By Benjamin Braun, University of Kentucky

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.

]]>(This is the first of two of our most popular Blog posts that we repeat for the month of July. )

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.*

But first, with the end of the school year 2018-19, we would like to announce several changes in this Blog’s editorial board:

We bid farewell to Art Duval, who has been on the editorial board for five years, and was one of the founding board members for this blog. His service and guidance have been indispensable to me in guiding the blog, and his voice will be missed.

We welcome two new editorial board members:

Yvonne Lai received her S.B. in Mathematics from MIT and Ph.D. from UC Davis, specializing in geometric group theory and hyperbolic geometry. Following a post-doctoral position at the University of Michigan in mathematics, she took a second post-doctoral position, this time at the University of Michigan School of Education. There, she began doing research in the area of mathematical knowledge for teaching, in the group led by Deborah Ball and Hyman Bass. Lai is now an associate professor in the Department of Mathematics at the University of Nebraska-Lincoln. She is founding chair of the MAA’s Special Interest Group on Mathematical Knowledge for Teaching, a member MAA Committee on the Mathematical Education of Teachers (COMET), and a member of the writing team for the NCTM publication Catalyzing Change.

Ben Blum-Smith received a B.A. in anthropology from Yale University in 2000, an M.A.T. in mathematics teaching from Tufts University in 2001, and a Ph.D. in mathematics from NYU in 2017, with a thesis in representation and invariant theory of finite groups. He worked as a middle and high school teacher in public schools in Cambridge, MA and New York City, and then as a mathematics professional development specialist for high schools and as a faculty member of Bard College’s teacher training program, before beginning his training as a research mathematician in 2011. He is currently a part-time faculty member of Eugene Lang College’s Department of Natural Sciences and Mathematics, and has also taught in the Bard Prison Initiative.

His research interests lie in invariant theory, algebraic combinatorics, their applications to data science, and connections between mathematics and democracy. He was a 2018 TED Resident, developing a TED talk about the relationship of mathematics and democracy, and is a founding organizer of the Mathematics and Democracy Seminar at the NYU Center for Data Science. He remains involved in teacher professional development through Math for America, an organization devoted to the career-long professional growth of teachers. He is also engaged in mathematical outreach. He has led math circles with students and teachers at the School of Mathematics, the New York Math Circle, the Westchester Area Math Circle, the LREI Summer Institute, the Center for Mathematical Talent at NYU, and the MathLeague International Mathematics Tournament, and is regularly a faculty member and faculty mentor at the Bridge to Enter Advanced Mathematics, an organization focused on creating a realistic pathway for underserved students to enter the mathematical sciences.

* * *

The theme of these two vignettes is how the teacher must value the student, must see as much value in his or her ideas—correct or incorrect—as in our own. Because sometimes they are correct.

I: Cecelia and the Grapes

This anecdote took place in a high school remedial class. For many years, I would take the 10 or so students who were not on any grade level at all and teach them together in an ungraded classroom. These were students who had struggled with mathematics, and had been failing, for many years. Many had learning disabilities. Some had significant troubles at home. All hated or feared mathematics. It was my job to un-teach them this fear.

Of course, the worst thing you can do in teaching remedial students is the same thing over again, even if you go slower and talk louder, even if make the definitions crystalline and the logic pristine. It won’t help. The students will tune out, will continue to hate and fear mathematics—and worst of all, will revert to the defensive learning habits that caused their failure in the first place.

Remember that definition of mental illness?

So in this class I used activities. Games. Manipulatives. Students measured and weighed to learn fractions. They walked the corridors to learn geometry. They went up and down stairs to add and subtract signed numbers. (Many of them would confuse left and right, but almost never up and down.) They analyzed card tricks—performed with different sized decks—to develop algebraic representation. My administrators, whose support of my work was vital and unflagging, were kept busy apologizing for my students being in the corridors or stairwells or shuffling playing cards.

I love this kind of teaching. It requires enormous flexibility, creativity, spontaneity. It means that you have to be right on top of the students’ cognition, reading their minds as best you can. And since their minds were full of ideas so different from my own, this was a challenge.

The other kind of teaching I like best is with gifted children—and for much the same reason. You cannot give them just more of the same, and faster. Sure, they will appear to succeed. But eventually they will hit a wall, will find some material that they cannot just read and understand. They need to experience early that understanding can be the result of struggle, or they will not have the means to surmount that wall. The teacher needs the same flexibility, creativity, spontaneity that the remedial classes require.

