A remarkable event took place a few weeks ago at the Alexandria, Virginia headquarters of the American Statistical Association. Leaders from five professional associations whose missions include teaching in the mathematical sciences came together to guide future progress to incrementally improve education in our fields. It is the first time that all five — the American Mathematical Association of Two-Year Colleges (AMATYC), the American Mathematical Society (AMS), the American Statistical Association (ASA), the Mathematical Association of America (MAA), and the Society of Industrial and Applied Mathematics (SIAM) — are working together. Our focus is the collection of credit-bearing mathematics courses a student might take in the first two years of college. We examine the undergraduate program using a wide-angle lens, inclusive of modeling, statistics, and computational mathematics as well as applications in the broader mathematically based sciences.
Why now?
Each year approximately 50 percent of students fail to pass college algebra with a grade of `C’ or better.[1] Failure rates under traditional lecturing are 55 percent higher than the rates observed under active learning.[2] Undergraduate education in the mathematical sciences is in crisis in the United States. This crisis will affect all mathematical scientists at post-secondary institutions, regardless of each individual’s level of interest in education.
The crisis in mathematical sciences education is well documented in high-profile reports such as the U.S. government’s PCAST report on STEM education and the National Academies’ report on The Mathematical Sciences in 2025. In response (or in some cases, in anticipation of) these reports, various mathematical science associations have on their own or in collaboration released reports such as
There have been, and continue to be, many successful initiatives aimed at addressing the challenges identified. However, we believe it is time for collective action. We can no longer say, “I don’t teach those classes,” or “I don’t teach those students,” because students are now more mobile than ever, transitioning between multiple postsecondary institutions. For example, the National Student Clearinghouse Research Center’s Two-Year Contributions to Four-Year Degrees report found that 46 percent of all students who completed a degree at a four-year institution in 2013-14 had been enrolled at a two-year institution at some point in the previous 10 years. Research on “collective impact” suggests that, in achieving significant and lasting change in any area, a coordinated effort supported by major players from all existing sectors is more effective than an array of new initiatives and organizations.[4]
To maintain a viable workforce for our country, to continue the expansion of scientific knowledge, and to remain relevant, we must update our curricula, make current our pedagogical methods, connect more strongly to other disciplines, and perhaps even evolve the culture of our own discipline. Many in our own community predict that if we do not achieve large-scale improvement in undergraduate education on our own, then markets, governments, or other structures will force change upon all of us. We believe it is better to have agency in making the necessary changes.
Ben Braun’s recent blog post, which gives an account of the October 2014 AMS Committee on Education (CoE), states that “the most prominent theme of the meeting was the critical role of collaboration and cooperation at many levels: among department members, at the institutional level among departments and administrative units, among professional societies with common missions, and at the national level to ‘scale up’ successful models for effective teaching.” It is very good news indeed that important stakeholders are involved. A group of prominent mathematicians has come together to form Transforming Post-Secondary Education (TPSE Math) and they have recently published their first report. The umbrella organization for professional associations in the mathematical sciences, the Conference Board of the Mathematical Sciences (CBMS) held its forum on the first two years of college math, and is discussed by Diana White in her November 2014 blog post. Common Vision brings together the five professional associations whose missions include teaching in the mathematical sciences; it is our view that bringing association leadership together to work on undergraduate education is critical for lasting change.
Collective action to improve teaching and education in the mathematical sciences appears to be gaining traction.
Who was at the workshop?
The Common Vision 2025 project encourages action by highlighting existing efforts and draws on the collective wisdom of a diverse group of stakeholders to articulate a shared vision for modernizing the undergraduate mathematics program. We embrace the diversity of experience of our members.
Workshop participants included AMS President Robert Bryant, as well as several current and past presidents of all five associations. Participants also included faculty members from large departments at research universities; a statistician working at Google; a mathematician working at an HBCU; a vice president from the New York Hall of Science; faculty members from liberal arts colleges; faculty members from large comprehensive universities; the Executive Vice President of the APLU; a chemist working at the American Chemical Society; and an Achieving the Dream project director.
What can you do?
In reaching out to the membership of the five associations (including through this blog post) we hope to galvanize our colleagues and spur on a grassroots effort to improve education in the mathematical sciences.
Read the reports listed above. Read the Common Vision report, which will appear later this year and identifies common themes found in the above reports in order to provide a snapshot of the current thinking about undergraduate mathematics and statistics programs. Our report will also include a list of project ideas generated at our workshop. For example, you might identify a part of your curriculum that you would like to change in some way (like the calculus sequence, or the collection of upper level analysis courses, or the courses that do not require calculus and are intended for non-majors), and organize a meeting this summer with your colleagues about it; in advance, start a Google document where you can share ideas. Small changes, including more care and intention about our curriculum, can help our students have a better classroom experience. The activities are ones where we deem “small wins” are realistic, and are aimed at updating the mathematical sciences curriculum, updating pedagogical methods to align with best practices, and changing the culture of our discipline.
Please, do something. Do something. Do something.
The Common Vision website: http://www.maa.org/common-vision
[1] Mathematical Association of America (2012). Partner Discipline Recommendations for Introductory College Mathematics and the Implications for College Algebra. Retrieved from Mathematical Association of America website: www.maa.org/sites/default/files/pdf/CUPM/crafty/introreport.pdf.
[2] Freeman, S, Eddy, S., McDonough, M., Smith, M., Okoroafor, N., Jordt, H., and Wenderoth, M.P., Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences. Vol. 111. No. 23. June 10, 2014.
[3] See Martha Siegel’s blog post.
[4] Kania, J. and Kramer, M. (2011). Collective Impact, Stanford Social Innovation Review, Winter 2011.
]]>When you return to the classroom as an adult student, a big perk is that what seemed like an unreasonable demand back then from the instructor suddenly makes sense, because maturity means you’re better able to fit it into the bigger picture. For me, a longtime journalist who decided to retake high school math at a community college after decades of hating and fearing it, that demand was “show your work.” As a teen, I’d always sighed when the teacher marked me down for not showing how I’d worked out a problem on an exam or in the homework. Why was it necessary to take eight steps to show a triangle’s angles added up to 180? What a bore.
But 20 years later, going from pre-algebra to calculus, I finally understand why, and I credit dance.
Huh? Let me explain.
To get to the math building on my community college’s campus, I’d usually take a shortcut through the dance department. I’d walk down a long corridor lined with mirrored studios, and no matter what kind of music was blaring out the doors – salsa, tap, jazz – an instructor would always count out a beat before the students began.
“And a one, two, three, four, five, six, seven, eight!”
Hearing this every class day, I not only realized that numbers were everywhere, but also that learning how to solve a math problem was a lot like learning how to dance. In both, there’s choreography involved, going from step one to step two to step three. And, at least in the case of ballroom dances like the fox-trot or waltz or cha-cha, there is a strict order of operations.
You may be Please Excusing My Dancing Aunt Sally, not my Dear one (PEMDAS), but missing a step or doing it out of order will really mess up the end result. Or else, it will turn the dance into something completely different.
