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.

]]>The 2014 American Mathematical Society (AMS) Committee on Education (CoE) meeting took place on October 16-18 in Washington, D.C. I attended as a member of the AMS CoE. In addition to the committee members, there were many attendees from academic institutions, government, other professional societies, and the private sector. Like the recent CBMS forum that Diana White discussed in a blog post earlier this month, the focus of the CoE meeting this year was the first two years of postsecondary mathematics education. In this post, I will reflect on some of the key themes that stood out to me during the CoE meeting.

*The importance of collaboration*

The most prominent theme of the meeting was the critical role of collaboration and cooperation at many levels: among department members, at the institutional level between departments and administrative units, among professional societies with common missions, and at the national level to “scale up” successful models for effective teaching.

In talks about department-wide efforts to improve mathematics education in the first two years of college, Matthew Ando (Univ. of Illinois Urbana-Champaign), Stephen DeBacker (Univ. of Michigan), and Dennis DeTurck (Univ. of Pennsylvania) all emphasized the importance of collaborative support for department- and college-wide initiatives. This support includes faculty participation in specific programs, but was also more broadly framed through such lenses as thoughtful academic advising, working with members of other departments (such as engineering), and establishing clearly defined cooperative roles for departments and administrative units. I believe that this is particularly challenging at large universities, many of which are facing problems such as declining state support, increasing undergraduate enrollment, and severe constraints on instructional resources. For teaching environments such as these, where educational environments are increasingly large-scale, the message was clear that faculty need to work together in teams to create effective solutions to local challenges. However, these messages came with the caveat that collaboration is time-consuming, difficult, and requires sustained commitment from faculty.

It was interesting that, in addition to faculty and administrators from colleges and universities, participants at the CoE meeting included representatives from organizations involved in K-12 mathematics education, namely the National Council of Teachers of Mathematics and Achieve. This reminded me of the inherent connections between our challenges at the postsecondary level and the current national discussion regarding K-12 mathematics education (largely inspired by the widespread adoption by states of the Common Core State Standards). Institutions of higher education have increased their engagement in this discussion through the creation of advocacy organizations such as Higher Education for Higher Standards, demonstrating the type of collaborative efforts that are taking place at a national level. Another important aspect of post-secondary mathematics education that was pointed out during the meeting was the interaction between community colleges and institutions offering four-year degrees; the transition between these types of institutions is a rocky one for many students, and addressing this problem requires institutions to effectively work together.

*The importance of student-focused teaching*

Multiple speakers emphasized the necessity of broad adoption of student-focused teaching methods. Talks by Michael Starbird (Univ. of Texas Austin) and Ryota Matsuura (St. Olaf College) provided interesting perspectives on alternatives to college algebra and on Hungarian problem-based pedagogy, respectively, with emphasis on creating engaging courses for students. In the talks by Ando, DeBacker, and DeTurck mentioned previously, the need for clear learning outcomes for students was also emphasized, with teaching methods selected directly in support of these outcomes. All of these speakers emphasized the key role that active learning environments play in student development; however, the implementation of active learning environments they described was varied. For some, this meant having calculus recitations be organized around a carefully-crafted worksheet, with teaching assistants serving as “coaches.” For others, this meant largely eliminating lectures from classes, capping class sizes at 30-35 students, and using extensive group work carefully guided by the course instructor. The main message on this theme that I took from the meeting was that while student-focused teaching methods are critical, there is no “one size fits all” method that works best.

Many of the discussions during the meeting, both formal and informal, centered on the core question of “what do we want our students to know and to be able to do?” Without a well-articulated answer to that question, it is challenging to decide which teaching methods faculty should adopt. A phrase that stood out to me, mentioned by Herb Clemens (Ohio State Univ.) during his introduction to Bernard Hodgson’s (Université Laval) talk about post-secondary mathematics education in Quebec, was that we need “systemic caring” for students to be embedded in our institutions. Regardless of their specific form or implementation, the articulation of student learning outcomes and the purposeful use of student-focused teaching methods are important components of systemic caring for students.

*The coherence of education initiatives in the mathematical sciences*

There is a remarkable coherence among current educational initiatives in the mathematical sciences, broadly defined. This was especially apparent during the talks by Mark Green (UCLA), Karen Saxe (Macalester College), and Nicholas Horton (Amherst College) about the Transforming Post-Secondary Mathematics Education group, the Common Vision for Undergraduate Mathematics in 2025 project, and the American Statistical Association Guidelines for Undergraduate Programs in Statistics, respectively. Other important initiatives and reports that are worth mentioning along with these are the 2015 Mathematical Association of America Committee on the Undergraduate Program in Mathematics Curriculum Guide, the National Council of Teachers of Mathematics report Principles to Actions: Ensuring Mathematical Success for All, and the National Research Council report The Mathematical Sciences in 2025.

These initiatives and reports share a strong focus on increasing the number of pathways for students into the study of the mathematical sciences, and on reducing the number of barriers for students to cross along the way. Speakers at the CoE meeting emphasized the important role that evidence-based teaching practices can play in this regard, and the need that faculty and departments have for professional societies to make such practices easily identified and accessed. I view the coherence of the recommendations arising from these non-coordinated efforts in the mathematical sciences as an extremely positive sign, as it provides encouragement for us to join efforts in pursuit of common goals; further, these reports provide a reasonably common language through which to do so.

*A final observation*

A recurring phrase used by participants through the meeting was “The Time Has Come,” hence the title of this article. I agree that the time has come for all of us involved in the mathematical sciences to work together to improve mathematics education at all levels.

]]>In early October, approximately 150 educators and policy makers gathered together in Reston, Virginia for the fifth Conference Board of the Mathematical Sciences (CBMS) Forum entitled *The First Two Years of College Mathematics: Building for Student Success*. Participants came from almost every state in the country and represented higher education institutions ranging from two-year colleges to top-ranked research universities. We spent two days reflecting, learning, and in some cases planning how to improve the last year of high school mathematics and the first two years of college mathematics.

As is often my reaction at these types of conferences, I found the two days both sobering and energizing — sobering because of the sometimes harsh realities and challenges we face, energizing because of the good work participants report on and the many people gathered together who care so passionately and who dedicate so much of their time and energy to moving us forward. For those who could not join us in Virginia, this blog post will present a few key highlights from the Forum, in an effort to open a broader conversation about the future of the first two years of collegiate mathematics instruction.

The Forum began by emphasizing many ways in which the mathematical sciences are thriving. The National Research Council report *Mathematical Sciences in 2025* notes that there have been many major research advances, both theoretical and in high-impact applications, with clear benefits to other STEM areas and to the nation. As the foundation of many recent STEM advances, the role of the mathematical sciences has expanded.

