Recent calls to bring Computer Science to K-12 schools have reached a fever pitch. Groups like Code.org and Girls Who Code have become household names, having raised tens of millions in funding from Silicon Valley luminaries and small donors alike. In February of 2016, President Obama announced the “CSforAll” initiative, and asked for $4 billion of funding from Congress to pay for it. Even in today’s divided climate, this initiative found bipartisan support, and mayors and governors from coast to coast have made sweeping commitments to bring CS Education to all students.

This effort has serious consequences for math education. Adding a new subject is easier said than done: recruiting, training, hiring and retaining tens of thousands of new CS Teachers will take decades and cost billions, and the finite number of hours in the school day and rooms in the school building make it difficult to find space for these courses. To meet these commitments, many schools and districts have employed three strategies: (1) take time out of existing math classes for CS, (2) take math classes out of a teacher’s schedule, and instead have them teach a CS class, and/or (3) have CS classes count as a math credit [1]. All of this is done because there’s a widespread misconception that “computer science is just like math”, and that skills from one will transfer to the other. Unfortunately, most of the programming languages being taught in these classes have little to do with mathematics, and embrace concepts that are explicitly *math-hostile.* In this article, I will discuss some of the challenges and opportunities faced by K-12 mathematics educators in our efforts to develop an authentic incorporation of CS into the K-12 curriculum.

**Some Challenges**

A core example of a challenge facing math teachers is that numbers themselves behave differently in most programming languages. Math places no limit on how small or large a number can be, yet programming languages frequently truncate values without warning, leading to unpredictable results. Any 5th grader should know that 2 ÷ 4 equals ½… but in Java the teacher will have to explain why the same expression evaluates to zero!

Making matters worse, programming languages like Java, JavaScript, Python, Scratch and Alice all rely on the concept of *assignment*. Assignment means that a value is “stored in a box”, and that the value in that box can be changed. Here’s a simple JavaScript program that demonstrates this:

x = 10

x = x + 2

The first line of code *assigns* the value 10 into a box named “x”. The second line reads the value back out, adds 2, and assigns the new value back into x. When the program finishes, x contains the value 12. Unfortunately, the semantics and syntax are completely incompatible with mathematics! In math, names are given to *values*, not boxes. In fact, there’s no notion of “boxes” in algebra (or “assigning” values into them) at all! Moreover, the written syntax of “x= x + 2” translates to a statement that is mathematically wrong. Adding insult to injury, computer scientists refer to x as a *variable*, despite the fact that it behaves nothing like a variable in math. The problem is made even worse when it comes to *functions*. In most programming languages, functions can (and often do) fail the vertical line test, producing different values for the same input or perhaps *no *value for *any* input. Students typically struggle with the concepts of function and variable when they get to algebra. Now, they are confronted with incompatible definitions of the same terms – in a class taught by a math teacher, for math credit.

It should come as no surprise that there is little evidence supporting the proposition that programming leads to higher performance in critical classes like algebra. Asking math teachers to cut back on math to make room for programming is problematic in and of itself. When numbers, variables and functions behave contradictory ways, all in the context of a “math-credit class”, the problems are far greater.

**Some Opportunities**

While the risks of bad integration are significant, the opportunities for an *authentic* integration are tremendous. I would argue that an “authentic integration” between math and programming has three characteristics:

*Tools*– The language itself must include (and enforce) basic mathematical concepts like Numbers, Variables, and Functions. At the very least, we need to get our tools right (within reasonable limits).*Curriculum*– The curriculum offered alongside the tools must be aligned to national and/or state standards for*mathematics*, with a clear scope and sequence that addresses the needs of a mainstream math teacher. It should include homework assignments, rubrics, assessments, and handouts that address mathematical concepts. Demanding that a math teacher find the time to figure out the alignment and make these resources on their own is a non-starter.*Pedagogy*– There is more to great teaching than having a great curriculum. A CS course that aims to address math content must also address pedagogical techniques that matter in a math class. How is an activity differentiated? How is a concept scaffolded? How should student break down a word problem broken down? The answers to all of these questions (and more) must be explicit, and must also fit within recognized best-practices for math instruction.

I firmly believe there are ways to do it right, and there’s tremendous potential for teachers who are able to do so. Authentic alignment of mathematics and computer science requires significant time to develop materials and integrate them with existing math curricula, and significant intersectional experience between computer science, mathematics, math instruction, curriculum development, software engineering, and teacher professional development. And while there are almost certainly multiple pathways to get here, I can speak from experience about one of them.

I’m a former math teacher, math-ed researcher, and the co-director of an organization that has spent nearly a decade researching this challenge and developing evidence-based solutions. Bootstrap (http://www.BootstrapWorld.org) is a research project at Brown University that offers a series of curricular modules built around *purely mathematical programming*. Our introductory module is carefully aligned to standard algebra, and after a decade of research has been shown to significantly improve students’ performance on standard, pencil-and-paper algebra tasks (http://www.BootstrapWorld.org/impact). The win for students is twofold: they’re learning real algebra, and they’re doing it in a way that is 100% hands-on and applied. Bootstrap gives math teachers a chance to teach algebra in a new way, and to makes their experience teaching math an asset rather than a liability when it comes to teaching programming. By leveraging the experience math teachers already have, Bootstrap makes it possible for math teachers to deliver rigorous programming education without years of re-training. And since every student takes algebra, it allows schools to bring computer science to *every child *without having to find room in the budget for a new teacher or room in the schedule for a new class.

**Conclusion**

Computer Science is coming, most likely in a form that finds its way into math classes across the country. As members of the math-ed community, we have a responsibility to make sure this integration happens authentically, and in a way that supports math instruction instead of undermining it. Doing this takes careful attention to the tools we use, the curricula we teach, and the pedagogical techniques we employ. If we withdraw from this conversation, it will happen without us – and recent history shows that it is likely to happen in a way that risks harming math education. If we are active participants in the conversation, the enthusiasm and energy surrounding CS education bring enormous potential to math classrooms everywhere.

[1] – Kentucky counts CS as a math credit, Georgia counts CS as a math credit, Pennsylvania counts CS as a math credit

]]>During my freshman year of high school, my geometry teacher came into class one day and challenged us to trisect an angle with a compass and a straight edge. Anyone who was successful would receive an A in the class for the rest of the year. We wouldn’t have to do any more homework or take any more tests. Nothing. Of course, this should have seemed too good to be true. But I was in ninth grade and didn’t know any better, so I set off to solve this seemingly innocent problem.

I came up with a dozen or so false proofs, all of which included reasoning like “well, now you just move the compass a bit over here and then you draw this line, and it works!” Of course it didn’t work, but this is the kind of non-proof you would attempt to make if you had only just learned what a proof is.

But rather than simply tell me I was wrong and insist that I was doomed to failure, my teacher let me share the ideas behind every failed proof so that I could see the shortcomings in my arguments. He sat with me and we talked more broadly about what does and does not constitute a proof. He knew I was going to be wrong. He knew this was an impossible assignment. But he still listened.

My teacher’s openness to hearing my ideas inspired me to keep working and to keep trying new approaches. As I learned more math, I kept coming back to this problem. I tried using trigonometry. I tried using calculus. I tried making up a unit distance that I would call “1.” After watching *Good Will Hunting*, I decided that it would probably help if I drew all of my diagrams on mirrors. None of these things helped. Along the way, I learned about quantifiers. I learned about proofs. I learned to identify the errors in my attempted proofs on my own. Ultimately, I think I shed a tear of joy when I finally saw the proof of impossibility in my graduate algebra class.

This story can lead to a lot of different discussions. Ben Braun wrote a beautiful article for this blog about the value of having students work on difficult and unsolved problems, which I highly recommend. Instead, I’d like to explore the value of talking about mathematical ideas informally, especially when they are ill-formed and possibly incorrect; the value of encouraging our students to share such ideas with one another; and the value of participating in these discussions with our students.

The practices of active learning, inquiry based learning, project-oriented group learning, and others have become quite popular as means of addressing the fact that simply lecturing about math is not effective for students (Deslauriers 2011), (Freeman 2014), (Lew 2016). Encouraging mathematical communication is a byproduct of many of these methodologies, and I would like to start by arguing in favor of talking about math because there are added educational and cultural benefits to encouraging open discussion of mathematics both inside and outside the classroom.

*By discussing mathematics together, our students develop their own language and intuition for mathematical ideas.*As I circle around the classroom listening to my students as they work, I sometimes hear conversations that can best be described as the opposite of the game of “telephone.” One student tries to describe a solution in a way that, to me, is completely incoherent. But his groupmates kind of get the idea and someone else chimes in with a more coherent explanation. Then someone else adds more clarity. And by the end they have a solid basis upon which we can introduce more formal language and definitions. If I had simply interjected with a litany of corrections and edits after the first incoherent attempt, I would be robbing my students of the opportunity to learn by developing their own ideas. I think we would be fooling ourselves if we claimed that our private research moments and meetings with collaborators did not follow this similar “telephone-in-reverse” phenomenon.*Mathematical conversations encourage multiple ideas, multiple perspectives, and different solutions.*It seems fair to say that most of us, as educators, want our students to appreciate that a single problem may have many possible solutions. Through traditional lecture, we may only present one such solution, leaving students who had different approaches to wonder if their solutions are correct — or worse yet, to believe that their different ideas are wrong.By giving our students time to work together and talk together, we give them the time to learn from one another by discussing different solutions and approaches to the same problem. Students are more likely to ask the question “I did the problem in such-and-such different way. Does that still work?” in a small group or private conversation than they are in front of an entire lecture section.*By discussing our students’ ideas, we can provide more personal attention to their learning.*When we talk with our students we can very quickly assess the difference between someone who can complete all but the hardest problem on a homework set and someone who is struggling with the first problem. Talking about the first problem with the former student or talking about the hardest problem with the latter student is a waste of everyone’s time. By talking individually with our students, we can concentrate on what they need to learn based on what they already understand.

Most of all, we empower our students’ learning by giving value to the questions and mathematical ideas that are at the front of their minds, rather than by taking a scatter-shot approach and hoping that we address something that is meaningful to everyone at some point during each lecture.

William Thurston wrote an article “On Proof and Progress in Mathematics” for the Bulletin of the AMS (Thurston 1994) where he says:

Mathematicians have developed habits of communication that are dysfunctional…we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models.

He goes on to explore this idea further in an example. If Alice and Bob are researchers within a given subfield, Alice may be able to communicate the overall ideas behind a recent research development to Bob over coffee. But in contrast, Bob may struggle to glean similar insights from an hour-long colloquium talk or over the course of several hours of reading Alice’s paper. Thurston continues:

Why is there such a big expansion from the informal discussion to the talk to the paper? One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use gestures, they draw pictures and diagrams, they make sound effects and use body language. Communication is more likely to be two-way, so that people can concentrate on what needs the most attention.

In contrast, research talks and written papers require far more formalism, and they prevent the audience from interacting with the material in such a personal and intuitive way.