And for either group of outliers, you need the same ability to read minds. These students will come up with things you haven’t thought of yourself, which are correct or incorrect in the most amazing ways.

Cecilia was a fifteen year old girl in my remedial class. She was not angry, not stubborn, not resistant to school. Yet she had trouble with mathematics. I was trying to get her to pass a first-year algebra exam—without cramming her full of test tricks and meaningless technique. In this classroom, I had the time to do it.

I had given the class a challenge problem:

Marty ate ten grapes on September 1. Then he ate twelve grapes on September 2. Then he ate 14 grapes on September 3. He kept eating two more grapes each day than he ate on the previous day. How many grapes had he eaten at the end of September 10?

Remember: this problem, for these students, had nothing to do with an arithmetic progression. It was a simple arithmetic problem. We had been working on how multiplication for natural numbers is repeated addition, how the distributive law could shorten computation, how to recognize ‘complementary numbers’ that added up to 10 or 100 or 1000. In short, we were busy taking arithmetic off the paper.

But this problem stubbornly remained on the paper. Students tried this and that. They got wrong answers, thinking this was yet another multiplication problem. They tried to fit the problem into a pattern they had seen before. Nothing worked.

They worked in pairs, but I had an odd number of students that day. So Cecilia was part of a group of three. And she was not interested. She rearranged her books. She stared out the window. She powdered her nose.

My job was to keep the wheels turning. So I came over to her group and asked what they were doing. One student, Tim, had a good idea, although he didn’t know it. He drew lines to represent the numbers of grapes for each day:

xxxxxxxxxx

xxxxxxxxxxxx

xxxxxxxxxxxxxx

xxxxxxxxxxxxxxxx

. . .

. . .

. . .

xxxxxxxxxxxxxxxxxxxxxxxxxxxx

Tim could have observed that the first and last line, the second and next-to-last line, etc, complemented each other to form a rectangle. But he didn’t. I encouraged him and made a mental note to come back to this idea later on.

Other students, of course, were busy adding things up by hand. Students younger than these are perfectly content to solve a problem using such busywork. They don’t see it as compulsive or boring. But teenagers do—and that was lucky for me. They don’t learn much from tedious computation.

Cecilia saw me come over and wanted to look busy. So she picked up her calculator. I always let students use calculators, if they could tell me their plan for using it. If I sensed abuse, I would forbid them the calculator and discuss what they were going to do with it. Only after they had a coherent plan would I let them have the machine.

Cecilia’s calculator was pink and heart-shaped. Each of the keys was a different colored rhinestone, and each was also in the shape of a small heart. She began pressing the keys.

“What is your plan?” I asked. She had none, of course, and was busy looking busy.

I said: “I’ll talk to Chris. When I come back, I want to hear your ideas.” She looked at me as if the word ‘ideas’ was not in the English language. I fought hard the temptation to lecture her about paying attention to the task. That never works.

I went over to Chris, who was more or less randomly multiply numbers together—by hand, to impress me. I asked why he was doing this or that. He had the (correct) idea that somehow the small numbers made up for the big numbers. But couldn’t express this arithmetically. He had missed the essential circumstance that the numbers were evenly spaced—formed an arithmetic progression. He knew that 10×10 would be too small, and 10×28 would be too big. I let him work a bit more.

Then I came back to Cecelia. She was staring at her paper, but not vacantly. I could see on her face that something had happened while I was gone from her desk. On her paper was written

5 x 38 = 190,

and on her face was a look of relief—not quite a smile—but a look that told me that she knew what she was doing.

“How did you do it?” I asked.

“With my calculator,” she replied. This was not mere adolescent backtalk. She really thought I was asking about the arithmetic.

“The small numbers make up for the big ones, so you can shortcut multiply.” And she started to explain. As I write this, I cannot recall how she explained it. Her words made no sense to me, whose mind was full of formulas for general terms and partial sums. I tried to listen, but quickly got lost in her verbal explanation.

No matter. She clearly understood the problem. She had figured out that pairs of numbers added to 38. I was delighted, but she was merely relieved. I got her to explain her ideas to the others in her group, who also didn’t quite understand her words. But I gave them another similar problem, and they could do it—and certainly used Cecelia’s ideas. So something had been communicated.

I never recovered Cecelia’s words. To this day I don’t know how she thought of the solution, nor how she managed to communicate it to others in her group.

Years ago, I saw a film called *Defending Your Life*. It took place mostly in heaven(!), and some of the characters were angels. It was explained that angels are not really different from humans, that humans only really use 10% of their brain’s capabilities, but angels use 90%. Maybe. My point is that for 10 minutes I had an angel in my classroom.

