A Trajectory in Math

A story about what students are capable of doing … and how resistant prior learning can be.

In our applications course, we took a test recently on numeracy and finance; of course, I did not call it ‘numeracy’ for the students.  They saw phrases like ‘percents’, ‘scientific notation’, and significant digits.  One of the non-standard problems on this test was:

My age is 5 · 10^9 seconds.  Is this reasonable?

We had done a couple of problems involving changing quantities in scientific notation to different units.  However, this combination had not been seen before — notice the unstated ‘change this to years’ part of the problem.  The majority of students did a good job with the problem.  Since this ‘age’ is about 159 years, the answer is ‘no’.

On the same test was a percent question:

The retail cost of a computer is 27% more than its wholesale cost.  Which of these statements is true?(A) The retail cost of the computer is 127% more than the wholesale price.
(B) The retail cost of the computer is 27% of the wholesale price.
(C) The retail cost of the computer is 127% of the wholesale price.
(D) The wholesale cost of the computer is 73% of the retail price.

Notice that the stem of the question is a direct conflict with choice (A).  Sadly, choice (A) was the most common incorrect choice; most students did not select the correct response (C).  Even though we had explored percent relationships in different ways, pre-existing knowledge seemed to trump recent learning.

So, here is the question:

Will students have significant long-term benefits from the college math experiences?

In other words, are we lining up trajectories in math … or are we just enjoying a shared experience with no impact of importance?  I would like to think that our courses are building reasoning, understanding, and structure; that we are aligning trajectories.  Of course, yes, I know — this is unlikely; perhaps I am hoping for too much.

I’m reminded of all I have read and studied about memory formation related to organized learning.  The human brain does like to organize information about the world; unfortunately, it seems like much of this ‘information’ is really an oral narrative related to experiences.  Perhaps this is due to the high emotional load many people in our culture experience in ‘mathematics’.

And, I think about all of the effort on ‘remediation’ of arithmetic and algebra.  The students who need the most tend to have strong connections to past stories with good endings, stories that contain bad mathematics.  [Cross multiply … PEMDAS … LCD … and other tag lines for stories.]

Perhaps we would be wise to focus on what mathematics students will need in courses they are likely to take.  Perhaps we can have some success in dislodging prior ‘learning’ if we create more intense environments for learning — with a focus on reasoning and connections.  Remediation might be possible given enough time and enough resources; a more reasonable goal would be building capacity and quantitative functioning.

If we focus on basic themes for the course or two with students, perhaps we can help our students get a positive trajectory in math.

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Applications — Starting on Correlation

In our Applications course, we develop some basic statistical reasoning.  I always start key topics (like what is statistics … what is correlation) by having students work in groups on discussion questions.  I recently changed the ‘correlation’ activity, and thought other people might be interested in seeing it.

My goal with statistics in this course is to develop statistical thinking (and caution!), in ways similar to the standards in GAISE (see http://www.amstat.org/education/gaise/GaiseCollege_Full.pdf)   We use real data whenever possible, and focus on reasoning first … computation later.

In the case of correlation, I originally used a variation of the “Cereal Plots” activity like that used in Statway™.  In the cereal plots, students are presented with various nutrition values plotted against the ranking of cereals by Consumer Reports.  Conceptually, this is really nice.  However, in practice, students have too much overhead — they don’t know about rankings in general, and certainly not cereal rankings by Consumer Reports.  We ended up spending about half of the class discussion time on secondary issues.

This semester, I created my own sample of graphs.  These scatterplots deal with contexts familiar to almost all students, such as cars (price, mileage, etc).  Here is my activity for this semester

So, this was the initial part of class yesterday.  Students were in groups of 3 to 5, and answered the 10 questions on the sheet.  The context for these graphs was much better — somebody in every group knew enough about cars to provide some additional wisdom, and everybody knew enough about cars to get started.  [The last graph, on grip strength versus arm strength, is accessible to all students.]

Overall, this new activity works much better.  All of the discussion was about the graphs and statistics.

This does not mean that students magically understood what correlation is … or how to judge it from a scatter plot.  We are still working on unlearning ideas about cause & effect.  However, we did make progress on judging a linear pattern in these graphs; when I say “negative correlation”, most students can connect this to a pattern like we visualize from the phrase.

In case you are curious, the application course is pretty limited in the statistical topics included.  We include some reasoning topics (samples, population, bias, correlation, and describing distributions); we also include some displays (frequency tables, bar graphs, etc) and a few calculations (mean – median – mode, standard deviation, rule of thumb for margin of error).  We do not calculate correlation coefficients, nor do we do regressions; we do not calculate actual margin of errors, though we do calculate 95% confidence intervals (using the 1/sqrt(n) rule of thumb).  Out of 16 weeks, we spend 3 weeks on statistics; 2 weeks are spent on basic probability.

By the way, all of the graphs on the correlation activity were taken from online searches.  I was honest with the class — we do not know how valid any of them are, though I believe that they are valid.

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Are You There, Mr. Gates? Flip that MOOC right over, guy!