As professional mathematicians we have all experienced this. We have all sat through talks without understanding anything after the first five minutes. We have all read the same sentence in a paper 20 times without understanding its meaning. And we have all asked a question over coffee to find illumination in a well-phrased answer from a colleague, collaborator, or friend. So if this is the case when we, the so-called experts, are trying to learn new material, how then can it *not* be the case for our students as they are trying to learn mathematics?

In theory this argument may resonate with a lot of people, but implementing these ideas may seem difficult for any number of reasons. Here are a few concrete tips that can be implemented anywhere:

- Set aside 5 minutes of each class for your students to work on an example problem. This example can be as simple as “What is the derivative of \((3x+1)^2\)?” Have them compare their answer with their neighbors and give each other a high five if they agree. If you have more time, set aside more time and give the students more problems. An example the students work out together is far more valuable than another example you do on the board.
- Encourage students to attend your office hours, your TA’s office hours, and a campus math help center. Remind them about these resources every day. Be open and approachable. Your students are human beings, and many of them are interested in doing cool things. If you engage with them on a personal level, they will feel more comfortable in asking you math questions.
- Share your mathematical struggles with your students. One reason that many of us have been successful as mathematicians is that we are willing to keep working on a problem that seems impossible at first. But in our students’ eyes, we can seem to be omniscient solutions manuals who know how to solve every math problem. We need to strive to bridge this divide.
- Solicit student input in helping you present solutions to problems. Ask them to articulate why they did certain things, and develop diplomatic reactions to incorrect ideas. Rachel Levy posts some great suggestions for accomplishing this here.

And Mr. Pelzer, if you’re out there reading this — thanks for letting me share my ideas with you. Failing to trisect an angle sparked a lifetime of mathematical curiosity.

Deslauriers, L. et. al. “Improved Learning in Large-Enrollment Physics Class.” *Science* 332 (2011): 862-864.

Freeman, S., et. al. “Active Learning Increases Student Performance in Science, Engineering and Mathematics.” *Proceedings of the National Academies of Sciences* 111, no. 23 (2014): 8410-8415.

Lew, K., et. al. “Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey.” *Journal for Research in Mathematics Education* 47, no. 2 (2016): 162-198.

Thurston, W. “On proof and progress in mathematics.” *American Mathematical Society. Bulletin. New Series* 30, no. 2 (1994): 161-177.

What would you do if you discovered a popular approach to teaching inverse functions negatively affected student understanding of the underlying ideas? Would you continue to teach the problematic procedure or would you search for a better way to help students make sense of the mathematics?

A popular approach to finding the inverse of a function is to switch the \(x\) and \( y\) variables and solve for the \(y\) variable. The strategy of swapping variables is not grounded in mathematical operations and, we will argue, is nonsensical. Nevertheless, the procedure is so ingrained in textbooks and other curricula that many teachers accept it as mathematical truth without questioning is conceptual validity. As a result, students try to memorize the strategy but struggle to “accurately carry out mathematical procedures, understand why those procedures work, and know how they might be used and their results interpreted” (NCTM, 2009; Carlson & Oehrtman, 2005). As we will illustrate, this common process for finding the inverse of a function makes it *harder* for students to understand fundamental inverse function concepts.

**Foundational Ideas about Functions and Their Inverses**

A function \(f\) describes the relationship between two covarying quantities represented by variables \(x\) and \(y\). Without loss of generality, let \(x\) be the independent variable for the function \(f\) and \(y\) be the dependent variable for the function \(f\). The inverse function \(f^{-1}\) also describes the relationship between the quantities represented by variables \(x\) and \(y\) except \(y\) is designated as the independent variable for the function \(f^{-1}\) and \(x\) is designated as the dependent variable for the function \(f^{-1}\).

The following properties hold:

**Concept #1: **The domain of a function \(f\) is the range of its inverse function \(f^{-1}\) and the range of the function \(f\) is the domain of its inverse function \(f^{-1}\) (Wilson, 2007).

**Concept #2:** \(f^{-1}(f(x))=x\). In layman’s terms, the inverse function *undoes* whatever the function does (Bayazit & Gray, 2004).

These two concepts form the foundational ideas of the inverse function concept and hold true for functions represented in equations, graphs, tables or words.

**Problematic Conceptions Arising from the Switch x and y Approach to Finding Inverse Functions**

We define a conception as “problematic” if it describes an understanding that obscures connections to related ideas, introduces mathematical inconsistencies, and/or is likely to hinder students from developing powerful meanings of future topics. There are two problematic conceptions that emerge from the

**Problematic Conception #1:** The inverse of \(y=f(x)\) is \(y=f^{-1}(x)\).

In this statement, the independent variable for both \(f\) and \(f^{-1}\) is \(x\) and the dependent variable for both functions is \(y\). This problematic conception develops out of the procedure of switching \(x\) and \(y\) to find the inverse of a function, as illustrated in the following example.

Given \(f(x)=86x+15\), find \(f^{-1}\). \[\begin{align*} f(x) &= 86x +15\\ y &= 86x +15\quad \text{since}\ y= f(x)\\ x&=86y+15\quad \textbf{switch x and y}\\ x-15 &=86y\\ y &= \frac{x-15}{86}\\ f^{-1}(x) &= \frac{x-15}{86} \end{align*}\]

To some educators, calling this statement a *problematic conception* may seem like heresy. However, it is easy see the conceptual problem when the variables are assigned real world meanings.

In 2016 – 2017, tuition at the Maricopa Community Colleges was $\(86\) per credit hour. All students registering to take classes were also required to pay a $\(15\) registration fee. The function \(y=f(x)\) where \(f(x)=86x+15\) (introduced earlier) relates the number of credit hours, \(x\), to the total tuition cost (including the registration fee), \(y\). For clarity and emphasis, we change the variables in this equation to \(c\), for the number of credit hours assigned, and to \(t\), for the total tuition cost in dollars. The resultant equation is \(t=f(c)\) where \(f(c)=86c+15\). No matter what we do to mathematically manipulate this equation, the meaning of the variables \(t\) and \(c\) will remain unchanged. Suppose we are asked to calculate and interpret the meaning of \(f^{-1} (445)\). Using the *switch \(x\) and \(y\)* approach, we concluded earlier that \(y=f^{-1} (x)\) where \(f^{-1}(x)=\frac{x-15}{86}\). In terms of \(c\) and \(t\) this is \(t=f^{-1} (c)\) where \( f^{-1} (c)=\frac{c-15}{86}\). So \[\begin{align*}f^{-1}(445) &= \frac{445-15}{86}\\ f^{-1}(445) &= \frac{430}{86}\\ f^{-1}(445) &= 5 \end{align*}\]

What is the meaning of the result? Since \(c\) is credits and \(t\) is tuition cost in dollars, the result must mean that \(445\) credits cost $\(5\). This statement is false because credits cost $\(86\) per credit hour! To make sense of \(t=f^{-1}(c)\), we would have to change the meaning of the variables \(c\) and \(t\).

The confusion is easily remedied by applying an alternate strategy to finding the inverse. The strategy of *solve for the dependent variable* is demonstrated in the following example. As stated earlier, \(t\) represents the total tuition cost in dollars and \(c\) represents the number of credit hours assigned. For the inverse function \(f^{-1}\), \(c\) is the dependent variable so we solve the equation for \(c\).

Given \(f(c)=86c+15\), find \(f^{-1}\).

\[\begin{align*} f(c) &= 86c+15\\ t &= 86c+15\quad \text{since}\ t=f(c)\\ t-15 &= 86c\\ c &=\frac{t-15}{86}\\ f^{-1} (t) &= \frac{t-15}{86} \end{align*}\]

Note that \(t\) is the independent variable and \(c\) is the dependent variable for the inverse function. \( f^{-1} (445)=5\) implies that when \(t=445\), \(c=5\). In other words, when the total tuition cost (including registration) is $445, then 5 credits are purchased. This statement is true.

By referring to basic inverse function concepts, we can also detect the fallacy in the statement, “The inverse of \(y=f(x)\) is \(y=f^{-1}(x)\).” Let \(x\) be the independent variable and \(y\) be the dependent variable of a function \(f\). Then \(y=f(x)\) . We know \[\begin{align*}f^{-1} (f(x)) &= x\quad \text{Concept 2}\\ f^{-1} (y) &= x\quad \text{since}\ y=f(x)\end{align*}\]

Notice that the independent variable for the inverse function \(f^{-1}\) is \(y\) and the dependent variable is \(x\). So the inverse of \(y=f(x)\) is \(x=f^{-1}(y)\) not \(y=f^{-1}(x)\).

The tuition example represents a traditional exercise where students focus only on a memorized procedure. Carlson and Oehrtman warn that “this procedural approach to determining ‘an answer’ has little or no real meaning for the student unless he or she also possesses an understanding as to why the procedure works (2005).” The conceptual weakness of the problematic approach to finding the inverse becomes clearly evident with functions representing real world contexts.

Keeping track of the meaning of variables is essential when working with exponential and logarithmic functions. Understanding that \(y=b^x\) is equivalent to \(\log_b y = x\) is key to understanding logarithms conceptually. The *switch \(x \) and \(y\)* approach to finding inverses obscures the inverse relationship between exponential and logarithmic functions. For example, suppose that \(f(x)=3^x\). Find \(f^{-1}\).

\[\begin{align*} \textit{Switch x}\ & \textit{and y}\ \text{approach} & \textit{Solve for the}\ & \textit{dependent variable}\ \text{approach} \\ f(x) &= 3^x & f(x) &= 3^x\\ y &= 3^x & y &= 3^x\\ x& =3^y\quad \text{switch}\ x\ \text{and}\ y & \log_3 y &= x\\ \log_3 x &= y & f^{-1}(y) &= \log_3 y \\ f^{-1}(x) &= \log_3 x & & \\ \end{align*}\]

Using the *switch \(x\) and \(y\)* approach, it is common for students to conclude incorrectly that \(\log_3 x=3^x\) because of the statements \(\log_3 x=y\) and \(y=3^x\) included as part of the problem solving process. No such confusion exists when the *solve for the dependent variable* approach is used.

**Problematic Conception #2:** With the horizontal axis representing the independent variable and the vertical axis representing the dependent variable, the graphs of \(f\) and \(f^{-1}\) may be drawn on the same axes. The resultant graphs are symmetric about the line \(y=x\).

It is true that the graphs of \(y=f(x)\) and \(y=f^{-1} (x)\) are symmetric about the line \(y=x\) but, as established earlier, there are inherent issues with saying that \(y=f^{-1} (x)\) is the inverse function of \(y=f(x)\). The result \(y=f^{-1} (x)\) comes from *switching the \(x\) and \(y\)* variables in the inverse function. In fact, switching the variables in any mathematical relation will create a graph that is symmetric about the line \(y=x\). The practice of graphing \(f(x)\) and \(f^{-1}(x)\) on the same axes should be avoided (VanDyke, 1996) because it muddles the concept of inverse. Instead \(f(x)\) and \(f^{-1}(y)\) should be graphed on separate axes labeled appropriately with \(x\) or \(y\) on the horizontal axis.

