Conversation I: Herb & Jack on the Future of Teaching

Over the past few weeks, I have been having a great conversation via email with Herb Gross, the founding President of AMATYC and long-term inspiration to many of us working in the first two years of college mathematics.  With Herb’s kind permission, I will share some of our conversations here.

Herb’s comment

In my opinion, the face of education is going to change dramatically very shortly  when we begin to use the Internet to clone “master teachers”.

Herb has a good point.  In fact, many readers will respond by saying that we are already doing just this.  Jack’s comment

Yes, the internet allows for distribution and re-use of good materials.  The issue we need to focus on is “what makes a good teacher” or a “master teacher”.

In recent years, we have seen a couple of forces that shift our focus away from a clear image of a good teacher.  One force is the “sage on the stage … guide on the side” image, where a teacher is a coach or trainer responding to students as they learn mathematics.  Another force is the huge emphasis on online learning components, which is sometimes implemented as a replacement for teaching in some redesign models.

In my view, being a good teacher is a very personal experience with a group of students.  When it works, students will say something like the following:

Thank you for a great semester.  You have been an amazing teacher.  Thank you for not giving up on any of us; you pushed me (at least) to do my best.

That is a note written to me, handed over before this student took her final exam.  She did not have the best average in class, but she did work hard and was engaged during every class.

Jack’s description of good teaching:

Good teaching involves the articulation of ideas in a way that is understandable to students, creating a learning environment that encourages learning, and inspiring students to work harder than they intended.

Only a small portion of this can be cloned, or distributed across the internet.  Herb has some great videos (like this one on You Tube:; this video is part of Herb’s work on math as a language, which you can see more about on his web site

One of the many things I like about Herb’s work is that he takes a lot of time.  We sometimes think that students do not have the attention span to deal with a deliberate approach; that is not generally true, and Herb gets a large amount of unsolicited positive feedback on his videos.

As good as these ‘clones’ (videos) are, they do not make a good teacher; having a master teacher articulate ideas is wonderful.  This is not sufficient to be a good teacher.  We need to provide clear statements concerning what the practice of teaching involves.  Our colleagues need this, our leaders need this, and certainly policy makers need this.

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Statistical Doors Into Mathematics

That’s really a question — does statistics create a door into mathematics?  Or, is statistics (for most students) an alternative off-ramp from the mathematics highway?

The question is perhaps trivial.  In terms of the bulk of our work, we are dealing with students required to take specific courses for their program.  Every math course becomes a common off-ramp for students.  Perhaps we should be satisfied with a curriculum consisting of terminal courses for students interested in everything else but mathematics.

One of my colleagues began her higher education as a fairly typical community college student at our institution.  She reports that a turning point for her was a particular computer science course that she decided to take; after this course, she changed her major and got a degree in computer science (and later a masters in math).  There was something of beauty in that computer science course that connected with her, and changed her life.

I would be interested in any research on the question:

Do students change their career path to mathematics after taking a statistics course?

I am sure that there are students who change their path to statistics after a statistics course, though I wonder if the rate is equal to that of ‘math program after a math course’.

Like most of us, my students are just interested in passing this math course so they can get their degree or that job.  I am fine with helping them along that trail; in fact, I am happy to do so.  I teach because I find that rewarding.

However, I am also a professor is in “affirms a faith in something”.  I think I have a responsibility to show students in each course something about the beauty of mathematics; something wonderful should show in every class.  Partly, this is needed to encourage more positive attitudes about mathematics; partly, this is needed to encourage a more accurate view of the nature of mathematics, that mathematics is much more than processes to generate answers.

To me, however, the largest reason for what I try to do is “opening doors”.  A major reason for lowering expectations for a given student is mathematics; lower-skill programs are selected because they require less mathematics (or none).  Students even avoid occupations that they would love to be in … just due to mathematics.  To me, every mathematics course should be a STEM magnet drawing students towards higher skilled jobs and more security.

I do not think that statistics operates as a STEM magnet.  Of course, there are many math courses in our institutions that are not STEM magnets; however, almost all math courses could be strong attractor points drawing students towards mathematical sciences.  I think the problem with statistics is that we teach statistics as a practical discipline without a core mathematical structure.  We focus on the innate appeal of statistics, on its utility; perhaps we need to show the mathematics supporting statistical methods when possible.  If there is no mathematics supporting a method (the ‘plus 4′ rule type of thing), perhaps we should question the presence of that method in a general statistics course.