Then she went back to powdering her nose.

II. Cold Weather: An Unfinished Story

Here is another incident that occurred in a remedial classroom. The students in this class were studying linear equations, starting with a story and generating mathematical models for the situations. They had worked on stories about cars traveling at constant velocity, about Mary working at the grocery and saving her money, about Bob spending the contents of his piggy bank at a constant rate, and so on. Then I gave them what I knew was a hard example for them, just to see what they would make of it:

At 50 degrees Fahrenheit, 30 people will complain about the temperature of a building. For every drop of 10 degrees in temperature, five more people will complain. How is the number of complaints received related to the temperature in the building?

I had expected a table of values something like this:

D |
50 | 40 | 30 | 20 | 10 |

C |
30 | 35 | 40 | 45 | 50 |

… and eventually the equation *C *= − (1/2) *D *+ 55.

This was not meant to be a realistic situation. I have found that such things do not trouble students. In this case, they had fun thinking of the occupants of the building shivering at their desks. It was in fact a cold January day.

The students understood the situation well enough to make a table of values. But they could not write an equation. At first, they could not decide which variable should be independent and which dependent. I described to them how an historian might look at the number of complaints to infer the temperature, but most of us would think the other way around. They had no trouble with this, once it was pointed out.

They were thrown by the fact that the table did not start at 0, although some of them had learned to extrapolate to get the value at 0. They were confused by the fact that the temperature went ‘down’, not ‘up’. And I had not yet talked about what to do when *x *jumps by more than 1 in a table.

As they worked, I observed. They were still not secure with the concept that the equation must be true for every pair of values they knew. They had somewhere learned to follow the ‘key words’ of the problem, so they had various ideas about how the words themselves generated the equation. And all the equations were wrong. This gave me the opportunity to show them that substituting values, rather than looking back at the words of the story, was what would tell them if their equation is correct.

Work was proceeding as I had expected, until Selma stopped me in my tracks. Selma was a vivacious 13-year old, the kind who seems to want to cling to her childhood. She must have weighed about 75 pounds sopping wet, all sinew and energy. And delighted with life.

Selma, among others, gave the equation C = 30 + 5D. Many students had realized that 30 and 5 play roles in the equation and were simply guessing about where to put them. One reason I selected this problem is that the slope is not an integer, and so it is less likely that they would get the correct answer by guessing. When asked, the class quickly saw that this equation was wrong.

But Selma persisted.

“Do I have to do the equation your way? Can’t I do it another way?”

There is only one answer that a teacher can give to this question, and I gave it. It turned out to be the best question I’d received all week.

“Well, what’s another way to do it?” I asked.

Selma came up to the board, and wrote the following table:

d |
0 | 1 | 2 | 3 | 4 | 5 |

C |
30 | 35 | 40 | 45 | 50 | 55 |

“See,” she said, “*C *= 30 + 5*d*.”

I was about to repeat my tiresome argument about plugging in values, but — just in time — I noticed the top line of her table.

“What is *d*?” I asked.

“Oh,” said Selma. “My *d *is different from yours. My ‘*d*’ stands for ‘drops’. One ‘*d*’ is one drop of 10 degrees. So when the temperature drops 10 degrees, we have 35 complaints: the 30 we had at first, plus five more. And for every drop, we add 5 complaints.”

I was speechless. But the class wasn’t. “That’s wrong!” “That’s right!” I had no trouble engaging them, but I myself didn’t quite know what to say.

So I played it safe. I told Selma that I understood her reasoning, and her representation. Could she use her ideas to get an equation in terms of the temperature Fahrenheit? She understood what I meant, and she also understood that she was right.

But what should I have done next? I might have exploited this idea of changing variables that Selma stumbled on. What are some fruitful directions? What Selma had discovered, without knowing it, was that a linear change of variables does not affect the degree of a polynomial function, so that a linear relationship remains linear. The trick of choosing your variables wisely is an old one. It lies behind much of the work of the Renaissance algebraists, and was made into an art form even earlier by Diophantos. I am still not sure what the best pedagogical strategy might have been on this occasion, but I feel that there is more here I could have done.

I often structure lessons so that a problem remains open at the end. This time, life structured the open problem for me.