The conceptual problems which occur when graphing \(f(x)\) and \(f^{-1}(x)\) on the same axes are evident when modeling even the simplest real-world context. The weekly earnings, \(y\), of an employee earning $\(10\) per hour who works \(x\) hours in a week is given by \(y=10x\). The independent variable for the function \(f\) is \(x\) and the dependent variable is \(y\). For the inverse function \(f^{-1}\), the dependent variable is \(x\) so we solve \(y=10x\) for \(x\) and get \(x=\frac{1}{10} y\). We have \(y=f(x)\) with \(f(x)=10x\) and \(x=f^{-1} (y)\) with \(f^{-1} (y)=\frac{1}{10} y\). If we switch the \(x\) and \(y\) variables in the inverse function equation, we get \(y=f^{-1} (x)\) with \(f^{-1} (x)=\frac{1}{10} x\) . We graph \(f(x)\) and \(f^{-1} (x)\) on the same axes and label the axes with the variables \(x\) and \(y\) as is customary. We include the units associated with the variables \(x\) and \(y\).

From the graph, we see that \(f^{-1} (20)=2\). The \(x\)-axis is labeled *hours worked weekly* and the \(y\) axis is labeled *weekly earnings (dollars)* so this must mean that when the employee works \(20\) hours the employee earns $\(2\). But this doesn’t make sense because we know the employee makes $\(10\) per hour! We could remove the labels from the axes, but this does not help someone understand a function’s graph as a visual representation of a relationship between two quantities and is likely to make it even harder to comprehend the meaning of a point on the graph. Graphing \(y=f(x)\) and \(y=f^{-1} (x)\) on the same axes created confusion and did nothing to help us understand inverse functions.

There are two equally viable strategies for representing functions and their inverses graphically. The first strategy is to graph each function on its own pair of coordinate axes with the horizontal axis representing the independent variable and the vertical axis representing the dependent variable of the function.

From the graph of \(f\), we determine that \(f(2)=20\) means that working 2 hours weekly results in weekly earnings of $\(20\). From the graph of \(f^{-1}\), we determine that \(f^{-1} (20)=2\) means that when weekly earnings were $\(20\) the number of hours worked was \(2\). Both results make sense in the real-world context.

The second strategy for graphing a function and its inverse comes from changing the way we think about graphs. With this approach, we use the same graph to represent a function and its inverse but designate the horizontal axis to represent the independent variable for \(f\) and the vertical axis to represent the independent variable for \(f^{-1}\) (Moore, Liss, Silverman, Paoletti, LaForest, & Musgrave, 2013). Observe that to determine \(f(2)\) we start at \(x=2\) on the horizontal axis and move vertically until we touch the graph of \(f\). We then move horizontally until we touch the vertical axis at \(y=20\). We conclude \(f(2)=20\). To determine \(f^{-1} (20)\) we start at \(y=20\) on the vertical axis and move horizontally until we touch the graph of \(f^{-1}\). We then move vertically until we touch the horizontal axis at \(x=2\). We conclude \(f^{-1} (20)=2\).

This way of thinking can be powerful for students who recognize the equation \(f(x)=30\) is equivalent to \(x=f^{-1} (30)\). The student finds \(30\) on the vertical axis and determines the corresponding value on the horizontal axis is \(3\). The student concludes that the solution to \(f(x)=30\) is \(x=3\) because \(f^{-1} (30)=3\).

Bayazit and Gray (2004) claim that learners with a conceptual understanding of inverse functions were able to deal with the inverse function concept in situations not involving formulas whereas learners limited by a procedural understanding of inverse functions (e.g. *switch \(x\) and \(y\)*) were less likely to be successful in a context without a formula.

A side benefit of discarding the *switch \(x\) and \(y\)* approach is that it frees learners from the \(x\)-addiction – the notion that only \(x\) can be the independent variable. In graphing, the \(x\)-axis becomes the *horizontal axis* and the \(y\)-axis becomes the *vertical axis*. The reality is that disciplines outside of mathematics rarely use \(x\) to represent the horizontal axis and \(y\) to represent the vertical axis. Rather, they use variable names (perhaps even complete words) that make sense in the context of the situation. Since, as we propose, the axes are no longer tied to \(x\) and \(y\), learners think more deeply about the concepts of independent and dependent variables when graphing real world data models such as \(p=f(t)\) where \(f(t)=298,213,000(1.009)^t\) and \(\textit{height}\ = f(\textit{time})\) where \(f(\textit{time})=-8.99 \cos(\frac{\pi}{6}\cdot \textit{time})+12.74.\)

When students understand the concept of inverse function in the context of a real world situation, they engage in reasoning (the process of drawing conclusions on the basis of evidence or stated assumptions (NCTM, 2009)) and sense making (developing understanding of a situation, context, or concept by connecting it with existing knowledge (NCTM, 2009)). This connects directly with the Standards for Mathematical Practices – specifically Math Practice #1 (make sense of problems and persevere in solving them) and Math Practice #2 (reason abstractly and quantitatively) (National Governors Association, 2010). The Mathematical Association of America encourages similar ways of thinking in their Committee on the Undergraduate Program in Mathematics Curriculum Guide (MAA, 2015). Cognitive Recommendation #1 states that *Students should develop effective thinking and communication skills*. All such connections help students understand and retain new information, something that is more challenging if students are not engaged in reasoning and sense making (Hiebert et al., 1997).

**Summary**

A correct understanding of inverse functions empowers learners mathematically. By eliminating the *switch \(x\) and \(y\)* approach and implementing the *solve for the dependent variable* approach, teachers can reduce confusion and enhance student understanding. By recognizing that the inverse of \(y=f(x)\) is \(x=f^{-1}(y)\), learners can make sense of inverse functions in multiple mathematical contexts including real world data analysis and modeling.

*Adapted from an article by the same authors, listed in the references below.*

**References**

Bayazit, I. and Gray, E. (2004, July). Understanding inverse functions: the relationship between teaching practice and student learning. *Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education: Vol. 2*. (pp. 103–110).

Carlson, M. & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Mathematical Association of America Research Sampler, No. 9, March 2005.

Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., Murray, H., et al. (1997). *Making sense: Teaching and learning mathematics with understanding*. Portsmouth, NH: Heinemann.

Mathematical Association of America (2015). *2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences*. Carol S. Schumacher and Martha J. Siegel, Co-Chairs, Paul Zorn Editor. Washington, DC: MAA

Moore, K. C., Liss II, D. R., Silverman, J., Paoletti, T, Laforest, K. R., and Musgrave, S. (2013). Pre-Service Teachers’ Meanings and Non-Canonical Graphs. In Martinez, M. & Castro Superfine, A. (Eds.), *Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (pp. 441-448). Chicago, IL: University of Illinois at Chicago.

National Council of Teachers of Mathematics (2009). *Focus in High School Mathematics: Reasoning and Sense Making*. Reston, VA: NCTM

National Governors Association Center for Best Practices, Council of Chief State School Officers (2010). *Common Core State Standards – Mathematics*. National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C.

United States Census Bureau. (2006). Table 96. *Expectation of Life at Birth, 1970 to 2003, and Projections, 2005 and 2010*. (NTIS No. PB2006500023)

Van Dyke, F. (February 1996). The inverse of a function. *Mathematics Teacher*. 89, pp. 121 – 126.

Wilson, F. (2007). Finite mathematics and applied calculus. Boston: Houghton Mifflin Company.

Wilson, F.C., Adamson, S., Cox, T., and O’Bryan, A. (March 2011). Inverse functions: What our teachers didn’t tell us. *Mathematics Teacher*. 104, pp. 500-507.

World Health Organization. (2006). *World Health Statistics 2006*. WHO Press. Geneva, Switzerland.

This article is intended to serve as a rough “proof” of the statement, “There exist many resources and opportunities supported by the National Science Foundation (NSF) to improve the teaching and learning of undergraduate mathematics.” We present a curated, annotated list of projects funded by the NSF’s Division of Undergraduate Education (DUE) that readers of this blog might be interested in. Additionally, we demonstrate the remarkable diversity of projects and institutions that are funded by DUE to improve the teaching and learning of mathematics, and share professional opportunities for people who share these goals.

The NSF is an independent federal agency tasked by the United States Congress to “promote the progress of science.” With a budget of 7.5 billion dollars in fiscal year 2016, NSF received approximately 50,000 proposals and made almost 12,000 awards. NSF is organized into seven Directorates that support research in various disciplines in science, technology, engineering and mathematics (STEM) as well as in education. Each of the Directorates is further organized into Divisions. For example, the Division of Mathematical Sciences (DMS) is situated in the Directorate for Mathematical and Physical Sciences (MPS). The Directorate for Education and Human Resources (EHR) houses DUE, which manages the awards that are the primary focus of this article.

DUE’s current signature program is Improving Undergraduate STEM Education (IUSE). IUSE is the latest incarnation of DUE’s programmatic efforts to actualize its mission “to promote excellence in undergraduate STEM education for all students.” Former DUE programs include “Transforming Undergraduate Education in STEM” (TUES) and “Course, Curriculum and Laboratory Improvement” (CCLI). The current IUSE solicitation is 15-585, and the next deadline for full proposals is January 11, 2017.

In the next section, we will describe (primarily using excerpts from publicly available abstracts) several active IUSE awards that illustrate variation in topics, institutions, budget size, grant duration, and project type supported by NSF.

*Progress Through Calculus***(1430540****), $2,250,003, PI David Bressoud, Mathematical Association of America. **

This project is a follow-up to the project *Characteristics of Successful Programs in College Calculus* (DRL-0910240) which “undertook a national survey of Calculus instruction and conducted multi-day case study visits to 20 colleges and universities with interesting and, in most cases, successful calculus programs.” ** Progress Through Calculus** has two focal areas of research guided by the following questions: (1) What are the programs and structures of the pre-calculus to calculus sequence as currently implemented? How common are the various programs and structures? How varied are they in practice? What kinds of changes have recently been undertaken or are currently underway? (2) What are the effects of structural, curricular, and pedagogical decisions on student success in pre-calculus to calculus? Success will be assessed on a variety of measures including longitudinal measures of persistence and retention, performance in subsequent courses, knowledge of both pre-calculus and calculus topics, and student attitudes.

** Collaborative Research: Data-Driven Applications Inspiring Upper-Division Mathematics (**

This is a collaborative project (which consists of linked awards at multiple institutions) involving investigators at , Hendrix College, Kenyon College, and Lewis Clark State College. The goals are to (1) introduce current cutting-edge research and practical data problems from science, industry, and government to students in undergraduate upper-division mathematics courses and (2) lead these students to develop the problem-solving, collaborative, and research skills that are so crucial in today’s work environment. The focus of this project is to create a body of applied data-driven instructional modules to motivate student research as well as to generate a deeper understanding and appreciation of the mathematical theory needed to solve these problems.

