Every year, at the beginning of the school year, a group of about two dozen mathematics instructors gets together from the University of Texas at El Paso (UTEP) and El Paso Community College (EPCC). For most of a Saturday, we put on a workshop for ourselves about teaching courses for pre-service elementary and middle school teachers. We have no incentive other than a free breakfast and lunch. While we have enjoyed putting together and participating in the workshops, we did not think it was especially noteworthy. But then several outsiders pointed out to us that working across institutional lines like this, between a university and a community college, is not so common. But maybe it should be more common, because we have found our partnership to be valuable to our respective institutions and to our students.

**How did we get started? **

In March, 2012, UTEP’s College of Education initiated a joint UTEP-EPCC meeting to discuss alignment of math courses for undergraduates in the teacher education program. Issues discussed at the meeting include aligning two lower-division math courses, selecting a textbook, creating a wiki to share resources/information, advising students, and improving communication. Although we were brought together by administrators in Dean’s and Provost’s offices, we the faculty quickly took ownership of the effort.

At the second meeting, we identified a need for professional development to help instructors structure their classes to increase student’s mathematical thinking. We discussed questions like: Why should instructors attend? What should they expect to get out of it? What topics are appropriate (e.g., lesson planning, problem solving, what mathematics thinking look like)? We chose “Helping Students Become Mathematical Thinkers” to be the topic for our first workshop, which was held in August, just a few months after we got started. We chose to hold the workshop less than two weeks before the start of the fall semester because we wanted to offer instructors some ideas and resources for their courses. We called our workshop Teachers Teaching Teachers (TTT).

The first workshop went well and we had another five:

- Summer 2012 –
*Helping Students become Mathematical Thinkers*(28 attended at EPCC) - Fall 2014 –
*Fostering Mathematical Thinking*(25 attended at UTEP) - Fall 2015 –
*Big Ideas in Statistics, Insights from Practicing Teachers, and Task Analysis*(26 attended at EPCC) - Fall 2016 –
*Active Learning*(20 attended at UTEP) - Fall 2017 –
*Reading Math Textbook*(24 attended at EPCC) - Spring 2019 –
*Proportional Reasoning*(27 attended at EPCC)

**What does a typical workshop consist of? **

We offer breakfast at 8:30 a.m. to encourage participants to come early and register. Our program starts at 9:00 a.m., with an ice-breaking activity. Most of our workshops consist of two main parts: a learning-and-sharing session before lunch and a working session after lunch. The learning-and-sharing session might have two or three activities in which attendees are participating as active learners or problem solvers. In some years, these activities were facilitated by the workshop organizers: math tasks that challenge participants to think (2012), hands-on-approach in understanding areas (2014), active learning (2016), and 10 essential understandings of ratios and proportions (2019). In other years, invited guests presented topics like big ideas in statistics (2015), perspectives of elementary or middle school teachers (2015), and three-levels of reading—stated, implied, and applied (2016). In the working sessions, attendees worked in small groups to create math tasks (2012, 2016), analyze textbook tasks (2015), generate questions to guide reading of math text (2017), and analyze and respond to student written work (2019). We typically close our workshop by encouraging the participants to take an online survey.

**What does the workshop-organizing process entail? **

Organizing the first TTT workshop took the most effort because we had to start from scratch. That first summer, a group of about 6-8 of us met almost every week to plan the content and the logistics. Subsequent workshops have involved less effort and coordination because we knew our duties (e.g. logistics, food ordering), and because there has been very little turnover among the organizing committee members. We now typically have 3-6 planning meetings in the summer to prepare for the workshop in the fall semester. Some years, it may take 2-3 meetings just to decide on a topic because we allow ideas to emerge and evolve. For example, the Spring 2019 workshop started with creating a model lesson on a topic that “all” instructors could adapt and implement in their courses. We then selected proportional reasoning to be the topic, then the topic evolved into identifying key concepts necessary for understanding proportional reasoning deeply and analyzing students’ misconceptions involving ratios and proportions.

Expenses are minimal, and consist mostly of breakfast and lunch for participants. This is covered by the hosting department (UTEP or EPCC) and/or a textbook publisher.

**How did participants respond to these workshops? **

One measure of our success is that many participants keep returning to the TTT workshop each year. The workshops seem to meet the needs of the participants. The participant comments revealed that they liked the topics, activities, and student-centered discussions. They enjoyed working with peers, sharing experiences, and learning from others in a friendly atmosphere. Participants from each institution were eager to interact with their colleagues from the other school. And they enjoyed the food.

The comments also revealed participants’ growing awareness of the importance of student thinking and engagement. One participant “learned about my own classroom practice, learned to stop and reflect between activities.” Another participant acknowledged that “it’s challenging to create activities that engage students.” Other comments included “thinking must be present in the class,” “you must create a need for students to be engaged in learning,” and “it’s important to work with others in solving problems.”

**What factors contributed to successful collaboration? **

We attribute the success of TTT workshops to the mutual respect and collegiality among UTEP-EPCC math faculty in the organizing committee. We are comfortable and enjoy each other’s company. Many of our meetings are held in the evening at a restaurant where we have a chance to dine, chat, and connect in addition to work and plan the activities for a workshop. We are open to ideas and willing to try new things. For example, we ran with the suggestion of having school teachers share their perspectives for the 2015 workshop because most of our workshop participants, who are college instructors, do not teach elementary and/or middle school students. We are reflective and adaptive. For example, our first workshop lasted 7 ½ hours and our second was 5 hours. We eventually found that 6 ½ hours is most appropriate.

Even well-meaning faculty (and staff) from a community college and a university working together face challenges arising from competing institutional demands and constraints. For instance, a community college has strong incentive for its students to complete the associate’s degree, while a university has a strong incentive for its students not to put off courses in the major for too long. We at UTEP and EPCC have a built-in advantage in that we are the only 4-year university and community college, respectively, in the area. This almost forces us to work together on issues such as articulation and enrollment. Many students from EPCC eventually transfer to UTEP and many students at UTEP started at EPCC. Some students are even taking courses at both campuses at the same time. To capture this reality, we repeated to each other, “Our students are your students are our students,” sometimes modifying it to “Our students are your students are **all of our **students.”

Working together year after year, faculty from both institutions have come to appreciate that we have more in common than we have differences. Both UTEP and EPCC faculty work towards a common purpose; that is, to increase the quality of our math courses for prospective teachers who would in turn improve the math courses they teach when they become teachers. We think an element of kindness among the organizing committee members underlies our success. That is, we care for each other, our fellow instructors, and our students.

** ****What have we learned? **

In our earlier workshops, each institution (UTEP and EPCC) was responsible for different sessions. We later realized that our collaboration would be stronger if each activity is co-facilitated by one faculty member from each institution. We learned that one or two ice-breaking activities help improve the workshop atmosphere. This year, we began by having workshop participants share how they implemented the ideas learned in the previous workshop and the outcomes. We encouraged them to implement the ideas and activities of this year’s workshop, on proportional reasoning, in their courses and then share their findings at the next workshop.

** ****What are some of our challenges or unfulfilled dreams? **

Our success in offering an annual workshop may be limited to creating awareness and may not have lasting impact on changing the way our instructors teach mathematics. We were very motivated at the end of our first workshop but we were not successful with follow-up efforts once the semester started. Workshop participants were not very responsive after the workshops and the organizing committee members were busy with their own routines. Effective professional development projects require follow-up activities and support throughout the academic year, and possibly over two to three years. If we could secure funding to support such an expansion to a full-blown project with more extensive follow-up activities, we would all benefit. On the other hand, the group dynamics and motivating forces would be different, and there is the risk that our current collaboration would dissolve at the end of the funded project.

When we started TTT, we thought one of the easiest things to accomplish would be to agree on a common textbook for both institutions. It is a surprise that we still have not done this. So students who transfer from EPCC to UTEP must buy a different textbook when they take the third (upper-division) course at UTEP. We have also had difficulty attracting adjunct faculty who teach only one or two courses to attend our workshop.

**Conclusion**

We cannot say for sure which parts of our story would work at other universities and community colleges. Some of what is going on here may be very specific to El Paso, or to the people who happen to be here. But one part that is perhaps universal is the value of just getting the faculty at the different institutions together to talk to each other, and seeing what they come up with.

]]>John Ewing

American education is in crisis… I’m told. Want evidence? Look on the Internet. Search for “education crisis in America” and you will find millions of articles, essays, and (yes) blogs, all describing, explaining, and lamenting the crisis in American education. The Internet confirms it—an education crisis.

The crisis has been brewing for some time. For example, in 2012 the Council on Foreign Relations published a report from a task force chaired by Joel Klein and Condoleezza Rice. Alarmingly, it tied the crisis to national security. The forward begins:

It will come as no surprise to most readers that America’s primary and secondary schools are widely seen as failing. High school graduation rates,… are still far too low, and there are steep gaps in achievement …and business owners are struggling to find graduates with sufficient skills in reading, math, and science to fill today’s jobs. (p. ix)

https://www.cfr.org/report/us-education-reform-and-national-security

The report assumed education failure as a premise. (The actual evidence was compressed in a mishmash of NAEP scores, international comparisons, and common wisdom.)

This wasn’t new. Roughly three decades before, President Ronald Reagan’s education task force produced the famous *A Nation at Risk, *which proclaimed an education crisis, again tied to national security.

Our Nation is at risk. Our once unchallenged preeminence in commerce, industry, science, and technological innovation is being overtaken by competitors throughout the world. …… The educational foundations of our society are presently being eroded by a rising tide of mediocrity that threatens our very future as a Nation and a people. … If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war.

Again, the crisis was self-evident. The evidence was largely common wisdom (most of which was shown wrong by a subsequent report from the Department of Energy).

https://www.edutopia.org/landmark-education-report-nation-risk

These are two examples of a rich tradition—many thousands of committees, task forces, and individuals, lamenting our education crisis, cherry-picking evidence to confirm its existence, and predicting doom.

Well, I say …poppycock! The evidence is scant and often ambiguous. Test scores on international exams? Yes, not good. But the U.S. has never done well on international comparisons, and the data are more complicated than the public is led to believe. (Who takes the exams? How do tests align with curricula? How are students motivated to apply themselves.) Are NAEP scores plunging? Hardly—we wring our hands because they are stagnant or not rising fast enough. Are graduation rates falling? Nope, going up. Are more high school graduates going to post-secondary school? The fraction has tripled over the past few decades … and so forth and so on.

Let me be clear—there are plenty of things wrong with American education. I’m not suggesting for a minute that everything is wonderful, that we should revel in success. It’s not; we shouldn’t. But a crisis? A turning point? An instability portending imminent danger and ruinous upheaval? Does that describe American education today?

I suspect that most people, on reflection, will admit “crisis” isn’t quite right. But in the age of cable television and breathless breaking news, they believe, a little education hyperbole is an innocent way to capture the public’s imagination. But it’s not, and shouting “crisis” is not only wrong—it’s disastrous.