By thinking of math problems that way, I was better able to tolerate my instructors’ endless insistence that I show all my work, especially on tests. I finally appreciated that they needed to know I truly grasped the elements of the problem, and that I respected the strategy needed to solve it. True, I still find it tedious to prove in eight steps that a triangle’s angles add up to 180 degrees, but I now know it’s good practice for way more complicated proofs, where thoroughness is key. I also appreciate that precision is vital to math, and if eight steps is what it takes to be precise in a triangle proof, so be it.
That said, a major peeve of mine, especially as I got further from applied math and closer to pure, was when instructors, while solving a problem, would take a sudden leap. This might entail doing quick factoring in a polynomial, going from 6x +6 to 6(x + 1) without explaining why it was necessary, or assuming students had memorized an obscure trigonometric identity, then making the substitution in a long equation without mentioning it.
I realize these are very simple examples, but depending on where I was in my math education, to me this was the dance equivalent of doing a two-step, then suddenly getting spun and landing on my butt. It would always take me a moment to regroup, and by then, I’d been left behind, standing against the wall and watching as everyone else whirled by. At least in class, I could try and stop the instructor and ask him or her to explain. But I always felt guilty about this, since we never seemed to have enough time to really get into the material. That guilt was spurred by the fact that every professor I had, from pre-algebra on, complained about class time never being enough to really go into depth on anything, especially if students didn’t grasp the material right away. And yes, all of these instructors had office hours for those slower students, but I discovered those hours were just as chaotic as they were in class, only now students were cramped into a tiny office, craning their necks to see what the professor was writing in a notebook. But that’s another discussion for another day.
It was worse when such a leap happened in the solutions manual. For the record, I’ve never much cared for these manuals, preferring to puzzle things out on my own. But sometimes I would come across a problem I just couldn’t solve, where it was all a blur, and I couldn’t pick out one step from the next. Looking at the worked out solution was a way to slow things down and get a guide.
However, when that guide skipped a step without explanation, there was no lecture to interrupt, no office to stalk. I was usually able to fill in the blank, but the time I spent doing so always had a cost. Sometimes it was not being able to get to all the other problems I needed to practice before the next test, or, more important, it dented my still fragile math confidence, making me unsure when I had to perform. And that anxiety sometimes led to failure on exams because I couldn’t relax enough to solve harder problems without second-guessing myself. Then I would make silly arithmetical mistakes on the other problems because I was rushing to catch up.
Now, I know that my inner demons were never my instructor’s problem. But that didn’t stop me from asking a few math professionals I came across why they skip steps while taking students through a problem.
Not enough class time, one said.
Including every step gets very tedious, said another, and you can lose sight of the bigger picture.
It is the student’s job to fill in the blanks, and doing so is the best way to retain the material, said several, though at least one added that this method worked best in classes more advanced than calculus. It can really backfire before then.
And these are all excellent reasons. But they didn’t help that jarring feeling of being spun, of falling, of landing badly, that I experienced when I revisited math after 20 years.
I understand that in math, as in dance, you have to get up and dust yourself off. And I did. But far too many of us don’t, which is why so many of us give up. And giving up on math has far worse repercussions, not just individually, but for all of society, than not becoming proficient at the fox-trot. And unlike me, most self-proclaimed math haters never return to the classroom.
So I ask the instructors reading this to consider shifting their perspective as I did mine. I accepted how important it was not to skip steps, to respect the choreography, so that you could see that I understood what was going on. You may have the best of reasons, but when you skip steps without explaining why, people like me, unused to the elaborate choreography, will fall down. We’re still learning. Don’t assume we can see how you did that leap. And hopefully, we’ll soon be dancing as gracefully as you.
]]>
One of my favorite assignments for students in undergraduate mathematics courses is to have them work on unsolved math problems. An unsolved math problem, also known to mathematicians as an “open” problem, is a problem that no one on earth knows how to solve. My favorite unsolved problems for students are simply stated ones that can be easily understood. In this post, I’ll share three such problems that I have used in my classes and discuss their impact on my students.
Unsolved Problems
The Collatz Conjecture. Given a positive integer \(n\), if it is odd then calculate \(3n+1\). If it is even, calculate \(n/2\). Repeat this process with the resulting value. For example, if you begin with \(1\), then you obtain the sequence \[ 1,4,2,1,4,2,1,4,2,1,\ldots \] which will repeat forever in this way. If you start with a \(5\), then you obtain the sequence \(5,16,8,4,2,1,\ldots\), and now find yourself in the previous case. The unsolved question about this process is: If you start from any positive integer, does this process always end by cycling through \(1,4,2,1,4,2,1,\ldots\)? Mathematicians believe that the answer is yes, though no one knows how to prove it. This conjecture is known as the Collatz Conjecture (among many other names), since it was first asked in 1937 by Lothar Collatz.
The Erdős-Strauss Conjecture. A fascinating question about unit fractions is the following: For every positive integer \(n\) greater than or equal to \(2\), can you write \(\frac{4}{n}\) as a sum of three positive unit fractions? For example, for \(n=3\), we can write \[\frac{4}{3}=\frac{1}{1}+\frac{1}{6}+\frac{1}{6} \, . \] For \(n=5\), we can write \[ \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20} \] or \[\frac{4}{5}=\frac{1}{2}+\frac{1}{5}+\frac{1}{10} \, . \] In other words, if \(n\geq 2\) can you always solve the equation \[ \frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\] using positive integers \(a\), \(b\), and \(c\)? Again, most mathematicians believe that the answer to this question is yes, but a proof remains elusive. This question was first asked by Paul Erdős and Ernst Strauss in 1948, hence its name, and mathematicians have been working hard on it ever since.
Lagarias’s Elementary Version of the Riemann Hypothesis. For a positive integer \(n\), let \(\sigma(n)\) denote the sum of the positive integers that divide \(n\). For example, \(\sigma(4)=1+2+4=7\), and \(\sigma(6)=1+2+3+6=12\). Let \(H_n\) denote the \(n\)-th harmonic number, i.e. \[ H_n=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n} \, .\] Our third unsolved problem is: Does the following inequality hold for all \(n\geq 1\)? \[ \sigma(n)\leq H_n+\ln(H_n)e^{H_n} \] In 2002, Jeffrey Lagarias proved that this problem is equivalent to the Riemann Hypothesis, a famous question about the complex roots of the Riemann zeta function. Because it is equivalent to the Riemann Hypothesis, if you successfully answer it, then the Clay Mathematics Foundation will reward you with $1,000,000. While the statement of this problem is more complicated than the previous two, it doesn’t involve anything beyond natural logs and exponentials at a precalculus level.
Impact on Students
I’ve used all three of these problems, along with various others, as the focus of in-class group work and as homework problems in undergraduate mathematics courses such as College Geometry, Problem Solving for Teachers, and History of Mathematics. An example of a homework assignment I give based on the Riemann Hypothesis problem can be found at this link. When I use these problems for in-class work, I will typically pose the problem to the students without telling them it is unsolved, and then reveal the full truth after they have been working for fifteen minutes or so. By doing this, the students get to experience the shift in perspective that comes when what appears to be a simple problem in arithmetic suddenly becomes a near-impossibility.