*Some of this Blog post will appear in the forthcoming book, edited by Hector Rosario: *Mathematical Outreach: Explorations in Social Justice. Singapore, World Scientific Publishing, 2019. *Other portions have appeared in “Anecdotes and Assertions about Creativity in the Working Mathematics Classroom” (with Mark Applebaum), in Leikin, R., Berman, A., and Koichu, B.,* Creativity in Mathematics and the Education of Gifted Students. * Rotterdam: Sense Publications, 2009.*

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- Completing the Square

This first story started when I got a terse note from the high school guidance office about James:

“James has a difficult situation at home. Any leeway you can grant him about deadlines, tests, or quizzes would be greatly appreciated.”

Well: my classroom was run with very few deadlines. Students could re-take quizzes and tests whenever they learned the material, except that I had to report to their parents quarterly about their progress, at which time they got a grade.

So it was easy for me to meet James’ needs.

What was the ‘difficult situation’? I didn’t know, and it didn’t really matter. From the cryptic note, I assumed it was a divorce, and that the family was not anxious for everyone to know. But it could have just as well been a marriage: a new step-parent or step-sibling can take some getting used to. Or it could have been the birth of a new, much younger sibling. Any one of dozens of such circumstances looms large in an adolescent’s life. It didn’t matter to me–James needed to be cut some slack, so I cut it.

James was surly, a sure sign of instability in his life. He made little irritating comments, usually addressed to me–within the bounds of adolescent propriety, but on the rebellious side. Something’s going on, I thought. Of course, it wasn’t really me he was angry at. I was an authority figure and he needed a target. I was willing to play that role for him. I have found that the most constructive way to deal with this situation is to ignore the barbs and engage whenever the student shows a more positive side.

So I found ways to dodge his anger and get at the person it was hiding.

James had a habit of wearing bright orange gym shorts. I took to teasing him about them. “You’ve got to get rid of those shorts!” I would say, in various ways. James loved the attention and developed a variety of snappy comebacks. Eventually, he mellowed, promising me a pair of orange shorts as a gift at the end of the year.

James often asked to leave class during the lesson. I knew full well that he didn’t have to use the rest room and was hanging out with friends in the hall. I went a step further, asking myself why he was not engaged in school, and what he might need in his life in order to continue attending school. I knew there was an answer, thanks to the note from guidance. But I didn’t know what it was.

James was getting a straight C in the class. He wasn’t closing doors to mathematics, but couldn’t find the resources in his life to take the next step. I waited to see what I might do.

In April, his father called, angry. Why is James getting a C? He has to go to a good college. I went to Harvard business school. A C is simply not acceptable. Why wasn’t I notified earlier that James was doing so poorly?

In fact he was notified, quarterly, of James’ progress, but didn’t respond. It turned out that he had a lot on his plate. During the conversation I found out what the matter was. His wife, James’ mother, was dying. She had been in and out of the hospital for treatment, and was getting ready to leave her family, and this world, forever. Now I understood fully James’ anger, and his father’s anger.

We all feel helpless in these situations. There’s not much you can do or say. I told James’ father what a wonderful son he had, how James was doing a great job keeping his life together during this crisis, how he might not get the greatest grade, but that he would eventually be able to recoup his academic losses. In this most difficult yet of James’ years on earth, I said, it is amazing that he is able to keep up a C average in a difficult subject. These comments dulled the father’s hostility. I was part of the solution, not part of the problem.

After this call, things got better between me and James. He knew that I was aware of his problems, and sympathetic to them. He looked me in the eye when he asked to leave the room and made arrangements to retake quizzes he had done poorly on. Luckily, James learned mathematics easily, and a few sentences from me set him back on the right path when the work got tough.

I was moving from one classroom to another the next year, and part of my daily routine was searching the school building for useable cartons to pack away a 30-year accumulation of books and materials. One day, I found two empty cartons in the delivery room, and was carrying them down the hall when I passed James.

He was stretched out on a bench in the hall. “Do you need any help with those, Dr. Saul?”

I really didn’t, but he clearly wanted to engage me. I decided to accept the invitation. “I’d love some help, James. Would you like to help me pack some books?”

We went down to my classroom. James and I talked about the books we were packing, how heavy they were, who had written them, who had given them to me–and eventually how it feels to be living in a house with a dying parent.

Then he said, “My mom is going into the hospital again, for a week.”

I gave James what I could–mostly the same words I had for his father, and especially how a C average is not so bad, given what he had to handle outside of school. He couldn’t stay too much longer, because he had an appointment with the dean at 3:00. I told him I was glad for whatever help he could give me, and not to be late for his appointment.