This creates a need to revisit many aspects of our mathematical training of students to ensure that we are meeting the needs of our diverse constituents. In addition, the overall student success rates in mathematics are concerning. The President’s Council of Advisors on Science and Technology (PCAST) report *Engage to Excel: Producing One Million Additional College Graduates with Degrees in STEM *notes that “Reducing or eliminating the mathematics preparation gap is one of the most urgent challenges — and promising opportunities — in preparing the workforce of the 21^{st} century.” PCAST considers the situation so dire that they suggest an experiment in which faculty from mathematics-intensive disciplines such as physics, engineering, and computer science design and teach college mathematics courses. The PCAST report has received a lot of attention from the mathematical community, see the blog post from David Bressoud for some historical context and the summary response from the Joint Policy Board for Mathematics. The full response is here.

Multiple speakers at the Forum called on the mathematical community to wake up and heed the call for change this proposal implies. They called for a broadening of localized efforts to respond, while taking note of illuminating data and promising programs that may make this possible. Speakers at the Forum emphasized that we do not lack demonstrably successful and promising programs to meet many of the challenges the first two years of college mathematics present. However, widespread adoption of these programs has lagged.

*STEM Careers: Building for Success in Calculus*

The *Characteristics of Successful Programs in College Calculus* study from the Mathematical Association of America provides a large-scale, evidence-based understanding of who takes calculus, why they take it, and what happens in the course.

Students come out of calculus courses with greatly decreased confidence and frustration at their lack of understanding. Further, many students who initially declare or express interest in Science, Technology, Engineering, and Mathematics (STEM) majors change their mind in calculus class. The lack of persistence in STEM trajectories is sufficiently dire that we can currently view a moderate decrease in overall student interest in pursuing a STEM career after taking calculus as a success.

However discouraging this may seem, it is also encouraging that we now have such extensive data, as it allows the mathematical community to transition from small scale studies and so-called autobiographical reasoning — based on personal experience at our own institutions or throughout our own careers — to evidence-based reasoning from a large scale study.

*Affecting Student Success in General Education Mathematics*

As noted by Tony Bryk, President of the Carnegie Foundation for the Advancement of Teaching, “developmental mathematics is where aspirations go to die”. Yet the vast majority of first year community college students require some form of developmental education. Lack of student success in these developmental mathematics courses often hampers degree completion, as less than half successfully go on to complete a transfer level mathematics course (defined as a course that meets the general education mathematics requirement at a four year institution). The likelihood of persisting to successful completion of a transfer level course decreases drastically as the number of developmental courses needed as a prerequisite to a transfer level course increase, dropping to approximately 10% for those needing three or more developmental courses.

The* New Mathways Project*, an initiative of the Dana Center and the Texas Association of Community Colleges, is thus far showing remarkable results in both decreasing the time required for students to be ready for a transfer level course, as well as the success rates in such courses. It consists of three primary pathways — the statistics pathway, the quantitative pathway, and the STEM pathway.

The statistics pathway, known as Statway, prepares students in disciplines such as nursing, social work, and criminal justice for the college level statistics course their disciplines require. Now in its fourth year of implementation, it has a success rate (defined as earning a C or better in a college level statistics course within two years) of approximately 50%, as opposed to approximately 15% for those in a comparison group.

Results in the other pathways are also promising, and scale-up efforts are in place. The New Mathways Project is now being implemented in additional states, including my own state of Colorado, with optimism based on solid data that it will positively affect degree completion.

*Moving Forward*

This blog post cannot possibly do justice to the scope of topics addressed throughout the two days. For example, I have not addressed the Common Core and secondary to post-secondary articulation, faculty instructional approaches, diversity and gender concerns, or the reality of the fiscal challenges associated with these efforts. The complete list of abstracts, many of which contain the corresponding slides, give a sense of the scope of the Forum.

There remain many challenges related to entry-level mathematics, and to put the importance in context, the mathematics community would be wise to keep in mind that the number of students taking these courses, including the calculus sequence, by far eclipses the number of mathematics majors. College mathematics instructors would benefit from increasing their awareness of the extensive developments related to curriculum and teaching, especially at this lower level. Several Forum speakers also noted that the reward structure for mathematicians in academia, especially those at research universities, is another obstacle that we need to address to further increase participation, especially amongst active research mathematicians.

The intensive two days ended with a final challenge. Noting that the Forum would be a failure if participants only used what we learned as information for ourselves, we were encouraged to talk broadly with colleagues about what we learned and more broadly about the first two years of college mathematics education. Toward this end, I presented a summary of this meeting to our undergraduate committee, and I am going to share a copy of *Mathematical Sciences in 2025* with both my department chair and dean. It doesn’t feel like much, but it’s a start. This blog entry is also a contribution to that discussion, and an attempt to reach out to colleagues beyond my institution.

What are your thoughts about the issues the Forum addressed? How can we improve the first few years of undergraduate mathematics instruction, either locally or nationally? What do you view as the barriers to wider implementation of programs with demonstrated success? What are you willing to do to contribute to this effort?

I encourage readers to contribute to this discussion in the comments section and/or under the announcement for it on the AMS Facebook page.

]]>*Comment from the Editorial Board: We believe that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. This article is our first such contribution. We feel it provides a window into many of the subtle challenges students face as they transition to advanced postsecondary mathematics courses, and that it mirrors many of the themes discussed in previous posts. We thank Ms. Mattingly for being the first student to contribute an essay to our blog.*

In previous math classes, I was the quiet worker who kept to herself and didn’t know when or how to ask questions. After improving my skills in a problem solving class, that has changed. The group work we did each day allowed me to be around other people who think significantly differently than I do. Being in this environment was difficult at first because I actually had to work through problems with other people, which was somewhat unfamiliar to me. My classmates and I were not just sitting down and reading information about specific math problems. We had to analyze and make sense of the best methods and strategies to use and present our ideas to each other. Confusion would set in when other students introduced different approaches. The only way I could understand their ways of thinking was to ask them to explain. Asking questions in math initially intimidated me, especially because my questions had to be directed to my peers. I did not want them to think that I could not keep up with the material or that I did not belong in the class. But I also did not want to misunderstand major mathematical concepts as a consequence of not asking questions. So I started asking my group members each week what strategies they used in their solutions. Although it may have seemed repetitive to them or obnoxious to have to explain their approaches, it helped me immensely. Through my question asking, I was able to talk and think about math in a unique way. I could compare my peers’ techniques to my own, which further stimulated my interest in the particular subjects that were covered in the class. This skill has been and will continue to be essential in my future relationship with mathematics.

