Category: math reasoning and applications

Math – Applications for Living XVI: The Evils of Line Charts

In our Math – Applications for Living class, we just finished our first chapter on statistics, which included some of the standard graphical displays (frequency charts, bar graphs,  and line charts) as well as scatter diagrams for bivariate data.  Based on observing students working with line charts, I commented to the class that line charts have a risk — human perception may suggest a much stronger pattern than is really present in the data.

Here is an arbitrary example of a line chart (pseudo-random data):

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The same data presented in a bar graph looks like this:

 

 

 

 

 

 

 

 

A quick search on research related to this yielded a study done by Jeff Zacks and Barbara Taversky at Stanford (1999); see      http://dcl.wustl.edu/pubs/ZacksMemCog99.pdf.  Briefly, their study found that bars tended to processed discretely while lines tended to be processed as a pattern.  My examples above are not ideal examples … the correlation is fairly high, perhaps higher than the bar graphs suggest. 

It seems that scatter diagrams have some of the same risk as line charts, in the sense that my students want to visually connect the dots as in a line chart.  We are working on identifying patterns correctly in bivariate data, though my expectations are not high.  Human perception might have a strong desire to establish a pattern, even when some data needs to be excluded.

 

Math – Applications for Living XV

Our Math – Applications for Living class is moving on to other material, much to the relief of the group.  We had more difficulty than usual this semester with our work with units and percents (the stuff on the first test). One problem in particular seemed to be more challenging:

A computer modem uses about 7 watts even when the computer is not on; the modem is left on constantly (“24/7”).  A home in Lansing pays about $0.1137 per kilowatt-hour.  For one year, how much does it cost to keep the modem ‘on’?

I know that one challenge was that the book gave the official definition of a watt (1 joule per second), which led a few students to think that they needed to calculate how many seconds in a year.  Given this, there might be a temptation to not provide that information.  However, ignorance is not good when it is by design; the course includes a variety of information that applies to life and to science, and is one of the strengths.  It does not help that the book does a similar problem by converting watts to joules (no change) then calculating the energy use for the time period in joules (for seconds in the period), and then changes the result back to watts per hour.

The correct calculation is not that complicated:

7 watts * 24 hr/1 day  * 365 days / 1 year =  61320 watt-hours per year.       This is 61.320 kw-h

61.320 kw-h * $0.1137 / kw-h = $6.97 (rounded)

A few students had the right idea, but failed to change to kilowatt-hours — they ended up with a cost about $7000 for this modem.  The “answer desperation” of students led them to record this answer, even though they could see that it is not reasonable.

This problem was similar to one we did as an example in class … the cost of leaving phone chargers plugged in constantly.  Phone chargers are a lower power drain than modems, but there are so many more of them that their energy use is an issue.  Other ‘vampire loads’ contribute to our huge appetite for electricity (most electronics have a constant power usage, even when ‘off’).

Hidden in this modem problem is a strange thing: students knew that ‘km’ is ‘kilometers’, but are stumped by ‘kw’.  This problem, I think, is caused by the mechanical way that metric conversions are presented (‘just move that decimal point as you move from the old to the new prefix’).  We connected that strategy to the standard method (dimension analysis in this class).

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Math – Applications for Living IV

Percents!  Percent discount!  Percent tax!  Percent increase!  What do our students learn about percents?  Not nearly as much as we would like.

One of my students made a comment about percents and taking his family to a restaurant.  As a result, this problem appeared on a worksheet last week in our Math119 class:

I have $20 to go out to dinner.  All restaurant purchases in my state have a 6% sales tax, and I like to leave a tip that is 15% of the total including sales tax.  How much can I spend on food (menu prices) to stay within my $20?

Every one of my students has shown that they can calculate a 6% tax, and find the total price.  Every one of them can find the 15% tip on a dinner, and the total cost.  Less than 10% of them knew what the ‘base’ for the percent is in the problem.  We had already been talking about the net result of a 10% increase (110%, or 1.10 times the base) as well as a 20% decrease (80%, or 0.80 times the base).  In fact, we started off our work with percents by the classic story:

Boss: Bad times; sorry, everybody gets a 10% pay cut.
Boss (next year): Good times are back; everybody gets a 10% pay raise.

