Never DO Mathematics in a Math Class?

Within our efforts to make major improvements for our students, both in developmental and gateway college math courses, we have been looking at our content and our methodologies.  I want to connect the over-used phrase “Do the math!” with our phrase “doing mathematics”, within this process of building a better future.

I’ve commented before that the phrase “do the math!” is frequently used as a propaganda technique, to imply that everybody would reach the same conclusion as the speaker (and often used when the ‘math’ in the statement involves a small set of numbers, far removed from meaningful evidence for any argument).  I wonder if our phrase ‘doing mathematics’ serves a similar purpose within the profession.

Part of our problem is that we assume that there is a single meaning for ‘doing mathematics’.  Historically, the phrase seems to have grown out of the view of mathematics as seeking patterns, often in a constructivist approach; this ‘doing mathematics’ just means to immerse people in a situation involving quantities with a goal to establish patterns and statements that make sense to the learners.  Within professional mathematicians, we have two contrasting meanings — occupational (actuaries, for example) using systems of mathematical knowledge, and researchers using a variety of tools to establish new knowledge or applications.

As a learning tool, the “doing mathematics” has very limited benefits for the student.  Most research I have read suggests that learners need a very directive structure for the learning to occur by discovery; this guidance takes the process out of the original meaning of ‘doing mathematics’ into the more appropriate ‘learning mathematics’.  Doing mathematics, with the goal of learning mathematics, is a very advanced process — it is what some experts can do; expecting novices to engage in this process is a bit like expecting novices to become good piano players by having them sit at the piano (without any technique, without any theory).  Doing mathematics to learn mathematics does happen, often not by design, frequently with great excitement by those involved.

Perhaps the question is:

If students can experience what we experience when we ‘do mathematics’, they will be motivated to learn more mathematics.

Now, motivating students is one of the central roles in our classrooms.  Sometimes we focus so much on content and skills that we provide no information on our world of mathematics.  If we are the ones doing mathematics, presented in a way that novices can follow, then I can see some real benefits.  I have tried to do this in all of my classes; with colleagues, I use the phrase:

Students will see and perhaps do some beautiful and useless mathematics in every math class.

I include ‘useless’ in the description, and actually focus on that.  Why?  Because it is not reasonable to expect students to understand the eventual usefulness of the mathematics which they can appreciate; to them, it will likely seem “useless” even though it is not to a mathematician.  When I do this, I am walking a little beyond the limits of what students currently understand; I am in a beautiful field on the other side of a path, and want to share this perspective with my students.  My honest answer for ‘why I do this’ is simple:  It is fun!  I also have pedagogical reasons; walking a little beyond the current level helps to create an atmosphere of respect and one where learning for learning’s sake is appreciated.

So, my advice is: never have students to mathematics in a math class.  Help them learn mathematics, and you should do some mathematics in class so they know why we are mathematicians.  We don’t do mathematicians just for the money, or the fame.  We do mathematics because we want to.

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Typing Versus Learning

We are all trying to help students learn so that they can succeed; we invest great amounts of time in designing classroom work and other components, and many of us are willing to spend ‘free time’ helping individual students.  Our work will be improved if it is informed by research and theory on learning in general, and learning mathematics in particular.

I ran into an item that referenced a research study conducted on using a laptop to take notes compared to hand-written notes in class.  The reviewed article has not been published yet  but you can read a summary at  https://www.psychologicalscience.org/index.php/news/were-only-human/ink-on-paper-some-notes-on-note-taking.html 

The key findings:

Taking notes on a laptop tended to be verbatim and less useful than hand-written notes.

Students who took notes on a computer memorized the same amount of information as those using hand-written notes.

Students who took notes on a computer performed more poorly on test items dealing with the ideas involved in their notes.

There also appears to be a pattern of verbatim notes (little processing) even if students are directed to not take verbatim notes; the device seems to encourage this type of behavior.

I do not see many students using a computer to take notes in my math classes (though it does happen).  What we all see more often is the use of another machine — calculators in particular, possible internet with a browser.  Would these activities also tend to be at the ‘verbatim’ level of processing?

To a large extent, I think the answer is yes — typing on a calculator tends to be done as a keyboarding activity with little processing of ideas.  I am not about ready to give up the learning advantages of using a calculator, nor am I going to pretend that those internet resources do not exist.  However, I need to be aware of the potential impact on the quality of learning when students do a lot of keyboarding of any kind.

Like many others, I tend to use a graphing calculator as a tool to explore properties and relationships.  This usually involves entering an expression or function, and then looking at some type of results.  Although I believe this is a “good thing”, the results of my efforts have consistently been disappointing — students seldom get the idea in a way that sticks with them.  It’s not like there is no gain; it’s more a sense that the calculator process is creating some type of opposing force that makes the learning more difficult.

We, as a group, may tend to equate “getting something, making it visible” with “learning the ideas”. When we use calculators or other keyboarding technology, the research cited above suggests that students may be processing the activity primarily as one of the keyboarding itself — like transcribing a conversation.  Processing of ideas, and looking for connections, might be more difficult when using calculators or computers — not impossible, just a challenge.  We need to provide a structure will pulls student attention away from the required keyboarding to the level where they think about ideas and connections.

I think our results will improve if we keep these factors in mind as we design instruction and experiences for our students.

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SOAP, PEMDAS … Is there some MATH here?

