Re – Writing the Job Description for a Math Instructor

[A guest blog post from Larry Stone]

In the two-year colleges of the near future, what will a math instructor do to earn his/her pay? What, exactly, will his/her job involve?

The model I see emerging is: setting some software to deliver a standardized list of prefab learning items (perhaps checking a few boxes to add or delete some items),
scheduling a few automated assessments, then letting each student follow an individualized path at an individualized pace (taking individualized assessments).
Occasionally, the “instructor” should check in to view the dashboard, just to make sure everyone’s been logging enough hours. Intervening in the actual learning process is only
necessary when the software seems to be struggling in guiding some student towards the correct responses – but we may expect this to become less necessary as the software
continues to improve year after year.

So, what is the math instructor of the near future? In essence, a software jockey with some tutoring ability.

Now consider: how much skill and training does that require? Presently, we hire experts with master’s degrees to teach almost all of the classes: teaching is a profession.
Instead, one can picture a small group of programmers and content developers at the center, with lesser trained software jockeys (to be nice, let’s call them “student support
specialists”) distributed among the schools. The huge potential savings to higher education, where the cost of high-credentialed labor is the largest expense, makes it easy
to see why we are inexorably moving towards this model: it’s individualized instruction for all, which makes it sound good, but it’s cheap, which is apparently the ultimate good.

But will we lose something that we could never get back?

When I proudly entered the profession in 1999, it was still a mostly traditional environment. It felt like a perfect fit for me, because it gave me the opportunity to
exercise two of my professional strengths: I love to write-write-write, crafting and re-crafting materials to make them fit together and flow ever more naturally, and I love to
put on a show, sharing my enthusiasm for math and engineering and the great fun that comes from understanding how the world works. Instructors at that time were expected
to be heavily involved in developing their learning objectives, lectures (not a naughty word if done well), exercises, projects, and the like; and as for “putting on a good show,”
that was the main reason for teaching at a two-year college instead of a four-year college: good teaching, not voluminous research, was what mattered.

I now see how fortunate was my timing: I’ve had a great run for twenty years. What has surprised me the most is that, having a human mind interacting with a human
world, I still continue to have sudden insights about how to make things even better! It keeps the job fresh, fun, interesting, and in tune with an evolving world. Best of all, I am
free to immediately incorporate my ideas into my curriculum, assessments and all, without having to worry about how it messes up some software’s learning item
connectivity database. The master plan is entirely in my own head, and that I can easily adjust. I feel like a craftsman at work.

I even dare say, I’d be pretty good instructional software if I could be downloaded — but we’re not really there yet with the technology, are we? Are we even close? Perhaps,
before we ditch the master craftsman model in order to adopt the factory automation model of education – before we lose the generation that understands what teaching as
craft is all about, and find ourselves dissatisfied with the skin-deep, stimulus-response McEducations that will result — we should ask ourselves: how easy will it be fix THAT
situation?

Instead of sliding down that road, we should refocus the original question. What SHOULD a math instructor’s job involve, in a perfect world? I’ll offer just four ideas:

1. Writing good learning objectives and lessons, hand-crafting exercises and assessments, and using classroom experience (and other experiences, such as
   from teacher conferences, etc.) to continually improve these materials over the course of one’s career. Besides having the basic drive to produce quality work, the
   instructor should delight in finding new ways to communicate ideas that seem to open up possibilities for ever deeper learning and insight.
2. Close, daily grading of student work, in order to hand-write custom feedback and advice for each student, while also learning which areas may need to be re
   addressed in the main class (which can be amazingly different from class to class and term to term). It takes time, but this, in my experience, is by far the best way
   to take the true pulse of your classes. Certainly, it provides a richer feel than turning to summary statistics on a computer.
3. Using one’s own professional and life experiences to show how learning content relates to the “world out there.” Nobody measures this, but as a student in college
   I always felt it was truly worth something to be coming into contact with so many different content experts, each applying his/her own unique background and style
   to the subjects at hand. You learn things that aren’t in the book/aren’t in the software. Believing one has a unique and valuable perspective to share that may
   inspire some students to go further is part of what motivates one to become a teacher.
4. Getting to know each student as a person. Besides putting students in a receptive mood, it helps one know how to be personally supporting and encouraging, in
   ways indescribably more effective than pop-up messages from the software saying “your hard work is paying off!” for the sixth time this week.

Computers are great for taking over tedious, repetitive calculations, but this is not what math education involves. If you view any of the above tasks as potentially tedious
then, historically speaking at least, you’re in the wrong profession. Meanwhile, show me a computer that loves teaching this stuff and maybe it can learn to take over my job —
but the technology isn’t there yet, and may never be.

by Larry Stone; February 26, 2019

Easy or Worthwhile?