Worker: Am I back to where I was?

Many percent situations in the world involve a chain of percent increase or decrease factors operating on a moving base.  In my dinner example, the goal is to see the situation as

1.15(1.06n)=20

The solution here ($16.41) is pretty good, as the rounding happens to work out well; in general, this method is a ‘good approximation’ — an idea that is not brought up in this class.  We are still going through a lot of struggle to identify the base in percent problems.  Later in the semester, we will connect this repeated percent concept to exponential functions; identifying the base correctly will continue to be an issue then.

Whatever you do with percents in your courses … please focus on identifying the base. Being able to calculate a tax or a decrease is nice, but of limited usage.  Percent change is all around us, and we often deal with an unknown base (frequently hidden within the context of the problem).  We don’t need to disguise problems to the point that finding the base is a horrendous exercise for students.  On the other hand, creating cookie-cutter exercises where no thought is needed is a self-defeating practice.  “Percents” and “thinking” go together in the world around us, as they should in our math classes.

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Practical Math — or Not

Last week, I spent several days with faculty who are working with the Carnegie Foundation’s Pathways — Statway and Quantway, at their National Forum (summer institute).  I continue to be impressed with the quality of these professionals; Carnegie is fortunate to have them involved.  One comment from a faculty member has been stuck in my thinking.  In the context of Quantway, this faculty member said:

Everything in this course has to be practical.  The math students see has to be practical.

I recognize that there is a high probability of head nodding and agreement with this sentiment among people reading this post.  Can we … is it reasonable or desirable … to shift from a ‘nothing in this course is practical’ to ‘everything in this course is practical’ position?

First of all, we need to recognize that ‘practical’ is a matter of perception, communication, and culture.  Our students will not see the same ‘practicality’ that we do.  For example, if we have a series of material looking at the cost of buying a car including operating and finance, many students will definitely not see this as practical.  The majority of my students are not able to consider this situation in their real life now, nor for several years; for some, they can not even imagine having a real choice to make about a car.  What we often mean is that math needs to be contextualized, not practical — context is a simpler matter to establish than practical.

Secondly, the ‘practical’ or ‘contextual’ emphasis reminds me of the old school approach to low-performing math students:  If a student was not doing well in math, put them in an applied math course (business math, shop math, personal finance), as a way of being polite about lowered expectations.  I realize that many of our students are initially happy with the lowered expectations of ‘practical math’; however, this approach does not honor their real intelligence, nor does it recognize the capacities in our students to understand good mathematics just because it is enjoyable to do so.

More important than these two points is the learning implications of ‘practical math’.  I’ve been reading theories of learning and research testing these theories … for close to 40 years now.  Nothing in the theory suggests that learning in a practical context is better than learning without the context; without deliberate steps to decontextualize the learning, the practical approach often inhibits general understanding and transfer of learning to new situations.  I do not believe that ‘all is practical’ is a desirable approach to learning mathematics.

However, context and practicality can be very motivating.  Motivation is the most elemental problem in developmental mathematics.  Therefore, it is reasonable to provide considerably more context for students than the traditional developmental math courses with its ‘train problems’.  I also would add that most students are motivated by learning mathematics with understanding when they can see the connections; true, our students need some extra support for this process, and it conflicts with the approach emphasized with them in the past (primarily memorization without understanding).

I have summarized my view on the ‘practical’ issue with this statement:

I will always include some useless and beautiful mathematics in all of my math classes.

Education is about expanding potentials and creating new capacities; practical learning is the domain of ‘training’ (which is also critical … but it is not education).  I encourage all of us to help our students learn mathematics in different ways: sometimes practical, sometimes in a context, sometimes imaginative, and sometimes logical extensions.  The mix of these ingredients might reasonably shift as a student progresses; developmental math courses might be more practical than pre-calculus.  Diverse learning is better than limited learning.  Diverse learning respects the intelligence of our students and maintains high expectations for all students.

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