Declaring a crisis ensures that education reform starts from a deficit model. Focus on everything that’s wrong. Fix what’s broken. Concentrate on the bottom. What should we do about failing schools? How do we get rid of ineffective teachers? Which subjects are weakest? This has been the underlying model for American education for the past few decades, and it does great harm.

A deficit model guarantees regression to the mean. Focus on the worst, ignore the best, and education drifts towards mediocrity. More importantly, it draws the public’s attention only to what’s wrong, so people see education through distorted lenses. All that’s wrong is brought into sharp focus; all that’s excellent is blurred. The people responsible for that excellence become demoralized and eventually give up.

Teachers are especially vulnerable to this, and one of the goals of Math for America (the organization I lead) is to counteract this phenomenon. In our New York City program, we seek the best math and science teachers—the ones who are excellent in every way (content knowledge as well as craft). We offer them a renewable 4-year fellowship providing an annual stipend ($15,000). Most importantly, we offer them a community of similarly accomplished teachers, who take workshops or mini-courses, on topics from complex analysis to cell motility, from racially-relevant pedagogy to the national science standards. They get to choose which workshops they attend (no one needs fixing!). They also create and run about two-thirds of the workshops themselves, and they are respected—really respected—as professionals. In New York City, we have over a thousand of these outstanding teachers and offer almost 800 two-hour workshops each year. MƒA master teachers form a pocket of excellence (about 10% of math and science teachers in the City) that models what K-12 teaching could be like if we truly treated teachers as professionals. And they stay in their classrooms, at least a while longer, teaching and inspiring about 100,000 students each year.

New York State has a similar program with about the same number of teachers outside New York City. Los Angeles has another, smaller. We advocate for such programs in other places, but the details of the model are less important than the principle: To build excellence, you focus on excellence. That’s true in every walk of life, but it’s especially true in education. We have ignored that principle for several decades in American education, focusing instead on failure—on the “crisis” in American education.

Why is it so hard to move away from this crisis mentality? Mainly because of incentives. For politicians, steady progress doesn’t capture the popular imagination—a crisis does, and when it involves voters’ children, it makes for good politics. (Reagan discovered this.) For the media, especially the education media, a crisis generates readership and guarantees a livelihood. For education experts and researchers, a crisis makes their work critically important and worthy of support. For education providers (think Pearson and standardized tests), a crisis sells products. Even for people who run education non-profits, a crisis helps to secure funding. (I was once told by a board member I should add “crisis” to our marketing.) I don’t mean to suggest that these groups or individuals deliberately prevaricate, but societal incentives make a crisis advantageous. In fact, nearly everyone in education benefits from the notion of a crisis … everyone, except teachers … and students.

Acolytes of the education crisis will denounce my blasphemy. We have lots of problems, they say, and we need to mobilize our nation to solve them. Even if we’re not in crisis (that is, a turning point), a crisis is sacred; challenging the notion is tantamount to giving up. This is a profound mistake—one we’ve been making for the past 30 years.

A crisis in American education? Poppycock. We are more likely to improve American education without histrionics. And we should try.

**References**

U.S. Education Reform and National Security, report from a task force of the Council on Foreign Relations, chaired by Joel Klein and Condoleezza Rice (2012).

https://www.cfr.org/report/us-education-reform-and-national-security

A Nation at Risk: The Imperative for Educational Reform, report from the president’s Commission on Excellence in Education (1983).

https://www2.ed.gov/pubs/NatAtRisk/index.html

Education at Risk: Fallout from a Flawed Report, by Tamim Ansary, Edutopia (2007).

https://www.edutopia.org/landmark-education-report-nation-risk

Google Ngram Viewer. http://go.edc.org/failing-schools

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As a graduate student working in algebraic geometry, I was often star struck at the impressive speakers who attended the local seminars I frequented. While many of these memories are faded and vague, one instance stuck with me. About three minutes into a talk, one famous algebraic geometer in the audience stopped the speaker and asked “Why do we care about this problem?” Watching such an exchange, it occurred to me that everyone needs motivation, even top mathematicians involved in abstract research. We all need purpose. Why should our students expect any less?

I have since gained a great deal of respect for the question “When are we ever going to use this?” when asked by students. These students recognize that learning mathematics takes a nontrivial amount of effort, and they are looking for purpose. The mathematician at the seminar was no different: knowing that the speaker was going to embark on a journey that took effort to follow, they wanted purpose too.

Many of our students, whether they are majors or non-majors find meaningful purpose in realistic applications. The emphasis should be on the word realistic – students will (and should) roll their eyes if a person is buying 68 cantaloupes at a grocery store in a problem!

This is where interdisciplinary collaboration comes in. It can be challenging to find realistic applications for mathematics. What’s more, you have to figure out how much to teach about the application and how much that obscures the mathematics. When working with collaborators from outside mathematics, not only do you find great applications, you get to experience being a student again. This helps you determine how much a student might need to know or learn about your applications and contexts, as well as how much a particular context makes the mathematics harder to learn.

Over the last three years, several institutions across the country have been part of a NSF-funded grant to support collaboration between mathematics and the partner disciplines to improve the teaching of mathematics in the first two years. The project is called SUMMIT-P [1] (see http://www.summit-p.com). The work of the consortium rests on research conducted by a committee of the Mathematical Association of America culminating in a series of reports called the Curriculum Foundations. [4][5][6]. There is a variety of institutions, including a community college, small liberal arts colleges, comprehensive public universities, and large research-oriented universities. There is also a variety of partner disciplines, from engineering to biology to psychology. The mathematics courses addressed include quantitative reasoning, college algebra, introductory statistics, calculus 1, 2, 3, and differential equations. Our goal is to establish collaborative, interdisciplinary communities at our institutions that facilitate the inclusion of realistic partner discipline contexts into mathematics while incorporating mathematics into partner discipline courses.

At my institution, Ferris State University, we are working with a faculty member from social work (Mischelle Stone) and another from nursing (Rhonda Bishop) on a 2-semester hybrid quantitative reasoning/algebra course (for the connections between quantitative reasoning and algebra, see [7]). The course sequence originated out of collaboration with business faculty (see [8]). Almost every lesson in the class is couched in some application that comes from the partner disciplines. However, the strongest and most meaningful applications come in case studies that students work on at the end of each chapter. So far, we have created case studies addressing human trafficking, genocide, a disease outbreak, and construction and management of the death star. Each case study requires a brief writing assignment framed as recommendations to a supervisor or board of directors. As an example, some of the tasks involved in the human trafficking case study are:

- Examine human trafficking data to prioritize resource allocation,
- Prepare a budget for the medical needs of human trafficking victims in a location,
- Forecast fundraising needs for a program to combat human trafficking in hotels,
- Prepare an annual budget broken down by months for a shelter for human trafficking victims (based on assumptions about how the number of guests per month changes), and
- Determine how much food to order for a human trafficking victims’ shelter from two different suppliers while minimizing the environmental impact.

For more detail, let’s look a little closer at the last item: determining how to determine which supplier to select to purchase food for a human trafficking victims’ shelter. In discussions with Mischelle about human trafficking, Mischelle shared the challenges associated with ordering supplies, especially at scale, while balancing a concern for the environmental impact of shipping the supplies. This leads to a desire to buy local and solicit donations. For smaller shelters, this is reasonable. For larger shelters, there are nontrivial logistical problems.

Our discussion led to a linear programming problem where human trafficking shelter managers have to make a constrained decision about how much food to order from different suppliers while minimizing environmental impacts measured by the total mileage involved in shipping. After a couple of preliminary questions in which students determine that they need at least 2700 meals per month (assuming a 30-day month) and want to spend less than $7500 per month, they are confronted with the following problem:

We need to decide how many trucks of food to order from each supplier while minimizing the total number of miles driven by all of the trucks from the two suppliers.

Our data is as follows:

- Our first supplier is 250 miles away. The new supplier is 400 miles away.
- Trucks from the first supplier carry 300 meals each. The trucks from the new supplier can carry 900 meals each.
- Each truck of food costs $1,500 (from either supplier).

We determined the minimum number of meals and maximum costs earlier. Within these constraints, how many trucks should we ask for from each supplier in order to minimize our environmental impact?

The problem could show up in a standard textbook as:

Minimize subject to:

The problem could also show up with some context, such as determining an optimal bundle of CDs and DVDs to purchase. But students find the human trafficking context much more compelling. They *care* about human trafficking, and they might also care about the environment. The problem feels like a legitimate professional decision they could run into. They don’t care about optimal bundles of CDs and DVDs for many reasons. First of all, such a problem is woefully out of date. Commercially published textbooks adapt slowly. But even with a more up-to-date context, students would dismiss the problem as artificial since they have been making these kinds of decisions for much of the lives without resorting to mathematical techniques such as linear programming. In addition, one may also object that such a problem promotes consumerist values, but I recognize that this is not a universal concern.

One unexpected byproduct of this problem that Mischelle and I came up with is that it can be adapted to other contexts. I had brief conversations with one of Ferris’ history professors (Barry Mehler) who studies genocide about the Shoah Visual History Archive (https://sfi.usc.edu/vha). This archive contains recorded testimonials of genocide survivors from all over the world. Ferris has recently obtained access. In my discussions with Barry, I learned that providing food for refugees fleeing genocide raises a similar problem (with different parameters). In particular, this problem could be applied to current refugee camps in Bangladesh for Rohingya fleeing genocide in Myanmar.

For both the human trafficking and genocide problems, students are asked to watch a video prior to working. For human trafficking, Mischelle provided us with a video about management challenges at a human trafficking victims’ shelter in Tampa Bay. For genocide, students watch a testimonial in the Shoah archive from a refugee discussing food distribution at a camp. While neither of these videos is mathematical, they enrich and humanize the context. This allows us to tap into the “caring” and “human dimension” components of Fink’s taxonomy of significant learning [3, pg. 2], each of which are easy to miss in a math class.

The point of this is that I would never have come up with this problem without collaborating with Mischelle. I probably wouldn’t have even thought of using human trafficking in a math class. And I would not have been able to extend the problem to genocide refugees.

What’s more, once you have one problem, you can generate more by asking students to go back and reconsider the original parameters. For example, in the human trafficking problem, the solution is to order all of the food from the second supplier. One could ask whether the first supplier could lower their price sufficiently to change the outcome in their favor. This leads to deeper mathematical reasoning beyond just solving a linear programming problem. In addition, it asks students to put themselves in a different role, allowing them to see further complexity in human society.

In addition to the case studies, Mischelle, Rhonda, and I have designed role-playing simulations that open and close the second course in the sequence. The first is based on a fictional budget crisis at a rural health clinic and has few mathematical prerequisites (see http://bit.ly/RuralHealthClinic) while the second is based on the Flint Water Crisis and uses most of the content learned in the class. One unexpected consequence we have noticed is that students see more than the connections between mathematics and the other disciplines, they see connections among the partner disciplines as well!