Some of us have been thinking about the influence that foundations (and rich people) are having on education.  What was once an influence of the national science foundation is now the influence of the gates foundation, with a smaller group of people making decisions based on priorities that are not open to public review or political approval.

A recent article described how Mr. Gates suggested to community college trustees that a ‘flipped MOOC’ might be a good solution — especially for developmental mathematics.  [See http://chronicle.com/article/MOOCs-Could-Help-2-Year/142123/].   I suspect that the article is misquoting the ‘doctor-not’ (Mr. Gates); an intelligent person would not use an oxymoron like ‘flipped MOOC’.  (Flipped means ‘lectures’ happen outside of class time; MOOC’s do not have class time.)

However, that minor detail (that is is not possible) will not make any difference.  Because it was Mr. Gates saying it, many of our leaders (college trustees) will be confident that it is true.  I expect to hear from my College’s trustees within a few weeks, as they wonder whether we would like to try a flipped MOOC model at our college to solve our dev math problem.

Coincidentally, I saw a very good presentation on an inverted design for instruction — a better name than ‘flipped’.  This presentation was at my state conference (MichMATYC) — a talk given by Robert Talbert (Grand Valley State University); a reference is http://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1183&context=colleagues .  I was impressed by the amount of analysis done by Dr. Talbert to create the inverted calculus classroom; the process is much more complicated than ‘lecture outside of class time’.

To some extent, the ‘flipped MOOC’ phrase illustrates the linguistic process for the evolution of word usage: the initial use of a phrase is specific, becomes accepted, and then is applied in usage to unrelated objects in order to imply something positive (or at least ‘current’).  As educators, we have been damaged by this “phrase drift” many times over the years (mastery learning, back to basics, applications, calculator friendly, collaborative, student centered, and others).  The difference in this period is that our future is being heavily influenced by people who have less understanding of curriculum and instruction … rather than more.

There was a time when ‘trends’ in education were declared by top-level academicians and national policy heads.  These people (generally office holders of some kind) were deeply networked in the collegiate life.  No more; we are spending most of our time either agreeing with or arguing with people ‘on the outside’ — foundations in the completion agenda, philanthropists, and legislators.  It’s not that we should ignore the concerns of outside stakeholders.  The problem is that the outsiders have taken control from us; we react to them.

So, I ask:  Mr. Gates, are you there?   When do we get to have a productive conversation about the problems we are trying to solve?  We could look for problems where we agree on solutions … problems where we agree on the problem but not on solutions … and problems where we see the problem differently.  I know this, Mr. Gates — the process being used so far has put a lot of money is promising practices and technology without much sustained benefit; your return-on-investment is not so good.

When do we have a productive conversation?  Until we have real conversations with the people and groups trying to solve the problems (with the best of intentions) … until we work together, and not in reaction … until we accept both the worthy and not-so-good about the old system … only then do we have any hope of building something that will both solve problems and be capable of surviving in our world.

If you want to ‘solve the developmental mathematics problem’, Mr. Gates, I suggest you start by collecting a team of the 10 best thinkers and practitioners in the field who work with you over a 10-year period.  We want to solve problems; we strive to have students succeed and complete.  Can you recognize the need to have us as partners?

Are you there, Mr. Gates?

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Should College Algebra Exist?

Once in a while, I do say something that I regret.  During a recent discussion about general education, I commented that college algebra is intermediate algebra with more fractions.  I knew that was not accurate; I wish I had said that college algebra is intermediate algebra with bigger words.  [Still not that accurate; however, the statement probably makes sense to non-mathematicians.]

“College Algebra” is one of the most common math courses on our campuses.  As a profession, we don’t agree on what this thing is … except that college algebra is one of those courses identified by the title ‘college algebra’.  In some cases, the name is given to a good math course because it will help get transfer credit when ‘general math’ would not.  In other cases, ‘college algebra’ is the title of a remedial pre-calculus course (prerequisite for pre-calc).

If the college algebra course is a STEM-focused, math intensive course, then ‘introduction to analysis’ or ‘analytical methods’ might be a better name.  Whether there is a reason to have this course as well as ‘pre-calculus’ is unclear to me.  If we mean precalculus, perhaps we should always say that.

One thing I do know — college algebra is not a good title for a general education math course.  If we have one course to develop mathematics within college students, I am sure that we can deliver more than symbolic algebraic methods and function theory.  [Some of my colleagues remedy this situation by doing modeling in gen ed math courses; I think this has limited benefits, by itself.]

When I looked for definitions of college algebra, I was pleased to find one that referenced introductory analysis.  Sadly, that source cited no references; I do not think we typically define the course this way.  I see little evidence that we have responded to the calls to modify the course to meet student needs (MAA CRAFTY, AMS Client Disciplines, AMATYC Right Stuff).

Currently, developmental mathematics is under great pressure to change … and we should.  However, the zone of the great unchanging curriculum seems to be college algebra.

No, ‘college algebra’ should not exist.  Let’s build something better in its place.

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