*Collaborative Research: Improving Conceptual Understanding of Multivariable Calculus Through Visualization Using CalcPlot3D ***(****1524968****), $456,993.00. PI Paul Seeburger, Monroe Community College. **

Three investigators at a community college (Monroe Community College), a public 4-year college (State University of New York at Buffalo), and a private 4-year college (Robert Morris University) are collaborating with faculty across the United States and Mexico to: (1) design and test a series of new visual concept explorations and applications in CalcPlot3D to improve student understanding of multivariable calculus; (2) expand the features of CalcPlot3D to accommodate the new concept explorations and address applications in differential equations, linear algebra, physics, and engineering; (3) create new visualization apps, including a new version of CalcPlot3D, that work on more platforms, including tablets and phones; (4) conduct and publish research investigating how student understanding of multivariable calculus concepts changes through the use of visualization and dynamic concept explorations; and (5) extend and diversify the user base by disseminating project materials through papers, workshops and conferences, by creating a Spanish language version of project materials, and by promoting the exchange of user feedback and research.

*Collaborative Research: Professional Development and Uptake through Collaborative Teams (PRODUCT): Supporting Inquiry Based Learning in Undergraduate Mathematics***, (****1525058****), $2,842,393. PI Stan Yoshinobu, California Polytechnic State University at San Luis Obispo. **

This collaborative project between California Polytechnic State University at San Luis Obispo and University of Colorado Boulder intends to greatly expand the capacity of faculty to implement the specific active learning strategy of inquiry-based learning (IBL). PRODUCT will conduct 12 four-day intensive IBL workshops, as well as 15 short workshops and five Professional Development (PD) Preparatory Meetings, and will host a PD Summit for mathematics faculty developers. Through these activities, PRODUCT will directly provide professional development for 320 undergraduate mathematics faculty, adapt and improve IBL PD materials, develop multiple new teams of faculty developers who will be prepared to engage additional faculty in the future, and develop a framework for building professional development capacity. A research-with-evaluation study will provide formative feedback, study the process and outcomes for development of the professional development teams, gather data to benchmark workshops led by new teams against a model known to be effective, and investigate the classroom practices of workshop participants to understand how the professional development experience shapes their teaching.

*MATH: CONFERENCE: Active Learning Approaches in Mathematics Instruction: Practice and Assessment Workshop (***1544374****) , $25,000, PI Ron Douglas, Texas A&M University.**

This project is an example of a workshop award (which are typically less than $50,000 and are submitted at any time after communicating directly with a program officer). 1544374 supports the implementation of a workshop entitled Active Learning in Mathematics Instruction that was held in conjunction with the Mathematical Association of America’s 2016 Mathfest conference. The workshop was designed to survey and investigate the characteristics, challenges, and evaluation of active learning approaches to collegiate mathematics instruction and to expand the community of individuals who are knowledgeable about both the methods and important questions involving active learning. Participants in the workshop included experts in education and social science research methods as well as active learning mathematics practitioners and departmental leaders.

*MATH: EAGER: Developing a Learning Map for Introductory Statistics (***1544481***)***, $299,832. PI Angela Broaddus, University of Kansas. **

This award is an example of a special funding mechanism at NSF called EAGER (Early-concept Grants for Exploratory Research) that is intended to support potentially transformative research that is considered “high risk high payoff.” The goals of 1544481 are to create and validate a “learning map” (Stat-LM) for the content of undergraduate introductory statistics. This learning map will be a graphical representation of statistics concepts with connections among the concepts suggesting effective learning sequences. Use of Stat-LM is intended to improve undergraduate learning by providing diagnostic information to instructors about students in their statistics courses, informing professional development for undergraduate statistics instructors, and modeling how critical prerequisites taught in high school connect to postsecondary learning expectations.

*Assessing the Impact of the Emporium Model on Student Persistence and Dispositional Learning by Transforming Faculty Culture (***1610482***), ***$299,999, PI Kathy Cousins-Cooper, North Carolina Agricultural & Technical University.**

This is an example of a project co-funded between two programs found in two divisions of EHR. 1610482 was submitted to the IUSE program in EHR/DUE but also received funds from the Historically Black Colleges and Universities – Undergraduate Program (HBCU-UP) in EHR/HRD (Division of Human Resources Development). The investigators will employ, study, and assess an instructional and student learning model, called the Mathematics Emporium Model (MEM), to improve students’ performance in introductory mathematics courses. These gatekeeper courses are normally taken during an intense and often difficult transition for students, from high school to college. The MEM eliminates lecture and uses commercially available interactive computer software combined with personalized on-demand assistance and mandatory student participation. The underlying principle of the Emporium Model is that students learn by doing. Research reveals that the shift to student-centered instructional practices enhances students’ attitudes and beliefs about learning in mathematics courses and increases student-learning gains. The project will directly reach a combined annual enrollment in traditionally low-pass-rate courses of more than 4,000 students, who will be mostly from underrepresented groups.

*Professional Development Emphasizing Data-Centered Resources and Pedagogies for Instructors of Undergraduate Introductory Statistics*** (StatPREP)**

This project responds to a recommendation found in *A Common Vision for Undergraduate Mathematical Sciences Programs in 2025*, a report funded by an award from EHR/DUE (1446000) and issued jointly by the American Mathematical Association of Two-Year Colleges (AMATYC), American Mathematical Society (AMS), American Statistical Association (ASA), Mathematical Association of America (MAA), and Society for Industrial and Applied Mathematics (SIAM). Specifically, StatPREP will catalyze the widespread use of data-centered methods and pedagogies in undergraduate introductory statistics courses. It will work directly with 240 college-level instructors by (1) offering an extended professional development program for mathematics instructors, particularly at two-year institutions, who teach introductory statistics; (2) establishing regional communities of practice to support instructors who teach introductory statistics; and (3) establishing a national online support network comprising instructors who teach introductory statistics and statistics education experts.

The above awards are just a fraction of the dozens of active awards we manage in EHR/DUE. We are excited about the work of the DUE-funded projects and their impact on improving the future of teaching and learning in undergraduate mathematics. We encourage readers to examine the full list of EHR/DUE active awards in undergraduate mathematics and mathematics education. If readers of this blog have ideas or suggestions for proposed activities that could improve the teaching and learning of mathematics, feel free to contact us.

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I think that mathematics draws in some people and repels others in large part because of the distinctive role of authority in our discipline and teaching, especially when we act as content experts and discussion leaders in the classroom. For instance, consider the following phrases from students, distilled from my interactions with college students over the past 15 years.

I’m not a math person. I learn best when you show me a bunch of examples and then I practice them. It’s true, so why do I have to prove it? That’s just how my last teacher told me to do it. I always liked math because there was one right answer. I just want to teach high school; why do I have to learn this? Wait, what – you want me to ask my own question!? Do I have to simplify my fractions? Well, that’s what the computer said was the answer. The test was unfair because it had problems we didn’t discuss in class. ~silence~

I expect that these comments are also familiar and painful to the reader. I think that each of these comments is in part a symptom of ways students have internalized a relationship with authority from our teaching. In this post, I will illuminate the role of authority in mathematics teaching, argue that taking a more overt stance toward it can better support both the students we repel and the ones we attract, and offer a handful of strategies for taking such a stance.

Depending on whom you ask, the truth of a mathematical conclusion can stand independently of a human authority or based entirely on the word of an authority. Mathematicians will often claim (e.g., [8]) that we depend only on proof to develop reliable knowledge and will dismiss student efforts to use empirical evidence or the word of an expert, but researchers [13] are showing that mathematicians use these strategies as well. I certainly agree that deductive reasoning occupies a special place in our discipline, but the absence of methods, evidence, and theoretical frameworks in the discussion of mathematics quietly places the math itself in the position of perfect (Platonic) authority. I do not take issue with this perspective except when our silence about authority leaves students to grapple with it alone, often painfully. This post will focus on these implications for our students.

**Models of Student Development**

My thinking about authority begins with Williams Perry’s work from the 1950s and 60s on the epistemological development of college students [14]. In broad strokes, Perry’s scheme describes a sequence of positions from which the students he studied viewed truth or knowledge; I will use a condensed version of this scheme with three main positions. From *Dualism*, students view knowledge as binary, and Authorities know the difference between true and false. From this position, students equate learning with memorizing any information that these Authorities pass on to them. From *Multiplicity*, students notice that much of knowledge is context-dependent and come to believe that any perspective is a valid source of knowledge. As a result, while in this position, students become less interested in the perspectives of others, including authority figures, instead focusing on their own authority. From *Relativism*, students begin to demand that other perspectives be justified; as a result, knowledge becomes the result of argument and evidence. In this position, students initially focus on learning HOW authorities want them to think, rather than WHAT. Eventually, these students acknowledge that their arguments must be grounded on accepted assumptions, and they become concerned with establishing appropriate and personal precepts. Common elements of college programs, such as first-year writing seminars and introduction-to-proof courses, encourage our students to adopt a relativistic position in their collegiate work by helping them practice appropriate modes of argumentation. For the purposes of this post, I will group students as though they have a predominant or preferred position.

My dualist students often find enticing or comforting the idea that mathematics is a field in which truth is absolute and completely known because they feel that it allows them to avoid pesky ambiguity, especially in their course assignments. These same students regularly say dismissive things about other disciplines to me, most commonly about literature courses. They seem to believe that the work of engaging literature reduces to forming a personal opinion and that all opinions are equally valid; they will often go further to rail against grading in these courses because they see it as relying on whether their valid opinion happens to match the one held by the teacher. To be clear, these are deleterious conceptions of both disciplines; I think that one of the most important goals of a college (liberal arts) education is helping students change these stereotyped views of disciplines. While these dualist students sometimes come to college liking math, they have been set up for a painful bait-and-switch when their math work shifts suddenly from execution of provided algorithms to generation of original arguments, leading to their comments about the good old days of Calculus. Perry saw evidence in his data of students retreating to earlier positions when facing difficulty, which can explain student resistance in their first proof courses, not to mention the first time they are asked to do something original and creative in math. I’m not surprised that many of these students consider switching to engineering or economics because they believe the math used there is aligned with their earlier perceptions of mathematics.

The other large group of incoming students, those for whom multiplicity is the preferred epistemological position, curiously say essentially the same things about mathematics and literature to me, with exactly opposite emotional value attached to the descriptions. They seem to hate that math has no room for their individual perspective and prefer discussion-oriented courses in the humanities and social sciences because they offer spaces in which they are encouraged to consider their own perspective. The departure of this group from studying mathematics is one of the leaky joints in the mathematics pipeline discussed by CSPCC [2] and even the Obama administration [7]. Perry’s data suggested that some students in this position aligned with authority figures and others against and that this alignment impacted their trajectory. I think that the importance of this split path can be understood by considering research that responds to one of the major critiques of Perry’s work. The population on which he based his scheme, namely students at Harvard at the time, was overwhelmingly male and non-representative of today’s college students in many other ways.