With this question-asking skill came a respect for other students’ speed of thinking. As I was learning to think like my group members in certain situations, I also was learning that these students were thinking at different speeds than I was. In many instances, I would attempt to understand what the problem was asking for and in the meantime, my peers were already halfway to a solution. I knew that I wasn’t misunderstanding anything. I simply was not making connections as quickly as others in the class. It constantly surprises me how swift some students are in accurately assessing problems and understanding what is needed to get to a solution. After working with all different types of students, I have learned to respect and accept that others may be working at faster cognitive speeds than I am. In previous classes, I thought that being quick with my math skills was most important. I have now seen that understanding the material is essential to becoming a fast thinker in mathematics.

Through the homework problems and quizzes in the problem solving class, I realized that math problems require perseverance. The problems that are actually difficult, that actually require a student to think about how to apply his or her mathematical knowledge to the solution of a problem, are the problems that take time. I found myself working hours on different math problems, trying to get closer and closer to the right answer. Oftentimes I went through the trial and error process. Failing in math is not as scary as it once was, because of my experience with this problem-solving course. Sometimes I would successfully solve a problem, while other times I didn’t come close to the solution. Either way, I was able to see that simply working with math was enhancing my problem solving skills. I will always remember Paul Zeitz’s quote in his book, *The Art and Craft of Problem Solving *[1], that says, “*Time spent thinking about a problem is always time [well] spent. Even if you seem to make no progress at all,*” (pg. 27). As I encounter future math classes and harder math problems that seem unsolvable, I will keep this quote in mind. Any time spent toying with a problem is enriching my mathematical knowledge.

Understanding Zeitz’s important quote about mathematical thinking prompted me to see the open mind that math problems require. When I took a number theory class in my first year, I was under the impression that all solutions to problems were very obvious and the methods to solve them were evident. Going through a geometry class as a sophomore challenged this belief because I began to see that not everyone knows which method to use immediately to solve or prove a problem. With the problem solving class, this belief was completely put to rest. I have seen firsthand that it occasionally takes experimentation to figure out which method or tool to use in problem solving. Through discussion and group work with my classmates, I noticed that it is not always blatantly obvious that we should draw a picture or use induction or reformulate a hypothesis to find the crux move in a solution. After discovering this, I attempted to open my mind when reading assigned problems. Instead of honing in on one specific method or strategy, I have accepted the fact that one specific method or strategy might not be the only way to achieve my goal.

An open mind in problem solving has allowed me to experience mathematical thinking outside of the classroom. My high school math experiences bred the idea that students don’t necessarily need to think about math outside of the classroom. I brought this idea to college and had no problem passing my classes. But in this math problem solving class, where we were challenged to think in different ways and to explore math on our own, thinking about math strictly inside the classroom was insufficient. One incident that had a deep influence on my problem-solving experience occurred on a walk home from class. In small groups, my classmates and I were trying to determine the difference between permutations and combinations. After working for an entire class period, I did not fully understand the difference between these two basic combinatorics concepts. As I walked home, I reviewed the strategies and methods that I used and compared them with the explanations of my peers. I was still stuck and still frustrated. Upon emailing my professor about my misunderstanding, I realized that although I had encountered an obstacle with the content, I was gaining an invaluable way of thinking. I was extending my mathematical thought processes and interests to outside of the standard classroom environment. I am still constantly left wondering why something in math works. By pondering on my own, whether it is with pencil and paper or not, I am able to see the impact of this class. No class before had prompted me to take my own time to figure out why I did not understand the material. This was a huge win for me as a math student. I was and am still experiencing the effects of struggling with math in a way that will always benefit my problem solving skills.

One of the most interesting proofs that I saw was dealing with the number of subsets of a set with \(n\) elements. Earlier that semester, my probability professor had mentioned that the number of subsets of an \(n\)-element set is \(2^n\). I didn’t think I would be seeing this anymore, so instead of trying to understand why this was so, I just accepted the information. Later on, in my problem solving class, we had a question about the number of subsets in an \(n\)-element set. Obviously I knew it was \(2^n\), but I had no idea why. I eventually understood after I showed interest in the problem through question asking, when another interpretation of all of the subsets of an \(n\)-element set was written up on the board. I saw that another way to write a subset of an \(n\)-element set is by exchanging the actual numbers in the set with 0’s and 1’s. A “0” indicates the absence of a specific element in the subset, while a “1” indicates the presence of a specific element in the subset. For example, the set \(\{1, 2, 3, 4, 5, 6\}\) has the subset \(\{2, 4, 6\}\). This subset can be written as 0, 1, 0, 1, 0, 1, where the 1’s correspond to the numbers found in the subset. Since each space has either the option to be in the subset or not, then each space has two options. Since there are *\(n\)* spaces, then there are \(2^n\) subsets. By asking questions, expressing curiosity, and actually attempting to understand why the answer was \(2^n\), I was able to see a portion of my growth as a problem solver.

I am left with questions regarding my future mathematical experiences. Instead of simply thinking about mathematics outside of the classroom, I am now wondering how to discover and develop new insights about mathematical concepts on my own. Instead of learning about how to use certain strategies, I am wondering how to present these strategies to a group of peers in an orderly and effective manner. Instead of asking questions that do not necessarily prove to be productive, I am slowly learning how to ask the *right* questions.

**References**

[1] Zeitz, P. (2007). *The Art and Craft of Problem Solving*. Hoboken, NJ: John Wiley & Sons, Inc.

*By Oscar E. Fernandez, Assistant Professor in the Mathematics Department at Wellesley College. *

Mathematics is a beautiful subject, and that’s something that every math teacher can agree on. But that’s exactly the problem. We math teachers can appreciate the subject’s beauty because we all have an interest in it, have adequate training in the subject, and have had positive experiences with it (at the very least, we understand a good chunk of it). The vast majority of students, on the other hand, often lack *all* of these characteristics (not that this is their fault). This explains why if I’d start talking to a student about how exciting the Poincare-Hopf theorem is, I probably wouldn’t see anywhere near the same excitement as if I were to, say, let them play with the new iPhone. This may seem like a silly hypothetical, but I believe it brings up all sorts of important points. For one, what does it say about our culture (and our future) when young people would rather be playing games on iPhones (or watching Youtube, or being on Facebook, etc.) than studying math or science? What causes our culture to be the way it is? How did companies like Apple and Facebook get students so interested in these activities? What are they doing that we math teachers aren’t?

First, let me admit that there are many, many differences between getting exciting about the new iPhone and getting excited about math*, but I’m interested in one of them in particular: you can see, feel, *interact with*, and *experience* the iPhone. Moreover, Apple thinks *very carefully* about *every aspect* of the user experience *well before* they release their next phone (there are, after all, *billions of dollars* at stake).

