*By Art Duval, Contributing Editor, University of Texas at El Paso; Kristin Umland, Associate Professor, Department of Mathematics and Statistics, University of New Mexico (on leave), and Vice President for Content Development, Illustrative Mathematics; James J. Madden, The Patricia Hewlett Bodin Distinguished Professor, Department of Mathematics, Louisiana State University; and Dick Stanley, Professional Development Program, University of California at Berkeley*

At the 2016 Joint Mathematics Meetings in Seattle this past January, an unusual mix of mathematicians and mathematics educators gathered for an AMS special session on Essential Mathematical Structures and Practices in K-12 Mathematics. This was the fourth consecutive special session at JMM organized by Bill McCallum and other folks at Illustrative Mathematics that focused on work in mathematics of mutual concern to mathematicians, mathematics educators, and K-12 teachers. The theme this year was inspired by a conversation between Dick Stanley and Kristin Umland about ratios and proportional relationships, and the talks were selected and ordered to highlight the development of mathematical ideas that are both upstream and downstream of this terrain.

Academic mathematicians are able to describe mathematical ideas in an efficient way. Across specialties, they share tools of language and habits of communication that have been shaped in order to facilitate the exchange of abstract knowledge. One purpose of the special session was to apply this cultural skill to selected topics in K-12 mathematics. The participants sought to create clearly expressed and easily understood descriptions of topics that are rarely developed clearly in the K-12 curriculum, such as measurement, number systems, proportional relationships, and linear and exponential functions. Although many people have been working in this area in recent years, much more needs to be done.

The mathematical community can—and should—contribute to the mathematics of the K-12 curriculum by applying to it the same principles of logical clarity that are used in the best expositions at the college level and beyond, so that teachers can have the best possible curricular materials and support. The coming of the Common Core State Standards in Mathematics has created a new opportunity for positive change in all subjects. Still, old habits change very slowly, and the path in this new direction is going to need many more signposts. This means that it is more important than ever for mathematicians to partner with teachers and mathematics educators to lay out a mathematically coherent path that fills in the outline of the standards.

To consider an example for which this is especially true, let’s take proportionality, the subject of two of the talks (Madden and Umland) at the special session. Mathematicians who look in detail at sources on this subject written over the past, say, 30 years will find a disconcerting jumble. Much of what is written is a throwback to procedures that were practiced in the middle-ages—neither wrong, nor useless, but out of sync with contemporary mathematical practice. Some things are needlessly obscure, ambiguous, confusing. It is easy to identify the difficulty here. Traditional treatments of “ratio and proportion” in school mathematics are rooted in a tradition that goes back to Euclid and entered the school curriculum in the Middle Ages. They are based on the concept of two equivalent ratios relating four fixed quantities.

There is no pathway through the traditional curriculum formulated on the basic idea of the modern understanding of proportionality: One (changing) quantity is proportional to another if their ratio is always the same. To understand why this is so, one must understand the different definitions for a ratio implicitly used by different people. In the traditional curriculum, people talk of a ratio \(a/b\) as a comparison between quantities \(a\) and \(b\), by which they mean that when making a multiplicative comparison between \(a\) and \(b\), the scale factor is \(a/b\) (or \(b/a\) depending on which direction one wishes to compare). So “the same ratio” in this context means “the same scale factor in a multiplicative comparison of two quantities of the same type,” and when people speak of a ratio, they sometimes mean the whole notion of a multiplicative comparison, and they sometimes mean just the scale factor.

