Word Problems and Reasoning … Rule 42

In my introductory algebra class, I gave a quiz today; this quiz included this question:

Some milk having 1% fat is being mixed with milk having 4% fat; the mixture will be 100 gallons, and have 2% fat. How much fat is in the mixture?

Now, I always review the quiz right after we complete … so sometimes I get to see some interesting reactions.  Please understand that all of the items on this quiz dealt with ‘puzzle’ word problems (not much real context), so students were not feeling mellow … many were feeling quite a bit of stress.  Besides the stated reactions about the question being ‘tricky’, I got to see students respond when they realized what they were supposed to do.

A typical wrong response to that question was to start working on solving the ‘problem’ they expected to see … how much of each type is supposed to be mixed?  Quite a few of these students wrote the correct equation including the “0.02(100)” for the milk fat in the mixture.  However, not many of these students realized that they had the answer to the question.  Our entire approach to these mixture problems in the class prior centered on “value = rate(quantity)” as a basic concept.

I was pleased that some students just wrote down the answer (2 gallons), perhaps with a note “0.02(100)”.  These students got that basic idea that value = rate(quantity).  I’d say that this 20% compares with the 40% who tried to ‘solve’ the problem but never realized that they had the answer … and the 40% who had no idea what to do.

One of the culprits for the difficulties is the inadequate way percents are done in math classes.  We focus so much on correct answers that we do not make it clear that percent is not ‘how much’ … and that every percent is a rate which is multiplied by a base.  For my question on the quiz, just knowing “percent times base” is sufficient to get the right answer (and show some understanding).

The other culprit is based on the high-anxiety suffered by students when faced with “word problems”.  I’d like to think that my class presents word problems as a reasonable use of language and algebra, even if the problems are either trivial or uninteresting.  Further, I’d like to think this positive approach helps students be more comfortable dealing with these problems.

Some readers might wonder “why do those puzzle problems at all” … perhaps we should “make the content relevant to the students”.  With all of the focus currently on ‘alignment’ and ‘context’, those are reasonable questions.  Based on my understanding of the learning process (along with some sociology), the question is not easily answered.  I am pretty sure that covering ONLY relevant applications is not a good idea for a mathematics course serving a general purpose; it might work in occupational math, or specialized math, but not so much when there is so much diversity among the students.  One student’s relevant problem is another student’s puzzle problem, and another student’s life survival issue; in addition, high context in problems can localize the learning and interfere with general reasoning and understanding.

So, I will continue to work with quite a few puzzle problems in our introductory algebra course — and keep a focus on the basic ideas that allow us to understand and solve them.  My goal is to help students develop a deeper understanding and develop connections.

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Jump Start in Math

We just had our Michigan developmental education conference (“MDEC” see http://www.mdec.net/conference/2015/program.html for details).  One of our colleagues at Schoolcraft College did a presentation on their “Jump Start” program for math.

The Jump Start program has two components (each 2 hours long).  The first component is on math study skills; since the person conducting the workshop is a professional in the learning assistance center, she has a good background to provide clear direction to students on being successful in math.  Within this study skills component, she also deals directly with motivational issues — her goal is to provide HOPE for all students.

The second component is the content, where students choose the one that matches their course for the upcoming semester.  Since Jump Start is offered within a few days of the start of the semester, this part of the workshop reviews content needed to be successful in that course.  The college offers a Jump Start option for the first 4 or 5 courses.

You can get some information about their Jump Start program at http://www.schoolcraft.edu/a-z-index/learning-support-services/learning-assistance-center/student-success-seminars-and-workshops/jump-start#.VRKngeFuNyE  with the current schedule at http://www.schoolcraft.edu/docs/default-source/lss—jumpstart/jumpstart-winter.pdf?sfvrsn=0 .

Overall, the Schoolcraft math curriculum is quite traditional; they still offer a basic math class, and do not yet have a mathematical literacy course.  However, I like their Jump Start program; in particular, the 50% (2 hours) invested on study skills (and motivation) is very appropriate for most students.  The professional doing the workshops has a math degree; in fact, she was originally a developmental math student who had to work very hard … and became a math major because “math chose me” (as she says).

The 50% (2 hours) on content would not be sufficient to correct for basic gaps in understanding, and the content done focuses quite a bit on procedures.  However, even this part likely is a good thing for students — the workshop covers a half dozen topics with multiple examples in each, which might help students develop accurate expectations for college math classes, as our pace can be quite an adjustment from high school.

