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.
By Jo Boaler, Professor of Mathematics Education, Stanford University, and co-founder of youcubed.org
(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.
For this month’s blog post, I offer two more vignettes from my classroom experience. My intention, as in the last column, is to communicate what I think of as the essence of teaching, which is the emotional—not just intellectual—bond between teacher and student.
But first, with the end of the school year 2018-19, we would like to announce several changes in this Blog’s editorial board:
As the Spring term ends, I thought I’d share with readers two vignettes from my teaching career. The intention is for us to remember how much of teaching is the emotional connection between student and teacher. For me, this is the reality of the experience, and is what makes possible the communication of mathematical ideas.
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.
By Ivette Chuca, El Paso Community College; Art Duval, Contributing Editor, University of Texas at El Paso; and Kien Lim, University of Texas at El Paso
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. Continue reading →
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)
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).
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).
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.
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.
By A. Gwinn Royal, Ivy Tech Community College of Indiana
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.
a step in the direction of enhancing mathematical insight
for teachers and the students they teach
What is the real value of interactive manipulable mathematics software?
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.
On Models and Exploration
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.
Understanding understanding
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 their thinking about questions they can pose to students, but the applets themselves could be used by students as well as teachers. Mathematical problems posed using Interactive Images elicit performance and provide students with the opportunity to make their own assessment of their actions.
In particular, I like to think of three forms of performance – mapping, constructing and deconstructing.
Mapping across Interactive Images
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 x2+px+q—is plotted in the {x,y} plane and depicted as a point in the {p,q} plane.
x2+px+q, plotted in the {x,y} plane and represented as a point in the {p,q} plane
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 polygon drawn in the Cartesian plane and represented as a point in the {perimeter, area} plane.
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
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.
Build a triangle.
-> 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]:
Build a parallelogram.
-> 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.
• 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]:
Area and perimeter of a rectangle.
• 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
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.
• 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]:
Another deconstruction in geometry.
• 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 2π.
•• 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?
• 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]:
Another deconstruction in algebra.
• 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?
Ergo…? What does this mean for us as teachers?
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 teachers, teachers may find many of them useful in working with students. The decision to do so should depend on the teachers’ judgment.
[3] “FOIL” (First, Outer, Inner, Last) is a common school-mnemonic for (but limited to) expanding products of two binomials.
The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.
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