Some thoughts about epsilon and delta

by Ben Blum-Smith

The calculus has a very special place in the 20th century’s traditional course of mathematical study. It is a sort of fulcrum: both the summit toward which the whole secondary curriculum strives, and the fundamental prerequisite for a wide swath of collegiate and graduate work, both in mathematics itself and in its applications to the sciences, economics, engineering, etc.[1] At its heart is the notion of the limit, described in 1935 by T. A. A. Broadbent as the critical turning point:

The first encounter with a limit marks the dividing line between the elementary and the advanced parts of a school course. Here we have not a new manipulation of old operations, but a new operation; not a new trick, but a new idea.[2]

Humanity’s own collective understanding of this “new idea” was hard-earned. The great length of the historical journey toward the modern definition in terms of \epsilon and \delta mirrors the well-known difficulty students have with it. Although it is the foundation of calculus, it is common to push the difficulty of this definition off from a first calculus course onto real analysis. Indeed, mathematicians have been discussing the appropriate place for the full rigors of this definition in the calculus curriculum for over 100 years.[3]

There is also a rich vein in the mathematics education research literature studying students’ engagement with the \epsilon\delta definition. Researchers have examined student difficulties coming from its multiple nested quantifiers[4] as well as its great distance from the less formal notions of limit with which students typically enter its study,[5] and have also made an effort to chart the paths they take toward a full understanding.[6]

This blog post is a contribution to this conversation, analyzing in detail three learners’ difficulties with \epsilon and \delta.[7] If there is a take-home message, it is to respect the profound subtlety of this definition and the complexity of the landscape through which students need to move as they learn to work with it.

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Understanding in Calculus: Beyond the “Sliding Tangent Line”

By: Natalie Hobson, Sonoma State University

If you give calculus students graphs, they are going to draw tangent lines. As instructors we often encourage students to rely on tangent lines so heavily that discussions about rates of change become lessons about sliding lines along graphs, rather than about understanding the relationships that these graphs represent in the context of a given problem.

So how do we help students develop a deeper understanding of these relationships? Let’s consider an exercise that you might give your own students during a lesson on tangent lines and rates of change.

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Precise Definitions of Mathematical Maturity

[This contribution was originally posted on April 15, 2019.]

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.

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Everyone Can Learn Mathematics to High Levels: The Evidence from Neuroscience that Should Change our Teaching

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.

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Posted in Active Learning in Mathematics Series 2015, Classroom Practices, Communication, Education Policy, K-12 Education, Mathematics Education Research, News, Research | Tagged , , , , | 1 Comment

Two More Teaching Vignettes

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:

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Posted in Active Learning in Mathematics Series 2015, Classroom Practices, Communication, Faculty Experiences, K-12 Education, Mathematics Education Research | Tagged , , , , , | 1 Comment

Two Teaching Vignettes

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.

  1. 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.

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Our Students Are Your Students Are Our Students: a University-Community College Collaboration

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

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The Crisis in American Education

 

The Crisis in American Education

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.

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

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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Interdisciplinary Collaboration, Teaching, and Purpose

By Victor Piercey, Ferris State University

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.

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Precise Definitions of Mathematical Maturity

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.

Continue reading

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