Without fail, my undergraduate students, most of whom are majors in math, math education, engineering, or one of the natural sciences, are surprised that they can understand the statement of an unsolved math problem. Most of them are also shocked that problems as seemingly simple as the Collatz Conjecture or the Erdős-Strauss Conjecture are unsolved — the ideas involved in the statements of these problems are at an elementary-school level!
I have found that having students work on unsolved problems gets them engaged in three ways that are otherwise very difficult to obtain.
One of the most interesting aspects of using unsolved problems in my classes has been to see how my students respond. I typically ask students to write a three-page reflective essay about their experience with the homework in the course, and almost all of the students talk about working on the open problems. Some of them describe feelings of relief and joy to have the opportunity to be as creative as they wish on a problem with no expectation of finding the right answer, while others describe feelings of frustration and immediate defeat in the face of a hopeless task. Either way, many students tell me that working on an unsolved problem is one of the noteworthy moments in the course. For this reason, as much as I enjoy witnessing mathematics develop and progress, I hope that some of my favorite problems remain tantalizingly unsolved for many years to come.
]]>By the end of every workshop and conference session on Inquiry-Based Learning that I’ve attended, someone has raised a hand to ask about coverage. “Don’t you have to sacrifice coverage if you teach this way?” Of course coverage took center stage for many of my professional conversations long before I tested the IBL waters; it’s important. But an equally important question is this: What do we sacrifice when coverage dominates? It may well be conceptual understanding; it’s possible to cover more ground, albeit thinly, if we settle for procedural understanding instead. More than once I’ve settled for even less, delivering a quick lecture just so that my students will have “seen” a particular idea. How do we strike a balance between coverage and other considerations when we are so practiced at reducing a course description to a list of topics?
Strong arguments for striking that balance have been made elsewhere. For example, Stan Yoshinobu and Matthew Jones offer a close examination of the “price of coverage”. “Coverage versus depth” is a “false dichotomy,” they say; racing through material makes for a passive student experience, which affects student understanding of what it is to learn mathematics. “Implied messages are sent to students through classroom experiences,” and some of those messages may have unproductive consequences, including overreliance on mimicking the instructor and memorization, and significant difficulties with non-routine problems.
Is there, on the other hand, a price of demoting coverage? Does a more comprehensive view of student learning get in the way of content knowledge? Recent research done by Marina Kogan and Sandra Laursen, brought to my attention by Yoshinobu and David Bressoud, suggests that students don’t necessarily suffer, and may be helped, from a holistic approach. From the conclusion to the Kogan and Laursen paper:
College instructors using student-centered methods in the classroom are often called upon to provide evidence in support of the educational benefits of their approach—an irony, given that traditional lecture approaches have seldom undergone similar evidence-based scrutiny. Our study indicates that the benefits of active learning experiences may be lasting and significant for some student groups, with no harm done to others. Importantly, “covering” less material in inquiry-based sections had no negative effect on students’ later performance in the major. Evidence for increased persistence is seen among the high-achieving students whom many faculty members would most like to recruit and retain in their department.
Still, it’s often difficult to prevent concerns about coverage from hijacking day-to-day teaching practice, regardless of course format. Here are some approaches I am using to keep coverage in perspective.
Regard conceptual understanding, mathematical writing and speaking, and other learning goals as integral parts of the “coverage” list, on an equal par with specific topics. Yoshinobu points out that we have a “systemic” issue, in that our institutions define coverage as no more than the list of topics. Hence I have to make a conscious effort, in planning each course, to weave all of the goals together, and to recognize that procedural skills won’t last without conceptual understanding, which in turn won’t happen if students don’t routinely speak and write mathematics.
Include learning objectives, not just a topics list, on the syllabus. Whether or not all of my students read the syllabus, it’s my way of formalizing my intentions and expectations. It’s also an invitation to consider the course in its entirety. This is especially important in mathematics, where students don’t understand many of the terms in a catalog description until after they’ve taken the course.
Have conversations with students, early and often, about the learning goals for the course. On the first day of linear algebra this semester, I devoted the entire hour to a class activity adapted from a model offered by Dana Ernst. The students’ responses to “What are the goals of a liberal arts education?” included “critical thinking” and “to experience the freedom to explore.” To “What can you reasonably expect to remember from your courses in 20 years?” I heard, “NOT details or the stuff you’re tested on,” but rather “how to figure out what’s relevant.” My own students understand the big picture; surely I can keep it in mind!
Halfway through the term, I had my students read this blog post from Ben Orlin and then fill out a survey online. I asked: to what extent are you practicing in the Church of Learning, as opposed to the Church of the Right Answer? Once again, the students reinforced my choices. Many of them also noted that their pre-college experiences, especially Advanced Placement Calculus, leaned heavily toward the Right Answer doctrine. In at least some cases, I’m working against students’ most recent experience of mathematics learning, so I need to be persistently transparent.
Gather data frequently on student understanding. Formative assessment isn’t just for elementary school teachers. I’m fortunate to teach small classes, so I can learn a lot just from classroom conversations. In an earlier post, I explained how recent research on learning has influenced my teaching. If I hear someone struggling to use “linearly independent” accurately during small group work, I can offer corrective feedback immediately. My students often show their work using a document projector. Anonymous surveys are useful as well; it only takes a few minutes for students to write down what’s puzzling them at the moment. I’ve never used clickers, but I’m intrigued by Eric Mazur’s methods. Most importantly, I try to design homework assignments that ask for deeper understanding. (It takes several weeks to convince students that homework is for formative, not summative, assessment, and that the graders’ job is to give constructive feedback.)
Bring student graders and teaching assistants in on the plan. I handpicked my graders this term, and made it clear that I want homework solutions to be clear and well-written, not just correct. They know that I’ve encouraged the students to show their attempts and partial solutions to more challenging problems. They let me know what misconceptions they see. The student tutors are also aware of my intentions.
It may be that I am especially sensitive to questions about coverage because my semester includes only twelve weeks of classes. My department colleagues and I agree that this poses a particularly vexing challenge in multivariable calculus. Getting to Green’s Theorem is challenging enough, and a thorough treatment of Stokes’ Theorem, which would add coherence to the entire semester, seems a worthy goal. Yet even here, I remind myself, what’s important is not only what I cover; it’s also what the students can retain.
]]>Making fundamental changes to the way you teach is a difficult task. However, with a growing number of students leaving STEM majors, instructors’ dissatisfaction with student learning outcomes, and research indicating positive avenues for improving undergraduate mathematics instruction, some instructors are ready and eager to try something new. In this post, we describe some promising research-based curricular materials, briefly identify specific challenges associated with implementing these materials, and describe a recently funded NSF project aimed at addressing those challenges.
Teaching Inquiry-Oriented Mathematics: Establishing Supports (TIMES) is an NSF-funded project (NFS Awards: #143195, #1431641, #1431393) designed to study how we can support undergraduate instructors as they implement changes in their instruction. A pilot is currently being conducted with a small group of instructors. In the next two years, approximately 35 math instructors will be named TIMES fellows and will participate in the project as they change their teaching of differential equations, linear algebra, or abstract algebra. As project leaders, we will study how to best support these instructors, as well as how their instructional change affects student learning. More details about the project follow later in this blog post.