Clearly, I may be demonstrating levels of ignorance vast and wide.  I wonder, though … do we share a view that math courses in the first two years should have a property of ‘STEM magnet’?  Can a statistics course be such a magnet?

Before the reader decides that I am far too optimistic about our mathematics courses — yes, I know that we fall far short of a STEM magnet in our current courses.  We tend to cede our territory, and deliver service courses; we focus on the practical at one extreme … or the totally useless on the other.  In between is the zone needed to be a magnet for students; a magnet can not be unidimensional.

Perhaps the question is more general than statistics; my concern is with the contemporary move towards requiring statistics as the typical general education course.  Perhaps the loss is trivial.  I do wonder if there is an innate qualitative difference between statistics and mathematics that results in statistics being far less able to contribute towards larger goals such as raising student goals and drawing students towards STEM.

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Fragile Understanding … Building a Foundation

Our beginning algebra class is taking a test on ‘exponents and polynomials’ today; this chapter is about as popular as a math chapter can be for my students.  The processes are fairly easy, and with some extra effort in class, most students do well on this test.  All is not good, however.

Students tend to have a fragile understanding.  For whatever reasons, the symbols in front of them do not have full meaning.  Here are two examples of what I am talking about.

Subtraction versus “FOIL”:
Seeing a problem like (5x + 4) – (2x – 3), many students convert it into (5x + 4)(-2x+3) and multiply.  They know the rule about ‘changing the signs’ when we subtract, but lose the little ‘plus’ between the groups.

Negative exponents versus polynomials:
Seeing a problem like (6x² – 9x)/(3x²), many students convert (2 – 3/x) into (2- 3)/x to get -1/x.

As teachers, we feel good when students show a process that fits with a good understanding.  Showing a process does not depend on a good understanding.  The relationship works one way cause and effect (understanding leads to good processes); a good process does not lead to, nor is evidence of, good understanding.

So, we give assessments to students and say “they know exponents” because of the processes and answers.  In the extreme form, we have a module on exponents and polynomials and certify “mastery” because of a high score on the module assessment.  We do not do enough assessments that do a compare and contrast — opportunities for us to see if a student has a fragile understanding, identify the weakness, and then build up a stronger understanding.

I continue to work on this problem.  In the case of ‘subtraction versus FOIL’, I use problems like the one shown on assessments early in the semester, during our first class on ‘FOIL’, and later in the chapter.  That helps; no magic, but the opportunity to discuss with an individual student is powerful.

I believe we need to work on two components of our instruction if we have any hope of building a strong understanding in place of fragile understanding.

  • Combination of active and direct instruction on the concepts, with a focus on “what choices do we have?”
  • Assessments that determine the presence of confusion of concepts (aka ‘fragile understanding’)

Our professional expertise is needed, since we can not assess for the presence of specific confusions unless we know what the common types are.  To make this even more challenging, we have no assurance that the confusions are global versus local — do students in beginning algebra courses tend to have the same confusion regardless of locality?

The best resource we have is the students in our classes.  Having purposeful conversations (oral assessments) is a critical source of information about both a specific student and zones of confusion.  These conversations provide insights, and form a way to validate our more convenient forms of assessment (paper & pencil, or computer test).  When I grade today’s test on this chapter, I will be comparing what I thought they understood to what I see being shown on the test; just like my students, there should not be any surprises to me on the test.

Of course, there is a good question … does it matter at all?  We have a pride in our work and profession, so we respond with an automatic ‘yes’.  We should be able to articulate to other audiences why it does matter.  Does a fragile understanding enable or prevent a student from completing a math course?  How about a science course?  Can we develop quantitative reasoning in the presence of fragile understanding?  Does a modular design support sufficiently strong understanding?  Do online homework systems provide any benefits for understanding concepts?

The issue of fragile understanding is critical to the first two years of college mathematics, whether in a developmental math class college level.  I have heard colleagues suggest that the prerequisite for a certain class be raised to calculus II, not because any calculus is needed but only because students have a stronger understanding after passing (surviving) calculus II.  We often cover this problem with a vague label “mathematical maturity”.