In modern times, we have expanded and abstracted the definition of a ratio. We talk of a recipe as a ratio and we are not limited to two ingredients, and ratio equivalence is characterized by multiplying all quantities by the same scale factor rather than taking their pairwise quotients. In addition, we have extended our conception of a ratio to include quantities of different types, like distance and time. Formally, the traditional definition of a ratio \(a \mathbin{:} b\) is limited to quantities of the same type (like length and length) and by definition the ratios \(a \mathbin{:} b\) and \(c \mathbin{:} d\) are equivalent if and only if \(a/b = c/d\). The modern definition does not restrict us to quantities of the same type, and equivalence classes are characterized as \[\{sa \mathbin{:} sb \mid \text{for all}\ s > 0 \in {\mathbb R}\}\] it follows from this definition that ratios of two quantities are equivalent if and only if their related quotients are equal. This extension of the idea of a ratio is what allows us to define proportionality as we do in modern times: two variable quantities (i.e. quantities that take values from a specified subset of the real numbers) \(x\) and \(y\) that are related to one another in such a way that the ratios \(x \mathbin{:} y\) (for all values of \(x\) and \(y\)) are always equivalent, from which it follows that \(y/x\) is constant. This may be expressed in the familiar form \(y = kx\).

Unfortunately, the classical/medieval treatments of proportionality do not lead to this view. In fact, in many traditional treatments, the phrase “proportional to” does not appear at all. (Just as, in many traditional treatments of “ratio”, the idea that equivalent ratios have equal quotients is not mentioned.) What is unfortunate is that many textbooks do not go beyond the classical/medieval paradigm, and many teachers are unaware of the need to do so. An earlier post showed in a striking way that many teachers and others in the mathematics education community are not able to bridge that gap and demonstrate a conceptual understanding of proportional relationships. This is true even though they could readily solve procedural problems involving four quantities that were proportional.

What is surprising (and this was the point of the earlier post) is the “learning curve” that separates the classical/medieval paradigm from the modern formulation. Teaching experience shows that going from one perspective to the other is not a natural transition, but a discontinuous conceptual change. Many people simply do not think in terms of variables. In the meantime, those who have acquired the ability to do so regard it as easy and natural—so much so that they appear to have as much difficulty understanding how anyone could fail to grasp it, as those who do not grasp it have in acquiring it.

There is another approach to proportionality that aims to fix this problem, outlined in the Common Core State Standards. Still, the key idea related to variable quantities is very brief and easy to miss. (Four lines in 7.RP 2c, page 48.) In fact, this is the only place where the phrase “proportional to” occurs in the treatment of proportionality in the standards. So it is easy for people to fall back on the old habit of looking at the subject in a static way in terms of four numbers that form two equivalent ratios.

As a result, the old habits of the traditional ratio and proportion curriculum are still firmly entrenched. It will take a coordinated effort by mathematicians and mathematics educators working together to bring the middle school treatment of proportionality into the modern era. We need not jettison the “Rule of Three” (the old manner of reasoning with two four quantities in two equivalent ratios), but the curriculum needs a clear pathway joining this to the \(y=kx\) paradigm. This work is not so different from something mathematicians do all the time: Find the right definitions that lead to the best explanations and descriptions of mathematical phenomena.

We should make sure that ideas are accompanied by grade-level appropriate definitions, so students and teachers have something reliable to refer to when they have questions or doubts. We should provide grade-level appropriate explanations of where formulas and other results come from. We should develop ideas in a way that leads smoothly to subsequent topics and supports them. No one wants to lead students into computational dead ends. In short, this means considering what we often call “grade school mathematics” as a legitimate part of the field of mathematics and treating it as such.

This will not be easy. The culture and practices of school mathematics are not familiar to most mathematicians, but understanding them is essential for success. Moreover, for the most effective work to be done, mathematicians need to get together and talk to each other and to mathematics educators in detail about how we can most fruitfully participate in efforts to improve this situation. Although there has been much work by educational researchers on ratios and proportions (see here and here for instance), we have not yet come across any that works with the modern view of proportional relationships of varying quantities, instead of solving equations about equivalent ratios.