Inquiry-Oriented Instruction
The curricula we utilize in the project are each examples of inquiry-oriented instructional materials. Inquiry-oriented instruction is a specific type of student-centered instruction. Not surprisingly, different communities characterize inquiry in slightly different ways. In the inquiry-oriented approach we describe here, we adopt Rasmussen and Kwon’s (2007) characterization of inquiry, which applies to both student activity and to instructor activity. In this approach, students learn new mathematics by: engaging in cognitively demanding tasks that prompt exploration of important mathematical relationships and concepts; engaging in mathematical discussions; developing and testing conjectures; and explaining and justifying their thinking. Student inquiry serves two primary functions: (1) it enables students to learn new mathematics through engagement in genuine exploration and argumentation, and (2) it serves to empower learners to see themselves as capable of reinventing important mathematical ideas.
The goal of instructor inquiry into student thinking goes beyond merely assessing student’s answers as correct or incorrect. Instead, instructor inquiry seeks to reveal students’ intuitive and informal ways of reasoning, especially those that can serve as building blocks for more formal ways of reasoning. In order to support students, instructors routinely inquire into how their students are thinking about the concepts and procedures being developed. As instructors inquire into students’ emerging ideas, they facilitate and support the growth of students’ self-generated mathematical ideas and representations toward more formal or conventional ones. The instructor’s role is to guide and direct the mathematical activity of the students as they work on tasks by listening to students and using their reasoning to support the development of new conceptions. Additionally, instructors provide connections between students’ informal reasoning and more formal mathematics.
With an inquiry-oriented instructional approach, instructors use mathematically rich task sequences, small group work, and whole class discussions in order to elicit student thinking, build on student thinking, develop a shared understanding, and introduce formal language and notation.
Curricular Materials for Undergraduate Mathematics Education
The TIMES project is organized around three sets of post-calculus, research-based, inquiry-oriented curricular materials.
· Inquiry-Oriented Abstract Algebra (IOAA), developed by Sean Larsen under the NSF grant Teaching Abstract Algebra for Understanding (#0737299), http://www.web.pdx.edu/~slarsen/TAAFU/ (User:AMSBlog; Password:teacher). These materials are designed for an introductory group theory course and include units on groups and subgroups, isomorphisms, and quotient groups. Supplementary materials for rings/fields are available upon request.
· Inquiry-Oriented Linear Algebra (IOLA), developed by Megan Wawro, Michelle Zandieh, Chris Rasmussen, and colleagues under NSF grant numbers 0634074/0634099 and 1245673/1245796/1246083, http://iola.math.vt.edu (must request login & password). These materials are designed for an introductory linear algebra course and include four units on span, linear dependence and independence, transformations, and eigenvalues, eigenvectors, and change of basis. Tasks for determinants and systems are also available upon request.
· Inquiry-Oriented Differential Equations (IODE), developed by Chris Rasmussen and colleagues under NSF grant number 9875388, website coming soon. These materials are designed for a first course in differential equations and include the following topics: solving ODEs; numerical, analytic and graphical solution methods; solutions and spaces of solutions; linear systems; linearization; qualitative analysis of both ODEs and linear systems of ODEs; and structures of solution spaces.
For each of these three curricular innovations, the student materials have been developed through iterative stages of research and design supported by grants from the NSF. In the early stages of these respective projects, the developers carried out small-scale teaching experiments focused on uncovering students’ ways of reasoning and developing tasks that evoke and leverage productive ways of reasoning. Instructional tasks then went through additional cycles of implementing, testing, and refining over a series of whole class teaching experiments. In the last stages of research and design, instructors who were not involved in the development implemented the materials and provided feedback.
Over the course of the last 10+ years, these extensive and ongoing research projects have produced many results, including: instructional sequences comprised of rich problem-solving tasks, instructor support materials, research showing positive conceptual learning gains (e.g., Kwon, Rasmussen, & Allen, 2005; Larsen, Johnson, & Bartlo, 2013), insights into how students think about these concepts (e.g., Larsen, 2009; Wawro, 2014; Keene, 2007) and the identification of specific challenges that instructors face as they implemented these materials. Some of the difficulties experienced by instructors implementing the materials include: making sense of student thinking, planning for and leading productive whole class discussions, and building on students’ solution strategies and contributions (e.g., Johnson & Larsen, 2012; Speer & Wagner, 2009; Wagner, Speer, & Rossa, 2007).
TIMES Project
The TIMES grant will allow us to better understand how to support instructors as they work to implement these three inquiry-oriented curricula materials. We have a three-pronged instructional support model, consisting of:
(1) Curricular support materials – These materials, created by the researchers who developed the three curricular innovations, include: student materials (e.g., task sequences, handouts, problem banks) and instructor support materials (e.g., learning goals and rationales for the tasks, examples of student work, implementation notes).
(2) Summer workshops – The summer workshops last 2-3 days and have three main goals, 1) building familiarity with the curricula materials, including an understanding of the learning trajectories of the lessons; and 2) developing an understanding of the intent of the curricula in particular and inquiry-oriented instruction in general.
(3) Online instructor work groups – The online instructor work groups have between 4 and 6 participants, each currently implementing the same curricular materials. Each group meets for one hour a week and works on selected lessons from the curricular materials. For each of the focal lessons, we discuss the mathematics and plan for implementation. Then, after instructors have taught the lesson, the group watches video clips of instruction with a focus on student thinking. The goal is to help instructors develop their ability to interpret and respond to student thinking in ways that support student learning. Every meeting also has time dedicated to address specific and immediate needs of the participants (e.g., difficulty with managing small group work, a particularly challenging task, strategies for getting students to share ideas).
Over the course of this three-year grant, we will offer these supports and investigate their impact. Our research will focus on the relationships and interactions among the supports, the instructors, and their instructional practices. In addition to assessing the impact of the support model, project data will be analyzed to identify aspects of the supports and instruction that have a positive impact on students’ learning.
We hope that this post provided a useful description of the inquiry-oriented instructional approach that can help instructors think about how they might (or already do) incorporate some of these ideas into their teaching. For instance, regardless of how you currently teach, really inquiring into your students’ thinking (not just their answers) can provide you with very valuable insights. We also hope that, after reading this post, you will be encouraged to see that some tangible, practical steps are being taken toward scaling up and supporting inquiry-oriented instruction.
If you are interested in learning more about the curricular materials or this project please visit http://times.math.vt.edu/. If you are interested in learning more about becoming a TIMES fellow, please contact Estrella Johnson (strej@vt.edu) for Abstract Algebra, Christy Andrews-Larsen (cjlarson@fsu.edu) for Linear Algebra, or Karen Keene (kakeene@ncsu.edu) for Differential Equations. We are the principal investigators on the project and would be glad to hear from you if you are interested in learning more.
References
Johnson, E. M. S., & Larsen, S. (2012). Teacher listening: The role of knowledge of content and students. Journal of Mathematical Behavior, 31, 117 – 129.
Keene, K. A. (2007). A characterization of dynamic reasoning: Reasoning with time as parameter. The Journal of Mathematical Behavior, 26(3), 230-246.