Belenky, Clinchy, and Goldberger, using a population of female interviewees whose ages varied widely, developed a scheme called “Women’s Ways of Knowing” [1]. These two schemes share many common features, but WWK illuminates two important observations. First, some of the women in this study talked about ending educational experiences because they perceived educational environments as placing no value on their voices and offering no pathways to expertise. In contrast, it seems important that the students in Perry’s population were male and enrolled at an elite college in a time when going to college was a rarer and more intentional choice than today; in other words, Perry’s population was likely selected to contain participants who already envisioned themselves as future authorities, an attribute that may help a person persist in (collegiate mathematics) courses when they experience friction between their epistemology and their course work. Combining this with the under-representation of women in our discipline, it’s hardly surprising that female students switch out of math and STEM more than male students [6]; they must endure years of study in an environment that seems not to value any student voices, and when students begin to find their voices, their male peers may have more support for identifying as future authorities themselves. Second, the WWK scheme also adds an important, fifth position (before the others) called Silence, from which interviewees experienced a world in which they had no access to knowledge or truth. I think the multiplicitous and silent students are driven from math similarly, though the “myth of genius”, which correlates to heavily with gender representation in STEM fields [10], might be a silencing factor well before college.

There is a third, much smaller group of students who view mathematics as relativists. I believe I was in this group; I liked how I could avoid memorizing almost anything in math because I knew that everything could be derived from the definitions. Importantly, this perspective seemed to maintain but reframe the dualist and multiplicitous things I liked about math. Yes, our knowledge and truth seemed to be of a different sort than in other disciplines, but this characteristic came from the kinds of arguments that are possible about abstract objects, not by declaration. And yes, there wasn’t much room to disagree with a theorem, but I felt empowered to make my own arguments given our egalitarian access to definitions, to try to extend algorithms or make them more efficient, and to have my own personal way of thinking about problems. I conjecture that most of us who have persisted in mathematics made similar transformations of our love for math that helped us persist and that many who did not persist did not make. Significantly, I believe that math classrooms can be set up to be welcoming to students in all of these positions while also helping them move forward.

**Using these Models to Inform Pedagogy**

Fortunately, Perry’s research also illuminates at least three kinds of experiences that impel students to move to later positions: encountering questions without known answers or about which Authorities disagree, engaging a pluralism of ideas among peers, and rigorously justifying claims and questioning assumptions. To me, this feels like a description of a classroom organized by inquiry. I further claim that mathematics makes it particularly easy to include these experiences in our classrooms: each new definition gives us instant access to an unlimited supply of perfect copies of objects into which we can explore, often very quickly and cheaply, while entertaining open conjectures that require justification. In contrast, work in the natural and social sciences seems to require data collection that can be slow and expensive, work in the arts and humanities can require nuanced analysis that uses a holistic perspective that is not immediately accessible, and work in applied fields often interacts with a great deal of information from its context. I’m not suggesting that these aspects of other disciplines are negative; in fact, I think each discipline has opportunities to highlight different facets of authority and epistemology more easily and that contrasting these themes across disciplines is a key mechanism for supporting student growth. I am saying that I see a way that our disciplinary context allows me to put the epistemological work of education front and center in the math classroom, in contrast to placing it in an invisible role.

Building a classroom that supports these epistemological themes can be challenging because it requires course materials and teaching skills that are consistent with this goal. A complete discussion of such course materials and general teaching skills is beyond the scope of this post, but I would suggest the reader start with the writings coming from these two threads [17,4]. Instead, I would like to offer two suggestions for strategies that engage authority in your classrooms overtly but don’t require an immediate reimagining of your teaching practice. First, I suggest that you read and discuss some mathematics education literature with your students — here are some of my favorites. Harel and Sowder [9] describe categories of students’ perspectives on mathematical justification called “proof schemes”; these categories align extremely well with Perry’s described above, and discussing this paper (or another that uses it) can help students understand professors’ expectations in advanced mathematics courses. Weber and Alcock [18] discuss student and expert behaviors when validating proofs; this paper always leads to discussion of subtle but important points including the role of communal expectations for proofs. If your students don’t yet have experience with proof, then articles or videos about Carol Dweck’s work on “growth mindsets” [15,5] or Paul Lockhart’s provocative (and informal) “A Mathematician’s Lament” [3] are both strong choices; these papers are not overtly about authority and epistemology, but they help students talk about the discipline in a way that allows an instructor to engage their perspectives. And while it’s not about mathematics, I also enjoy discussing summaries of Perry’s scheme with students in many courses, most commonly using Chapter 1 of this reference [11]. You may worry that these activities would take away time from other learning objectives, but the opposite has been my experience; through these reflection opportunities, students are able to see course activities in a new light that deepens understanding of the past work and makes future work more effective and efficient.

Second, I suggest that you humanize the mathematical practices in your classroom. I think that small changes in our language can help students adapt a more productive stance toward authority. Using “we” and “us” communicates that the people in the room are engaged in mathematics and have some local authority. Tagging ideas, questions, and conjectures with student’s names when we reference them highlights the fact that individuals are impacting the development of the mathematics; asking students to use their peers’ names suggests that these people have perspectives that we will consider. This language makes space for the psychological and social aspects of our work that we may crush with our silent authority otherwise. In addition to investing students with some authority, I think it is important to humanize ourselves to resist becoming the abstract authority. As a first step, we can talk about moments inside and outside of the course during which we are or were uncertain mathematically. However, I think that telling students that they can only know mathematical things about us as people communicates a preference to be seen as a distant authority. Personally, I feel an ethical obligation to go further and be a multifaceted person with my students. For example, I end up coming out in some of my courses because doing otherwise feels hypocritical given the level of openness I’ve asked of students about their lives and minds [12]; I also appreciate that they are now aware of at least one queer mathematician as a potential role model.

**Conclusion**

I would like to reiterate the importance of mathematics educators taking a more overt stance toward authority. I think we are compelled to use the distinctive interaction of mathematics with authority to help students mature generally in their lives, and I think the lenses discussed here help me see opportunities for this work. For example, Perry noticed changes in the most common starting position of first-year students in the 1960s, which he connected to the marked changes in the national dialog and the role of Authority therein. Analogously, I believe that these schemes are going to be key tools for understanding the waning public faith in higher education and for responding to the changing needs, perspectives, and skills of students entering college during the next decade, especially regarding the authority of hands-on parents and of information technology. In the context of our classrooms, I have argued above that when we vest all authority with mathematics in the abstract, we seem to create a vacuum that leads students to engage with us as Authorities instead, in ways that attract some students while setting them up for struggle and make other students feel unwelcome. I have made some conjectures about how this might perpetuate the under-representation of women in mathematics, and I would make similar connections and conjectures about under-representation of people of color (e.g, [16]). In summary, I think that being intentional and overt about the role of authority in our teaching can transform our classrooms into more inclusive and equitable spaces, and I hope you feel empowered to use the ideas in this post to do this work.

**References:**

[1] Belenky, M. F. (1986). *Women’s ways of knowing: The development of self, voice, and mind*. Basic books.

[2] Characteristics of Successful Propgrams in College Calculus: Publication & Reports – Retreived from http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus/cspcc-publications

[3] Devlin, K. (2008, March) *Lockhart’s Lament*. Retreived from https://www.maa.org/external_archive/devlin/devlin_03_08.html

[4] Discovering the Art of Mathematics – Retreived from https://www.artofmathematics.org/

[5] Dweck, C. (2014, November). *The Power of Believing that You Can Improve (TED talk)*. Retreived from http://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve

[6] Ellis, J., Fosdick, B.K., and Rasmussen, C. (2016). *Women 1.5 times more likely to leave STEM pipeline after calculus compared to men: Lack of mathematical confidence a potential culprit*. PLoS ONE 11(7): e0157447. doi10.1371/journal.pone.0157447

[7] Feder, M. (2012, December 18) *One Decade, One Million more STEM Graduates (based on a statement by the President’s Council of Advisors on Science and Technology)*. Retreived from https://www.whitehouse.gov/blog/2012/12/18/one-decade-one-million-more-stem-graduates

[8] Fischbein, E. (1982). Intuition and proof. *For the learning of mathematics*, *3*(2), 9-24.

[9] Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. *Research in collegiate mathematics education III*, 234-283.

[10] Leslie, S. J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. *Science*, *347*(6219), 262-265.

[11] Love, P. G., & Guthrie, V. L. (1999). *Understanding and Applying Cognitive Development Theory: New Directions for Student Services, Number 88* (Vol. 27). John Wiley & Sons. (Chapter 1)

[12] Lundquist, J. & Misra, A. (2016, October 18) *Establishing Rapport in the Classroom*. Retreived from https://www.insidehighered.com/advice/2016/10/18/how-engage-students-classroom-essay

[13] Mejia-Ramos, J. P., & Weber, K. (2014). Why and how mathematicians read proofs: Further evidence from a survey study. *Educational Studies in Mathematics, 85*(2), 161-173.

[14] Perry Jr, W. G. (1999). *Forms of Intellectual and Ethical Development in the College Years: A Scheme. Jossey-Bass Higher and Adult Education Series*. Jossey-Bass Publishers, 350 Sansome St., San Francisco, CA 94104.

[15] Popova, M. (2014, January 29) *Fixed vs. Growth: Two Basic Mindset that Shape Our Lives*. Retreived from https://www.brainpickings.org/2014/01/29/carol-dweck-mindset/

[16] Robinson, M. (2011, Spring). *Student Development Theory Overview*. Retrieved from https://studentdevelopmenttheory.wordpress.com/racial-identity-development/

[17] Teaching Inquiry-Oriented Mathematics: Establishing Supports – Retreived from http://times.math.vt.edu/

[18] Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. *For the Learning of Mathematics*, *25*(1), 34-51.

Given five minutes, can you turn to the person next to you and describe your research? How about over 15 minutes in front of a class of 10th graders? Thinking of one of your research graduate students, how would you prepare her/him to make such an activity equally beneficial for her/him and the 10th graders? For many of us, these are skills only nurtured through conference talks and time within the profession. The __SF-State (CM)²: San Francisco State University Creating Momentum through Communicating Mathematics program__ worked to change this, creating a program that developed mathematics graduate students who could have this conversation and were better engaged in why they were studying mathematics and what role they wanted to have in the future of our profession. As an NSF GK-12 program, (CM)^{2} ran from 2009-2014, working with master’s level students in mathematics to engage them in mathematical discourse while also supporting their research and professional development. Over the course of the five-year program, a total of 43 SF State mathematics graduate students were involved in the project, spending considerable time and energy on K-12 activities in 13 schools in the greater San Francisco area. A key goal was to strengthen the graduate students’ communication, teaching, outreach, and teamwork skills by immersing them in mathematics classrooms and the San Francisco Math Circle. A second key goal was to make mathematics, especially algebra, and its career connections more relevant and explicit for 6-12th grade teachers and students. This post will share successes, lessons learned, and resources for you as a faculty member to build aspects of outreach, teaching, and professional development programs for your own students.