To carry our collaboration further, we are facilitating a faculty learning community (see [2]) with three mathematics faculty, two business faculty, two nursing faculty, and two social work faculty. The members of the faculty learning community are split into three teams. Each team has one mathematician and two faculty from different partner disciplines. The teams are currently developing scenarios that will be translated into course materials for both the quantitative reasoning sequence and in the partner discipline courses. The scenarios they have developed are:

- Managing a hurricane shelter for low-income families that includes several individuals with chronic illnesses.
- Managing a 50
^{th}wedding anniversary banquet, following which contaminated food leads to an outbreak of food-born illnesses. - Examining local police-stop data for racial profiling and preparing a budget to implement recommendations to the police department.

While rich and realistic applications appeal to students’ practical desires, they may strike you as too utilitarian. There is much more to mathematics than how it is used. There is the thrill of problem solving, and there is beauty (see e.g. https://www.artofmathematics.org/). However you frame it, though, you are addressing a purpose to mathematics, even if that purpose is more intrinsic than extrinsic. These purposes can also be served by interdisciplinary collaboration, whether with those in the fine arts or those in game-design.

Collaborating effectively requires a great deal of listening. Find out what your colleagues teach in their courses. Find out what they know about what is taught in your mathematics courses. You will be quite surprised! Be patient with one another, and avoid disciplinary microagressions. One of the activities that the SUMMIT-P institutions engaged in is a fishbowl conversation: partner disciplines sit in the middle of the room and discuss questions from a protocol while the mathematicians sit along the perimeter of the room and don’t speak.

You will find language and conventions are very important when collaborating across disciplines. Create a dictionary of terms used in the partner disciplines and their mathematical equivalents. For example, economists refer to the derivative of a function as a marginal quality (marginal cost, for example). Sharing that dictionary with students will help them to see the connections between the mathematics and the application in economics.

To be clear, the kind of collaboration I am talking about happens behind the scenes, in the design of a course or course materials. There are other forms of collaboration in teaching and learning, such as team-teaching or teaching a student learning community. However you approach it, interdisciplinary collaboration can help you to define mathematical purpose for your audience, whether it is the student who wants to know why they have to learn implicit differentiation or the star professor listening to your talk who wants to know why your problem is interesting.

[1] Collaborative Research: A National Consortium for Synergistic Undergraduate Mathematics via Multi-institutional Interdisciplinary Teaching Partnerships (SUMMIT-P); proposal funded by the National Science Foundation (NSF-IUSE Lead Awards 1625771 and 1822451).

[2] Cox, M. D. (2004). Introduction to faculty learning communities. In Cox, M.D. & Richlin, L. (Eds.),* Building faculty learning communities *(pp. 5-23). *New directions for teaching and learning*: No. 97, San Francisco: Jossey-Bass.

[3] Fink, D.L. (2005). Integrated Course Design. IDEA Paper 42. Available at https://www.ideaedu.org/Portals/0/Uploads/Documents/IDEA%20Papers/IDEA%20Papers/Idea_Paper_42.pdf

[4] Ganter, S.L. and Barker, W. (Eds.) (2004). Curriculum Foundations Project: Voices of the partner disciplines. MAA Reports, Mathematical Association of America, Washington, DC.

[5] Ganter, S.L. (2009). The Curriculum Foundations Project: phase II. MAA Focus, February/March, Mathematical Association of America, Washington, DC.

[6] Ganter, S.L. and Haver, W.E. (Eds.) (2011). Partner discipline recommendations for introductory college mathematics and the implications for college algebra. MAA Reports, Mathematical Association of America, Washington, DC.

[7] Piercey, V (2017). A quantitative reasoning approach to algebra using inquiry-based learning. *Numeracy*, Vol. 10, Issue 2, Article 4.

[8] Piercey, V and Militzer, E (2017). An inquiry-based quantitative reasoning course for business students. *Problems, Resources, and Issues in Mathematics Undergraduate Studies*, Vol. 27, Issue 7, pgs. 693 – 706.

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The phrase “mathematically mature” is frequently used by mathematics faculty to describe students who have achieved a certain combination of technical skills, habits of investigation, persistence, and conceptual understanding. This is often used both with a positive connotation (“she is very mathematically mature”) and to describe struggling students (“he just isn’t all that mathematically mature”). While the idea that students proceed through an intellectual and personal maturation over time is important, the phrase “mathematically mature” itself is both vague and imprecise. As a result, the depth of our conversations about student learning and habits is often not what it could be, and as mathematicians we can do better — we are experts at crafting careful definitions!

In this post, I argue that we should replace this colloquial phrase with more precise and careful definitions of mathematical maturity and proficiency. I also provide suggestions for how departments can use these to have more effective and productive discussions about student learning.

**Three Psychological Domains**

As I’ve written about previously on this blog, a useful oversimplification frames the human psyche as a three-stranded model:

The intellectual, or *cognitive*, domain regards knowledge and understanding of concepts. The behavioral, or *enactive*, domain regards the practices and actions with which we apply or develop that knowledge. The emotional, or *affective*, domain regards how we feel about our knowledge and our actions. All three of these domains play key roles in student learning, and when we talk about “mathematical maturity”, what we usually mean is that students have high-level functioning across all three of these areas.

As a first version of a better definition of mathematical maturity, we can specify that students who are mathematically mature have highly developed intellectual, behavioral, and emotional functioning with regard to their mathematical work. When we replace our colloquial phrase with this refined three-domain language, then we can clarify more precisely the distinction between students who have good technical skills but give up too easily (i.e. mature intellectually but developing in their behaviors), or who are persistent problem solvers yet are not confident about any of their results (mature behaviorally but developing emotionally), etc.

**The Five-Strand Model of Mathematical Proficiency**

Once we have become more familiar and fluent with using language that distinguishes between the intellectual, behavioral, and emotional domains, it is useful to further specify proficiency within those domains. One means of achieving this can be found in the 2001 National Research Council report *Adding It Up: Helping Children Learn Mathematics*, where a five-strand model of mathematical proficiency was introduced. While this model was motivated by research on student learning at the K-8 level, in my opinion it is an excellent model through at least the first two years of college, if not beyond. In this model, mathematical proficiency is defined through the following five attributes (see Chapter 4 of *Adding It Up* for details).

*conceptual understanding*— comprehension of mathematical concepts, operations, and relations*procedural fluency*— skill in carrying out procedures flexibly, accurately, efficiently, and appropriately*strategic competence*— ability to formulate, represent, and solve mathematical problems*adaptive reasoning*— capacity for logical thought, reflection, explanation, and justification*productive disposition*— habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

The five-strand model and the three psychological domains weave together well. In particular, one can view the first two strands as refinements of the intellectual domain, the third and fourth strands as refinements of the behavioral domain, and the fifth in alignment with the emotional domain.

In my experience teaching students in their first two years of college mathematics, most significant stumbling blocks for students fall clearly within one of these five strands. For example, when students are able to compute a derivative correctly, but are unable to use that information to find the equation of a tangent line, then this student is succeding in strand #2 but struggling with strand #1. As another example, suppose a student is able to do routine computations and is able to explain how formulas are derived, e.g. the quadratic formula from completing the square, but is challenged by multistep modeling problems such as a max/min problem that requires both introducing and solving an appropriate quadratic function. In this case, a reasonable argument exists that the student “knows the math”, i.e. is proficient with strands #1 and #2, but is struggling to develop mastery of the strategies to apply those skills, i.e. strand #3. As a third example, for students who have a negative view of mathematics and their mathematical capabilities, as related to strand #5, it is challenging to develop the persistence and self-efficacy required to do mathematics successfully.

Much like our mathematical conversations benefit from having clear definitions, our conversations about student learning benefit from having clear and agreed-upon language to describe key components of proficiency. The five-strand model provides an excellent starting point for more clear discussions on this topic.

**Mathematical Proficiency for Majors**

For students studying advanced mathematics, whether they be mathematics majors or math minors in math-intensive major programs, the five-strand model is not a sufficient foundation for articulately discussing mathematical proficiency. In this setting, I feel that one of our most useful resources is the 2015 MAA CUPM Curriculum Guide. Specifically, the following two recommendations copied directly from the Overview to the guide provide an articulate description of some advanced behaviors and intellectual knowledge that majors should attain.

*Cognitive Recommendation 1: Students should develop effective thinking and communication skills. *Major programs should include activities designed to promote students’ progress in learning to:

- state problems carefully, articulate assumptions, understand the importance of precise definition, and reason logically to conclusions;
- identify and model essential features of a complex situation, modify models as necessary for tractability, and draw useful conclusions;
- deduce general principles from particular instances;
- use and compare analytical, visual, and numerical perspectives in exploring mathematics;
- assess the correctness of solutions, create and explore examples, carry out mathematical experiments, and devise and test conjectures;
- recognize and make mathematically rigorous arguments;
- read mathematics with understanding;
- communicate mathematical ideas clearly and coherently both verbally and in writing to audiences of varying mathematical sophistication;
- approach mathematical problems with curiosity and creativity and persist in the face of difficulties;
- work creatively and self-sufficiently with mathematics.

* **Content Recommendation 6: Mathematical sciences major programs should present key ideas from complementary points of view: *

- continuous and discrete;
- algebraic and geometric;
- deterministic and stochastic;
- exact and approximate.

At the major level, the 10 items in the CUPM Cognitive Recommendation and the four items in the CUPM Content Recommendation provide a framework that further extends both the three domains and five strand model. The Cognitive Recommendations are primarily focused on the behavioral and emotional domains and on the third through fifth strands. The Content Recommendations further refine the idea of procedural and conceptual understanding in the first two strands by emphasizing that at an advanced level, students need to understand not only the techniques and concepts themselves, but how those techniques and concepts fit together within a broader vision of mathematics.

**Putting These Into Practice**

I will end this article with a few suggestions for how departments or faculty working groups can put these ideas into action.

- Have two or three faculty jointly present these frameworks/definitions of proficiency during a department seminar or colloquium.
- Gather a team of faculty to review the structure and content of a course for first-year students using the three domain and five strand model. Which of these domains/strands are targeted for development by assignments or activities in the course? Are there any that are being unintentionally omitted from the course curriculum or structure?
- Conduct a similar exercise for a major level course or sequence, this time using the language from the MAA Curriculum Guide. Which of these goals are students being explicitly trained toward? If any of these goals are not treated within that particular course, are there other required courses within the major where students are provided the opportunity to develop in that direction?
- Design a short activity/survey for students in a particular class based on this language. Have the activity introduce the language from one of these frameworks, and ask them to identify activities or experiences in their course that they felt helped them develop with regard to those domains/strands/goals. Discuss the results of this activity/survey with a team of faculty or at a department meeting.