Kwon, O. N., Rasmussen, C., & Allen, K. (2005). Students’ retention of mathematical knowledge and skills in differential equations. School Science and Mathematics, 105(5), 227-239.
Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. The Journal of Mathematical Behavior, 28(2), 119-137.
Larsen, S., Johnson, E., & Bartlo, J. (2013). Designing and scaling up an innovation in abstract algebra. The Journal of Mathematical Behavior.
Rasmussen, C., & Kwon, O. N. (2007). An inquiry-oriented approach to undergraduate mathematics. The Journal of Mathematical Behavior, 26(3), 189-194.
Wawro, M. (2014). Student reasoning about the invertible matrix theorem in linear algebra. ZDM, 46(3), 389-406.
Speer, N. M., & Wagner, J. F. (2009). Knowledge Needed by a Teacher to Provide Analytic Scaffolding During Undergraduate Mathematics Classroom Discussions. Journal for Research in Mathematics Education, 40(5), 530-562.
Wagner, J. F., Speer, N. M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. The Journal of Mathematical Behavior,26(3), 247-266.
]]>When I first started teaching, I was mystified (and, frankly, at times panicked) at the thought of having undergraduates work with me on research. I realized this was part of the job, part of my institution’s mission, but I just couldn’t figure out how it would be effective. Sure, these students were bright, eager and motivated to learn, but how much could they contribute with such limited time? A typical research experience might be 8-10 weeks during the summer (full time) or 10 hours a week during a semester; best case, I might find a student who would work with me for a couple years in this way. I had just finished six years in grad school and still felt like I knew nothing. On top of that, my research is at the intersection of computer science and math with applications in the domains of engineering and biology – would I be able to find students with experience in even two of these fields? As it turns out, I would soon discover how powerful research with undergraduates can be, and I’d like to share some of the lessons I’ve learned over the years.
I remember thinking I should come up with a list of very specific problems, solvable with limited time and background, before trying to find students. Looking back, I think I was trying to mirror the familiar classroom experience, where a careful syllabus provides clear expectations to students with specified prerequisites. It turns out that specific problems, while useful in giving students an idea of the research area, almost never provide the direction we end up moving toward. I suppose I should have seen that coming, as research never does go as planned. It can’t be clean and predictable just because undergraduates are involved. I’ve learned to embrace the prospect of the unknown, instead looking for students with more broadly defined interests, such as computational biology or robotics.
Once I’ve found students, the most successful approach comes from guiding them along paths that suit their own passions and interests. Many times, they don’t really know these in advance, so I view the first part of my time with them as a chance to let them play with different types of problems. This may mean coming up with examples that fit a given set of combinatorial properties, reading and presenting a research paper on an algorithm we hope to generalize, or building modules of Mathematica code to explore properties of certain matrices. One summer, my students built little robots with microcontrollers and old VCR boxes; that activity resulted in one student determined to continue working on hardware and another determined to work only on software (a surprise to her). This “discovery period” can be truly transformative for some, and the reward of knowing that I helped a student find out a little more about herself is one of the main reasons I became a faculty member.
After identifying her interests, the student begins to get a glimpse of how research feels by facing the energizing and terrifying prospect of defining her own problems and pathway. I have the students pitch their own projects and timelines (which are always too ambitious) and work with them to create several milestones along the way. I usually let them start off on their overly optimistic timeline, but know they generally won’t make it past the first milestone. The students track their own progress by maintaining a website with blog updates on their work. This serves two purposes: (1) it helps me understand what they have done, what they understand and where they are stuck, and (2) it gives them something to reflect on at the end of the experience. Throughout this time, I am very conscious of each student’s confidence level. For some, the unfamiliarity of not having lectures, assignments and a textbook can cause them to doubt their own ability. Explicitly telling them that research is coupled with a feeling of the unknown and relating imposter syndrome stories of my own and of other researchers often gets them back on track. This is one of the things I enjoy the most, mentoring students who are excited to work on problems related to my research and helping them find the confidence to jump-start their own research careers. It is an amazing feeling when they tell me years later that it was that seed of a research experience which grow into their passion, whether it is pursuing a graduate degree with an NSF fellowship or becoming a teacher who will inspire new generations or working on cutting edge technology at an industry leader.
I used to worry that working with students on problems that interest them might be a distraction from my own research. I had, in earlier years, been asked by a local roboticist to help advise students on a project of his. I had no experience in robotics, but saw the excitement on the students’ faces. As the only faculty member positioned to co-advise, I knew that my saying “no” would crush their hopes. As I became more involved in their projects, helping to build a 3D printer from a kit, I became enamored with microcontrollers and the “maker” movement. At that point, it was just fun to build stuff and create an interactive project with a few lines of code; in my mind, it was completely decoupled from research. Then, two years ago, I began thinking about applying for an NSF grant, but was stumped as to what exciting research pathway I could propose. Serendipitously, a roboticist, whom I’d met through this robotics work, sent me a link to a TED talk with quadcopters cooperating to catch a ball in a net. A light bulb went off, and I saw a connection to the theoretical core of my research. I took a risk and proposed this robotics-based research program. To my delight and surprise, my proposal was funded!
This was not the only time a surprising connection came from working with a student. In fact, my first undergraduate researcher impacted my career in a way she may not even know. She sought me out one day as she was double-majoring in computer science and mathematics and had been told that my research straddled both. I felt completely unprepared as I had no list of specific problems; instead, I described my research on the fly, and my work in computational biology piqued her interest. She has since become a co-author with another undergraduate and two biochemists, but perhaps her most unexpected gift to me was a new collaboration, which I value deeply. As part of writing up her thesis, this student wanted to provide background on Lie groups, and she sought out the expertise of a mathematics professor. This professor saw a connection between the thesis work and the research area of another math professor. She encouraged us to start talking and thus began a collaboration for which I will be forever grateful.
As faculty at a research liberal arts institution, involving undergraduates in research is a core part of what we do. These budding researchers may not always be able to produce significant original contributions, but I can genuinely say my research path has been dramatically transformed for the better because of them. This is the biggest lesson I have learned: don’t underestimate the power of undergraduate researchers. They might directly contribute to your research, becoming co-authors on your next publication, or provide context and intuitions from things you’ve never thought about. And one day, those student interactions just might result in a connection that will transform who you are as a researcher. That connection could lead to a fantastic new collaborator or even a successful grant proposal. And, to top it all off, you get the amazing reward of knowing you played a small role in helping them in their own journey of discovering where to go next.
]]>I serve as chair of the Mathematical Association of America’s (MAA’s) Committee on the Undergraduate Program in Mathematics (CUPM). Approximately every ten years, CUPM publishes a new curriculum guide, with the primary goal of assisting mathematics departments with their undergraduate offerings. Over five years in the making, the 2015 Curriculum Guide to Majors in the Mathematical Sciences encourages departments to engage in a process of review and renewal, by examining their own beliefs, interests, resources, mission, and particularly their own students in designing or revising a major in mathematics or, more generally, in the mathematical sciences. In the remainder of this blog post, we discuss the history, development, process, and key characteristics and recommendations of the 2015 Guide.