Later I met him in the hall, where he told me that he his appointment was actually with a ‘cute girl’. I told him that I knew what that was like, and that we all shared certain feelings in life. And that we are not alone in handling them. He looked at me and smiled. He knew that I wasn’t just talking about cute girls.

The next day, I met James in the hall again. He had come to math class, but left in the middle, and hadn’t returned. We were working on completing the square, a challenge for this group, but a challenge they must meet if they are going to attach any meaning at all to the quadratic formula.

After an exchange of pleasantries, I asked James, “Do you know how to complete the square?”

“Yeah, it’s not that hard, except when you get fractions.” It was clear, from the way he answered, that he knew how to do it, and had practiced it enough to know where the difficulties lay.

“So come and take a quiz when you’re ready,” I said.

“Okay.” James looked down. Then he looked up at me. “Dr. Saul,” he said, “Give me a hug.” I quickly obliged.

Sometimes it is as important to complete a circle as to complete a square.

- The Chain Still Holding

“That building–that’s what scares me.” It was Ivan who said this, pointing to the Hancock tower in Boston, across the river. I was sitting with him on the MIT campus, talking about his problems adjusting to America. A brilliant mathematics student, he had come from Bulgaria for six weeks of study with equally gifted American high school students at a summer program I ran here. He was 17 years old, and the year was 1993.

I had noticed that Ivan was too much in his room, not playing enough Frisbee, not bonding with other students. My job was to draw him out. So we were sitting, late one night by the banks of the Charles River, and talking heart-to-heart.

“That building scares me,” he repeated. “America goes too fast, too far. The people move quickly. The cars goes fast. The food tastes like…nothing.”

We took our meals in a student cafeteria.

“But don’t you have trouble getting food back home?” I queried. The American experience was overwhelming him, and I hoped to turn him towards its positive aspects.

“Yes. It is difficult. Prices have risen and salaries have not kept up.” He paused before taking the bitter medicine: “I think, wherever Communism has come, it leaves garbage behind.” His face got hard, and he was silent for a bit.

Then, “I dream each night of having breakfast at home.” Home was Dobrich, a town in an agricultural area of Bulgaria which Ivan considered “not too small.”

His monologue was strangely compelling. There was some reason why I had to listen, had to respond, to the expression of bewilderment, of events running away with him, and of his own country and his own upbringing, betraying him.

For a while I didn’t understand my own reaction. And then, slowly and quietly, understanding came over me. I suddenly felt like a link in a chain, a stitch in a fabric.

There are moments when the meaning of life overwhelms you, forces itself into your consciousness, and thrusts still deeper, into your unconscious mind. Suddenly, Ivan’s confusion and awe, and even the tears he was clearly holding back, were mine as well. It wasn’t a tall building that scared him. It something greater.

In 1913, my grandfather Froim arrived in America from a small town in east Europe. Mother Russia had become more of a warden than a guardian to her Jewish children, who were leaving in droves to build America. Stories of his confusion have become family legends. When someone showed him the subway to the Bronx, he thought he was going down into a root-cellar. His cousin, who had arrived two years earlier, had to teach him how to drink liquids through a straw. Seeing the Woolworth building from afar, he tried to walk over for a closer look. But he didn’t realize how big it was, and how far away, and spent his whole lunch hour getting nowhere.

I grew up with these stories. But why did I react to them so strongly? Why did my soul vibrate, hearing them? Could it have been because I knew I would be having this conversation in Boston? Was it for this moment that my grandfather repeated them to me, these stories that took place in 1913, and were told me in 1963? Are these messages sent across the years, from one struggling immigrant to another, with myself as the medium?

It was more, though, than giving back to Ivan what my grandfather had given to me. More: I was Ivan, I was Froim, and I was at the same time a father and grandfather to both. They say this happens, that the roles reverse as one grows older, and you begin to care for your parents as once they cared for you. But I didn’t expect these feelings at forty-something, with both my parents in good health. I wasn’t ready.

“The child is father to the man,” says the poet. I hadn’t felt the feeling to these words so strongly before, and was overwhelmed.

And I wasn’t ready for the brush with eternity that these feelings would bring, of a contact with events that happen over and over again, and so occur outside of time. It’s not just the experience of the immigrant, or the foreigner. It’s the experience of the child and the parent. Somehow, in this conversation, with a youth I hardly knew on the banks of a dark river, I felt a contact with the future as well as the past. My own children, and their children, will feel these feelings, will go through these experiences. And my adventures will become their legends. The chain is still holding, the fabric unrent.

]]>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/

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