The Jump Start model might be a good alternative for many colleges who can not commit resources to week-long boot camps or similar programs.

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Just for Fun … Creative Factoring

I’m teaching a class focused on individualized learning and flexible pacing.  One student in that class took a test on factoring in our intermediate algebra course.  In the process, I experienced something very enjoyable — a creative way to factor a polynomial.

Here is the situation:

Problem:   Factor r^4 – 16

Student:   (r – 2)(r³ + 2r² + 4r + 8)

Initially, I found this a bit confusing; I was not expecting to see a proposed factor with 4 terms.  In the materials, we focus on patterns to factor binomials involving the difference of squares.  So, I asked the student why he did this; his answer was “it checks”.  [This is exactly what I tell students when they ask WHY we factor a polynomial in a specific manner.]

After a quick transition from confusion to mathematical thinking, I looked more closely at the cubic factor.  Sure enough, it factors to produce:

Correct answer:   (r – 2)(r +2)(r² + 4)

This particular student (planning to be an engineer of some sort) had a creativity I would like to see more of.  The only negative feedback I had to deliver was “Finish the factoring”.

I found this to be just a lot of fun (though I doubt this student enjoyed it as much as I did, though he did enjoy it).  Mathematical fun is meant to be shared.  In 40 years, I’ve not seen a student do this; it’s too good of a thing to keep to myself.

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Statistical Doors Into Mathematics

That’s really a question — does statistics create a door into mathematics?  Or, is statistics (for most students) an alternative off-ramp from the mathematics highway?

The question is perhaps trivial.  In terms of the bulk of our work, we are dealing with students required to take specific courses for their program.  Every math course becomes a common off-ramp for students.  Perhaps we should be satisfied with a curriculum consisting of terminal courses for students interested in everything else but mathematics.

One of my colleagues began her higher education as a fairly typical community college student at our institution.  She reports that a turning point for her was a particular computer science course that she decided to take; after this course, she changed her major and got a degree in computer science (and later a masters in math).  There was something of beauty in that computer science course that connected with her, and changed her life.

I would be interested in any research on the question:

Do students change their career path to mathematics after taking a statistics course?

I am sure that there are students who change their path to statistics after a statistics course, though I wonder if the rate is equal to that of ‘math program after a math course’.

Like most of us, my students are just interested in passing this math course so they can get their degree or that job.  I am fine with helping them along that trail; in fact, I am happy to do so.  I teach because I find that rewarding.

However, I am also a professor is in “affirms a faith in something”.  I think I have a responsibility to show students in each course something about the beauty of mathematics; something wonderful should show in every class.  Partly, this is needed to encourage more positive attitudes about mathematics; partly, this is needed to encourage a more accurate view of the nature of mathematics, that mathematics is much more than processes to generate answers.

To me, however, the largest reason for what I try to do is “opening doors”.  A major reason for lowering expectations for a given student is mathematics; lower-skill programs are selected because they require less mathematics (or none).  Students even avoid occupations that they would love to be in … just due to mathematics.  To me, every mathematics course should be a STEM magnet drawing students towards higher skilled jobs and more security.

I do not think that statistics operates as a STEM magnet.  Of course, there are many math courses in our institutions that are not STEM magnets; however, almost all math courses could be strong attractor points drawing students towards mathematical sciences.  I think the problem with statistics is that we teach statistics as a practical discipline without a core mathematical structure.  We focus on the innate appeal of statistics, on its utility; perhaps we need to show the mathematics supporting statistical methods when possible.  If there is no mathematics supporting a method (the ‘plus 4’ rule type of thing), perhaps we should question the presence of that method in a general statistics course.

Clearly, I may be demonstrating levels of ignorance vast and wide.  I wonder, though … do we share a view that math courses in the first two years should have a property of ‘STEM magnet’?  Can a statistics course be such a magnet?

Before the reader decides that I am far too optimistic about our mathematics courses — yes, I know that we fall far short of a STEM magnet in our current courses.  We tend to cede our territory, and deliver service courses; we focus on the practical at one extreme … or the totally useless on the other.  In between is the zone needed to be a magnet for students; a magnet can not be unidimensional.

Perhaps the question is more general than statistics; my concern is with the contemporary move towards requiring statistics as the typical general education course.  Perhaps the loss is trivial.  I do wonder if there is an innate qualitative difference between statistics and mathematics that results in statistics being far less able to contribute towards larger goals such as raising student goals and drawing students towards STEM.

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