CUPM has a long and distinguished history, going back to curriculum reports in the 1950’s. The CUPM recommendations of 1991 were limited to the mathematics major and can be found in the report, Heeding the Call for Change. The CUPM Curriculum Guide 2004 addressed the entire undergraduate mathematics curriculum, with a particular emphasis on service courses of the first two years and pedagogical methods. In 2011, when we began thinking about preparing this guide, we decided to revisit the undergraduate mathematics major, as almost 25 years has passed since the MAA has done an in-depth study of the mathematics major. The 2004 Guide included important basic principles that apply to all mathematics courses. However, the applications of mathematics have expanded so much of late that CUPM felt it important to address the full scope of majors in the mathematical sciences.
At national meetings over the last five years, CUPM has held numerous focus groups, presented ideas to groups of chairs and coordinators of undergraduate mathematics programs, and solicited advice from directors of graduate programs, at both Master’s and doctoral levels. Several years of these discussions of content and cognitive goals have led to a consensus of what might be considered the fundamental components of any mathematical sciences bachelor’s level program. The draft of the 2015 Guide was submitted to the member organizations of CBMS, and many CBMS members created Association Response Groups that sent us valuable suggestions and comments. We were pleased with the response and have incorporated almost all of their advice.
The recommendations and principles in the Curriculum Guide are meant to drive a process of intentional design within a framework of basic cognitive and content goals that have been agreed upon by the mathematical community across many types of institutions. As the title reflects, this report is intended as a guide, not as a prescription. It is not an accreditation document. The members of CUPM and the Curriculum Guide Steering Committee found fundamental agreement as to what should be the core of any mathematical sciences major. From the many applied fields to the doctoral programs in pure mathematics, we heard much the same basic ideas. Although we spent quite a lot of time refining the words in the recommendations, there was very little disagreement on the principles. Nevertheless, CUPM emphasizes that each department is expected to examine its own program(s), mission, students, resources, and interests as it considers its own goals and pathways. At its August meeting in 2014, the Board of Governors of the MAA approved the Cognitive and Content Recommendations discussed in the Overview of the 2015 Guide and also reaffirmed the Principles of the 2004 Guide.
The real planning and intentional structure of the mathematics major derives from the cognitive and content goals and from recognizing that a department must design courses in the core to advance students toward the mathematical maturity that is the sum of the stated goals.
We believe that a successful major program offers a program of courses to gradually and intentionally lead students from basic to advanced levels of critical and analytical thinking, while encouraging creativity and excitement about mathematics. This requires that students be expected to do more each semester to develop and grow so that they achieve the goals eventually. This is the essence of the generic mathematics major. Each department should incorporate the core goals by offering their own intentional design of appropriate experiences and electives for their students. The Overview section of the 2015 Guide is the closest we get to “defining” the major.
There are many parts to the current Guide. There is a downloadable brochure, which briefly explains the contents and can be used to discuss the main points with administrators and others. A printed abbreviated version of the 2015 Curriculum Guide will be sent to all department chairs. The bulk of the details on all the course areas and program areas will appear online only. Carol Schumacher of Kenyon College co-chaired the Curriculum Guide Steering Committee and is co-PI with me and Michael Pearson of the MAA on the NSF grant [DUE-1228636] and the Educational Advancement Foundation grants that made this project possible. Paul Zorn is the editor, and has generously offered wise counsel to Carol and me in preparing all of the disparate parts for distribution.
The Introduction and the Overview are essential reading. They describe the context and essential components of the report and the recommended goals for all mathematical sciences. Although this Guide makes recommendations about a great variety of programs that center on new and developing applications of mathematics, CUPM urges departments to continue to offer an option for a major in mathematics that is perhaps considered traditional in scope and purpose. Some departments call this the “pure” mathematics track; some call it the preparation for doctoral programs in mathematics. Such programs should rely on educational processes that lead to mathematical maturity, should cultivate appreciation for mathematics for its own sake, and, while incorporating some applications of the theory, they are centered on the theoretical nature of the subject. All undergraduate major programs in mathematics/mathematical sciences have common elements however, and the cognitive and content goals reflect those common threads.
Focus groups and MAA SIGMAAs (MAA’s special interest groups) generated many ideas for an in-depth view of undergraduate course areas, and of course, provided a modern look at the many applied areas that beckon mathematicians. The Mathematical Sciences in 2025, a publication of the Board of Mathematical Sciences and Its Applications of the National Research Council, provided CUPM with many examples of fast-growing career opportunities for mathematicians. The INGenIOus Report in 2014 added to the call for career-readiness in college graduates. These resources and the members of the Curriculum Guide Steering Committee were instrumental in the selection of the many Course Areas and Program Areas featured in the report. Invitations to the members of the nearly 40 Study Groups were informed by the Steering Committee, as well.
Some MAA committees provided a natural resource as, for example, the American Statistical Association-MAA Joint Committee on Statistics or the CUPM Subcommittee on Mathematics Across the Disciplines. Of particular importance are the reports on the Preparation of High School and Middle School Teachers, which are reports from the MAA’s Committee on the Mathematical Education of Teachers. CUPM is pleased that more than 200 individuals have contributed to these group reports. The willingness of the mathematical community and many of our colleagues in allied fields to commit their time to this project has been very gratifying. I urge everyone to read these reports. They represent the views of a large and varied group of mathematicians and scientists who teach undergraduates.
Because a successful department is more than just its course and program offerings, the Guide contains a large section on matters “beyond the curriculum.” These are essential departmental and institutional policies that advance and enhance undergraduate major programs. These include articles on recruitment and retention, assessment, and placement, for example.
In some ways, the most interesting CUPM discussions were focused on Review and Renewal, the “how” to go about making change, particularly focused on meeting our cognitive and content goals. We believe that a healthy mathematical sciences program should incorporate intentional evolution and continual improvement. Every mathematical sciences department should have and follow a strategic plan that acknowledges local conditions and resources, but is also informed by recommendations from the greater mathematical community. The process of planning and renewal should be guided by consultation both within the department and with outside stakeholders at the institution and beyond. Departments should assess their progress in meeting cognitive and content goals through systematic collection and evaluation of evidence. The process of program review and departmental responsibilities in renewal are discussed in our report.
The most radical feature of this CUPM Guide is that it is meant to be dynamic in that we expect CUPM will add resource material to the Course and Program Areas regularly and do a thorough review of several course and program areas each year. Rather than trying to write a guide to the undergraduate curriculum and the major every decade, CUPM hopes to maintain an on-going revision and renewal process of its own that is directed at all of the course and program areas. We invite members of the mathematical community to submit ideas such as projects, resources, and videos, so that they can be posted online appropriately. We are preparing a submission procedure at this time. CUPM’s mission is to provide a current, meaningful, and useful resource for the design and improvement of the undergraduate major, and we welcome your input.
Feel free to contact me with your comments and suggestions at msiegel@towson.edu.