Sadly, the way math is taught in many places, students’ experience with mathematics is often confined to a blackboard or piece of paper. They also spend the majority of their time interacting with math in a very different way, e.g., trying hard to get the right answer before the homework is due as opposed to playing around with the content to discover something new, as a first-time iPhone user might do. And what about the Steve Jobs or Jony Ive of the class—the instructor—who is supposed to make it all magical? Oftentimes that person follows the “definition, theorem, proof” style of teaching, which is likely only “magical” to already math-inclined students. My point: *we* (the math teachers) are the most important drivers of our students’ interest in and excitement about mathematics. Collectively, we are the Apples and Samsungs of the math world. And if we teach math like *we discuss it amongst ourselves***, we’re likely to continue losing the vast majority of students to other careers.

So, what should we do? I say we look to Apple, Samsung, and all the other companies that have successfully hooked our students on their ideas and products. Sure, they have hordes of people whose sole job it is to make their products* fun, cool, and relevant*, but why can’t we do that, too? Why can’t we, for example, give out a survey the first day of class that asks students about their hobbies and interests, and then, at the very least, choose examples and applications for the rest of the course that align with those interests? In fact, why don’t we just structure our courses to make mathematics something that our students can *directly experience* and is *directly relevant* to their lives?*** Let me call this the *Everyday Mathematics* (EM) approach.

Here’s an example. Instead of reviewing the graph of a sine function by drawing a sine curve, explaining what the frequency, amplitude, or period are, showing examples where these parameters change, and finally discussing a Ferris wheel, picture this instead. You pull up a chart of human sleep cycles, you explain that the average cycle length is 90 minutes, that there are four stages of sleep—with Stage 4 being “deep sleep.” You ask your students to find the formula that best fits the sleep chart. Then you ask them: at what times should you wake up to avoid feeling groggy (which happens when you awake near the bottom of a sleep cycle)? You would then guide them to the revelation that they can now use their formula to predict these times and other interesting things, too. Presto! Sine and cosine have now become *relevant*; they are now concepts that help explain *every student’s* sleep cycle and can help them avoid morning grogginess. In other words, this EM approach has made this particular topic at least *relevant* to your students’ lives. I wouldn’t be surprised if, when you move on to tangent, some of your students would start wondering “Hey, what can tangent do for us?” (By the way, how often have you heard a student ask that?)

In general, the EM approach begins with a topic or phenomenon directly relevant to your students’ lives. Then, you (the instructor) build a lesson that slowly guides students through the math you would have taught anyway, except that now there is context, that context is personal for each student, and there is a point to all of it that students can buy into (in the example, helping them sleep better and explain morning grogginess).

From an instructor’s perspective, the EM approach may seem like a lot more work than a more traditional approach. However, I myself was able to generate enough of these EM-like examples (pertinent to Calculus I topics) to write an entire book about it, mainly by just spending a few days being very observant about everything going on around me and then putting on my mathematician hat to see the math behind it. Granted, this approach might not be appropriate for all courses—it probably wouldn’t work in a course on cohomology—but that’s okay, because by that point that student is probably more interested in how that subject relates to other areas of mathematics.

The EM approach may not be the answer to our national crisis in math, but I think it is a step in the right direction. At the very least it realigns our presentation of the content with our students’ interests. It also attempts to emulate the successful efforts of corporations to get people excited about their products, since the approach puts our students—and their interests—first, and then scaffolds on our content goals (as opposed to the other way around). In my experience using the EM approach, I have received some of the most enthusiastic responses I’ve ever gotten after teaching certain concepts. I would love to hear about your own ideas to make math fun, relevant, and something students can directly experience.

_______________________________________________________________

* There are, after all, people who spend weeks in line waiting for the new iPhone; I’ve never heard of a student camping out outside a classroom for weeks waiting for a course to start.

** This would be like Apple unveiling its iPhone by talking *mostly* in technical jargon—after all, that’s how the designers, engineers, and programmers think. I doubt their press events would be so well attended were this the case.

*** No more talking about the largest area a farmer can enclose with a given amount of fencing, or about a ladder falling down the side of a building, for example.

*Editor’s Note: Carl Lee is a recipient of the 2014 Deborah and Franklin Tepper Haimo Award from the Mathematical Association of America. This essay is based on his acceptance speech at the 2014 Joint Mathematics Meetings.*

**My place.** I was born into a family littered with academics, teachers, and Ph.D.s, including a grandfather who was an educational psychologist at Brown serving on one of the committees to create the SAT. My early interest in things mathematical was nurtured in a home stocked with books by Gardner, Ball and Coxeter, Steinhaus, and the like. With almost no exception my public school teachers were outstanding. I was raised in a faith community, Bahá’í, that explicitly acknowledges the presence of tremendous human capacity and the high station of the teacher who nurtures it. I played and experimented with, and learned, mathematics in both formal and informal settings. Thus I grew up in a place in which I was able both to feed my mathematical hunger as well as to have a clear idea of what it was like to teach as a profession. I thrived.

I recount this not to present a pedigree to justify personal worthiness, but rather to emphasize that I enjoyed a perfect match between my personal mathematical inclination and my learning environments. Because of this background, it took me a while to understand the sometimes profound gap between others’ mathematical place, and the consequent care required to pay attention to that place, when designing an effective realm for learning. As a K–12 student I often engaged in math classes at a high cognitive level merely as a result of a teacher’s direct instruction (“lecture”). As a teacher I quickly learned that I engaged few of my students by this process. Not all developed their “mathematical habits of mind” or “mathematical practices” through my in-class lectures and out-of-class homework (often worked on individually). I now better appreciate the significant role of personal context and informal education in the development of students’ capacity.

**The student’s place.** There is an entire discipline of “place-based” teaching and learning, focused on recognizing and making explicit connections with the student’s physical location and social community (an “outer place”). Mirroring and linked with this is a student’s personal cognitive place (or “inner place”)—here, I recall Vygotsky’s writings on the “ZPD,” the zone of proximal development, which he describes as “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers.” That is to say, learning can be promoted when the material is above the student’s current state, but not so far above to be unattainable even with scaffolding and assistance. Identifying these outer and inner student places, and making wise and deliberate instructive choices, are major challenges of the teacher.

With respect to the student’s outer place we are all well aware of the encouragement to teach mathematics through “real-world” problems. The Common Core State Standards for Mathematical Practice encourage modeling with mathematics: “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” My work with teachers in central Appalachia has convinced me authentic and locally placed problems can provide powerful stimulus and support for mathematical learning.