It is important to keep in mind that the best way to be more effective in our considerations of student learning is to frame our discussions within clear and precise definitions of mathematical proficiency. For some courses or departments, the three domain model will be sufficient for this, and for others the five strand model or MAA Curriculum Guide goals will be needed. In any event, we need to move beyond overly-vague discussions of “mathematical maturity” and toward a more sophisticated language to discuss student learning.

]]>Currently, I am focusing on mitigating “learned helplessness” with respect to the study of mathematics. According to an article on the APA website (https://www.apa.org/monitor/2009/10/helplessness.aspx), newer research on learned helplessness suggests that the real issue is (lack of) control. This leads me to believe that by affording my students greater control over their own learning (within the bounds of mandated curriculum and instruction), I can deliver them from helplessness to a place where they acquire a keen sense of agency in their academic endeavors. Many of the students I teach are in my courses because somewhere along the way, their study of mathematics has primarily concerned learning to fail. I teach them how to *fall*.

Teaching Students How to Fall

On an ice-skating outing, a parent of a toddler wants the child to enjoy the experience. There are several approaches to this scenario: the parent can just let the toddler have free reign on the ice, the parent can hold hands with the child, or the child can use a skate trainer. Suppose the toddler is free to explore. This could be dangerous, as there may be no safe place for the toddler to learn to skate by trial and error. Now, suppose the parent and child hold hands to skate. Put yourself in the position of the toddler for a moment—you’re doing your best to keep up with someone whose strides are far longer, smoother, and faster than yours, you’ve got to keep one arm up at an uncomfortably steep angle with the other one frantically waving around, and losing your balance means you’ll just get dragged along. This is less than ideal. Enter the skate trainer: this solves a lot of problems for the toddler because it now becomes a situation within which our inexperienced skater has some measure of control (slower speed, ability to take breaks when frustrated or fatigued, the separation of balancing skills from skating skills). The use of the skate trainer reminds me of Amanda Serenevy’s description for the Traditional Math approach, which includes heavy scaffolding. The kind of helplessness that often results is one of dependency; math students who are almost completely reliant on the instructor to provide hints, cues, and prodding aren’t going to make much headway toward increasingly bigger ideas if they are not given the opportunity to become more metacognitive and confident in their ability to teach themselves how to learn. Moreover, the skate trainer has limited usefulness; the skill of skating is still yet to be fully exploited—the more fun and interesting maneuvers, such as jumping and spinning, would be hampered by the use of extra equipment. If the skate trainer is used moderately, tapered off, and given up during childhood, the child will learn to be resilient (i.e. comfortable with falling and getting back up) as the skills of skating emerge. However, if the child grows into an adult who is still dependent on the skate trainer, it’s much more difficult to separate the two. Adults have an affective filter that tends to inhibit the necessary risk-taking behavior that paves the way for learning. I mitigate this kind of lack of control by incorporating a different approach.

With the Conceptual Math approach, the learner is required to exert a measure of control over certain aspects of their experiences so that deeper understanding can be cultivated. For example, students may be expected to start with a lot more of their own thinking and they are encouraged to explore their own ideas and approaches as they stumble along the path of learning. This is also where I, the instructor, can practice the art of “be[ing] less helpful,” as proposed by Dan Meyer (https://www.ted.com/talks/dan_meyer_math_curriculum_makeover#t-613428).

For example, in solving a multi-step linear equation, such as 5x+24=2x+36, I enlist the help of the class in presenting the solving process. I take suggestions, and execute the orders of my students, whether it involves a mathematically legal move or not, and whether the move makes the problem easier or more difficult to solve. A student recalls the multiplication property of equality and advises that we divide both sides by 5 to rid ourselves of the 5 in the 5*x* term so that the variable is isolated. Initially, I try to cheat the system by dividing only the 5*x* term by 5, and the students call me on it right away. I praise them for catching this “error” and I continue with the division; we end up with x+24/5=2x/5+36/5. A few students look uncomfortable; this is clearly not what they expected, but they seem content to move forward. I solicit advice once more. Someone wants to move the constant term to the right-hand side. I attempt to follow the addition property of equality *literally* by *adding* 24/5 to both sides (it’s called the addition property, so I always add, right?); another student interrupts and states that, because the term is already positive, we have to *subtract* it on both sides instead. I (pretend to) protest this because it’s not called the subtraction property. A brave soul haltingly posits that we are really adding the opposite. Several students concur, and I concede the point. I take a moment to encourage the students to speak up even when they’re uncertain, as I expect the entire class to be supportive and helpful throughout our learning activities. We then have the following: x=2x/5+12/5. Someone declares that the problem is solved because the variable has been isolated on the left, but a counter-argument emerges, as there’s still another instance of a variable term on the right. After a few moments of constructive debate—usually, for this situation of variables on both sides of the equation, someone makes an analogy of using a word to define itself—the student who claimed that the problem was finished retracts the proclamation and insists that we continue solving. But we’re stuck; there is no apparent way to get the *x*-terms to combine. There seems to be the consensus that it is impossible to combine a plain *x* with a fractional *x*, and I intentionally allow this misconception to persist until we go back to analyze the problem after it’s completed. As we ponder a strategy, a student expresses consternation that the problem seems harder than it should be and suggests that we get rid of the fractions by multiplying the entire equation by the LCD. Most students nod in agreement, so we arrive at this: 5x=2x+12. Pleased to be free from the dreadful fractions, simultaneously two students suggest moving a variable term to the other side. I ask them if it matters which one I move, but I’m met with silence. I ask, “Can I just pick a term to move then?” I see more nodding, so I proceed to move the 5x to the right-hand side: 0=2x+12-5x. I “forgot” to combine like terms, and I’m promptly reprimanded for that oversight. I correct it to this: 0=-3x+12. A few students who have been quietly following along contribute that I should have moved the 2x to the left-hand side. I feign distress and slowly reach for the eraser to go back, but I’m told that I can just keep going by moving the -3x instead. I feign relief at being let off the hook for my wrong turn, and I arrive at 3x=12. From there, we find the solution x=4. We check the proposed solution by evaluating the original equation at 4, and we find that the solution checks out. Before we leave the problem, I step the class back through our process and suggest things to consider; sometimes, we re-work the problem using alternative strategies.

We learn to strike the balance between launching out into the unknown and making tentative plans to accomplish the learning task. We use our book and notes for basic information, but we allow ourselves to fall (and subsequently get back up) when we are actively engaged in learning. Our follow-up discussion includes labeling the places where we fell so that we can learn to recognize traps and create strategies to deal with them. For instance, our first decision led to falling awkwardly in our problem-solving technique because we got ahead of ourselves. In our re-work, we knew to hold off on the division by 5 until after like terms had been combined on both sides; we used our previous falling spells to think more critically and to make better decisions.

I am not a math major per se; I hold a master’s degree in Education (and I’m proud to wear baby blue). I stumbled into teaching mathematics after having spent many years tutoring students in various disciplines. Math by far was the most hated subject, and I was dismayed that so many people weren’t able to do the most basic calculations without experiencing anxiety on the level of PTSD. I thought that perhaps there was something I could do to help; I continued my studies by adding graduate math courses to my credentials, so here I am. I’m not a math genius (some areas of math are hard for me, too); I share my tales of struggle with my students to let them know that learning new things does not always come easily, and that it is OK to wrestle with a problem. I want them to become critical thinkers willing to ask questions that lead to interesting problems, and to confront those problems once they arise. How do I know they’ve changed? When I hear things such as, “It’s not as bad as I thought,” “I can help my kids with their homework now,” or “I can use this stuff.” I consider that a victory.

]]>a step in the direction of enhancing mathematical insight

for teachers and the students they teach

Many educators see value in hands-on learning. To me the essential attribute is the ability to manipulate the things one studies, letting the learner explore and tinker, gain experience and familiarity and build intuition.

However, the long-term goal of using *hands-on* is to reach *minds-on —*an understanding of, and appreciation for the abstract. One might say that the point of education is to get learners, in response to objects and events in the world around them, to continually ask of themselves, “What is this a case of?”

Normally, the move from *hands-on* to *minds-on* is difficult because it requires that one move from tangible and manipulable objects to intangible, and thus presumably, non-manipulable abstractions. Many of the mathematical objects and actions that secondary students encounter don’t have easy physical embodiments to manipulate; visual representations of abstractions that can be manipulated offer a means to experiment with ideas, tinker to adjust them, and build conjectures worthy of further investigation and proof. Seeing with the physical eye and manipulating with the physical hand can help in the transition from hands-on objects to minds-on ideas.

It is here that the computer enters. Artfully crafted software environments can present learners with visual representations of the abstractions they study. Moreover, these environments often allow the user to manipulate these representations, thereby mimicking on the computer screen the act of manipulating a tangible object that happens in the context of *hands-on* learning. Computer environments that allow users to display such images and manipulate them are giving the users a *hands-on*[1] experience with an intangible manipulable.

The larger point in all of this is that appropriately crafted software environments can serve to extend the reach of our minds, allowing us to manipulate in a sensory fashion that which we could hitherto only imagine. Further, the ability to manipulate and explore images and their interaction can well led to invention and innovation. It is these interactive images—pictures for the mind’s eye—that give this essay both its title and its impetus.

The teaching and learning of mathematics is intended, at least in part, to help us deal with the complexity of our surround. Doing so requires us as teachers and students to model that complexity and to use our mathematical tools to manipulate those models. Having built models we must also learn to cope with imprecision of these models and exercise good judgment in when and how to use them.

Models of intangible mathematical objects allow us to manipulate elements of the model to understand and explore the relationship(s) among these elements. Such models allow experimenting, interesting problem posing, the generation of ideas and conjectures. However, not everyone is comfortable manipulating symbols that act as surrogates for the objects in our surround. Many people claim to understand better when presented with a visual argument. Indeed we often hear people say “Now I see!” to indicate that they have understood something. This is probably what we mean by developing insight!

Should we consider implementing visual versions of our mathematical models? Mathematical models, *visually* expressed,[2] would consist of *images* that could be manipulated just as mathematical models, *symbolically* expressed, consist of *symbols* that can be manipulated. In many situations our current technology allows us to make such visual mathematical models. Suppose that as a matter of course we were to offer mathematical models in the form of images, screen objects that are reminiscent of, or evocative of the objects of the model in question and allow people to manipulate these screen objects in order to explore the relationships among them?

Consider the potential gains of both allowing exploration of mathematical models, both visual and symbolic, and providing teachers and students with the tools and the encouragement to explore. Students are rarely given the opportunity to control elements of their learning. Allowing students to manipulate and control the images that they use to explore the model of the situation being modeled may produce just the degree of engagement and provocation needed to get them to speculate and make conjectures. This, in turn, may lead them to a better understanding of the issue they are exploring. Further, and perhaps most importantly, it may lead us, their teachers, to a better understanding of what understanding a topic might be.

