Applications for Living — Geometric Reasoning

We are taking a test in our Applications for Living class, and I am struck by two things.  First, students have made major improvements in how they deal with converting rates (like pounds per second into grams per hour). Second, how bad geometric reasoning is, before and after our work on it.

Just about the simplest idea in all of geometry is ‘perimeter’.  Students have very little trouble with a rectangle as a stand alone object.  This problem created a speed-bump:

As a class, we ‘passed’ on that item (in terms of proportion with correct work).

However, we struggled with this problem:

We did not pass on this item, as a class.  The most common error, of course, was counting the ’12 inches’ (which is completely internal to the figure).  Not as many included the ‘8 inches’, which is also internal.  We always say that perimeter is the distance around a figure, but that is not internalized as strongly as the “2L + 2W” rule.

A bonus question on the test looks like this:

This problem combines the reasoning about perimeter with some understanding of right triangles as components of shapes.  A few students got this one right.

We spent parts of 3 classes working on our reasoning and problem solving.  These compound geometric shapes are common objects in our environment (at least in the USA).  I’d like to think that our students would be able to find the amount of trim or edging to install.

We are a bit too eager to pull out a formula for perimeter (where it is never required for sided-figures); when we talk about circles, it’s not connected well enough to other ideas like perimeter.  One of the problems we did in class caused a lot of struggle:

We used this problem as a tool to work on reasoning about perimeter (and area).  Much scaffolding was needed; since we only spent 3 classes on geometry, we did not overcome prior mis-conceptions in most cases.  Our better results with dimensional analysis (rate conversions) is due mostly to the fact that students had few things to unlearn.

Let’s do a little less variety in geometry, with more focus on reasoning.  Formulas are fine for area and all-things-circular, but have no business in the perimeter of sided figures.

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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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More Math Lit Materials Available

The second textbook for the Mathematical Literacy course is available.  Great news!!

The new book is by Brian Mercer and David Sobecki, published by McGraw Hill; the title is  Pathways to Math Literacy ( © 2014).  I had an opportunity to see some of the material prior to publication; you will find this text to be a good contrast with the other book currently available (some info below on that one).

This new book has a web site for instructors at http://successinhighered.com/pathways/   Take a look … they have information about the book that is actually helpful, and they have a resource page for faculty considering ‘pathways’ (whether you use their book or not).

Also available is the initial Math Lit book by Kathy Almy and Heather Foes, published by Pearson ( © 2013).  You can get information on that book at http://www.pearsonhighered.com/product?ISBN=0321818458  and at Kathy’s blog (http://almydoesmath.blogspot.com/ .  I noticed that a recent post on that blog is a recording of a webinar on implementing pathways; very helfpul.

As of this semester, we are up to about 170 sections of Mathematical Literacy across the country (this is not counting the Quantway™ sections).  Now that we have a second commercial textbook, even greater growth is possible!

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Is Mathematics a Science?

My college has recently completed a ‘reorganization’ of programs and departments.  As a result of this change, mathematics is now in the same administrative unit as science. Is this a good fit?

Although we share much, I have seen some interesting differences.  One striking difference is this:

Mathematics faculty are expected to be flexible generalists.

Science faculty are expected to be specialists.

We are likely to be posting one full-time position in mathematics, and at least 2 full-time positions in science.  As the programs talked about requirements for the positions, mathematics consistently kept flexibility as a top priority — to be able to teach a variety of courses.  Science faculty, on the other hand, consistently listed specific backgrounds — micro-biology versus biology, physics versus geology, etc.  I have asked about why this is the case, irritating a few of my friends along the way; the rationale basically boils down to ‘we need a specialist to teach x’.

In mathematics, we sometimes seek a specialist — like a math for elementary teachers course, or statistics.  The vast majority of math faculty (full-time) are qualified (in our view) to teach any of a dozen courses.  Science faculty seem to keep themselves in a box, where they may have 3 to 5 courses that they can teach.  I am not sure which approach is superior, but I do know that the situation is related to the other observation about math & science.

Science, in general, does not do developmental.

Students in K-12 have had a variety of science.  When students arrive at college, the college-level science courses they take are determined by their program — not by ‘deficiencies’.  Certainly, students who have struggled in science select programs that will provide them with lower-level science courses.  Every student begins chemistry with a college-level chemistry class; every student begins biology with a college-level biology class.  [My college had, at one time, a developmental science course — never a large population.]

Part of this is the acceptance of ‘science’ as a set of (almost) independent disciplines (sometimes competing disciplines).  Students will generally take courses in 2 science disciplines.

Mathematics is seen by policy makers as a single, continuous strand.  At the bottom is arithmetic; at the top, calculus … in between, lots of algebra, a little geometry, and some trigonometry.  There is “one mathematics”; there are “multiple sciences”.

Of course, this ‘one mathematics’ is an incorrect view.  First of all, that image confuses a sequence of prerequisites for a content structure; only parts of algebra are needed for calculus, as is the case for geometry and trigonometry.  Students in occupational programs are the ones who might get to experience the other parts of these mathematical disciplines.  We, the faculty, reinforce this incorrect view by testing and placing all students along this single continuum (including the requirement for remediation of arithmetic and algebra).

Secondly, there are mathematical disciplines that are relatively unrelated to calculus preparation … disciplines that are used extensively in the modern world.  Students are more likely to interact with network problems than they are common denominators.

As we talk with career experts and other programs about what their students need, what topics do we ask them about?  I suspect that 99% of the discussion focuses on the ‘calculus continuum’ (arithmetic to calculus, via algebra).  Do we ask about topics that are not in developmental math courses?  Topics that are not in introductory college courses?  I’ve not seen that done.

Could we envision a world where there really was no need for developmental mathematics (in the sense of repeating school mathematics)?  Unless students need calculus for their program, would it be possible to start with “basic quantitative reasoning” or “introductory statistics” or “math for electronics” for students less prepared?  Better prepared students, perhaps, could take “applied calculus” or “diverse mathematics for college” or “statistics and probability”.   Students needing calculus could take “general calculus” as a preparation for a calculus sequence. These questions, perhaps, are related to the nudge that some state legislators are giving us when they limit developmental education.

Although mathematics is the “Queen of the Sciences” (historically), our practice of mathematics is not so much a science.  A science is based on a collection  of methods applied to related sets of objects (like chemistry does); mathematics does consist of several disciplines.  However, we do not function like a science, nor do we provide students with preparation for scientific thinking within our math classes.

Mathematics in college is not a science.  Would we serve our students better if it were?  What would that look like?

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