. I would say it have a major impact on didactics and mathematical theory. For instance: Category theory tells you can compose arrows, which means you get expressions like “+2 -5 -3 +4”, where you compose elements with “bound” operators. The expression tells we have a composition of an increase by 2, a decrease by 5, etcetera. The outcome will be a total decrease by 2. (Incidentally, this way of regarding a sum of termes seemes easier for pupils to grasp. E.g. the interpretation “decrease” is more natural than “negative number”.)

But as this is a blog comment I am writing, not a book, I have better end here.