On the other hand, with respect to the student’s inner place, another mathematical practice advocates that students must “make sense of problems and persevere in solving them.” Research by the psychologist Carol Dweck, for example, confirms that praise focused on developing a growth mindset positively affects subsequent student achievement, while praise that reinforces fixed intelligence beliefs has the opposite effect. Further, fostering a growth mindset rather than a fixed mindset in the classroom *with the explicit knowledge and understanding of the students* appears to lead to increased academic achievement when students are aware of the value of the struggle. Polyá was an early advocate of the deliberate shift toward raising the explicit awareness of and cultivating mathematical practices among students. Seeing his film “Let us Teach Guessing” while in high school left a lifelong impression upon me. As a result, I feel I must promote *and **observe* struggle in my classroom—deliberately create opportunities in the classroom in which students grapple with mathematics and communicate with each other; carefully listen and use what I learn to shape what is to come; and provide an environment in which mistakes are opportunities for learning and not censure.

The student’s outer and inner places are, of course, deeply connected—where a student is mathematically is not isolated from his or her background and environment. And in Appalachia (as in many other places), struggle is a part of life. The preeminent Appalachian poet and writer Wendell Berry beautifully captures this notion in his essay “Poetry and Marriage” from *Standing by Words*.

There are, it seems, two muses: the Muse of Inspiration, who gives us inarticulate visions and desires, and the Muse of Realization, who returns again and again to say “It is yet more difficult than you thought.” This is the muse of form. It may be then that form serves us best when it works as an obstruction, to baffle us and deflect our intended course. It may be that when we no longer know what to do, we have come to our real work and when we no longer know which way to go, we have begun our real journey. The mind that is not baffled is not employed. The impeded stream is the one that sings.

**The place of community. **There is a continuum of participants and stakeholders in STEM education, including: P-12 students, school teachers, counselors, principals, superintendents, parents, community members, college students taking math and science courses, majors in STEM fields, aspiring STEM teachers, higher education faculty in content departments teaching all of these types of students, higher education faculty in education departments teaching courses for future teachers and engaging in teacher training programs, practicing teachers including those who supervise student teachers or are enrolled in graduate programs, higher education faculty engaging in STEM education research or in outreach to schools, and various local, state, regional, and national agencies and organizations, public and private, commercial and non-profit. There is a natural tendency for each of the diverse participants to operate within a somewhat limited sphere of activity. If we wish to build institutional and regional capacity, there is an imperative need for mathematicians to lend their expertise to this continuum, and for institutions to appropriately reward their contributions. The *Mathematical Education of Teachers II* (CBMS), for example, offers a call to action with explicit guidance and suggestions.

Reflecting on my work with others in these many roles in Appalachia, it is very clear to me that an appropriate understanding of place is essential. Many regard rural Appalachia with “deficit vision” and wish to come in and “fix things.” Yet Wendell Berry’s view is completely opposite—read his poem “The Wild Geese.”

Berry articulates his vision in “The Loss of the Future” from *The Long-Legged House*: “A community is the mental and spiritual condition of knowing that the place is shared, and that the people who share the place define and limit the possibilities of each other’s lives. It is the knowledge that people have of each other, their concern for each other, their trust in each other, the freedom with which they come and go among themselves.”

Bob Wells recalls Berry’s 2007 speech at Duke Divinity School.

“Whatever doesn’t fit a place is wrong,” Berry said. “It doesn’t matter if it is true or false. If it doesn’t belong, it is wrong.” Without a standard of “place” as a measure of real prosperity, Berry said, we will never know what to make of development, technology, research, education, modernization, religion and the environment, or ecosphere.

I have learned that to be more effective I must view place from the perspective of a partner rather than as a knowledgeable outsider (however well-intentioned). Consideration of place must be approached with an authentic attitude of partnership, setting aside such common barriers as “outsider-insider,” “knowledgeable-ignorant,” and “wealthy-poor.” The wealth and strength of Appalachia include rich experience and an abiding sense of community, both of which can significantly contribute to sustainable approaches to educational challenges.

Sentiments such as these were central principles in two recent large-scale NSF funded projects in Appalachia that I had the privilege to work on. ACCLAIM, an NSF Center for Learning and Teaching, focused on “the cultivation of indigenous leadership capacity for the improvement of school mathematics in rural places.” A highlight of this project was the creation of an interinstitutional doctoral program in mathematics education built around issues in mathematics, mathematics education, and rural sociology. Students in this program demonstrated a commitment to rural place and earned their degrees without having to quit their jobs. Their desire to remain in their communities helped sustain ACCLAIM’s impact on future teachers. The resulting dissertations were not required to address rural topics, but often did. I encourage perusal of https://sites.google.com/site/acclaimruralmath for uncovered understandings at the intersection of mathematics education and rural education.

AMSP, an NSF Mathematics and Science Partnership, was an ambitious Appalachian enterprise involving nine institutions of higher education and about 60 school districts. Lessons learned during the earlier years led later to community-based Partnership Enhancement Projects generated by groups of stakeholder partners based on local concerns. The place of the work (e.g., the school, district, or county) provided the explicit context in which participants evaluated challenges, assessed resources, planned, executed projects, and evaluated outcomes.

**The place of mathematics and the mathematics of place. **On the one hand many (including mathematicians) value mathematics precisely because it *transcends* place, even though it may be initially motivated by a particular context (mathematical, physical, or otherwise). On the other hand, the value of place (including rural or urban place, and personal place) offers a rich and meaningful setting in which to nurture the understanding of mathematics and make important connections that can promote mathematical learning and more effective teaching. My present understanding is that the latter view is important to support the former. In teaching and professional development I therefore try to work with others in a spirit of partnership — there are things that I know, and there are things that my partners know. If we abandon a sense of superiority as we approach classroom teaching, professional development, or community capacity building, striving to understand our place, we can dramatically increase the efficacy of our work together.

Chapter 1 of *Make It Stick: The Science of Successful Learning* [2] is called “Learning is Misunderstood.” That is an understatement, as demonstrated by the remaining chapters. The book has received several strong reviews ([3], [5], [8]), so rather than providing a critique, my aim here is to explore the ways in which its account of cognitive science research has validated some decisions I have made about my teaching and gotten me to reconsider others.

Since the early 1990’s, I have been using a form of what we now call Inquiry-Based Learning (IBL) in my Abstract Algebra course; more recently I’ve been doing so in Number Theory as well (using [6]). This all started when Professor Bill Barker of Bowdoin College described an Algebra course built around small-group work, and I was hooked. Surrounded here at Middlebury College by excellent immersion language programs, I realized that Bill was describing a mathematics immersion program. I modeled my course on his so that my students would learn mathematics by speaking mathematics with each other, while I roamed the room as consultant. That first post-conversion semester, there were numerous classes that went overtime before any of us noticed, so engaged were the students.