(CM)^{2} provided funding support for nine Ph.D.-bound graduate students per year, with the understanding that these students (the “Fellows”) would work 10 hours per week in an education-related environment. A lot of work went into preparing these students for that experience, with a year-long schedule starting with bi-weekly summer workshops focused on research and applying for Ph.D. programs, culminating in a two-week intensive training to prepare graduate students for the classroom. We reinforced the lessons from the summer throughout the year with weekly graduate student meetings and larger monthly meetings of graduate students with their partner K-12 teachers. Our goal was to ensure that the graduate students spent the summer focused on making research progress prior to the intensive school-year schedule. The partner teachers attended the second half of the two-week training, meeting their mentor students and starting to create lessons and schedules for the upcoming year.

At the first workshop, (CM)^{2} Fellows were asked to turn and explain their research to the student sitting next to them. Their goal was to prepare a lesson for middle/high school age students to help them learn about the graduate student’s research, and the first step in this process was to describe their work to a colleague at the same academic level. This was a struggle, as everything you might expect happened with the biggest issue being Fellows’ use of precise language specific to their area of research. They were so focused on showing how much they knew about their field that they failed to concentrate on communicating their work in context. When first presented with the exercise, the Fellows struggled, but with practice their explanations were more prepared for a K-12 classroom. From there we scaffolded the progression by having the graduate students explain their work to their mentor teachers, whom they would be spending the majority of their classroom work time with. This mentor team then worked for several workshop days to create a 15-minute sample lesson to share with the group of 9 mentorship teams. With feedback from the other graduate student and teachers, they worked over the next year to create grade-level appropriate mathematics activities that were related to their academic research. Over the year, the Fellows further developed these into 5-10 page lesson plans with an introduction to the mathematical content, an overview of the lesson structure and directions for implementation, worksheets, and a summary of how the lesson went, including recommendations for the future. These mathematics lessons were developed, shared between partnership teams, and papers to accompany them were posted online; you can still see many of these lessons at __http://math.sfsu.edu/cm2/materials.php__. Development of the lesson into such a report furthered the Fellows’ mathematical communications skills with writing practice.

Creating this lesson plan wasn’t the only aspect of the (CM)^{2 }program. Once classes started, the graduate students quickly fell into a routine of weekly commitments, which included 10 hours of GK-12 student interaction (in class and at Math Circle), a one-hour GK-12 seminar, their own coursework and research, and continued work on their Ph.D. applications. Several of the fellows completed NSF graduate research fellowship proposal (GRFP) applications while participating in the fellowship program.

__San Francisco Math Circle (SFMC) __was key to the success of the (CM)^{2 }program, providing the ideal stepping stone for graduate students to get involved with mathematics activities for middle and high school students. The program had been set up so that the graduate students worked in some of the schools we were using for our satellite programs (including Mission High School, Thurgood Marshall Academic High School, Lowell High School, and June Jordan School for Equity). While in the classroom, the graduate students created a mentorship relationship with the younger students whom they encouraged individually to attend Math Circle. The graduate students then served as instructors at the high-school Math Circle programs. We believe this improved the quality of instruction at the high-schools programs because the graduate students had the opportunity to incorporate more of their research into their Math Circle presentations and were better able to moderate their presentations for the correct level of students based on their experience in the classrooms of the students they were working with. In addition, the (CM)^{2 }program sponsored nine teachers who were involved in the Math Circle. Overall, partnering with the (CM)^{2} program helped SFMC address not only the challenges in maintaining the diversity of the students in the Circle, but also that of increasing the number of qualified teachers willing to participate in a Circle with the students.

The dichotomy of our Fellows’ work, including both in-class and out-of-school Math Circle experiences, provided an informed balance that helped them become better mathematicians and teachers. The in-class work provided a model for how to work with students of this age, giving reinforcement to expectations of behavior and any aspects of student discipline. Here are three sample testimonies from Fellows that show the diversity of experiences and highlights this program provided for the participating graduate students:

“I’m serious about becoming an educator. Readings and discussions in 728 were terrific, and I learned more about teaching from working with […] than I did last year teaching on my own. My work at […] and SFMC were more valuable to me than the paychecks..”

“(CM)² has also enhanced my preparation to enter the Ph.D. programs. Being in this program has increased my confidence and excitement to start my Ph.D. program through the support from Matt and Brandy as well as the teachers and other fellows. In addition, the financial support has opened up opportunities by removing financial barriers […] As an underrepresented minority, this program has given me so much in terms of academic support, encouragements, and resources. Through (CM)², I have become more aware of the low numbers of underrepresented minorities in higher education and I’m ecstatic to be a role model to others.”

“In all honesty, I was not interested in working in the high school or in math circle to begin with but obviously took this fellowship for financial and moral support. Now, I’m looking sadly at the end of my time working at […] High and with […] and […] and the other teachers. I’ve learned so much, even if I never teach again, well, I may start up my own math circle someday, but even if I don’t, it has been a very personally rewarding experience.”

The out-of-school work provided creative outlets for the graduate students to discuss their research and higher level mathematics that didn’t need to be connected to curriculum goals or other K-12 benchmarks. We also saw success reflected in comments we received from partner teachers:

“Having another person in the room with deep math knowledge is good for the students and definitely helps with the amount of material covered. As to my pedagogy, I’m sure it has had an effect but I haven’t tried to define what it is. It has caused me to think more about what engages the students. I tend to push moving through the concepts, but now I am thinking more about ways to get the students involved. I don’t believe this necessarily has to be a show of how the concepts are relevant to the students lives. Coming up with intriguing problems where the students don’t feel shut out and turn off because of prior assumptions of them having a particular skill set seems to work well. Just having someone for me to interact with intellectually has been good, and I think it will help in improving course content.”

“Having another pair of knowledgeable eyes in the classroom has pushed me to think about and to be able to articulate what my motives and intentions are for each lesson. […] is always thinking beyond the math concept to its implementation and I have benefited from his sharing his insights. Also, the students love it when we come up with different interpretations on a topic.”

Our GK-12 program provided us with an invaluable opportunity to create a community of scholars, from elementary students to university faculty — vertical integration at its best. Working with our graduate students, their partner teachers, and their K-12 students was both great fun and an interesting challenge; everyone could learn something from the other participants (and that most certainly included us). We are somewhat heartbroken that the NSF cut the overall GK-12 program. The impact of our program went beyond the students; e.g., the San Francisco State mathematics department continues to offer a graduate-level writing-in-the-discipline course, which was initially developed as part of (CM)^{2}. Some of our current graduate students continue to be involved in the San Francisco Math Circle, including teaching and leadership positions. We hope we’ve inspired you to bring an aspect of this or a similar program into your work and encourage you to contact us for further information. Finally, we thank the graduate students and teachers who we worked with for their contributions and support of our program.

Consider how you would respond to two different versions of a question. In the first, you are asked to solve a high school mathematics problem. In the second, some high school students’ solutions to that problem are shown to you. You are asked to assume the role of the students’ teacher and to evaluate the mathematical validity of the students’ different approaches. What knowledge, if any, do you need in the second situation that you don’t need in the first situation?

Some would argue that the second situation is just about knowing math. If you, yourself, can solve the high school mathematics problem correctly, and you are very capable in high school mathematics, then this should be enough to evaluate a high school students’ solution. Yet others might say that this question is about teaching. If you can’t interpret students’ work, you can’t judge it accurately. Still others might say that this question targets something in between straight math and teaching. We would say that this scenario assesses a blend of all of these things that previous scholars have named *mathematical knowledge for teaching *(MKT). We ask the reader to join us in considering, as some have argued, why MKT is a form of applied mathematics – and why mathematicians have a stake in thinking about MKT in this way.

To begin, consider this example:

This question illustrates a coordination of mathematics and teaching. The intended answer is that only Matt’s method is valid. Jing found the slope between two points and assumed the slopes between all points were the same, without verifying that the data were actually linear. Matt’s method, without producing the function, verifies that all the given points fall on the line. The key idea to note is that the assumption of linearity is not warranted by the information given in the problem. Jing makes this assumption, and Matt does not. Solving the question requires knowing that the linearity assumption is not warranted. Solving the question also requires making sense of what Jing’s and Matt’s written work might mean. The linearity assumption jumps out to an experienced teacher as the important point that is likely to trip students up.

The question asks the respondent to think simultaneously about mathematics and teaching in a way that taking conventional mathematics coursework may not prepare a prospective teacher to do.

In a recent validity study (funded by the NSF __[1]__) designed to support a measure of teachers’ MKT, we asked secondary mathematics teachers to respond to this question. We saw variations of all of the above perspectives in their responses. Some focused on the mathematics, others on digging into what the two students might have been thinking, and many on something in between. Notably, respondents who did not attend to the linearity assumption were likely to favor Jing’s response, often describing it as more “complete” or “sophisticated”. In a number of cases, respondents stated that Matt needed to check all three points because linearity was not given, but also indicated they would have given at least partial, if not full, credit to Jing’s answer. After all, Jing derived the equation, whereas Matt only verified it. Yet, from the perspective of strict mathematical correctness, these judgments seemingly hold the two students to different standards. Imagine these respondents bringing this understanding to real students in a real classroom. Students might deduce from this that Matt’s approach is mathematically wrong. They might generalize that only solutions that involve algebraic manipulation of formulas are valid. They might pick up the implicit message that apparent sophistication is more important than whether a method effectively and efficiently addresses a given problem. They might be genuinely confused by the apparently different standards of mathematical validity. And finally, they might take from this that linearity can be assumed when convenient, or simply not have an opportunity to learn that such assumptions need to be established and/or verified as a general practice. In other words, these responses indicated something important: the knowledge these teachers did or did not demonstrate was likely to *matter* in their teaching, for both mathematical and pedagogical reasons.

Scholars have acknowledged the importance of MKT at the elementary level, even mapping out programs of study designed to provide elementary teachers with an opportunity to acquire this knowledge. It has been less clear whether secondary teachers-in-training need to study anything beyond the content of conventional mathematics and pedagogy courses. The study from which the opening question was taken demonstrated that that there *is* MKT at the secondary level that (1) can be measured and that (2) differs from what teachers are likely to learn in conventional mathematics courses. And we suspect that secondary mathematics teachers-in-training have few opportunities to learn this MKT as part of their teacher preparation programs.

While it seems clear that MKT learning opportunities belong in teacher preparation programs, it is less clear whether they belong in mathematics courses, pedagogy courses, or elsewhere. One could argue, as some have done, that MKT is applied mathematics; applied in the context of teaching. We ask the reader to join us in considering why one might think of MKT as a form of applied math in this way and present three arguments for why mathematicians teaching mathematics to teachers-in-training might have a stake in thinking about it this way.