In response to a recent post, Herb Gross (AMATYC founding president) wrote a comment, in which he emphasized the “WHY” in the math classes he taught.  I totally agree with his comment, in which he said that students want the why — they want to understand.  Although a human brain can learn with and without understanding, there is a natural preference to learn with understanding.

A fragile understanding, lacking the ‘why’, leads to both short term and long term problems for students.  I think we waste their time in a math class if we accept correct answers for the majority (70%) of problems as a proxy for ‘knowing’.  Determining that a student knows mathematics is a complicated challenge, and forms a core purpose for having a strong faculty professional development.

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This is Hard Work … Why It is Important!

The AMATYC 2014 conference is almost over.  Many conversations and sessions have dealt with changing the curriculum in basic ways, whether shifting to a New Life model or Dana Center or Carnegie … in part, whole, or modified.  Some of us get so enthusiastic about this change that we don’t get slowed down by worries or concerns about the amount of hard work this will take; for us, it’s like nothing will stop us from reaching  our goal.

Most of us, however, are facing constraints.  We are intimidated by the work involved — the hard work of developing an idea, getting consensus or approval in the department, and building institutional support, all of which is required before we get to develop the teaching in the new courses.  Perhaps, we think, it would be best to take a small step like replacing one chapter in the current course with a new one, and see how the ideas work out at other institutions.

Take the biggest step you think you can.  In fact, take a step a little bigger than you think is realistic.

I could justify these statements by citing policy initiatives that are coming, by showing data on how ‘bad’ things are now, or by invoking the Common Core mythology.  Today, I want to take a different approach to the rationale for why it is important to make changes a little bigger than you think is realistic.

Call it personal, or perhaps religious.  My world view, informed by my beliefs, goes something like this:

  • Do no harm to others
  • Place the needs of others ahead of (some) of your own needs

For a large portion of students placing below calculus, we are doing harm to them.  They come to us with dreams and aspirations, and we place steps in front of them that are frequently and artificially difficult.  Yes, we bear some responsibility for students giving up on their dreams.  We have been doing significant harm, even though our work is driven by a desire to help.  We must stop doing harm to such a degree.

We should be placing the needs of students above our own needs.  We will have jobs, though different, even if we eliminate half of our developmental courses; sure, it’s not comfortable … it can even be threatening.  However, continuing what we have been doing is not putting students’ needs first.  Does any student need the 100 learning outcomes of my college’s developmental mathematics courses?  Absolutely not.  Those 100 outcomes are there because we there might be a need sometime for somebody to use them in some situation; much of this ‘need’ is driven by assessments within mathematics courses.

Herb Gross, founding president of AMATYC, gave a popular keynote speech here at the conference.  At one point, Herb said that we should re-phrase the golden rule; instead of “Do unto others as you would have them do unto you” it should be “Do unto others as they would do unto themselves.”  I appreciate the intent with this re-phrasing, but it totally misses the faith justification for the original … we are called to see ourselves as the other person; the ‘do unto you’ refers to what they could do to you or for you if your roles were reversed.

So, envision yourself starting at college, and your initial mathematics course is not very ‘advanced’.  Perhaps you are the person who had AP Calculus in high school who is told that you now must take two courses before taking college calculus.  Perhaps you are the person who passed two algebra courses with weak grades who now finds herself sitting in a class reviewing grade school arithmetic with 25 other students of color.  Perhaps you are the person who did not do well before, and really needs help building a mathematical base … and you find yourself in a math course which deals strictly with procedures, with some drive-by attention to concepts, and no real applications in sight.

The issues I am talking about are definitely not just in ‘developmental’ or remedial courses.  At all levels, we tend to have a mismatch between student needs and what we provide.  The harm is done in pre-algebra and pre-calculus, in algebra and calculus, and even in statistics.

I have never met a math teacher at any level who wanted to do harm to students.  Almost all of us have a sincere desire to help students, to provide the best mathematics possible.  Change can involve a risk to do harm; the profession has enough knowledge about mathematical needs and learning to avoid much of the risk.

Do no harm, as much as possible.  We must take the biggest step possible … and a little more … to reduce the harm we do.  We should take advantage of the external forces, and create the types of change we think are best — to put our students’ needs first.

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