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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Probem Solving in a Digital Age

Whether we ‘flip’ a classroom, use an online homework system, or refer students to Khan’s Academy … our students are using task-oriented videos.  In addition, students have a tendency to see ‘look it up online’ as a substitute for learning something.  As we become immersed in (and dependent upon) the digital age, can we still work on problem solving or critical thinking?

One of the sessions at this year’s AMATYC conference dealt with the topic ‘stop the assault on critical thinking’.  In the session, they played the roll of a short video on subject (related rates in calculus, maximizing a function in pre-calculus, or unit conversions in a liberal arts math class).  The audience experienced something like a typical 3 minute video on that topic, and then we talked about how this supported critical thinking (or not).

The next session was one by Jim Stigler on ‘using teaching as a lever for change’, though he talked more about the futility of identifying specific teaching activities as being ‘effective’.  Dr. Stigler did include 3 aspects of teaching that are connected to improved learning — productive struggle, connections, and deliberate practice.  Learning is a complex process, and the presence of these 3 factors in the learning environment are connected to improved learning.

So, there is a connection between this research-based observation and the concern about critical thinking, I think:

Discrete learning experiences like short videos focused on successful completion of a task, based on clarity and being easy to follow, are guaranteed to limit both overall learning and critical thinking.

Mathematicians hold critical thinking as a goal to be valued; we want students to be able to flexibly apply knowledge to novel situations and interpret results.  This seems to be a basic problem.  Our students expect math to not make sense, that they could not figure something out; task-oriented videos support this self-defeating belief.

We can not hide from the digital age, even if we wanted to.  However, we can improve our understanding of the factors that contribute to the learning opportunities for our students.  A balanced approach appropriate to each course can help students through the learning process — including the struggle, the connections, and the deliberate practice.  We might even see these digital resources as just-in-time remediation to be used occasionally, rather than seeing the digital material as the basic course.

We need a more subtle understanding of how our teaching can contribute to student learning.  A single belief or methodology will not succeed for our students, no matter how good of an idea we might have.

To see a presentation by Dr. Stigler similar to what he did at the conference, see http://www.salesmanshipclub.org/downloadables/scyfc-Stigler.pdf

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Liberal Arts Math … College Algebra … ??

Once upon a time, colleges and universities wanted a math course for students in non-science fields.  The initial math for liberal arts course was designed for this purpose … a little bit of this, a little bit of that, and light on formality.

Once upon a time, colleges and universities wanted a math course for students who might or might not need calculus.  Since the content focused on ‘algebra’ and it was not remedial, the resulting course was called college algebra.

We might be better off if both titles were banned from the collegiate landscape.

In a way, both courses have a ‘this is not the other math’ type of implication.  We should be able to articulate a positive statement (and title) for what the courses are about.  This is not to say that all courses with these titles are ‘bad’ in some way (though some are bad in some ways).  I know of a few liberal arts math courses which are contemporary in design, with a focus on reasoning and some formality.  Some college algebra courses are actually high quality pre-calculus courses.

Back in the day, there actually were many programs that were not scientific.  Even fields like biology were considered non-quantitative, as were social sciences.  This landscape has changed in fundamental ways over the past 30 years; the fields that require no quantitative background are small in number.  Instead of ‘liberal arts’ math, we should use variations on the more modern ‘quantitative reasoning’ title.

College algebra is a mess.  It’s defined by what it is not (not remedial), and the title is used for 3 basic types of courses (general education, pre-calculus, and prep for pre-calculus).  If we need a general education course, we have better alternatives available now than we did 30 years ago (think quantitative reasoning and intro statistics).  Using college algebra for general education ensures that the course will primarily be a barrier to students completing a degree, and likely makes the course very challenging for faculty to teach.  It’s not unusual, of course; a major university close to me uses college algebra as their primary gen ed requirement.

If a college algebra course is meant to be pre-calculus, then we use the better title — pre-calculus.  Calling it ‘college algebra’ when it is meant to get students ready for calculus implies that the primary factor in calculus success is algebra beyond the remedial level.  I hope that this is not the case!

And, if college algebra is meant to be preparation for pre-calculus, there are larger questions.  Is the course non-remedial?  Are we adding a course to the sequence to have more classes to teach and fewer students completing?  If there is a valid reason for having both college algebra and pre-calculus, I have never seen it … and would appreciate seeing such a reason elaborated.

No, I don’t think we need either title anymore.  They once served a purpose, but Liberal Arts Math and College Algebra are obsolete.  The sooner we stop using them, the better we serve our students.

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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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