I was walking by our copy machine this week, and saw a handout for the same material that I was about to work on in a class.  I took a look, and reacted a bit strongly to what I saw.

The basic idea on the handout was this:

The easy way to solve these equations is to enter one side as Y1, the other side as Y2, and have the calculator find the intersection.

I have to admit that using the calculator can be easy … not as easy as just looking up the answer, but sometimes easier than a human being solving the problem.  The question is:  Is it just easy, but not worthwhile?  Do students gain anything from using a built in program to solve a problem?

I face this issue in our Applications for Living class.  A bit later in the semester, we will talk about medians and then about quartiles.  Students discover that the calculator will find all of that for them.  Should students start to use the calculator to find the quartiles and median right away, to avoid the tedious work of ordering sets of 12 to 20 numbers?

In this statistics example, the material is worthwhile if the student can answer this question easily:

A set of 100 numbers has a median of 40, a lower quartile of 25 and an upper quartile of 70.  How many of those numbers are between 25 and 70?

A basic understanding of quartiles gives a good approximation (50); I’d be thrilled if a student said ‘about 50 but we don’t know for sure’.  In the practice of statistics, technology is always used to find the calculated parameters … and we need to know how to interpret those values.

The content for the handout I saw was ‘solving absolute value equations’, one of my least favorite topics because it tends to be hard to understand while there are a relatively small number of places where this needs to be applied.  However, the understanding of absolute value statements contributes to some common themes in mathematics — multiple representations in general, symmetry in particular.  Technology (as used for an ‘easy way’) avoids all of this stuff that makes it worthwhile.

A focus on the ‘easy way’ presumes that the only purpose for a topic is to get the corresponding correct answers.  To me, a student that uses the calculator to solve |x-5|=7 is just as dependent as a student who uses a calculator for 8 + 5.  The solution is simple enough that it can be done mentally; even writing out all steps gets it done quicker than a calculator process.  If all we do is show students how to obtain correct answers, what is the value that we have added to their education?  If we need to solve |25.8x + 4/3|=8.52, I will certainly tell students … ‘well, we understand how to solve this problem ourselves, so let’s set it up that way — and here is how to check that on a calculator’.  Of course, I know of no place, outside of an algebra textbook, where such a problem would be needed.

Easy is not the primary goal.  Worthwhile learning, and education, are the main things.  Every time we avoid learning we detract from our students’ education.  Technology has a role to play; ‘easy’ does not.  Understanding is a lot more valuable than a hundred correct ‘answers’.

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Some of My Best Friends are Calculators

Some years ago, we had an extended discussion about college credit for developmental courses (math in particular).  The proposal being discussed was eventually superseded by other policies; however, strong opinions were voiced.  During one commentary, a colleague was decrying students getting credit for such courses (though he had nothing against faculty who teach them.  Our Divisional Dean leaned over to me and said “some of my best friends are developmental math teachers”, which I thought was quite funny (though the situation prevented me from laughing at the time).

When I hear some colleagues talk about calculators, I am reminded of that comment rephrased … “some of my best friends are calculators”.  Calculators have their place, such colleagues say; calculators are not bad … it’s how students use them, so we need to prevent students from using calculators in a math class (as they say).  In fact, I once took the position that graphing calculators not be allowed in a first algebra course (back in 1993).  Since 1995, I have taught in an environment where graphing calculators are required starting with our first algebra course; although there are days when I find this frustrating, I have become a supporter of using calculators.

Unfortunately, the problem is much more complex than a ‘no calculator’ policy could solve; nor does a ‘required calculator’ policy solve these problems.  Here are some of the problems that we can avoid discussing by focusing on a calculator policy issue:

1. Students want a calculator for basic operations for a reason — they feel ‘dumb’ at math; that’s a major issue.
2. Students view correct answers as being a valuable commodity, instead of seeing correct answers as suggesting good understanding
3. Numeracy leads to feeling smarter; having a sense of how quantities ‘behave’ is possible for almost all humans (just like language literacy).
4. Reasoning about quantities is a natural human endeavor, though we communicate this with language systems that are artificial (a necessary condition)
5. A single math class tends to be very ineffective at changing long-held beliefs and habits; data suggesting an impact normally are measuring temporary conditions.
6. The big picture ideas are more important than how a student calculates a particular value; the big picture includes their self-image about mathematics.

I like requiring a calculator in math classes, to provide a better venue to discuss these issues with students.  Sometimes, a student ‘gets it’ (what we are talking about) and they change their math trajectory; for most students, it’s not that much of an issue either way — it took them 12 or more years to get to this point, mathematically, and a short-term experience is not likely to hurt them any more.  Using the calculator, it seems, at least opens the doors to possible positive changes over a longer period.