As teachers we generally agree that assessing how well we have taught and/or how well our students have understood what we have taught is best done by posing a problem that elicits a *performance* of some sort on the part of the students beyond simply parroting what was said to them either orally or in writing. Such *performance* implies change—a situation is presented and the student is asked to transform it in some way that sheds light on the problem. Asking students for performances that involve change implies that the elements of the problem situation should be manipulable in some way by the student. I’ve created a collection of ** Interactive Images** with exactly this purpose in mind. My own use of the site, and therefore the style of many of the questions I pose on it, is for educating teachers and stimulating

In particular, I like to think of three forms of performance – mapping, constructing and deconstructing.

*Mapping* is identifying the correspondence of both mathematical *objects* and mathematical *actions* across at least two different complementary representations; specifically this means interpreting how each aspect of a mathematical *object* in one of the representations is represented in the others and how the *actions*—i.e., the tools for manipulating and transforming *objects*—in each representation are related to the *actions* of the other representations.

Here is an example __[click here to get the live app]__: A function of one variable presented in symbolic form—say *x ^{2}+px+q*—is plotted in the {

Here are some questions that can elicit mapping performance:

• Drag the point around the {*p,**q*} plane by sliding the large YELLOW tick marks on the p and q axes. What happens in the right hand {*x,y*} plane?

• What conditions make the point and the parabola change color? Where are they RED? GREEN?

• What is the shape of the red/green boundary in the {*p,q*} plane?

• In the {*p,q*} plane, the boundary can be thought of as a function *q*(*p*). What is this function?

• How is it related to the discriminant of the quadratic?

• The locations of the real or complex conjugate roots of the quadratic appear in the {*x,y*} plane as large gold dots. Trace the complex roots in the {*x,y*} plane. Can you formulate a conjecture about the path of the roots as you move the point in the {*p,**q*} plane along a horizontal line? Along a vertical line? Can you prove or disprove your conjectures?

And here __[click here]__ is a second example designed to elicit mapping performance.

A rectangle (or any polygon) is drawn in the Cartesian plane and is also depicted as a point in the {perimeter, area} plane. Here are some questions that can elicit mapping performance:

• Every point in the first quadrant of the {width,height} plane corresponds to a rectangle.

• The applet allows you to generate either

•• a family of rectangles by moving the GOLD point along a height = constant/width curve, or

•• a family of rectangles by moving the GOLD point along a height+width = constant curve.

• Can you explain the nature of the curves generated in the {Perimeter,Area} plane as you drag the GOLD point in the {width,height} plane? qualitatively? analytically?

• Can you find the region(s) in the {Perimeter,Area} plane that correspond to all rectangle with a 1:3 aspect ratio? with a 3:1 aspect ratio?

Constructing interactive images involves using the primitive elements of a mathematical topic—e.g. points, circles and lines in the case of geometry—or the constant function and the identity function in the case of algebra – to build more complex mathematical objects. These objects, the relationships among them and the way(s) in which they be manipulated constitute a mathematical model, visually expressed.

Here __[click here]__ is an example with sample questions.

-> Given: A line segment (purple) whose length is fixed and known.

-> Given also a line segment (blue) of fixed length drawn to its midpoint and a third line segment (green) of fixed length perpendicular to it.

• Is it {always, sometimes, never} possible to build a triangle which has one of the line segments as a side and the other line segments as a median and an altitude to that side?

A second example of construction in geometry __[click here]__:

-> The length of one side AB (purple) and the two diagonals AC (green) and BD (blue) of a parallelogram are fixed and known.

• Can you construct the parallelogram ABCD ?

An example of construction in algebra __[click here]__:

• Build a polynomial by multiplying and transforming products of linear functions.

• Enter a target polynomial of order *n* = 1, 2 or 3.

and a second example of construction in algebra __[click here]__:

• Drag the yellow dot in the left panel.

• If the curve in the right panel was a plot of the the function *f*(*x*), what would the algebraic expression of *f*(*x*) be?

• What questions could/would you put to your students based on this applet?

Deconstructing Interactive Images involves decomposing an image into component parts, e.g. hypotenuses of triangles that may be part of a complex geometric diagram in order to uncover relationships among and within the mathematical objects in the image. In cases where the image is a graph, with polynomials or rational functions for example, deconstructing can mean decomposing the functions into the linear functions that were combined to produce them. These more elementary objects, the relationships among them and the way(s) in which they be manipulated constitute a mathematical model, visually expressed.

Here is one example __[click here]__:

• A blue rectangle is inscribed in the green square.

•• What fraction of the area of the green square is occupied by the blue rectangle?

•• What fraction of the perimeter of the green square is the perimeter of the blue rectangle?

•• Drag the GOLD dot. Can you explain the shape of the curves in the right panel?

• Now let a blue square be inscribed in the green square.

•• What fraction of the area of the green square is occupied by the blue square?

•• What fraction of the perimeter of the green square is the perimeter of the blue square?

•• Drag the GOLD dot. Can you explain the shape of the curves in the right panel?

• What questions could/would you put to your students based on this applet?

And a second example of deconstruction in geometry __[click here]__:

• A circle of radius 1 circumscribes a regular polygon of *n* sides. Inside the regular polygon is an inscribed circle. In the limit of a very large number of sides the area and perimeter of both the inner and outer circles approach those of the polygon.

•• Write an expression for *A*(*n*), the area of an *n* sided regular polygon inscribed in a unit circle.

•• Write an expression for *P*(*n*), the perimeter of an *n* sided regular polygon inscribed in a unit circle.

•• Contrast the rates at which *A*(*n*) and *P*(*n*) approach their limits.

• Challenges:

•• The number n of sides grows while the length *S* of each side gets smaller and smaller.

•• How does the product of *n* and *S* behave? How do you know? Can you prove it?

•• The area of a UNIT circle is π and its perimeter is 2*π*.

•• How do you convince a student that the area of a circle is NOT half its perimeter?

• What other questions could/would you ask you students based on this applet?

An example[3] of deconstruction in Algebra __[click here]__:

• Choose factoring to factor a quadratic function *f*(*x*). Then enter your function *g*(*x*) in the form *a*(*x*+*b*)(*x*+*c*).

• What can you learn about possible errors in factoring by examining the difference function *f*(*x*) – *g*(*x*).

• What questions could/would you ask your students based on this applet?

A second example of deconstruction in algebra __[click here]__:

• Enter a function *f*(*x*) in the green box at the top center of the screen.

• Explain how the translation, dilation and reflection transformations of your function are all instances of composing that function with a linear function.

• What questions could/would you put to your students based on this applet?

The central question I have tried to address is How can we use interactive images to enhance and extend the ways learners (both teachers and students) use such interactive activities to scaffold invention and innovation?

Having devoted more than five decades of my professional life to the endeavor, I am remain optimistic about the future of computers and the “*pictures for the mind’s eye*” that can be generated with them in mathematics and science education.

One reason to be hopeful is the amount of attention and concern about the future of mathematics education that is currently being expressed in the media. Given this degree of concern one hopes that society will make the necessary investment of intellectual and fiscal resource necessary to address the issues that it regards as pressing. In an earlier blog[4] I wrote about the one of the reasons a society maintains an educational system that includes mathematics; to provide people with the intellectual tools to model the world they encounter in the practical, economic, policy and social aspects of their lives.

A reason that I’m pleased at the existence of this AMS blog is that public discourse about mathematics education, as well as the consequent question of how well the system we now have helps us attain our goals for educating people in mathematics will increase and become more substantive. I write in the hope that incorporating new visual approaches to mathematics more fully and richly into the educational process may help us move forward in attaining those goals.

ENDNOTES

[1] More properly a* hands-*[mediated by mouse]-*on* experience

[2] Some illustrative examples of what is meant by the notion of manipulable interactive images as well as all of the examples in this essay can be found in interactive form ** HERE**. While these examples were designed to enhance and deepen understanding and insight for

[3] “FOIL” (First, Outer, Inner, Last) is a common school-mnemonic for (but limited to) expanding products of two binomials.

[4] https://blogs.ams.org/matheducation/2018/12/01/a-physicists-lament/

]]>I have been worrying a lot about mathematics education for over a quarter century now. While many university mathematicians who get involved in mathematics education focus on the need for new teaching methods, I have been drawn to examples of failure of the US curriculum to deal properly with basic ideas.

One of the first such ideas I identified was place value, or to be more precise, the base ten place value notational system for whole numbers (and later, decimal fractions). This is the bedrock of school mathematics, and it is used in almost everything that is done day-to-day with mathematics. We ought to try to get this as right as possible, and to have students learn it as well as possible. Yet mathematics education research indicates that we fail rather badly to do so.

Susanna Epp and I wrote a lengthy discussion [Epp, Howe 2006] of the details of the principles and techniques that constitute base ten arithmetic. This was organized around what we called the *five stages of place value*. These are summarized by example as follows.

$$352 \ \ = \hskip 2.4 in$$

$$\ \ \ \ \ \ = \ 300 \hskip .6 in + \ \ \ 50 \ \ \ \ \ \ \ \ \ + \ \ \ 2 \ \ \ \ \ $$

$$\hskip .25 in = \ 3 \times 100 \hskip .38 in + \ \ 5 \times 10 \ \ \ \ + \ \ \ 2 \times 1$$

$$\ \ \ \ \ = \ 3 \times(10\times10) \ + \ \ 5 \times 10 \ \ \ \ + \ \ \ 2 \times 1$$

$$\ \ \ \ \ \ \ \ = \ 3 \times 10^2 \hskip .4 in + \ \ 5 \times 10^1 \ \ + \ \ \ 2 \times 10^0.$$

The first stage, $352$, is just the standard base ten notation for the number. The second stage, $300 + 50 + 2$, indicates that each digit in the number stands for a number of a special kind, and the number itself is a sum of these special numbers. These numbers have a mathematical description – they are digits times powers of 10 (as is made explicit in the fifth stage), but there has been no simple reference term for them in the mathematics education literature.

Recently, the textbook [Beckmann 2017] has used the name *place value parts* for these numbers, and we will adopt this term here. The place value parts are in some sense the atoms of the base ten system, and general whole numbers are like molecules, obtained by combining atoms. The lack of a simple term has impeded focusing on the place value parts as the basic building blocks of the system.

The place value parts have multiplicative structure, and the third and fourth stages make this structure explicit. The third stage displays each place value part as a digit times a *base ten unit *– a place value part with digit 1. The fourth stage exhibits the base ten units as also being products, of a certain number of factors of 10, which is the base of the system. The fifth stage then uses the standard notation of exponents to indicate each base ten unit by simply recording the number of factors of 10 used to make it. It exhibits the number as what might be called a *polynomial in 10*. To indicate the relation between the place value parts and the size of numbers, we also refer to the exponent as the *order of magnitude*.

Together, the five stages reveal the basis for the extraordinary power of base ten notation: it is using all the power of algebra – addition, multiplication, and exponentiation – simply to write numbers, and it employs the clever convention of place value, which allows each place value part of a number each to be indicated simply by its associated digit. This of course is mediated by 0, the symbol for zero, one of the all time great mathematical inventions.