Meanwhile I began making less drastic changes in my Calculus courses, devoting at most one class session (out of four) each week to small group work. The results were less satisfying; those sessions felt like an add-on rather than an integral part of the course. I assumed that I couldn’t abandon lectures completely because of the list of topics I felt compelled to cover. Last spring, however, considering data showing that few of our Calculus I students go on to Calculus II, I decided to ditch the massive textbook in favor of fewer topics and an interactive format. My goal had shifted from getting them through a fixed set of material to having them engage the ideas deeply enough that they thought differently about measuring change, whether in their economics and biology classes or when reading the news ten years from now.

*Make It Stick* confirmed my preference for an active learning model as soon as page 3: “Learning is more durable when it’s *effortful.* Learning that’s easy is like writing in sand, here today and gone tomorrow.” It’s easier for students to copy my problem solution from the blackboard and then imitate it in a bunch of similar homework exercises, but it’s no wonder that they don’t seem to retain much in that setting. “When you’re asked to struggle with solving a problem before being shown how to solve it, the subsequent solution is better learned and more durably remembered.” [2, p. 88] What I’m reconsidering is the way in which I choose problems; I want the particular struggle to be productive in ways that the authors describe.

Naturally some students resist a shift from passivity to activity. The student evaluations for my first IBL-ish course were quite positive, except for one that said “You’re the expert; you should tell us what to do. I learn better in a lecture,” an assertion that I continue to hear from a few students. According to *Make It Stick*, those students may well be misunderstanding their own learning: “*We are poor judges* of when we are learning well and when we’re not. When the going is harder and slower and it doesn’t feel productive, we are drawn to strategies that feel more fruitful, unaware that the gains from these strategies are often temporary.” [2, p. 3]

To confront such resistance, I put some effort into what I thought of as a sales job: “This way I can help you speak mathematics in real time, and it gives you practice collaborating for later in life, and aren’t we lucky to have small classes at Middlebury,” and so on. These days I think of such effort in the context of metacognition, which I first encountered in [7]. In being explicit about why I structure my courses the way that I do, I’m also encouraging my students to think more critically about their own learning, which is in itself an asset to that learning. This semester I’ve put a page on the course website with information about the science of learning.

The work of the social psychologist Carol Dweck ([1], [4]) comes up in *Make it Stick. *Perhaps I’m biased, but surely mathematics learners are particularly prone to the curse of the “fixed mindset” rather than having a “growth mindset.” This semester, my first assignment in Calculus was to read “Bad at Math is a Lie” [9] and then have a class discussion. First my students shared their “bad at math” moments in groups of three or four, and then we heard some in the full group. I know that one event won’t move everyone into a growth mindset, but it’s a start.

For some reason – the relentless “coverage” drumbeat? – a while back I stopped my practice of taking mini-surveys on Fridays in Calculus classes. These had three questions: (1) What were the important themes this week? (2) What concept(s) intrigued you? (3) What concept(s) are still muddy to you? They helped me know what students were thinking, and communicated to the students that I wanted to know what they were thinking. I’m reintroducing the surveys, not just for those purposes, but also because they ask students to reflect on their learning. “Reflection can involve several activities … that lead to stronger learning.” [2, p. 89]

On the other hand, I’ve always resisted quizzes because of the added stress. According to *Make it Stick*, however, frequent low-stakes assessments that require students to retrieve new knowledge can assist in the learning process. So this term I’ve scheduled weekly quizzes in which anything from the semester so far will be fair game.

The authors of *Make It Stick* suggest that instructors “be transparent.” [2, p. 228] One way in which I convey my intentions to my students is by including this quote at the end of my syllabi: “Trying to come up with an answer rather than having it presented to you, or trying to solve a problem before being shown the solution, leads to better learning and longer retention of the correct answer or solution, even when your attempted response is wrong, so long as corrective feedback is provided.” [2, p. 101] I am still trying to come up with the best ways to provide corrective feedback; that effort might be the subject of a future post. In the meantime, I am grateful to Bill Barker and many others who have been transparent about their pedagogy as I refine my own.

**References**

[1] Braun, Benjamin. Persistent Learning, Critical Teaching: Intelligence Beliefs and Active Learning in Mathematics Courses. *Notices of the American Mathematical Society, ***61 **(January 2014), 72-74.

[2] Brown, Peter C., Roediger, Henry L., and McDaniel, Mark A. *Make It Stick: The Science of Successful Learning.* Belknap Press, 2014.

[3] Christie, Hazel, in *The Times Higher Education*, April 3, 2014.

[4] Dweck, Carol S. *Mindset: The New Psychology of Success.* Ballantine Books, 2007.

[5] Lang, James N. Making It Stick, in *The Chronicle of Higher Education*, April 23, 2014.

[6] Marshall, David C., Odell, Edward, and Starbird, Michael. * Number Theory Through Inquiry. *Mathematical Association of America, 2007.

[7] National Research Council. *How People Learn: Brain, Mind, Experience, and School: Expanded Edition*. Washington, DC: The National Academies Press, 2000.

[8] Stover, Catherine.“For the most part, we are going about learning in the wrong ways.” *A Fine Line* blog, April 10, 2014.

[9] Waite, Matt. Bad at Math is a Lie. *Math Horizons. * September 2014, p. 34.

One of the highlights of my summer was attending a research conference, Stanley@70, celebrating the 70th birthday of my Ph.D. advisor Richard Stanley. Because it was a birthday conference, many of the speakers went out of their way to say a little something about Richard Stanley, with mathematical or personal anecdotes. One talk in particular, by Lou Billera, did an especially good job giving the history and context of the study of face numbers of simplicial polytopes, in which Richard played an essential role. (The slides don’t totally convey the breadth of the talk, but at least give you some idea of the mathematical story he was telling.) I really appreciated Lou’s talk, and I know (from asking them) that other participants did too. This got me thinking that the mathematical community could do more of this sort of thing, not just at conferences, but more importantly in courses for our undergraduate majors and graduate students. In these courses, we rightfully focus on the truth of mathematical results. Let’s also spend some time sharing with our students *why we care* about the mathematical objects and ideas that show up.

We’ve long been blessed in mathematics to have the freedom to not worry about applications of our discipline. My favorite expression of this attitude comes from Bernd Schröder whose slides at a recent talk jokingly answered the question “Who cares?” with “Who cares who cares? It’s cool.” This was his clever way of expressing his observation that enthusiasm can override pragmatism. (It is only fair to note that Bernd advocates for applications, and that he subsequently gave more specific reasons to care about the topic of his talk.) In other words, we are free to investigate whatever looks interesting to us, and I value that freedom just about every day. But what looks interesting to us, and why?