]]>Why are fractions hard to learn for so many people? There are many reasons for this, but I like to think about one in particular, a mathematical idea hiding in plain sight, from elementary school to college: equivalence relations. Consider the fraction sum 2/3 + 1/5, which we of course compute by using 2/3=10/15 and 1/5=3/15, arriving at an answer of 13/15. This raises a whole host of fundamental questions about equality: If 2/3 equals 10/15, why can we use one but not the other in evaluating the sum? Does this mean something is wrong with our idea of “equals”? Could we have used something else besides 10/15; or, in the other extreme, should we always use 10/15? This shows that often when we say “equals”, what we really mean is “equivalent”. Equivalence introduces a number of useful mathematical connections, but we must be careful in how we handle it with our students who just want to know, for instance, how to add two fractions.
We’ll see in a bit how these same ideas arise in mathematical topics from elementary school through college, and what potential difficulties they may cause. I’ll conclude by sharing some specific examples of what I do in my classes to confront these issues.
But first, a very quick primer for readers who haven’t seen how to define rational numbers with mathematical rigor: Declare two fractions a/b and c/d (a,b,c,d integers, b, d not 0) to be equivalent if ad=bc. We then verify this relation really is an equivalence relation (a relation satisfying the familiar reflexive, symmetric, and transitive properties of equality), which also means this relation partitions all the fractions into equivalence classes. It’s those equivalence classes that are the rational numbers. So the rational number 2/3 is actually the equivalence class of all fractions that are equivalent to the fraction 2/3, by our usual method of reducing to lowest common denominator.
We should also verify that this relation behaves nicely with respect to operations such as addition (defined by the formula we get from using common denominators: a/b + c/d = (ad + bc)/bd) so that anything equivalent to 2/3 plus anything equivalent to 1/5 gives something equivalent to 13/15.
All of this is more meaningful (and the proofs are much more straightforward) if we define two fractions to be equivalent if they correspond to the same length on a number line. Indeed, this more natural way to define the equivalence is why we care about it in the first place. But that doesn’t help when we want to do exact calculations, or when our fractions contain algebraic expressions. Then we really are confronted with issues of equivalence:
One more issue with some equivalence relations doesn’t show up with fractions, but does in geometry: there may be different equivalence relations on the same set. Sometimes, we want to consider geometric objects equivalent if they are congruent, or sometimes even just similar. But other times we need a finer partition that distinguishes between congruent objects. For instance, a left-hand glove is congruent to a right-hand glove, but not equivalent in most real-world settings, because they differ by a reflection; similarly, a triangle pointing up may be congruent to a triangle pointing to the left, but they convey different information, because they differ by a rotation.
Once you start looking, you can find equivalence relations in many mathematical ideas. Here is a sampling, showing that this notion of equivalence spans a wide variety of mathematical topics, from very elementary to advanced ideas. The four issues described above with fractions and geometry appear in various combinations throughout.
Well, finding equivalence relations is a fun game for mathematics people, but so what? What can we do differently after finding them? I now do two things differently in my classes based on my thinking about equivalence:
To conclude, no matter what level you teach (or learn) at, keep looking for those hidden equivalence relations lurking throughout the mathematics curriculum. With a little awareness, we can help students avoid the traps and make more sense of this powerful tool.
]]>
This two-part series is a summary of the longer paper, Textbook School Mathematics and the preparation of mathematics teachers.
TSM (Textbook School Mathematics) has dominated school mathematics curriculum and assessment for the past four decades, yet, in mathematics education, TSM is still the elephant in the room that everybody tries to ignore.
We will look at three examples of this phenomenon.
Example 1. The volume on The Mathematical Education of Teachers II ( MET2) rejects the temptation of teaching teachers only advanced mathematics and leaving them to find their own way in school mathematics. That is good. But its recommendation for the preparation of high school teachers, for example, is the completion of the equivalent of a math major and three courses with a primary focus on high school mathematics from an advanced standpoint. The suggested organizing principles for the three-semester course sequence are (MET2, p. 62):
Emphasize the inherent coherence of the mathematics of high school.
Develop a particular mathematical terrain in depth.
Develop mathematics that is useful in teachers’ professional lives.
There is no mention of teachers’ thirteen years of mis-education in TSM, much less what to do about it.
For example, we know how definitions are ignored or mangled in TSM: definitions are not important. A definition, teachers are told, is nothing more than “one more thing to memorize”. How then can they learn to start emphasizing the importance of definitions?
Take the concept of congruence: in K-8 TSM, students are taught that any two figures are congruent if they have the ‘same size and same shape,” but in high school “curvy” figures are forgotten and only polygons are considered: two polygons are congruent if corresponding sides and corresponding angles are equal. Now come the CCSSM which want congruence to have one definition in terms of reflections, translations, and rotations all through middle school and high school. Similar remarks hold for all other concepts, such as similarity, expression, graph of an equation, graph of an inequality, etc.
When we ask for such a sea change, would a genteel discussion in general and an in-depth investigation of a particular terrain be enough to bring it about?
I suggest that these three courses be used to give a systematic exposition of the high school mathematics curriculum at a level as close to the school classroom as possible, but in a way that is mathematically correct. Such an exposition will show teachers how definitions can be used productively in the school classroom as well as how school mathematics differs from TSM in terms of coherence, reasoning, precision, and purposefulness. If we want a sea change in teachers’ conception of mathematics, let us show them the way, from the ground up.
Few math departments have the resources to offer such courses, but one of them is at UC Berkeley: see the description of “Math 151–153″ in the Appendix of this article. Until we can provide teachers with a knowledge of correct school mathematics, the more esoteric recommendations in MET2—such as research experience for high school teachers—can wait.
Example 2. The CCSSM have made significant inroads in steering many topics away from TSM, but the CCSSM have also prefaced the content standards with eight Mathematical Practice Standards (MPS) for students. A confluence of unusual circumstances has created the misconception that equates the CCSSM with the MPS. The idea that, in order to implement the CCSSM, all it takes is to study the MPS has taken root. Let us take a reality check.
If teachers know correct mathematics, the substance of the MPS would be a natural side effect of this knowledge. Mentioning the MPS somewhere in the content stanards is definitely a good thing. Unfortunately, putting MPS front and center in the transition from TSM to the CCSSM puts the cart before the horse. Let us consider, for example, the second and third MPS that state:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
How would these work out for the task of writing down the equation of the line passing through two given points, \( (1,2)\) and \((3,4)\)?
TSM only teaches how to do this by rote, because the slope of a line is incorrectly defined in TSM as the difference quotient of the coordinates of two points that are a priori given on the line. But the CCSSM want slope to be defined correctly so that any two points on the line can be used to compute its slope. Then it is self-evident that both of the following lead to an equation of the line: For any point \((x,y)\) on this line,
\[ \frac{y-2}{x-1}\ = \ \frac{4-2}{3-1} \quad \mbox{and} \quad \frac{y-4}{x-3}\ = \ \frac{4-2}{3-1}\]
Now the MPS exhort students to explain how the equations come about and to critique each other’s reasoning. Given that we have only provided teachers with a knowledge base of TSM and that students continue to get TSM from their textbooks, studying the MPS will help neither the students nor the teachers in this task. Instead of encouraging this fixation on the MPS, how about first helping teachers to replace their defective knowledge (TSM) with correct mathematics?