Of course, mathematicians understand this very well, and translate almost unconsciously between the first stage and the last. However, there is evidence in the mathematics education literature that most students do not master this structure. In the 1980s, C. Kamii published several papers [Kamii 1986], [Kamii, Joseph 1988], showing that 3rd and 4th grade students did not understand the significance of the digits in 2- or 3-digit numbers. More recently, E. Thanheiser [Thanheiser 2009, 2010] has shown that many, probably the large majority, of students were arriving in college not understanding the third stage. This is relevant to U.S. mathematics education not only as a measure of current quality, but also for projecting future quality, since as Thanheiser points out, teachers must know the third stage if they are to teach addition and subtraction conceptually. The recent book [Newton 2018] gives examples showing how US K-12 mathematics instruction may fail to develop base ten structure adequately.

It took me several more years to appreciate that, in fact, although mathematicians may take the five stages as common sense, looked at from the educational point of view, each successive stage represents a significant conceptual development, that can take a year or more to get children to understand. The first stage starts in first grade, or even Kindergarten, when students are introduced to 2-digit numbers. In fact, the second stage (aka *expanded form*) is often stated at about this time, and may or may not be used in a conceptual fashion. The book [Newton 2018 ] shows that, even the meaning of the digits in two-digit numbers is often taught inadequately. The third stage has to wait until students can deal with multiplication, so at least until 3rd grade if you follow the Common Core State Standards for Mathematics (CCSSM), although it can be dealt with implicitly with manipulatives somewhat earlier. The fourth stage requires considerable comfort with multiplication, and in particular, should involve appreciation of the Associative Rule/Property for Multiplication, which is arguably the subtlest of the Rules of Arithmetic, so could probably not be readily absorbed before 4th grade, perhaps 5th. The fifth stage requires a grasp of powers and exponents. For powers of 10, CCSSM calls for using whole number exponents in 5th grade, but the general idea of whole number powers comes in 6th grade. Thus, preparing a student to grasp the five stages in a conceptual way requires most of elementary school. And the evidence is, that we largely fail to do that.

The CCSSM pay considerable attention to the issues of place value, and in so doing provide a considerable advance over previous versions of mathematics standards. In particular, one of the summary headlines for Number and Operations in Base 10 in 5th Grade is “Understand the place value system”. However, it does not explicitly formulate the five stages.

One advantage of working with the five stage scheme is that it helps one to focus on the place value parts. A crucial feature of the base ten system is that the operations of addition and multiplication are simply combinations of operations with two place value parts (supplemented with regrouping – replacing 10 of some unit with 1 of the next larger unit). This follows from the Rules of Arithmetic (aka, Properties of the Operations). For addition, the two parts should even have the same order of magnitude.) Moreover, the results of such operations are given by the basic number facts (addition facts or multiplication facts), times an appropriate base ten unit. But it hard to teach or understand these points if there is no name for the place value parts.

The place value parts are also the key to comparison and estimation. Here the salient fact is that any place value part of a given order of magnitude is larger than any part of a smaller magnitude. It is at least 10 times as large as any part of two or more orders of magnitude less. (In fact, although place value parts of only one magnitude difference may be approximately the same size, for example 1,000 and 900, about half the time, the larger part will be 10 or more times as large.) It follows that any number is well approximated by the sum of a few of its largest few place value parts, and for practical purposes, can (and should) often be replaced by this. This is of course the topic of rounding, but provides the perspective that the place rounded to is not so significant at the number of places kept (called* significant digits* in the context of scientific notation).

Perhaps the value of thinking in terms of place value parts is revealed most clearly in long division, the most troublesome topic in whole number arithmetic. Long division in the whole numbers is about the operation of division with remainder. Given a number $n$ and a divisor $d$, we are looking for a quotient $q$ that is the largest whole number such that $qd \leq n$. That is, $n = qd + r,$ with the remainder $r$ being less than $d$. How do we find $q$? If $p = a 10^{\ell} $ is the largest place value part of $q$, then $(a+1)10^{\ell} > q$, so that $(a+1) 10^{\ell} d > n$. That is, $p$ is the largest place value part such that $pd < n$. The converse also holds. So to find the largest place value part of $q$, we should look for the largest place value part $p$ such that $pd \leq n$. Inspection of the long-division algorithm will reveal that this is exactly what it does. The smaller place value parts of $q$ are then found by repeating the process with $n’ = n – pd$, and continuing in this manner. Being able to use the language of place value parts might make the process more transparent.

A common current mantra is that teachers should understand and teach mathematics conceptually. What does this mean for whole number arithmetic? I would contend that it mostly amounts to

- i) understanding the five stages of place value;

along with

- ii) knowing the sums (and differences, when appropriate) of two place value parts of the same order of magnitude; and how this is a consequence of the Rules of Arithmetic together with the structure of the place value pieces;
- iii) knowing the products of two place value pieces, and how this is a consequence of the Rules of Arithmetic together with the structure of the place value pieces;
- iv) understanding how the Rules of Arithmetic combine with ii) and iii) to allow multidigit computation; and
- v) understanding the relative size of place value parts, and how this enables efficient approximation of multidigit numbers with numbers having few non-zero place value parts.

This starts with knowing the five stages, so I would like to see this as a standard part of math preparation for elementary teachers. In seminars I have run with practicing teachers through the Yale Teachers Institute, I have been surprised with the alacrity they show in adapting the five stages to their classroom. One teacher has taken the trouble to write and testify how the five stages seem to have helped her (high school) students.

It seems possible that the five stages can provide an avenue to teaching whole number arithmetic (and beyond this, decimal fractions) in a student-friendly and conceptual manner. Let’s make the five stages of place value a standard part of teacher preparation.

**References**

- [Beckmann 2017]

Beckmann,S. (2017). Mathematics for Elementary Teachers with Activities (5th Edition), Pearson Education, London and New York, 2017. - [Epp, Howe 2006]
- [Kamii 1986]

Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education,. 1, 75-86. - [Kamii, Joseph 1988]

Kamii, C., & Joseph, L. (1988). Teaching place value and double-column addition. Arithmetic Teacher, 35(6), 48-52. - [Newton 2018]

Newton, X, Improving Teacher Knowledge in K-12 Schooling: Perspectives on STEM Learning, by Xiaoxia A. Newton, Palgrave Macmillan 2018 - [Thanheiser 2009]

Thanheiser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40, 251–281. - [Thanheiser 2010]

Thanheiser, E. (2010). Investigating further preservice teachers’ conceptions of multidigit whole numbers: Refining a framework. Educational Studies in Mathematics, 75(3), 241-251.

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For some writers, the reference may be to standardized testing (SAT, GRE, etc.). Certainly these are gatekeepers. Is this privilege ‘unearned’? I don’t know. That argument is for the College Board and the Educational Testing Service to make. I will argue, however, that the whole practice of judging a person’s fate in life by her or his performance on a single test, even the same test given multiple times, is not a good one (although the question of what such a test does select for is interesting). And this observation holds for any subject matter being tested, not particularly mathematics. So even if this is the ‘gatekeeper’ referred to, it’s not about our subject. And this form of gatekeeping is a matter of practice, of implementation, and not a widespread or deeply-held belief about mathematics. The deeply-held belief is about the nature of testing.

Maybe some writers are talking about textbook mathematics, mathematics as it is taught in a mediocre setting, as a set of rules and procedures. Well, this is not mathematics. This is rules and procedures, more and more imposed on teachers by the requirements of high-stakes state testing. Again, it seems to me that the gatekeeper is the testing, not the subject. And again, this observation is not at all specific to mathematics.

In fact it seems to me that mathematics is less guilty of ’gatekeeping’ than many other academic subjects.

There are many gatekeepers, in any culture. The use of language, both spoken and written, is a much stouter gate than knowledge of mathematics. The reader will shortly see how even one misspelled word can cast doubt on the vallidity [n.b.] of a thought, or even on the intelligence of the writer. We commonly value someone who speaks and writes standard academic English over someone who uses vernacular, or even who has a heavy local accent (and we all have local accents!). Just think of the effect, in a job interview or resume cover letter, of even a single mispronounced word or grammatical solecism. Conversely, we all know well-spoken imposters.

And language is unavoidable. We are constrained to speak in any social situation, and to write in any professional position. This gatekeeper appears unbidden. And often unconscious: we frequently don’t have control over judgments we make on the basis of language.

There are still other gatekeepers. Dress is the most obvious. And some very unfortunate ones: race, class, gender. It is quite human, but at the same time quite de-humanizing, to react unconsciously to people we don’t know by grouping them with others with whom they share external characteristics. Unlike language, these gatekeepers are not routinely addressed by formal education. They are almost always unconscious, hence powerful. And they are clearly ‘unearned’.

Is the privilege of mathematics ‘unearned’? Well, no. I think it is hard earned. Mathematics is the *locus classicus* for addressing logic, the derivation of statements from other statements. And this skill pervades human activity. Further, the better you are at this skill, the more valuable your activity to others. This is why I have argued (in several places) that the teaching of mathematics should be centered on logic, and not on algorithm. The latter, for me, should be a consequence of the former. Even for the 80% of our students (this figure is approximate and variable: see https://nces.ed.gov/programs/raceindicators/indicator_reg.asp) who don’t go into STEM related fields, this legacy of a mathematical education is central.

I have heard arguments about other ‘forms’ of mathematics, sometimes called ‘non-Western’. I would argue that the classification of logic-based mathematics as ’Western’ (or sometimes ‘Greek’) is a misnomer at best, and simplistic at worst.

But let’s not talk about mathematics for a minute. Let’s talk about pizza. Everyone thinks of pizza as Italian, and it is consumed worldwide. But tomatoes originated in America, wheat (probably) in the middle east, basil and pepper in India. The Italians just put it all together.

Similarly, the pieces of what we call “Western” mathematics were lying around for centuries, in different parts of the world. The Greek achievement was a synthesis of these human thoughts. That is, mathematics is ‘Western’ only in the sense that pizza is ‘Italian’. Everyone enjoys pizza. Everyone benefits from logic.

And like pizza, everyone wants mathematics. ‘Western’ mathematics. Algebraic topology. Bessel functions. Lie algebras. Intellectual domains in which a ‘non-Western’ culture had not penetrated, before a cultural influence from the West. I have personally worked on every continent except Antarctica, and everyone wants to learn mathematics the way it has developed ‘in the West’.

But notice that this form of mathematics is now being developed as much in Asia as in Europe. And in fact it always has been. Modern mathematics is ‘Eastern’ as much as it is ‘Western’. And, if Africa and Latin America develop as quickly as we all expect they will, modern mathematics will soon be ‘Southern’ as much as ‘Western’.

What about the other 20%, the students preparing for STEM fields? Let’s leave aside for now the fact that we don’t always know who these students are. Is the status of mathematics ‘unearned’ to this group? Again, no. A knowledge of mathematics is, in fact, an intellectual gatekeeper, or better yet, gateway, into STEM fields. For those who are going to make contributions in these fields, mathematics is vital. And it is growing in importance as the sciences, and even the social sciences, develop.