Of course, in many settings it is the application that makes a result or topic interesting, and I don’t mean to diminish this motivation in the least. Many studies [8, 9] recommend including more applications in mathematics classes at all levels, not just for the sake of the application, or its use for students who are in (or who will be going into) science and engineering, but to help students better understand the underlying mathematics itself. Indeed, the five strands of mathematical proficiency in [8], including “strategic competence” (problem formulation and solving), are specifically described as “interwoven and interdependent”. But these recommendations tend to point towards the K-12 classroom, or towards applied or lower-division undergraduate courses, such as calculus. This same principle seems to me no less relevant in upper-division pure mathematics courses for our majors, and even graduate courses: You can get a better handle on an idea if you know where it came from, or where it is going.

Here are some questions for students to ask or for teachers to answer, even in pure mathematics. The answers don’t need to be long or detailed. Why did people start looking at this topic? What were the motivating examples? How did the ideas develop? How is it used in other areas of mathematics or outside mathematics? Why is this topic in this textbook, or why is this course being taught?

Sometimes a topic, theorem, or definition is just inherently interesting for purely mathematical reasons, which will have resonance for mathematics students who already appreciate abstract thinking. Some quick examples from the mathematics I’m most interested in include symmetry of structures, large matrices whose eigenvalues are integers, and large polynomials that factor linearly. But even when something is inherently interesting to experienced mathematicians, it can be worthwhile to take the time and effort to point this out to students who are just beginning their careers as mathematicians or teachers, and who may not have yet developed that same appreciation.

The needs of future mathematics teachers in this regard may be a little different than those planning to go to graduate school in mathematics. For this cohort, by far the most important context is “How will this show up in my high school (or middle school) classroom?” (See [3] for more detail.) Here, some of the textbooks for capstone courses for teachers, for instance [2, 10], have good ideas, which can be incorporated into other courses as well. For instance, the plethora of structures introduced in an algebra class have important examples in high school, which may be helpful for other students as well: The reals and rationals are fields, integers are a ring, and polynomials and matrices each form an algebra, etc. One textbook we’ve used [10] illustrates the need for all the field axioms by showing that these axioms are exactly the rules we need to solve linear equations.

Students don’t have to wait for instructors to do this for them. Students can ask the questions above, or make up new ones. Students may consult good books with historical and contextual material. Teachers can strongly influence this type of self-study by recommending references, for example the mathematics history books listed in the “What to read next” chapter of [1] and mathematics biographies in the “review of the literature” in [4], and also some of the other references listed below. Teachers can find other ideas for how to guide students to mathematics history at Reinhard Laubenbacher and David Pengelley’s excellent website for teaching with original historical sources in mathematics, which has resources for single projects or entire courses [6, 7].

All this is not to say we should *stop* doing what we normally do in pure mathematics classes. Of course, the careful definition-theorem-proof development of topics is the backbone of mathematics. This is what lets us be certain of our results, which is the other blessing we have working in pure mathematics. But it can also lead to students’ misconception that mathematics is *created* in this order: First come the definitions, which cannot be changed once they are written down, and then theorems are stated, and subsequently proved. Those of us in the business know it doesn’t usually work this way! We should let students in on the secret that the process is a lot more circular than the textbooks let on.

A nice example of this messiness that also illustrates some other ideas here is the notion of an ideal in a ring. It is usually introduced in algebra books simply with the definition, and then some basic results about it are stated and proved. But why would you want to work with this definition? To summarize greatly (see [5] for much more detail), ideals started with Kummer’s introduction of “ideal numbers”, generalizing integers by considering the set of multiples of one number, or, more broadly, a set of numbers. But one reason (which I explicitly share with my algebra students) we see them so much is that if you need to make a quotient ring, then the definition of an ideal gives you exactly what you need in order for this quotient to be well-defined. (This is a good exercise if you haven’t thought about it before.) Each of these extra facts about ideals reinforces points you would probably want to make anyway.

Let me finish where I started, with Lou Billera. When I wrote to him about including a reference to his talk in this post, we had a nice email conversation about these ideas. I’ll give Lou the last word, from that conversation:

When I first started teaching here, I became aware of several older professors (in engineering) whose class lectures consisted of what I called “war stories”,

e.g., how they solved this or that problem for this or that company. I thought they were just wasting their time BS’ing and not “covering the material”, but the students loved it. In the end, for them, the “war stories” were probably much more useful in their professional lives as engineers than the “material” ever could be. (Besides, the “material” was in the book, and they all knew how to read.) To the extent we can get “war stories” into our own mathematical teaching, without sacrificing “the material” (too much), our students will be better off for it.

**References**

[1] Berlinghoff, W., & Gouvêa, F. (2004). *Math through the ages: A gentle history for teachers and others. *Farmington, ME: Oxton House Publishers; and Washington DC: Mathematical Association of America.

[2] Bremigan, E., Bremigan, R., & Lorch, J. (2011). *Mathematics for secondary school teachers. *Washington, DC: Mathematical Association of America.

[3] Conference Board of the Mathematical Sciences (2012). *The mathematical education of teachers II. *Providence, RI: American Mathematical Society; in cooperation with Washington, DC: Mathematical Association of America.

http://www.cbmsweb.org/MET2/met2.pdf

[4] Hersh, R., & John-Steiner, V. (2011). *Loving and hating mathematics: Challenging the myths of mathematical life. *Princeton NJ: Princeton University Press.

[5] Kleiner, Israel, (1996, May). The genesis of the abstract ring concept. *The American Mathematical Monthly*, 103(5), pp. 417-424.

http://www.jstor.org/stable/2974935

[6] Knoebel, A., Laubenbacher, R., Lodder, J., & Pengelley, P. (2007). *Mathematical masterpieces: Further chronicles by the explorers.* New York, NY: Springer.

[7] Laubenbacher, R., & Pengelley, P. (1999). *Mathematical expeditions: Chronicles by the explorers.* New York, NY: Springer.

[8] National Research Council (2001). *Adding it up: Helping children learn mathematics.* Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

http://www.nap.edu/catalog.php?record_id=9822

[9] Schoenfeld, Alan (2007). What is mathematical proficiency and how can it be assessed? In *Assessing mathematical proficiency* (pp. 59-73). New York, NY: Cambridge University Press.

http://library.msri.org/books/Book53/files/05schoen.pdf

[10] Usiskin, Z., Peressini, A., Marchisotto, E., & Stanley, D. (2003). *Mathematics for high school teachers: An advanced perspective.* Upper Saddle River, NJ: Pearson Education.