- MKT is, at least in part,
*math*. Although a teaching perspective makes the linearity assumption in the opening example jump out to an experienced teacher, attending to such assumptions is also*mathematically*crucial. Often, mathematical ideas are pedagogically key because they are mathematically key. Because students are still learning how to engage in mathematical thinking, teachers need to do more than just think mathematically themselves; they also need to know how to call attention to what they are thinking and why. This involves understanding the mathematics in more than tacit ways. It is not enough for a teacher to avoid the error of assuming linearity; a teacher needs to know to recognize the linearity assumption as important, pedagogically. But both kinds of understanding are grounded in the mathematical idea that linearity cannot be assumed. - As with other forms of applied math, learning components in isolation may be less useful than learning them in context. Finding solutions to differential equations doesn’t get you far as a mathematical biologist if you haven’t had a lot of practice using them to approximate natural phenomena. In the validity study, we saw striking cases in which respondents with strong mathematics backgrounds and who demonstrated strong, specific, but decontextualized mathematical knowledge in other parts of the interview did not apply that knowledge appropriately to contextualized problems. In other words, we should not assume that studying mathematics in traditional courses necessarily or automatically equips teachers to call on and apply that knowledge appropriately in their work.
- We all have a long term stake in improving teachers’ MKT. Teachers were once students and learned mathematics in school. Their students are future teachers. The authors of
*Mathematical Education of Teachers*2 urge us to remember Felix Klein’s “double discontinuity,” that a teacher-in-training doesn’t see how university and high-school mathematics connect, and upon entering the classroom, doesn’t see what they learned in university has to do with teaching. As part of the second discontinuity, prospective teachers who are able to use quite sophisticated reasoning or sound mathematical habits of mind, such as checking assumptions in advanced math classes, might be capable of applying those skills to high school mathematics that they will be teaching, but simply have never had the occasion to do so.

What might we do to create learning opportunities for preservice teachers to learn MKT as applied math? We recently convened a group of mathematicians and mathematics educators to take on this question. Participants considered questions like the opening one as examples of the MKT we would ideally expect teachers-in-training to have opportunities to learn in teacher preparation. Workshop materials included a set of 55 such questions. (We encourage you to ask us more about these questions! They were designed with the intention of giving more examples of what MKT is like, especially at the secondary level. Copies of these questions can be obtained by contacting Heather).

When we began this workshop, we had hopes that these questions could serve as potential in-class or homework questions in both mathematics and pedagogy courses. The workshop provided some encouragement that our hopes could be realized, and that as a bonus, the questions could seed productive conversations among colleagues. For instance, our participants suggested that questions could help in identifying mathematical ideas that might be missed by both mathematics and pedagogy courses because they do not fall squarely into either. They suggested using the opening example as a context to discuss what constitutes mathematical validity. Our participants further suggested that the questions could be concrete examples for conversations about the tension between the notion that in mathematics, one wants to “never say something you have to take back,” yet in teaching, one should not “say something a student can’t take in;” and that questions can be mini-cases that could develop into extended cases of teaching and mathematics. Some participants are making plans to use these questions in their programs.

Based on the results of our validity study, and the promising suggestions arising from the workshop, we are now turning our attention to supporting the use of these questions to supplement coursework, and to forming professional communities focused on learning and teaching MKT through the use of questions such as the opening example. We invite your thoughts and comments about your experiences teaching MKT as a form of applied mathematics, especially at the secondary level, and would welcome requests to share materials.

[1] This material is based upon work supported by the National Science Foundation under Grant #DGE-1445630/1445551 Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Anyone who teaches mathematics in the US knows that the quality of education could be better, but we also know that the problems are complicated and defy easy solutions. I grew up in Ontario, Canada, where I attended high school and completed an undergraduate degree in mathematics. Afterwards I completed a Ph.D. in the United States and I have now been teaching undergraduate mathematics here for over ten years. These experiences suggest to me a change that would improve college mathematics education in the US. It won’t solve every problem, but it is something concrete that we can do right now.

Suggestion: Replace the typical one-semester “introduction to linear algebra” course with a two-semester linear algebra sequence. This would be taken in the first year of college, in parallel with calculus. It would not have calculus as a pre-requisite.

In effect, this would place linear algebra and calculus side-by-side as the twin pillars of undergraduate mathematics. I believe this would have several immediate benefits for the curriculum. In this blog post I’ll describe three of these benefits and then I’ll explain how my experience as a student in Canada and as a professor in the US has brought me to this position.

**Three Fundamental Problems**

*Linear algebra is being undersold. *Linear algebra is the common denominator of mathematics. From the most pure to the most applied, if you use mathematics then you *will* use linear algebra. This is also a fairly recent phenomenon, historically speaking. In the 19th century, linear algebra was at the cutting edge of mathematical research. Today it is a universal tool that every user of mathematics needs to know. This becomes more true every year as algorithms and data play a bigger role in our lives. It seems to me that the current curriculum was fossilized in an earlier time when linear algebra wasn’t so useful. But times change; sometimes we need to re-examine the mathematics curriculum to see if it is still relevant. I believe that a two-semester linear algebra sequence in the first year will be a more honest representation of how mathematics is used today.

*Complex Numbers. *Complex numbers are currently an orphan in the undergraduate curriculum. According to the Common Core Standards, students are supposed to learn about complex numbers in high school; however, from my experience with US college students I know that they are not learning this material. At the University of Miami (where I teach), the basic ideas of complex numbers including de Moivre’s Theorem appear in our pre-pre-calculus remedial course, so it is not reasonable that a student who takes this class will have time to complete the math major in four years. Therefore, in practice, we are pretending that our undergraduates have seen complex numbers when this is not the case. It seems that many of our students are introduced to complex numbers for the first time in an upper-level complex analysis course. In my opinion this is way too late. If US students are not to learn about complex numbers in high school, then it seems to me that the first semester of a two-semester linear algebra sequence is an excellent place to introduce this material. Here they can see the 2-dimensional geometric interpretation of complex numbers via 2X2 rotation and dilation matrices. This would have the added benefit of teaching complex numbers to future high school teachers (who are usually not required to take a complex analysis course).

*Linear Algebra remediation eats up other courses. *At the University of Miami we require all math majors to take MTH 210 (Introduction to Linear Algebra). As is typical in many departments, this is a one-semester course that is usually taken in the second year, as it has Calculus II as a pre-requisite. Most of the students have never seen vectors or matrices before, so our goal is to get them from the basic ideas to the useful applications in one semester. This course is then required as a pre-requisite for many upper-division courses. However, most instructors find that the students’ subsequent linear algebra background is very shaky because one semester is not enough to absorb all of the material. In practice this means that many upper-division courses must include a crash course in the relevant ideas from linear algebra, which is hugely inefficient. A two-semester first year sequence would ensure that teachers of upper-division math courses could assume that the students are familiar with the basic ideas of linear algebra. This would save time and allow each of these classes to cover more material.

**This Has Been Done Before**

I attended high school and college in Ontario, Canada between 1993 and 2002. In those days we had 5 years of high school; the fifth year (called OAC for “Ontario Academic Credit”) was optional for the general population, but it was required for entrance to many university programs. The OAC mathematics curriculum consisted of three courses: Algebra and Geometry, Calculus, and Finite Mathematics. A student entering a Bachelor of Science program at an Ontario university was expected to take all three of these. The Calculus course was similar to the AP Calculus course that exists in American high schools today. The Finite Mathematics course was an introduction to probability and statistics with a discussion of induction and elementary combinatorics. The OAC Algebra and Geometry course was a novelty that has no parallel in American schools. (Since the OAC program was phased out in 2003 it no longer exists in Canada either.) To give you an idea of the course, here is the table of contents from the final version of the course textbook (now out of print).

The curriculum was mostly analytic geometry in 2 and 3 dimensions providing a thorough introduction to representing lines and planes in 3D. It also introduced Gaussian elimination, orthogonality, the language of vectors with applications to physics, matrices as linear transformations, and complex numbers. I remember this course very vividly as one of the experiences that led me to become a mathematician. Ontario universities required these OAC courses for entrance to certain programs. The undergraduate curriculum was then able to make good use of these courses. For example, when I entered Queen’s University as a mathematics and physics major I was required to take two full-year mathematics sequences in my first year. One sequence was Calculus and it required OAC Calculus as a pre-requisite. The other sequence was Linear Algebra and it required OAC Algebra and Geometry as a pre-requisite. Let me emphasize this situation:

I was required to take a semester of linear algebra in high school, which was then followed by a required two-semester linear algebra sequence in the first year of college.

The textbook that we used for Linear Algebra was the 3rd edition of “Elementary Linear Algebra” by Larson and Edwards. This text is indistinguishable from the standard textbooks used today for the “introduction to linear algebra” course in the United States. Thus, by this point I had seen the same material, but I had seen it over three semesters instead of one.

After the first year sequence was an additional one-semester course in linear algebra that was required for mathematics majors with a concentration in statistics or physics. (It was also required for many engineering programs.) This course gave a more abstract introduction to inner-product spaces, with applications to function spaces. The students in the course were diverse, coming from many different departments. This indicates to me that the material of this course was still seen as universal in the sense that it was relevant to all users of linear algebra, whether pure or applied. It was not a special topics course.

In summary, by the time I had completed a Bachelor of Science degree in mathematics with a physics minor, I had been required to take *four semesters of general-purpose linear algebra*. And you want to know something funny? As I proceeded to graduate school in pure mathematics at Cornell, I shortly came to feel that the most serious deficiency of my undergraduate education was that *I had not seen enough linear algebra!* Not only did I find my multilinear algebra background weak when I learned representation theory, I was also shocked when I learned about the Perron-Frobenius theorem and its amazing applications (e.g., to ranking webpages): Why had no one told me about this theorem?

**My Teaching Experience**

I joined the mathematics faculty at the University of Miami in 2009, and since then I have taught our Introduction to Linear Algebra course (MTH 210) four times. After this, many upper-division courses (such as Multivariable Calculus) list MTH 210 as a pre-requisite. We have two upper-division linear algebra courses in the curriculum (one more abstract and one more numerical), but these are regarded more as capstone courses and they do not serve as pre-requisites for anything else. Thus MTH 210 is a very important course.

Let me describe my experience teaching the course and how my approach to the syllabus has evolved. The official department syllabus reads as follows:

Vectors, matrix algebra, systems of linear equations, and related geometry in Euclidean spaces. Fundamentals of vector spaces, linear transformations, determinants, eigenvalues, and eigenspaces.

The majority of students in the course have never seen the language of vectors and matrices before, nor have they taken a physics course which could be used for motivation. In my opinion, it is completely impossible to cover the material from a standard textbook under these conditions, so my experience with the course has been a painful process of deciding what to leave out. After trial and error I have decided that the abstract notions of “vector space” and “linear transformation” had to go, so that I can get to some substantial applications by the end of the semester. The language of matrix arithmetic is already plenty abstract for students seeing it for the first time.