The AMS Special Session at the Joint Meetings is one example of the work that mathematicians are doing related to school mathematics. Several of the speakers are also involved in curriculum development efforts and teacher professional development, as are mathematicians at many institutions across the US. MSRI sponsors Math Circles, and AIM sponsors Math Teachers’ Circles and both have online resources to help mathematicians who are interested to get started. Jason Zimba’s recent article in the Notices about the Common Core also has many suggestions, including especially the final section, “What Mathematicians Can Do”.

To summarize, there is a lot more work to be done. Will you join us?

This is a topic that I’m really passionate about and have been trying to come up with ways to share math that mathematicians do with kids. A few of my ideas are in these two blog posts:

https://mikesmathpage.wordpress.com/2016/01/25/amazing-math-from-mathematicians-to-share-with-kids/

https://mikesmathpage.wordpress.com/2016/02/14/sharing-math-from-mathematicians-with-the-common-core/

Since writing those I’ve run across a few more fun ideas, too:

This is a great counting activity for kids suggested by Jim Propp:

https://mikesmathpage.wordpress.com/2016/03/05/a-fun-counting-excercise-for-kids-suggested-by-jim-propp/

and this minimal surface project is something that Dana Rowland from Merrimack college shared with me:

https://mikesmathpage.wordpress.com/2016/02/27/zometool-and-minimal-surfaces/

I hope that your push to get mathematicians involved with k-12 education is successful!

Mike, thanks for sharing all those really interesting problems we can give to even very young children! There are many ways mathematicians can be involved with K-12 math. In our post, we focused on improving the required curriculum, but your collection of problems shows how we can also help get kids doing interesting mathematics by expanding the curriculum.

Other readers are welcome to share their favorite ways of working with K-12, too.

I have seen this problem many times at my college. I ask my mathematics and physics majors what it means for y to be proportional to x. I almost never get the answer: “It means that there is a constant, say k, so that y = k x.” It is very strange. I get lots of complicated words.

So I wish to add an idea to the article. It seems that complicated phrases (sometimes obscure) pass for deeper knowledge than simple statements. I suspect, but cannot prove, that this is associated with the kinds of answers considered acceptable on essay questions for high stakes testing.

Thanks for this insight. Perhaps there are mathematicians who are willing to help people who write high-stakes tests think about how to assess what really matters? I only suggest this for mathematicians who are committed to the problem and willing to work to understand what drives the high-stakes testing industry. Otherwise, they will shut us out of the conversation because we “don’t get it.” But with commitment and patience, I believe we can make a difference.

I certainly understand the “lots of complicated words” problem you refer to! For many examples of this, see the earlier post “Proportionality Confusion” from November 2014. High stakes testing may be partially to blame for this situation, but testing generally follows curriculum. And the traditional middle school curriculum is certainly a culprit here. This spends a lot of time on “ratio and proportion”, but never seems to arrive at the big idea, namely what it means to say that y is proportional to x.

This post illustrates some of the reasons academic mathematicians are not that effective at teaching. One reason is that we not self aware. Nobody who reads the academic literature of mathematics or attends professional seminars should think we are good at communicating with each other. We stink at it.

Another reason is that others don’t want to learn mathematics as philosophy (the “right” definition of ratios), but as a tool to get some job done. Yet we math teachers usually don’t know what that job is or how we can help.

The way I understand a quantity being directly proportional to another quantity is: When quantity “a” increases or decreases, the corresponding quantity “b” also increases or decreases respectively. Otherwise when quantity “a” increases and quantity “b” decreases, this is an inverse proportionality.

This is an interesting and helpful comment. You have described a more general and very important idea – the idea of a mononically increasing or monotically decreasing function. A function that describes a proportional relationship is an important special case of a mononically increasing function, one that can be expressed as y = kx for a positive constant k. We want teachers and their students to see both the general idea and the special case.

Great issue for the math and I’m glad to read your post, it’s so interesting and yes this is the major problem faced by the students. It’s true that some teachers are not good at teaching as they should be, that’s the reason for students getting fewer marks nowadays.

Thanks for your post. Do you have suggestions for improving things?