So yes, it is a gateway for this group. Just as organic chemistry is a gateway for medical school. Do you want to be treated by a doctor who hasn’t studied it?

And do you want to travel over a bridge built by an engineer who hasn’t studied ‘Western’ calculus?

(I thank Paul Goldenberg and Al Cuoco for their help in writing this post.)

]]>2018 was an important year for the Letchford family – for two related reasons. First it was the year that Lois Letchford published her book: *Reversed: A Memoir*.^{[1]} In the book she tells the story of her son, Nicholas, who grew up in Australia. In the first years of school Lois was told that Nicholas was learning disabled, that he had a very low IQ, and that he was the “worst child” teachers had met in 20 years. 2018 was also significant because it was the year that Nicholas graduated from Oxford University with a doctorate in applied mathematics.

Nicholas’s journey, from the boy with special needs to an Oxford doctorate, is inspiring and important but his transformation is far from unique. The world is filled with people who were unsuccessful early learners and who received negative messages from schools but went on to become some of the most significant mathematicians, scientists, and other high achievers, in our society – including Albert Einstein. Some people dismiss the significance of these cases, thinking they are rare exceptions but the neuroscientific evidence that has emerged over recent years gives a different and more important explanation. The knowledge we now have about the working of the brain is so significant it should bring about a shift in the ways we teach, give messages to students, parent our children, and run schools and colleges. This article will summarize three of the most important areas of neuroscience that directly apply to the teaching and learning of mathematics. For more detail on these findings, and others, visit youcubed.org or read Boaler (2016).^{[2]}

The first important area of knowledge, which has been emerging over the last several decades, shows that our brains have enormous capacity to grow and change at any stage of life. Some of the most surprising evidence that highlighted this came from studies of black cab drivers in London. People in London are only allowed to own and drive these iconic cars if they successfully undergo extensive and complex spatial training, over many years, learning all of the roads within a 20-mile radius of Charing Cross, in central London, and every connection between them. At the end of their training they take a test called “The Knowledge” – the average number of times it takes people to pass The Knowledge is twelve. Neuroscientists decided to study the brains of the cab drivers and found that the spatial training caused areas of the hippocampus to significantly increase.^{[3]} They also found that when the drivers retired, and were not using the spatial pathways in their brains, the hippocampus shrank back down again.^{[4]} The black cab studies are significant for many reasons. First, they were conducted with adults of a range of ages and they all showed significant brain growth and change. Second, the area of the brain that grew – the hippocampus – is important for all forms of spatial, and mathematical thinking. The degree of plasticity found by the scientists shocked the scientific world. Brains were growing new connections and pathways as the adults studied and learned, and when the spatial pathways were no longer needed they faded away. Further evidence of significant brain growth, with people of all ages, often in an 8-week intervention, has continued to be produced over the last few decades, calling into question any practices of grouping and messaging to students that communicate that they cannot learn a particular level of mathematics.^{[5]} Nobody knows what any one student is capable of learning, and the schooling practices that place limits on students’ learning need to be radically rethought.

Prior to the emergence of the London data most people had believed either that brains were fixed from birth, or from adolescence. Now studies have even shown extensive brain change in retired adults.^{[6]} Because of the extent of fixed brain thinking that has pervaded our society for generations, particularly in relation to mathematics, there is a compelling need to change the messages we give to students – and their teachers – across the entire education system. The undergraduates I teach at Stanford are some of the highest achieving school students in the nation, but when they struggle in their first math class many decide they are just “not a math person” and give up. For the last several years I have been working to dispel these ideas with students by teaching a class called How to Learn Math, in which I share the evidence of brain growth and change, and other new ideas about learning. My experience of teaching this class has shown me the vulnerability of young people, who too readily come to believe they don’t belong in STEM subjects. Unfortunately, those most likely to believe they do not belong are women and people of color.^{[7]} It is not hard to understand why these groups are more vulnerable than white men. The stereotypes that pervade our society based on gender and color run deep and communicate that women and people of color are not suited to STEM subjects.

The second area of neuroscience that I find to be transformative concerns the positive impact of struggle. Scientists now know that the best times for brain growth and change are when people are working on challenging content, making mistakes, correcting them, moving on, making more mistakes, always working in areas of high challenge.^{[8, 9]} Teachers across the education system have been given the idea that their students should be correct all of the time, and when students struggle teachers often jump in and save them, breaking questions into smaller parts and reducing or removing the cognitive demand. Comparisons of teaching in Japan and the US have shown that students in Japan spend 44% of their time “inventing, thinking and struggling with underlying concepts” but students in the U.S. engage in this behavior only 1% of the time.^{[10]} We need to change our classroom approaches so that we give students more opportunity to struggle; but students will only be comfortable doing so if they have learned the importance and value of struggle, and if they and their teachers have rejected the idea that struggle is a sign of weakness. When classroom environments have been developed in which students feel safe being wrong, and when they have been valued for sharing even incorrect ideas, then students will start to embrace struggle, which will unlock their learning pathways.

The third important area of neuroscience is the new evidence showing that when we work on a mathematics problem, five different pathways in the brain are involved, including two that are visual.^{[11, 12]} When students can make connections between these brain regions, seeing, for example, a mathematical idea in numbers and in a picture, more productive and powerful brain connections develop. Researchers at the Marcus Institute of Integrative Healthhave studied the brains of people they regard to be “trailblazers” in their fields, and compared them to people who have not achieved huge distinction in their work. The difference they find in the brains of the two groups of people is important. The brains of the “trailblazers” show more connections between different brain areas, and more flexibility in their thinking.^{[13]} Working through closed questions, repeating procedures, as we commonly do in math classes, is not an approach that leads to enhanced connection making. In mathematics education we have done our students a disservice by making so much of our teaching one-dimensional. One of the most beautiful aspects of mathematics is the multi-dimensionality of the subject, as ideas can always be represented and encountered in many ways, such as with numbers, algorithms, visuals, tables, models, movement, and more.^{[14, 15]} When we invite people to gesture, draw, visualize, or build with numbers, for example, we create opportunities for important brain connections that are not made when they only encounter numbers in symbolic forms.

One of the implications of this important new science is we should all stop using fixed ability language and celebrating students by saying that they have a “gift” or a “math brain” or that they are “smart.” This is an important change for teachers, professors, parents, administrators – anyone who works with learners. When people hear such praise they feel good, at first, but when they later struggle with something they start to question their ability. If you believe you have a “gift” or a “math brain” or another indication of fixed intelligence, and then you struggle, that struggle is devastating. I was reminded of this while sharing the research on brain growth and the damage of fixed labels with my teacher students at Stanford last summer when Susannah raised her hand and said: “You are describing my life.” Susannah went on to recall her childhood when she was a top student in mathematics classes. She had attended a gifted program and she had been told frequently that she had a “math brain,” and a special talent. She enrolled as a mathematics major at UCLA but in the second year of the program she took a class that was challenging and that caused her to struggle. At that time, she decided she did not have a “math brain” after all, and she dropped out of her math major. What Susannah did not know is that struggle is really important for brain growth and that she could develop the pathways she needed to learn more mathematics. If she had known that, and not been given the fixed message that she had a “math brain,” Susannah would probably have persisted and graduated with a mathematics major. The idea that you have a “math brain” or not is at the root of the math anxiety that pervades the nation, and is often the reason that students give up on learning mathematics at the first experiences of struggle. Susannah was a high achieving student who suffered from the labeling she received; it is hard to estimate the numbers of students who were not as high achieving in school and were given the idea that they could never do well in math. Fixed brain messages have contributed to our nation’s fear and dislike of mathematics.^{[16]}

We are all learning all of the time and our lives are filled with opportunities to connect differently, with content and with people, and to enhance our brains. My aim in communicating neuroscience widely is to help teachers share the important knowledge of brain growth and connectivity, and to teach mathematics as a creative and multi-dimensional subject that engages all learners. For it is only when we combine positive growth messages with a multi-dimensional approach to teaching, learning, and thinking, that we will liberate our students from fixed ideas, and from math anxiety, and set them free to learn and enjoy mathematics.

*This blog contains extracts from Jo’s forthcoming book*: Limitless: Learn, Lead and Live without Barriers, *published by Harper Collins.*

[1] Letchford, L. (2018) *Reversed: A Memoir*. Acorn Publishing.

[2] Boaler, J (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[3] Maguire, E. A., Gadian, D. G., Johnsrude, I. S., Good, C. D., Ashburner, J., Frackowiak, R. S., & Frith, C. D. (2000). Navigation-related structural change in the hippocampi of taxi drivers. *Proceedings of the National Academy of Sciences*, 97(8), 4398-4403.

[4] Woollett, K., & Maguire, E. A. (2011). Acquiring “The Knowledge” of London’s layout drives structural brain changes. *Current **b**iology**:CB*, 21(24), 2109–2114.

[5] Doidge, N. (2007). *The Brain That Changes Itself*. New York: Penguin Books,

[6] Park, D. C., Lodi-Smith, J., Drew, L., Haber, S., Hebrank, A., Bischof, G. N., & Aamodt, W. (2013). The impact of sustained engagement on cognitive function in older adults: the Synapse Project. *Psychological science*, 25(1), 103-12.

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

[8] Coyle, D. (2009). *The Talent Code: Greatness Isn’t Born, It’s Grown, Here’s How*. New York: Bantam Books;

[9] Moser, J., Schroder, H. S., Heeter, C., Moran, T. P., & Lee, Y. H. (2011). Mind your errors: Evidence for a neural mechanism linking growth mindset to adaptive post error adjustments. *Psychological science*, 22, 1484–1489.

[10] Stigler, J., & Hiebert, J. (1999). *The teaching gap: Best ideas from the world’s teachers for improving education in the classroom*. New York: Free Press.

[11] Menon, V. (2015) Salience Network. In: Arthur W. Toga, editor. *Brain Mapping: An Encyclopedic Reference*, vol. 2, pp. 597-611. Academic Press: Elsevier;

[12] Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning*. Journal of Applied & Computational Mathematics*, 5(5), DOI: 10.4172/2168-9679.1000325

[13] Kalb, C. (2017). What makes a genius? *National Geographic*, 231(5), 30-55.

[14] https://www.youcubed.org/tasks/

[15] Boaler, J. (2016) *Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching*. Jossey-Bass/Wiley: Chappaqua, NY.

[16] Boaler, J. (2019). *Limitless: **Learn, Lead and Live without Barriers.*

What surprises you mathematically in your classes? When do you witness students’ creative moments? How often does this happen?

When instructors develop an environment where students are willing to put themselves “out there” and take a risk, interesting moments often happen. Those risks can only build one’s creativity, which is the most sought-after skill in industry according to a 2010 IBM Global Study.