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One of the challenges of teaching mathematics is understanding and appreciating students’ struggles with material that to the instructor, after years of thinking about it, may seem straight forward. Once we understand an idea, it may seem almost impossible not to understand if it is presented clearly enough. Yet experienced math teachers know that presenting mathematical ideas clearly, as important as that is, is generally not enough for students to learn the ideas well, even for dedicated and determined students. At the same time, students who struggle can have insightful and productive ways of solving problems and reasoning about mathematical ideas. Research into how people think about and learn mathematics reveals why this surprising mix of struggle and competence can coexist: learners can use what they do understand to make sense of new things, yet ideas that are tightly interconnected and readily available for an expert may be fragmented or inchoate for a learner.

Consider the ideas surrounding slope and rate of change, which are well known to be difficult for students. To the expert, a slope is a number that expresses a measure of steepness. It connects changes in an independent variable to changes in a dependent variable. This connection is multiplicative and explains why non-vertical lines have equations of the form *y* = m*x* + b. But even students who appear to be proficient—because they can calculate a slope and use it to find an equation for a line—may be missing some crucial connections. They might not see slope as a number, but instead think of it as a pair of numbers separated by a slash, basically “rise slash run.” If the “rise” is 3 and the “run” is 2, then even if they know that 3/2 is a number, they may not connect it to the geometry and algebra of the situation. They might not see this number as a measure of steepness, and if asked to describe steepness, might prefer to subtract the “run” from the “rise.” Students might not see the “rise” as 3/2 *of* the “run” and they might not connect this multiplicative relationship between the “rise” and “run” to the point-slope form of an equation for a line. Mathematics education research is examining the fine-grained details of how students think about ideas surrounding slope. It is investigating how certain ways of representing and drawing attention to ideas can help students extend and connect their ideas. Research-based instruction can then take into account known challenges and opportunities for learning.

We thought readers of this blog might be interested to learn a little about approaches to slope and linear equations that we are currently investigating. Proportional relationships—pairs of values in a fixed ratio—provide an entry point into the study of linear functions and are a focus in the Common Core State Standards for Mathematics at grades 6 and 7 (see [1] and [2]). So consider the proportional relationship consisting of all pairs of quantities of peach and grape juice that are mixed in a fixed 3 to 2 ratio to make a punch. When graphed, these points lie on a line. One way to think about the slope, 3/2, of this line is that for every new cup of grape juice, the amount of peach juice increases by 3/2 cups. This way of thinking is part of what we call a *multiple batches* view, a view that has received significant attention in mathematics education research. From this perspective, we may think of 1 cup grape juice and 3/2 cups peach juice (or 2 cups grape juice and 3 cups peach juice) as forming a fixed batch of punch, and we vary the *number* of batches to produce different amounts in the same ratio. This fits with the image in Figure 1a, which evokes repeatedly moving to the right 1 unit and up 3/2 units. But as indicated in Figure 1b, the general multiplicative relationship, *y *= (3/2)* x,* is less evident, especially for *x* values that are not whole numbers.

*Figure 1:* Slope from a multiple-batches perspective.

Another way to think about the punch mixtures in a fixed 3 to 2 ratio uses what we call a *variable parts* perspective. This perspective has been overlooked by mathematics education research, but we are currently studying how future teachers reason with it. In a variable-parts approach, for any point on the “punch line” (see Figure 2), there are 3 parts for the *y*-coordinate and 2 parts for the *x*-coordinate, and all the parts are the same size. From this perspective, we vary the *size* of the parts to produce different amounts in the same ratio. The parts expand or contract depending on the direction the point moves along the line. In a variable-parts approach, the slope 3/2 is a direct multiplicative comparison between the numbers of parts of grape and peach juice: The number of parts peach juice is 3/2 the number of parts grape juice. Put another way, the value 3/2 is the factor that multiplies the number of parts of grape juice to produce the number of parts of peach juice, regardless of amounts of juice in each part. Therefore the *y*-coordinate is 3/2 of the *x*-coordinate, so *y *= (3/2)* x*.

*Figure 2:* A proportional relationship viewed from a variable-parts perspective.

*Figure 3:* Slope and equations from a variable-parts perspective.

We don’t think there is any way to make the concept of slope easy for students. But we suspect that working with both the multiple-batches and the variable-parts perspectives should help students develop a more robust understanding of slope. In particular, the variable-parts perspective might help students connect the slope of a line and an equation for the line. References [3] and [4] discuss the multiple-batches and variable-parts perspectives in greater detail.

We are currently conducting detailed studies of how students in our courses for future teachers reason from both the multiple-batches and the variable-parts perspectives on proportional relationships*. Discoveries about how future teachers reason about the interconnected ideas of multiplication, division, fractions, ratio, and proportional relationships, and what is easier and what is harder to learn, will help us identify productive targets for instruction in courses for future teachers. But we also hope that others will try the variable-parts perspective with other groups of students. For example, we could imagine a group of college algebra instructors collaboratively designing lessons that use a variable-parts perspective to help students better understand slope and its connection to equations for lines.

We also think that the variable-parts perspective is potentially productive for trigonometric ratios. From a variable-parts perspective, we can think of the radius of a circle as 1 part of variable size, *r*. For a fixed angle, its radian measure, sine, cosine, and tangent can all be viewed as a fixed number of parts (although this number is often irrational). With this perspective, equations such as *x* = cos(θ)*r* and *y* = sin(θ)*r* arise from the very same reasoning that connects slope to the equation of a line.

We think that there are many useful findings of mathematics education research that could help improve mathematics teaching and learning, but that environments and cultures are often not conducive to using the knowledge that we have. We need professional environments and cultures that foster serious discussions about what to teach and how to teach it, where knowledge about teaching and learning mathematics is intertwined with the practice of mathematics teaching, and where knowledge and practice grow together. We applaud the editors and the AMS for starting this blog as a way to nurture and develop such a culture.

[*] We are grateful to the University of Georgia, the Spencer Foundation, and the National Science Foundation, award number 1420307, for supporting our research.

**References**

[1] National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). *Common core state standards for mathematics*. Washington, DC: Author. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

[2] Common Core Standards Writing Team. (2011). *Progressions for the common core state standards for mathematics (draft), 6–7, ratios and proportional relationships.* Retrieved from http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

[3] Beckmann, S., & Izsák, A. (in press). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. *Journal for Research in Mathematics Education.*

[4] Beckmann, S., & Izsák, A. (2014). Variable parts: A new perspective on proportional relationships and linear functions. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.). (2014). Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 2). Vancouver, Canada: PME. http://www.igpme.org

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We thought our readers might be interested to know that nominations are now open for several American Mathematical Society awards related to teaching and learning. The deadline for nominations for the following awards is September 15, 2014.

- Award for Impact on the Teaching and Learning of Mathematics.

- Award for an Exemplary Program or Achievement in a Mathematics Department.

- Mathematics Programs that Make a Difference.

More information about these awards and the nomination process can be found here: http://www.ams.org/profession/prizes-awards/prizes