This conversation with myself started when somebody reminded me of an article I wrote for the 1993 AMATYC journal; reading that article was an awkward experience, as I could see errors in my own thinking.  Perhaps this post will encourage readers to examine their own position on calculators in math classes from a different perspective, one reflecting my course correction on the use of technology in mathematics.

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Khan, Comfort, and the Doom of Mathematics

Perhaps you already knew this:

If students perceive instruction as clear, the result will be reinforcing existing knowledge (often not so good knowledge).

I recently ran into a reference to a fascinating item posted by Derek Muller, specifically about videos like the Khan academy; Dr Muller’s specific interest is science education (physics in particular), and you might find the presentation interesting http://www.youtube.com/watch?v=eVtCO84MDj8 (it’s just 8 minutes long).

In mathematics, even more than physics, students come to our classrooms with large amounts of prior experience with the material.  Of course, much of their existing knowledge is either incomplete or just plain wrong (whether they place into developmental math classes or not).  A ‘clear’ presentation means that the existing knowledge was not disturbed in any significant way.  Clear presentations make students even more confident in the validity of the knowledge they possess.  This is not learning.  Reinforcing wrong information is the doom of mathematics.

In Dr Muller’s study, two types of presentations were done.  The first were the ‘clear’ ones; students felt good about watching, but the result was absolutely no improvement in their learning.  The second type were ‘confusing’ ones, where the presentation deliberately stated common misunderstandings and explored them.  Students did not like watching these;  however, the result was significantly improved learning.

We see this in our classrooms.  This past Friday, a young man from my beginning algebra class came in to see me … he had left class in the middle, in a distracting way to other students.  Turns out that he left because he could not stand the confusion.  In talking to him, he believes he can do the algebra but he is getting very confused by the discussion in class about “why do that” and “here is another way to look at it”.  In fact, this student has a very low functioning level about algebra.  If he does not go through some confusion, his mathematical literacy will remain unchanged; that is to say … he won’t have any meaningful mathematical literacy.

Khan Academy videos are popular; I understand … I have watched some myself.  I consider them to be very clear and essentially useless for learning mathematics.  If a person already has good knowledge, they will not need them; if a person lacks some knowledge, they will not perceive what they lack from watching a video.  [Just like witness research in criminal justice, students perception is controlled by their understanding.]

The attraction of modules and NCAT-style redesign is often the clarity and focus.  Students do not generally see anything that might confuse them; the environment is artificially constrained to avoid as many confusing elements (inputs) as possible.  To the extent that students in these programs are ‘comfortable’ and the instruction ‘clear’, that is the extent to which existing knowledge is reinforced.  Learning can not happen if we primarily reinforce existing knowledge; confusion is an essential element in a learning environment.  [I sometimes tell my students that instead of being called a ‘teacher’ they should call me ‘confusion control expert’.]

I suspect some readers are thinking that “He has this wrong … I have data that shows that students do really learn.”  It’s true that I don’t have proof; I don’t even have my own research (though I would love to see some good cognitive research on these issues).  What I do know is that student performance on exams — especially procedural items — is a very poor measure of mathematical knowledge.  I suggest that you interview some average students that you think know their mathematics based on exam performance; have them explain why they did what they did … and have them explain the errors in another person’s work.  Based on what I have heard from students, I think that you will find that only the best students can show mathematical knowledge in an interview at a level equal to their exam performance; average students will struggle with the interview about their mathematics.

How do we avoid the doom of mathematics?  How do we prevent our classes from becoming reinforcers of existing knowledge?  I think we need to create environments for learning where every student faces some confusion on a regular basis … not overwhelming confusion, and not trivial confusion, but meaningful confusion about important mathematics.   Do we need an LCD to do that?  Must we move terms in an equation before we divide by the coefficient?  Is that distrubuting, or is that subtraction?   Confusion is where students bump into the areas of knowledge that need their attention.

Our students have a strong tendency to drive through our courses as fast as possible, without really dealing with mathematics.  They believe the myth that the experts always understand, that we are never confused.  We need to be comfortable in showing confusion to our students and model appropriate behavior to resolve it.  The appropriate response to confusion is figuring out where we went wrong … not running away for a comfortable explanation.  Confusion may call for some meta-cognitive efforts, or we may simply need to polish one particular mathematical idea.

Confusion is the fertile soil of learning.  Avoiding confusion creates a sterile environment without growth.  Comfort is fine, and we all need comfort; however, comfort never learned anything. 

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