The Dysfunction of Functions in Abstract Algebra

Kathleen Melhuish & Kristen Lew
Texas State University

“[Functions] are completely different, which is what makes this course so challenging.” – Abstract Algebra Student

Functions are hard for students, even students in abstract algebra courses. Even if students have seen the definition and worked with examples of real functions throughout high school and college, their understanding might be stretched to a breaking point when it comes to ideas like homomorphisms on groups or rings. Fundamentally, we might know students don’t understand functions, but the extent to which they don’t understand functions goes deeper than we might think. In this blogpost, we will share some insights from our projects on key places where students’ idea of function can be detrimental to learning concepts of abstract algebra, and what we as instructors might be able to do about this.

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What math professors and k-12 teachers think of each other

By Michael Pershan, St. Ann’s School


What do primary/secondary math educators think of the teaching that happens in colleges? And — the other way around — what do mathematics professors think of primary and secondary math teaching?

I’m nearing my tenth year as a primary and secondary classroom math teacher, and every once in a while I end up in a conversation with a graduate student or professor who suggests (politely, almost always!) that math education before college is fundamentally broken. A few weeks ago, a mathematician told me that PhDs are needed to help redeem secondary teaching from its “sins.” Once, at the summer camp where I teach, a young graduate student told me that there is simply no real math happening in American schools.

Well — I disagree! But how widespread is that view? And why does it exist?

The flipside phenomenon is also interesting. When a mathematician criticizes primary/secondary math education, primary/secondary educators sometimes lash back. Often we point our collective finger at pure lecture. Primary/secondary educators tend to think of pure lecture as uniquely ineffective. It gives the teacher no knowledge of whether students understand the material, and students no chance to practice new ideas in class. It is rarely used in primary/secondary math classes. Still, pure lecture was the main teaching mode in my own college classes, across subjects. We therefore bristle at mathematicians critiquing our work; “Let those without pedagogical sin throw the first stone!” I’ve even said this before, or something not far from it.

I have heard this “anti-lecture” critique expressed by some primary/secondary educators, but I wonder how widely held this view is. Is it held even by some math professors? And, in general, do primary and secondary educators tend to see flaws in the way math is taught in colleges?

In short, I wanted to better understand how mathematics professors and educators of younger students relate to each other’s teaching.

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Action is the antidote to despair

By: Steven Klee, Seattle University

After my day-to-day interactions with students, one of my favorite things about teaching is talking with other teachers.  There is no shortage of amazing teachers who are working hard to make their classes better and improve student learning.  Likewise, there are plenty of opportunities to find inspiration in our colleagues’ work, ranging from attending talks at conferences to simply getting coffee with coworkers to talk about how our classes are going.

A few years ago, I realized that the proportion of inspiring ideas that turned into measurable change in my classroom was essentially zero.  As I thought more about this, I realized that I was the biggest hurdle to this change.  There was a little voice in the back of my head with a constant and emphatic message: No. I can’t do that, and here are fifteen reasons why.

I know I’m not the only one who hears this voice.  Of course, the reason we have these thoughts is that they are often true.  No two people experience teaching in the same way. We have different personalities, different styles, and allow for organized chaos in different ways.  As a community, it is easy for us to despair in the challenges we face in our teaching.

Joan Baez said, “Action is the antidote to despair.” At the end of the day we are all mathematicians and we have been trained in solving problems.  To be apathetic in the face of the challenges put before us is antithetical to our training as problem solvers. And teaching, particularly teaching well, should be viewed as a problem that desperately needs to be solved.  Like many real-world problems, the problem (“What does it mean to teach well?”) is not clearly defined.  The data is messy. There is not one single correct answer.

In the rest of this post, I would like to discuss some methods for moving beyond the little voice that says “no” and changing your teaching without reinventing the proverbial wheel.  And, as with many real-world problems, I will not answer the question at hand (“What does it mean to teach well?”) and instead I will address a different question – How do I teach better?

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Is there a switch for “making sense” ?

By Elena Galaktionova

Elena Galaktionova sent us this article shortly before she passed away earlier this year.

Foreword by Cornelius Pillen

Elena Galaktionova received her first introduction to mathematics from her favorite middle school teacher in Minsk, Belarus, her hometown. After she had finished her education at the Belarusian State University she went on to receive a Ph.D. from the University of Massachusetts in Amherst. Her area of research was representation theory. She taught mathematics for many years at the University of South Alabama, after some earlier stints at the Alabama School of Math and Science and the School of Computing at USA. In Mobile, Alabama, she was one of the organizers and teachers of the Mobile Mathematics Circle. The Circle has been going strong for 20 years. Later she recruited local teachers and a middle school principal to participate as a team at an AIM workshop on Math Teacher Circles.  Upon return to Mobile she founded the Mobile Math Teachers’ Circle.  Twice she gave presentations at the Circle on the Road conferences. Her work with local middle schools and her interests in home schooling were motivated by her love for mathematics. She cared deeply about math education. Sadly, Elena passed away earlier this year after a long battle with cancer.

In all my classes I try to teach reasoning, writing and problem-solving skills.  I noticed that if a class is heavy on computations and dense in content, such as Calculus, the result of this effort is barely noticeable if at all.  I recall a memorable moment in a multi-variable calculus class.  The topic was optimization. My students knew just fine how to use the Lagrange multiplier method given a function and a constraint, thank you much.  But it turned out they were helpless in the face of even the simplest application  problems.  Some of these students were studying Calculus with me for almost three semesters and their grades were good and I tried so hard to teach them what matters in mathematics the most.   I remember a chilling realization at the moment, that we — the students and I —  wasted three semesters.

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Pre-Service Mathematics for Elementary (and Secondary) Teachers: a third essential element

by Paul Goldenberg and Al Cuoco

Content is essential; so are strategies and craft for teaching; but there’s more. It’s often said that “many elementary teachers don’t really know the content; the content they ‘know’ they don’t really understand; often they don’t realize that there is anything to understand.”

However true that characterization of elementary teachers is, we think it’s a distraction. There is no kindergarten teacher anywhere who doesn’t know how to count and add and subtract, which is most of what her children will encounter during the year. And if the teacher isn’t sure of the name of some particular geometric shape, that’s way down in the noise of what will matter for teaching. Lack of mathematical information—even a lack of understanding of why particular algorithms work—is not the biggest roadblock in the earliest grades. The remedy might involve more courses in mathematics, especially mathematics they will teach, but we think that the key issue is not more but different, even for secondary teachers.

As we see it, what hurts elementary mathematics teaching most—and hurts secondary teaching as well—is some of the ways in which teachers know “too much” math without a tempering sense of what the mathematical enterprise is and what not to teach. We will give two examples, one from fourth grade and one from first year algebra, to illustrate what we mean.

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Posted in Classroom Practices, Mathematics teacher preparation | Tagged | 2 Comments

Some thoughts about epsilon and delta

By Ben Blum-Smith, Contributing Editor

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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Posted in Faculty Experiences, Mathematics Education Research, Student Experiences | Tagged , , , , , , , , | 4 Comments

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

(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