Math: Applications for Living — The Chance of That

Our Math-Applications for Living course is finishing up our work with statistics and probability.  A couple of students commented that they realized that their thinking had to change when we talked about probability — what seems natural for experts is not natural at all for novices.   This was an issue for combining probabilities — either a sequence (multiply) or options for one event (add).

On today’s test, one item seems to be really confusing:

Determine the probability of meeting someone whose phone number ends in the same digit as yours.

Students are asking “what about the rest of the digits” and “how many digits are we including for the phone number”.  From my point of view, this question was an attempt to measure the basic rule about probability — for random events, the probability is the ratio of “yes” to “total”.  We had dealt with this in other contexts, but I wanted something new on the test to see if they would apply the idea.

Probability continues to be one of the more challenging parts of the course.  One of the ironic results is that students appreciate the algebra that we will do next, because they seem to ‘get that’ a little better.

Of course, many of us who deal with probability do so within the context of a statistics course (as opposed to my survey-type course).  I wonder if you find that students have similar difficulties in ‘probabilistic thinking’.

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Reasoning with Units: Correct Wrong Answers

In the world of problem solving (as an academic endeavor), we talk about non-routine problems … ill-defined problems … and we talk about problem solving strategies beyond specific content issues.  When facing these types of situations, many students find great difficulty in transferring content knowledge; as mathematicians, we sometimes see this problem solving as the core outcome of learning mathematics.

Unfortunately, the teaching of mathematics often discourages broader reasoning.

I have been running in to a consistent error in thinking about units, which has led me to think about how this happens.  Here is the situation:

A desk has an area of 5 ft².  How many square inches is that?

Two bits of knowledge (neither correct) get in the way of solving this routine problem.  First, students equate 1 foot with 12 inches, whether we are talking about length or area or volume (they get 60 square inches).  Second, students treat the exponent (square) as affecting the 5 as well as the feet (300 square inches).  The first issue was addressed in an earlier post (see https://www.devmathrevival.net/?p=1471).  How about the exponent issue?

Misapplying the exponent could be caused by an over-generalized property of exponents.  However, I think the more likely error is a combination of two practices in mathematics education:

“find the area of a 4 inch square”

“just use the numbers in formulas, and write the correct unit with the answer — area is always squared”

The first practice, extremely common in early work with area, leads students thinking that something needs to be squared when they see a square indicated (like an exponent).  The second, more to my point today, leaves students with no reasoning about units.

For example, in last week’s Math Lit class, I asked the group what the formula for distance is (and got D=rt).  I asked how we usually measured distance; once we agreed on a context (a car) we agreed ‘miles’.  The next question — how do we measure speed?  This was much tougher, even though students deal with speed limit signs every day (usually without units 🙁 )  Once we got to ‘miles per hour’, we then wrote a typical calculation showing what happens to the units.

The next step:

A car has a speed of 40 miles per hour.  How far do they go in 20 minutes?

Many students see this as a trick question, saying that we should always give the time in hours (we would say ‘consistent units’).  However, including the units in the calculation makes it more obvious that we just need to change minutes to hours (they could do that).

Back to the square feet situation, few of us show the units in calculating area.  If we consistently did include units in calculations, students would have more experience in seeing where the ‘square’ came from (in ft²), and would be less likely to apply the square to the feet.

We have another instructional practice which discourages reasoning with units: the degree sign for temperatures.  By itself, the degree sign is not the unit — the unit for temperature must include the scale involved.  When we require units for temperatures, we should not accept just the degree symbol — 40° F is much different from 40° C, and nobody wants a household temperature of 40° K.  Even the simple conversion of F to C temperatures does not make sense if the scale is not included — the process becomes a black box of non-reasoning.

It is certainly true that “reasoning with units” will slow us down.  Our work is ‘cluttered’ by non-numerical information.  However, numerical information is the easier part for our students — it is the ‘clutter’ that needs to be seen and reasoned through if our students are to have any lasting benefit from our courses.

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Problem Solving … and Learning Mathematics

Our Math – Applications for Living course is sometimes used as a last option; students try passing the intermediate algebra class, and (after 2 or 3 tries) an adviser says that they have another option.  This is not true for all students in the course, though it is a common path to my door.  The result is a class with some very anxious students, and many who doubt their ability to solve ‘word problems’.

Math – Applications for living is all about problem solving; all topics are verbally stated.  We had an interesting experience last week when we did an example with a simple statement:

The distance from the Moon to the Earth is 3.8 x 10^5 km.  A light-year is 9.5 x 10^12 km; in one second, light travels 3 x 10^8 meters. How long does it take light to travel from the Moon to the Earth?

The problem presents to issues to resolve: the operation to perform, and making the units consistent (meters and km).  A few students knew to divide distance by speed to get time; if they did not already know this, it did not help much to solve the D=rt formula for t.  We explored the problem by working with rates (as we have been doing for most unit conversions); this helped a little more.

We got frustrated, however, with the km and meter conversion in the same problem.  After about 10 minutes of discussion, some progress was made.

In working through these struggles, more than one student said something like:

Can’t you just show us how to solve these in a way that we already understand?

Of course, it is exactly this gap between current understanding and present need that causes learning to happen.  As a problem solving issue, this is essentially a statement of what problem solving is … as opposed to exercises.  In the most encouraging manner, I told the class that this tension they are frustrated with — is the zone where we will learn something.  I stated, with emphasis, that if I did not create situations where there was a gap like this that they would leave the course with the same abilities as when they started.

I’ve been talking with faculty in some other programs at my college about the mathematical needs of their students.  The first thing they say is always ‘problem solving’, and they don’t mean solving a page of 20 ‘problems’ using the same steps.  The second thing they say depends on their program, and a surprisingly large number of them say ‘algebra’ is the next priority — in spite of the fact that algebra is often de-emphasized outside of the STEM-path.  In the Math – Applications for Living course, we use algebraic methods when useful, as it is when solving problems with percents.

In the larger context, all learning is problem solving.  A learner faces a situation where existing knowledge is not sufficient, and the gap is completed by some additional learning.  I believe that this statement is true regardless of the pedagogy a teacher uses, whether active or passive for the learner.   I do not agree with a constructivist viewpoint, especially the more radical forms; however, there is a basic element in the constructivist view that is true, I believe — knowledge is built as a result of gaps.  I believe that teachers can (and should) model the process of filling the gaps, and explaining the reasoning behind ideas that can help.  Learning math does not need to involve students stumbling through to discover centuries of mathematics; we can both guide and be a sage in the process.

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Math – Applications for Living: The Point of It

In my Math – Applications for Living class, a couple of students did something humorous (and sad) with a problem on a quiz yesterday.

Here is the problem:

Computer sales for a certain company were reported to be $40.3 million in 2009.  This was stated to be a 12% increase over the prior year.  Find the computer sales for 2008 (round to the nearest tenth of a million).

We are working on translating a percent change to a multiplying factor (1.12 in this case), and most students are not there yet.  However, here is the thing these particular students did:

40,000,000.3   is ‘40.3 million’

I have not talked to these students about what led them to make this mistake; it looks like they think that ‘.3’ means that part is stuck after the decimal point.  Since we just started working with scientific notation, that type of error will be a large issue.

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