How do we get students to be creative? And how does that balance with the content we are required to cover? Below, past and present members of the Creativity Research Group present reasons on why and how we each teach for creativity. We all have different but synergistic teaching practices we engage in to foster creativity in our students. Gulden focuses on having students making connections, while Milos has students take risks through questioning and sharing wrong answers. Emily focuses on tasks that have multiple solutions/approaches; Gail emphasizes the freedom she gives students in exploring these tasks. Mohamed provides time for students to incubate their ideas. Houssein and Paul reflect on their teaching practices and how teaching for creativity has been integrated into theses practices. There is also the thread of opportunities for student self-reflection woven throughout these stories.

One common aspect is that we try our best to saturate our courses with chances for students to be creative from beginning to end. These stories are our attempts at being creative about fostering creativity. Enjoy!

**Gail Tang**

I recently watched an *Ugly Delicious* episode where world-renowned chefs talked about their definitions of pizza and what it meant to them to make pizza. These chefs fell into two types: those who stuck to traditional ways to make pizza and those who departed from these ways. Those who ended up leaving the traditional ways behind did so because they felt stifled to operate under the strict rules of making pizza; they felt their identities were being compromised. With the courage of their convictions, they left to forge their own pizza paths. These chefs rejoiced in their freedom; finally they made pizza in a way that paid tribute to their identities. Chef Christian Puglisi said “If you only look at how it used to be done, or how it’s supposed to be done, you don’t allow yourself to move it forward.” This episode really resonated with me; replace “pizza” with “math” and “make pizza” with “teach math,” – you get the same story of suppressing innovation in the name of tradition. The idea of teaching others in the same way I was taught suffocated me. I was not interested in producing generations of students who could mimic my every mathematical move.

I started with baby steps in Calculus. I found exercises with more than one solution path to the same answer and assigned these without any direction. Students wrote different solutions on the board. Students were not used to seeing each others’ creations let alone creating their own solution paths. The energy in the room was thrilling. Unfortunately, I did not collect this data at the time, but fortunately Houssein El Turkey (see his narrative below) has an example of three students’ work on computing the limit below.

Letting students try problems on their own with little direction has the opportunity to have a profound impact on their mathematical identities. For example, one student started my Calculus 1 as a biology major and ended Calculus 1 as a math major! She wrote in her Calculus 2 weekly reflection:

I think having math ‘done to me’ rather than getting to explore it and have fun with it in high school is the reason I didn’t enjoy math in high school. I love how in your classes we get to try problems our own way and don’t have to use your method. I also think it’s super cool that you encourage using different methods. I would never have considered myself a creative person until I started working with numbers. I’m not anywhere near as creative as I should be, but I feel like math is helping me become more creative. The other night I did a problem with a method that I knew was going to be wrong, but I just wanted to see what happened. It actually helped me understand why that method doesn’t work.

**Emily Cilli-Turner**

A turning point in my thinking about students’ potential for creativity and how to foster it in the classroom happened while I was teaching a Linear Algebra course using the inquiry-oriented linear algebra materials. One task asked for the solution to a system of three equations in two variables; if there was no solution, find the “best” approximate solution. This task purposefully did not define the word “best” so the students would be forced to think about what qualifications the best approximate solution would have. Every group graphed the linear equations (which bounded a triangular region) and most presented “best” solutions as averages of $x$ and $y$ values of intersection points. However, one student was able to find the exact least squares solution by using optimization of functions of two variables techniques from calculus. Once this student presented his solution to the class, the other students were intrigued and could see the drawbacks of their own solutions. This was very unexpected for me. It drove home the point that students can and will be creative when we give them the tools and the freedom to hone their creativity.

In my mind, teaching for creativity has two main components: task design and collaboration. A large part of teaching for creativity is providing students with tasks that involve multiple approaches/solutions. The above episode would have turned out very differently if I had given the definition of the least squares solution and then several problems finding the least squares solutions of a system. The student would have never had a chance to find his original solution because all of the mystery would have been taken away with a provided definition. Yet, if the students had not been working on the task in groups, bouncing ideas off of each other, and using the group whiteboard to do scratch work, I think this vignette would not have happened. Collective creativity can be greater than that of the individual, and the students’ discussions helped that individual student refine his ideas and come up with the idea of using optimization to solve the problem.

**Milos Savic**

I believe teaching for creativity addresses a lot of issues in mathematics education. When a student is trying to be creative, there are many side effects, including more saturation with the content, greater mathematical confidence, and the ability to manipulate mathematics in different or new ways. Also, I believe mathematical creativity allows a student to be more of themselves instead of more like me. Finally, solving problems in STEM fields requires creativity, and I strive to create authentic experiences so that students can engage in being creative. These beliefs either are expressed explicitly (“Class, I want you to play with this idea”) or implicitly through tasks, quizzes, tests, and other requirements.

To generate curiosity, I have students ask two questions for every homework assignment; one is intended to be about the concepts, and the other is about their mathematical processes. I also give many routine problems with little twists. For example, I pose problems that require a student to go backwards instead of forwards (e.g. what non-linear function, when integrated from 1 to 2, is 17?). I also ask them to provide their *own* definitions or theorems using what they know. For example, using what they knew about groups and semigroups, a student created an anti-identity (the additive inverse of the multiplicative identity) and anti-inverses. My actions support these tasks; I celebrate many wrong answers in the class in order to show the process of mathematics and the creative moments within it.

**Houssein El Turkey**

It is interesting how my teaching has evolved to focus on more than just covering the basic learning outcomes in the classes I teach. I have become aware that teaching mathematics can and should include discussions with students on the novelty and flexibility in problem solving (or proving). Now I seek different approaches from my students to show them that there are often multiple ways to solve a problem. I also point out to my students when we build on something we discussed a while back to show that making connections is crucial. Another action I have taken is to explicitly show how certain processes generalize to a bigger picture.

Seeing an AHA moment from a student is one of the best highlights from a class. For example, when I asked my students to factor $x-1$ many of them were baffled but with hints and guidance, some of them came up with: $x-1 = (\sqrt{x}-1)(\sqrt{x}+1)$. The majority were taken by the simplicity and/or originality of the solution and the look on their faces was priceless. These AHA moments have been occurring more than before and they are constant reminders that they don’t have to be ground-breaking in order to have significant impacts on students. To a first-year college student, these simple tricks generating AHA moments can be crucial to show the originality aspect of doing mathematics even though instructors might find these tricks standard.

I also noticed that teaching to foster creativity has lifted my expectations of my students. I now see more potential in them and I work harder to get the best from them as I challenge them with tasks that require incubation and effort. An example of such a task that I used this semester was finding the limit $\lim_{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$ in three (or more) different ways. Students struggled in their first attempt in class to see it as a slope of a tangent line but I left it for them to think about it at home and we picked it up the next class. I was happily surprised that some students had made that connection. Whether it was due to incubation or someone suggested it to them remains unknown to me.

Assigning challenging tasks was accompanied by emotional support through my continuous encouragement and emphasizing that it is OK to fail or not complete a task and it is OK to struggle because that shapes the learning process.

**Mohamed Omar**

Many students come to mathematics with an inherent curiosity, but traditional teaching practices tend to suppress this. It is essential that we tap into the curiosity that students have as a way to make mathematics come alive for them. Infusing creativity in the classroom fosters this in a way unlike any other.

To facilitate creativity, I ask open-ended or open questions on assignments. For example, a central theme in one course was counting the number of regions that a set of hyperplanes partition an $n$-dimensional space into, and determining what data about the set of hyperplanes are sufficient to answer that question. The related open-ended question asked how these results change if we used circles or other geometric objects instead of hyperplanes. I gave students plenty of time to play with these open-ended problems, and supported them along the way. The key to unlocking creativity was to create a structure where students were rewarded for their efforts and diversity of approaches, rather than their final output. To facilitate this, I required a three-page reflection using the Creativity-in-Progress Rubric (To read more about the rubric, please see our short article in MAA Focus Feb/Mar 2016). Students had to submit all their scratch work and all partial results, and subsequently use the rubric to reflect on their problem solving process. For instance, if a student used definitions and theorems from the course in conjunction definitions and theorems from outside resources, they could provide direct evidence for this in their work and comment on how the conjunction occurred. This allowed students reflect on their thinking processes, facilitating the pathway to creativity.

**Gulden Karakok**

Through active learning teaching practices, I plan learning situations that provide opportunities for discovery and making connections. Making connections has been an important component of my teaching, as I believe this process facilitates learning of the new topics and also allows students to transfer learning to other situations. Making connections comes in many forms in my courses — connections between definitions, theorems, various solution approaches, examples, and representations. I often ask my students if they have seen a similar topic, definition or example before. My goal is to discourage compartmentalization of ideas and topics. Unfortunately, our education system seems to train students to see ideas in disparate categories. To address this concern and foster creativity, I have been pushing for the process of making connections. I think asking students to find similarities and differences between ideas from their perspectives and background not only helps students to “own” these ideas but also develop sense of “usefulness” of them. With this ownership, students will be more equipped to be creative.

One example of how making connections promotes creativity comes from my preservice elementary math content course. During a class discussion on definitions of even and odd numbers, one student raised her hand and asked how we can determine “quickly” if a number in base 5 is even or odd (e.g., Is 123 base 5 even or odd?). This particular student was making connections to different bases discussed during the first week. Students worked on this problem and came up with several generalizations. We then discussed connections between those generalizations.

**Paul Regier**

Far too many students are afraid of math. I believe the antidote to their fears concerning math is experiencing mathematics by their own creativity. I suspect that in removing creative exploration from teaching mathematics, we run the risk of damaging our students. Just as the processing of modern food removes most of its nutritional value, removing creativity from math robs students of the most significant benefits they can gain from studying mathematics. Although a few students may appreciate the sugar rush of an already processed solution presented to them, it does not nourish them. What they gain does not last. It does not stick with them.

I teach for creativity by thinking creatively myself! When I lesson plan, after I have some kind of basic connection to the material and to past experience, I try to incubate for at least a day before I think about how I’m going to structure the class. Then I ask myself, “How little time can I spend presenting an idea, so that students are motivated and ready to start thinking about it themselves? What do I subconsciously withhold from students’ experiences (the joy of discovering and creating mathematics for themselves) that I can give conscious attention to?” In my experience, it is much easier to facilitate this kind of awareness by having students work together in groups with limited (but carefully planned) guidance and encouragement from me. It’s may be easy to give students quick answers, but this often takes away the student’s creative drive. Thus, I am learning to acknowledge students’ own thinking to be able to better provide opportunities for their own creative discovery.

**Conclusion**

As this blog ends, we hope that our stories serve as the beginning to your adventures in teaching for creativity! The Creativity Research Group has recently been awarded an NSF IUSE Grant (#1836369/1836371), “Reshaping Mathematical Identity by Valuing Creativity in Calculus”. To learn more about how to participate, or to communicate any of your ideas about fostering creativity, email us at creativityresearchgroup@gmail.com.

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