Example 3. A recent volume Principles to Actions (NCTM 2014) has the goal of describing “the conditions, structures, and policies that must exist for all students to learn.” We will refer to this volume as P-to-A.
P-to-A makes no mention of TSM or the need to help teachers overcome the damage done to their thinking by TSM. Nevertheless, it asks teachers to use “purposeful questions” to “help students make important mathematical connections, and support students in posing their own questions”, (P-to-A, pp. 35, 36). Given teachers’ immersion in TSM, what can they say when students ask for the purpose of learning the laws of (rational) exponents? In TSM, it could only be to ace standardized tests. P-to-A also says “effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding” (P-to-A, p. 42). Since TSM gets it all wrong even in something as mundane as solving a linear equation (see p. 22–25 of this article), teachers who know only TSM will be able to do nothing but transmit TSM’s pseudo-reasoning about the procedure of solving an equation to their students. There goes conceptual understanding out the window. And so on.
P-to-A enthusiastically recommends actions to realize these and other worthy learning goals seemingly without realizing that, given their damaged knowledge base, our teachers are not ready for these actions. On the issue of how to help teachers, all that P-to-A has to say is that they should be provided with all the necessary resources and professional development they need. Nothing about TSM. Since current professional development mainly recycles TSM, how can this possibly help?
TSM is the elephant in the room that everybody tries to ignore. This cannot go on.
Let us bring closure to this discussion. TSM comes from school textbooks, so why not just concentrate on getting rid of TSM by writing better textbooks? Two reasons: (1) The vicious circle syndrome: Staff writers in major publishers are themselves products of TSM. (2) The bottom-line mentality: In order to maximize the sales of their books, publishers do not publish anything teachers (products of TSM) don’t feel comfortable reading.
At the moment, the only hope of getting better school textbooks is for teachers to reject TSM-infested textbooks. Then, and only then, will publishers listen. Helping teachers to eradicate TSM is therefore not only imperative for improving their content knowledge, but it may also be the only way to get better school textbooks written.
The author is very grateful to Larry Francis for his many suggested improvements.
]]>This two-part series is a partial summary of the longer paper, Textbook School Mathematics and the preparation of mathematics teachers.
School mathematics education has been national news for at least two decades. The debate over the adoption of the Common Core State Standards for Mathematics (CCSSM) even became a hot-button issue in the midterm elections of 2014. This surge in the public’s interest in math education stems from one indisputable fact: school mathematics is in crisis.
From the vantage point of academia, two particular aspects of this crisis are of pressing concern: School textbooks are too often mathematically flawed, and in spite of the heroic efforts of many good teachers, the general level of math teaching in school classrooms is below acceptable.
Mathematicians like to attack problems head-on. To us, the solution is simple: Just write better school textbooks and design better teacher preparation programs. I will concentrate on the latter for now and will not return to the textbook problem until the end of Part 2.
An effective math teacher has to know the subject thoroughly (strong content knowledge) and be able to communicate with students (good pedagogy). However, my own involvement with the professional development of teachers in K–8 points strongly to the fact that, as of 2014, students’ nonlearning of mathematics has more to do with teachers’ content knowledge deficit than with deficiencies in pedagogy. If our short-term goal is to get schools out of this crisis as soon as possible, a focus on this content knowledge deficit must be our top priority.
Let me make clear at the outset that the blame for this deficit has to be laid squarely at the feet of the education establishment: the schools of education, statewide and district-wide administrators, and the mathematics community. Its systemic failure to provide teachers with the correct content knowledge they need for teaching mathematics is nothing short of scandalous. When teachers are themselves students in K-12, the mathematics they learn from their textbooks is the same fundamentally and seriously flawed mathematics they will in turn teach to their students. We call this body of knowledge Textbook School Mathematics (TSM). When they get to college as preservice teachers, they are (as of 2014) generally not made aware of the flaws in TSM, and are consequently not provided with any replacement for TSM. It therefore comes to pass that when it is their turn to teach students, they can only fall back on the TSM that they learned as K -12 students. This is how TSM gets recycled in schools, and this is how a body of unlearnable mathematical “knowledge” has come to rule the classrooms of our nation.
TSM has been around for at least four decades. Some may be taken aback by this statement, because didn’t the 1989 mathematics education reform correct all that was wrong with “traditional math”? Isn’t “reform math” at least different from “traditional math”? The short answer is that the two differ mostly in the packaging but not in their underlying mathematical substance: they both suffer from the same mathematical defects. How could this be otherwise when the people who brought us the reform were themselves victims of TSM?
Let us give a brief list of the defects of TSM. Note, however, that these defects are so deep and pervasive that they cannot be fully captured by a short list.
One can easily infer from this list that we consider it imperative for teachers to know the mathematics they have to teach in K–12 in a way that is consonant with the normal development of mathematics: the definitions leave no doubt about what is being discussed; the precise language minimizes misunderstanding; the reasoning and the coherence reduce learning by rote; and finally, knowing the mathematical purpose that a concept or skill serves gives students more motivation to learn it. This list is all about maximizing the learning outcome.
A typical illustration of TSM is the way the concept of percent is presented to students. The meaning TSM gives for percent is out of a hundred. Perhaps “out of a hundred” sounds clear in everyday language, but mathematically its lack of precision is unacceptable and unusable. Is percent perhaps a number? Because TSM gives no clear definition, percent has become one of the most feared topics in middle school. Without a definition for percent, students cannot reason their way to solutions of percent problems. Therefore, when they need to find the percent of a shaded area when 6 of 41 congruent squares are shaded, students are forced to resort to the rote skill of “setting up a proportion”: solve for \(x\) in \(6/41 = x/100\). In fact, rote memorization of procedures is a hallmark of what passes for learning in TSM.
Any hope of improving mathematics learning hinges on our ability to replace TSM in the school curriculum with correct mathematics. For a reason to be explained in Part 2, the starting point has to be helping our teachers shed their knowledge of TSM and learn how to do K–12 mathematics correctly. The mathematics community should take note that this cannot be accomplished simply by teaching future teachers good, advanced mathematics. University mathematics is fundamentally different from school mathematics. In short, the latter is an engineered version of the former. For example, while rational numbers can be dispatched in three lectures in a junior-level abstract algebra course, it would not do to teach ten-year-olds that a fraction is an equivalence class of ordered pairs of integers. Similarly, one cannot tell twelve-year-olds that constant speed means the distance function has a constant derivative. And so on. If we are serious about doing our share to resolve the education crisis, university campuses across the land will have to commit to teaching correct school mathematics to teachers until TSM is no more.
This will be a serious commitment. The absence of such a commitment thus far has frustrated many teachers who are hungry for correct content knowledge but have nowhere to turn for this knowledge. The education establishment has systematically let math teachers down. There should be no illusion, however, about the heavy responsibility that comes with this commitment: teachers do not shed the habits acquired over thirteen years of immersion in TSM without a protracted struggle and without a lot of help. The help they need translates into sustained hard work on our part. This is hardly glamorous work, but if mathematicians don’t do it, who will?
The author is very grateful to Larry Francis for his many suggested improvements.
]]>