For some reason, I have always found mnemonics to be irritating.  Perhaps this is based on a worry that understanding was being condensed to a ‘word’ that referenced a ‘phrase’ that had no connection to the mathematics involved.  Because we cover order of operations prior to algebra (questionable), we introduce one mnemonic to almost all students — PEMDAS.  Something about Aunt Sally (we all have one?) being excused.  The mathematical statement might be Grouping, Powers, Products and Quotients L to R, then Sums and Differences L to R.  Somehow, we don’t see “GP(P/Q)(S/D)”, even though it is better mathematically.

Another idea is reduced to a mnemonic — SOAP, for ‘same, opposite, always positive’ in factoring binomials involving two cubes.  This one at least refers to a memory process; this factoring is essentially a formula application.  The mathematics that is lost is ‘binomials of cubes’.  Perhaps this one should be ‘cubic SOAP’.  Of course, cubic-SOAP still is incomplete … it fails to capture the binomial going with the ‘Same’, and the trinomial first term (another always positive).

However, I wonder about the MATH mnemonic.  Perhaps you’ve heard it:

Man, Anything That Helps!!  (“MATH”)

Are we so desperate that we offer incomplete or inaccurate memory aids?  Perhaps we confuse correct answers with understanding mathematics.

Instead, I would like us to consider what this means:

Students should learn good mathematics in every math course.

A list of nice topics does not create a set of good mathematics.  In conversations, I usually find a good amount of consensus on the phrase ‘good mathematics’; we might have trouble articulating a single definition, but we have a good idea what it looks like at various levels of student mastery in various domains of mathematics.

Not everybody in the world uses ‘math’ as a label.  The label ‘maths’ is better, since our field has a plural nature; there is not one mathematic … there are fields of mathematics.  Perhaps if we kept using the word ‘mathematics’ instead of the inaccurate ‘math’ it would help us maintain our focus on why we are here … what we are helping our students WITH.  We are not here to get students to produce a minimal number of correct answers; we are here to help them learn mathematics with value.

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Should a Math Class be an Approximation?

I was tempted to title this post “should high school math teachers be allowed to teach college?”, because that is what I was thinking about recently; of course, that is not really the issue.  The issue is: should a math class, such as developmental math, be an approximation of the mathematics or should it be precisely the mathematics?

Here is the situation that got me thinking about it … While I was at the AMATYC conference (and Dev Math Summit) last week in Anaheim, I had substitutes in all four of my classes.  Two of them were former high school teachers, one a retired state worker.  Both of the former high school teachers confused my students by their mathematical presentation.  In one case, the teacher said that students should shade all regions for each inequality in a system including the ‘double overlap’; this is an approximation to the mathematics — only the overlap area should be shaded.  In the other case, the topic was the imaginary unit and complex numbers; this teacher did not appear to say anything ‘wrong’ but focused entirely on the mechanics.

It’s probably obvious that no math class can achieve precision in all topics during the learning process.  Approximations come from various factors, some more malleable than others.  One factor is linguistic in nature … precision is based on language, and deep understanding of language comes with experience.  We can not expect an expert understanding of the language from novice users; however, I would like to think that we design courses and curriculum so that students will move steadily towards the expert level.  This is complex, perhaps impossible … but I think it is critical to invest energy in this process for students.

Another factor for precision is created in the modeling process we provide to students.  In the ‘imaginary’ number case last week, the instructor emphasized correct symbol manipulation as a proxy for understanding the topic.  However, the human brain does not store information in a purely symbolic form … the process involves a verbal statement (sometimes called ‘unpacking’) from the symbols.  A novice student has no knowledge connected to the symbols; my substitute confused students by not supporting a verbal (conceptual) framework.

To re-state the title …

Should a math class be a deliberate or accidental approximation?

At this point, we should be thinking something like “Well, what is the problem if a math class is an approximation (deliberate or otherwise)?”

Here is a key problem:

Correcting prior knowledge is more difficult than creating accurate knowledge.

You may have noticed that students’ understanding of fractions is resistant to our efforts of ‘correction’ (same with algebraic faux paus such as distributing a power over a sum).  We spend millions of dollars on instruction partially as a result of math classes being a approximations at some prior stage(s) of the student’s math history.  Every time a student is required to take a standardized test in math, we are seeing the direct results of approximations in math classes and the harm they cause students.

I have no delusions that excessive ‘approximations’ are limited to K-12 teachers; I’m sure that many of our college instructors and professors do the same kinds of things.  I am guessing, however, that school teachers are more prone to running approximate math classes (based on interviewing experienced teachers across levels).  Also, policy makers who focus on ‘skills’ often provide indirect motivation to make math classes more approximate, as does a focus on ‘teaching for the test’.

Here is the tension we face:

Approximations result in inaccurate or incorrect learning.

Perfect precision results in no learning at all.

This is the math teacher’s paradox.  Like most paradoxes, this one serves to sharpen our problem solving.  The solutions lie along a path where approximations are deliberately limited and then refined towards perfection over time.

Many of us seriously underestimate the amount of work needed to learn mathematics — both professionals and policy makers.  Resolving the math teacher’s paradox depends upon appropriate conditions; the most basic of these conditions is time.  “Covering” the Common Core or “covering” the algebra curriculum will tend to doom most students to suffering the consequences of repeated approximations to mathematics.

We’d be better off working on precision for a lot less curricular content.

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