Since the students are starting from scratch, I begin the course with a thorough introduction to Cartesian coordinates and analytic geometry in 2 and 3 dimensions. Without the visual intuition that this provides I don’t think they’ll get very far. By the halfway point of the semester I want to cover the standard material on Gaussian elimination and to be able to explain the geometric intuition behind it.

After covering these minimum pre-requisites in the first half of the semester, in the second half we dive into the actual “linear algebra”. Here is where the painful cuts come. Since this is the only linear algebra course that most of these students will ever see, I need to cover the significant applications. But I also need to cover enough theory that these applications can be understood. With practice I’ve come up with a way to do this that I’m reasonably happy with. If you want to see the details, my lecture notes from two previous iterations of the course are available here and here. But on the other hand, I’m not happy at all. I know how useful linear algebra is, and I know that most of these students would benefit from a deep understanding of it. I also know that this one-semester class goes by too quickly for the ideas to really sink in. I can only hope that I’ve given them a roadmap so they can fill in the details later when needed.

**Conclusion**

It seems clear to me that a two-semester linear algebra sequence would vastly improve the undergraduate mathematics curriculum in the United States. This would give extra time to absorb the important ideas and give space to discuss neglected topics such as complex numbers and analytic geometry. It would also allow upper-division courses to use linear algebra in a deeper way.

This is natural from a mathematical point of view, but I also recognize that from an administrative point of view it’s pretty radical. I’ve sat on committees and I know how time consuming it can be just to change the title of an existing course. Adding a new course at the center of the curriculum could easily be a decade-long process, and it would be far from painless. Being realistic, I suspect that a change like this will only follow from a discussion at the national level. The MAA’s Committee on Undergraduate Programs in Mathematics has already made some recommendations for the role of linear algebra in the curriculum in their 2015 Curriculum Guide.

I generally agree with the CUPM’s remarks, but I think they are too shy in their recommendations. For example, the committee’s Content Recommendation 1 from the Overview says that “Mathematical sciences major programs should include concepts and methods from calculus and linear algebra.” This language seems to put calculus and linear algebra on equal footing (I assume they are listed in alphabetical order), but it doesn’t grapple with the reality that linear algebra is currently under-represented with respect to calculus. I would rephrase their recommendation in stronger terms:

Mathematical sciences major programs should include concepts and methods from calculus and linear algebra.

These two subjects should be introduced in parallel, and both should be studied in the first year.

Phrasing it this way emphasizes that a change is needed. Undergraduate programs in the US do not introduce linear algebra early enough and they do not place it beside calculus at the center of the curriculum. Other countries (such as Canada) have done a better job with this, and I hope that we in the US can learn from their example.

]]>It could be the punchline of a joke that at any given college or university, at some point, the administration will lean on departments to be more “efficient” by teaching classes in larger sections, or online, or with some technology or another. By the metric of student credit hour to faculty work hour, of course, large lectures are tremendously efficient, and scale admirably. One may argue that there is little difference between an instructor lecturing to 100 or to 200 students, and little difference between an instructor rendered small by the distance to the front of a large lecture hall and one rendered small in the pixels of a video screen. This is the Massive Open Online Course (MOOC) model, which extends this efficiency of scale from 200 to 20,000. Anecdotally, the MOOC tide seems to be receding, but the pressures that argue for this efficiency are not going away. Many departments are being asked to teach, with fewer resources and greater accountability, more students whose mathematical preparation is weaker than in the past [2].

The difficulty here is that the student credit hour metric is easy to measure, while student learning is not. Research says that our efficient passive lecture does not result in the student learning gains we can see with more active teaching techniques [6,7,8]. Indeed, through the Conference Board of the Mathematical Sciences, the presidents of fifteen professional societies in the mathematical sciences have recognized this conclusion and endorsed the use of active teaching methods [4]. But neither the research nor the endorsement provide us with a simple, usable measure by which to demonstrate the effectiveness of these techniques. So our endeavor of teaching remains by easily applied metrics an inefficient one, and I increasingly think one that is inevitably inefficient by even more measures.

This is probably true for all education, but there is a case to be made that it is particularly true for mathematics. For over twenty years, we have watched an accelerating rush to calculus in American high schools [3], and I recall a discussion with Project NExT [9] Fellows in about 1995 in which the observation was made that as calculus is pushed down into the high school curriculum, algebra is pushed up into college. But it feels worse than a one-to-one exchange, because while calculus is, more-or-less, well-defined―though this characterization deserves further evaluation―what bubbles up is not.

As a result, our efficient lecture, in which all students are assumed equally well-prepared and equally well-served by a uniform delivery method, becomes even more poorly suited to reach our students. We are stuck, again, with our most effective―and perhaps only effective―remediation being inherently inefficient: we need the instruction for these students to be individually responsive, not broadly scalable. Further, mathematics is not a field in which students’ understanding is built up from only a small number of physical laws. Thus, the responsive diagnosis and remediation must be nimble and able to evaluate individual students daily, as we navigate in class a varying mathematical terrain that requires varying prerequisite knowledge. As a discipline, the thinking and logic we demand of all practitioners, students included, is that of a science, but in the interconnected myriad details our subject may be more akin to the humanities. (The need for languages to be taught in small sections is rarely questioned; perhaps we may argue that the mathematics education research is really demanding the same for mathematics.)

So to be effective instructors, especially in our present environment, we may claim that we are inevitably inefficient. But I think that this premise extends beyond the classroom, infiltrating even the systemic support that effective instruction requires.

I wrote above that calculus is, more-or-less, well-defined, and I think this is true. Comparing two arguably different calculus textbooks―Stewart’s Calculus and the text of the same name by Hughes Hallett, et al.―reveals 28 sections that cover essentially the same material and only four that are demonstrably different in mathematical content. But calculus courses themselves are, by dint of institutional constraints (student preparation and needs foremost among them) far less uniform, and this difference in courses between institutions gets only more pronounced as we move on from calculus. To some degree this has limited import when students stay at one institution through graduation, but this is becoming less and less the case. The push, and need, for greater affordability of higher education means that increasing numbers of students may be transferring between colleges (especially two-year colleges) and universities. Thus there are increasing numbers of students who have taken courses at other institutions entering our classrooms, and if they have unexpected gaps in background knowledge we need to give the instructor time and contact to be able to diagnose and remediate that. For this to be at all a feasible undertaking, we need first to ensure that students are in the right course in the first place, which requires a “diagnosis” of any courses with which our students arrived. I think that this diagnosis is also one that is necessarily inefficient. It is not one that someone without knowledge of the subject can do reliably, and thus for it to be done well we must use expert faculty time to do it. This use of faculty time is also not efficient by any standard business model: we are using our most highly trained employees in the organization to evaluate on a student-by-student basis these course requests.

I won’t argue that this is the only way to evaluate transfer credit, but I think it is an effective way to do it, even as it is by some metrics inefficient. I’ve evaluated perhaps 200 courses for equivalency to courses at the University of Michigan in the past year. And while this doesn’t show up in the list of teaching duties that I have performed in that time, I believe it to be a service that allows our teaching to be effective.

I watched a similar, and similarly inefficient, evaluation unfold this summer in the course of several meetings of the faculty who are most directly involved in the administration of our Introductory Program (loosely, our course preceding calculus, calculus I, and calculus II). The University has a summer program to promote diversity in STEM subjects, and asked if we could designate some sections of calculus I for those students so that they could enroll in calculus with other members of their cohort, and in sections taught by instructors the students already know. From the perspective of this program these are obviously desirable outcomes. However, it also has the potential to isolate students in those sections who are not part of the cohort, and requires that the Introductory Program Directors establish these sections in advance so that this is possible. These latter outcomes are less desirable, and the former is a significant concern for the learning of other students in those sections. As a result, this evaluation was not a straightforward one, and it unfolded over the course of two or three meetings of the five to seven people involved in these decisions. There are perhaps 18 students involved in this program for the fall. To me this seems to fall in the category of inefficient processes, at least as measured by the time spent by the decision makers.

At any given college or university that is trying to do a good job in teaching mathematics I suspect there will be similarly inefficient systems supporting the inefficient work of the mathematics teaching itself. They may not―arguably, will not―be doing the tasks I’ve picked here. Because of the nearly infinite variation in the colleges and universities across the country and world, the systemic challenges at each will be correspondingly different and varied. But because the difficulties in dealing with differently prepared students in all of these environments is the one thing we are certain will be constant, it’s hard to imagine the systemic issues will ever be absent.

It is said of at least some theoretical mathematicians that they are proud that their chosen studies do not have (visible) practical applications. I think that we as mathematicians who are concerned with the effectiveness of our institutions at educating students in our chosen field should perhaps be similarly proud of our inefficiency by practical measures. Active learning really is better, and is better done on a scale at which there is significant student-faculty interaction [5]. And our students learn best when the systemic support for these active learning classrooms allows them to operate at their best. Insofar as these are inefficient, these inefficiencies are inevitable. Thus they are also an essential characteristic of the effective teaching of mathematics.

**References**

[1] ALEKS. (2016). Accessed Aug. 31, 2016.

[2] Bressoud, D. (2015) Calculus at Crisis I: The Pressures. Launchings. (May 1, 2015). Accessed Jul. 7, 2016.

[3] Bressoud, D. (2015) Calculus at Crisis II: The Rush to Calculus. Launchings. (Jun. 1, 2015). Accessed 16 May 2016.

[4] CBMS Statement on Active Learning in Post-Secondary Mathematics Education.

(July 15, 2016). Accessed 24 Aug 2016.

[5] Chickering, A.W. and Z.F. Gamson. Seven Principles for Good Practice in Undergraduate Education, New Directions for Teaching and Learning #47. Jossey-Bass (1991).

[6] Freeman, S., et al. (2014). Active Learning Increases Student Performance in Science, Engineering and Mathematics. Proceedings of the National Academies of Sciences, 111(23):8410–8415.

[7] Kogan, M. and S.L. Laursen (2014). Assessing Long-term Effects of Inquiry-based Learning: A Case Study from College Mathematics. Innovative Higher Education, 39(3):183-199.

[8] Laursen, S.L., M.L. Hassi, M. Kogan and T. Weston (2014). Benefits for Women and Men of Inquiry-based Learning in College Mathematics: A Multi-institutional Study. Journal for Research in Mathematics Education, 45(4):406-418.

[9] Project NExT. Accessed 24 Aug 2016.

This is an announcement to the mathematical community that the White House Office of Science and Technology Policy (OSTP) has issued a Call to Action for incorporating Active Learning in K-12 and higher education STEM courses. Their call includes a submission form where they ask: “What new (i.e., not yet public) activities or actions is your organization undertaking to respond to the Call to Action to improve STEM teaching and learning through the use of active learning strategies?” The deadline for submission of responses is Sept 23, 2016. It would be excellent for the mathematics community to be well-represented among the submissions, so I encourage our readers to submit their activities and to share this information with others.

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