Math – Applications for Living III

In class this week, we talked about ‘precision’.  Even though many of our math classes ignore this topic, students relate to it reasonably well.

One example — find the area of a rectangle that is 3.6 meters wide and 4.2 meters long.  The correct answer is ’15 square meters’, since each measurement has only 2 significant digits.  Calculating ‘15.12’ is only part of the story.  How this 15 square meters is used depends on the purpose for finding the area.  If we are estimating the amount of time needed to paint the area, it is fairly safe to work with 15 square meters … however, if we are buying the paint, we might do better with 20 square meters (1 significant digit).

Several students in class have been dealing with the concept of significant digits in their science class as well … isn’t it nice when people can see an immediate use for what we cover in math class?

The topic of significant digits is a natural whenever we cover geometry.  However, we tend to do a bad job with this; I suspect that we are too concerned about ‘keeping things simple enough’.  One of the larger errors on our part is the treatment of π.  In many books and courses, students are told to use ‘3.12’ as the value of this number regardless of the precision of the other numbers involved.  As you know, the correct process is to use all available digits and round the final answer to the appropriate number of digits.  The irony is that almost all students have access to the value of π to 10 or more digits (calculator or computer).  Let’s start doing good mathematics by having students use the built-in constant instead of the (always inappropriate) approximation.  [It’s always inappropriate because we are not supposed to round intermediate values.]

Another example from class: “A city has a deficit of 43.8 million dollars.  How much per person is this if the city has a population of 136,500?”   As a division, we can calculate any number of digits; many students would ‘naturally’ round the result to the nearest cent ($320.88), though this is not the correct value.  We often say ’round money to the nearest cent’, and this is quite appropriate with interest calculations.  However, it may not be appropriate in many other applications.

The topic of ‘significant digits’ (precision) is appropriate for most math classes, and is accessible to almost all students.

 
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New Life model – Compared to Statway & Quantway

I’m at the Statway winter institute (in Palo Alto, California), where we are getting some initial results from the Statway implementations going on this year — very encouraging, and the colleges involved have done a great job.  Makes me wish I was at a Statway institution.

I’ve also been having discussions in recent months about how the New Life model compares to the ‘Pathways’ (Statway & Quantway).   This has been addressed other places, but one aspect might help us understand a basic distinction.

First, let’s review — there are similar goals, and much common mathematics, between the New Life model and the Pathways.  The Carnegie Foundation has been gracious and inclusive in their work, which has enabled us to work on both approaches.  In both approaches, we seek to provide more appropriate mathematics for students and help them complete developmental mathematics more quickly.

So, a basic concern in Statway and Quantway is ‘recruitment’ — how do we identify the students who can take advantage of the Pathway?  The Pathways are designed to serve groups … Statway focuses on students whose ‘final’ course is an introductory statistics course, while Quantway is for students whose ‘final’ course is a quantitative reasoning-type course (with some variation in Quantway).  With the Pathways, a college has the existing sequence and then the Pathway alternative so that finding the students is a central concern.  This is especially the case with Statway, since it is a 2-semester sequence designed to be completed by each student.

In the New Life model, the vision is more general.  The first course, “MLCS” (Mathematical Literacy for College Students), is designed to connect with a variety of non-STEM college courses —  including intro statistics, quantitative reasoning, and others.  The second course, “Transitions” is used to connect to STEM-like college courses — such as college algebra, pre-calculus, and others.  The approach here provides flexibility to colleges and students.  Students identify their needed sequence of courses (hopefully 2 courses at most!) by looking at course prerequisites.

A difference between New Life and Statway is that Statway is a ‘2 semester set’ while New Life is ‘sequences for each student’ that follow patterns familiar to community college students — to take course X, the prerequisite is course A (which the student might or might not need).  Quantway is similar to the New Life MLCS, though the expectation is that Quantway students will proceed to take the quantitative reasoning course the next semester. 

In other words, colleges can build their own ‘stat path’ by implementing MLCS as a prerequisite to an intro statistics course.  Colleges can implement MLCS and Transitions as a replacement for the old courses.  Colleges can implement Transitions as a better bridge to STEM-like course; some might choose to do this for some STEM courses but not for ‘less STEM-like’ courses.  Transitions might be a better preparation for basic science courses than ‘intermediate algebra’.

New Life is all about flexibility at the local level to provide better mathematics preparation for their students.
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Why Change Developmental Mathematics?

Why should we make basic changes to developmental mathematics in communuty colleges?

The latest ‘Instant Presentation’ has been posted, and it is on this topic.  Take a look!   https://www.devmathrevival.net/?page_id=116

Math – Applications for Living II

Another day in “Math119-Land” (applications for living), though students were not enthusiastic about today’s work.

Percent are standardized rates; percents are ‘evil’ (in the way that people describe things that do not make sense).  We were doing some puzzles today, to prepare us for the good stuff.  For example, today we had “after an 8% discount, the price was $75.60”.  Today, this was a puzzle … meaning the answer was known by the person who wrote it, and our job is to find it.  Most students wanted to multiply 8% by 75.60 and then add … sadly, close in numeric value but awfully wrong in terms of relationships.

The whole point of these puzzles, in this class, is to connect a percent increase to a growth model and a percent decrease to a decay model.  Eventually, we will write the exponential models in this class … calculate various outputs, graph a bit, and even find the original value or the multiplier in limited types.  We went through the usual ‘8% less means take off .08 times the original, which gives us 0.92 times the original’.  A large portion of the class did not think this made any sense at all.

On days like this, I wonder if we should delay all percent work until we are in a setting where we can use algebra.  Students have learned one method (operations on the numbers given), and resist a transition to a formalized method.  This resistance handicaps their problem solving skills, which would show in other classes besides math (science in particular but including ‘social sciences’).

Earlier, I had a post on ‘stealth percents’, and today’s post is related — our students really struggle with percents.  Another example from today —
    “A survey last year gave the mayor an 84% approval rating, and the recent survey showed a 33% approval rating.  What is the relative change?”

Students easily subtracted the percent values, though some thought that the percents HAD to be converted first.  However, few of them saw how to make this a ‘relative change’ — which is often the only measure that makes a significant difference. 

Unfortunately, percents are commonly used in a variety of problems and situations; today’s troubles with percents will show again when we talk about finance formulas in a couple of weeks.  Later in the semester, we will bring up the exponential growth and decay, where we connect a percent change to an algebraic model.    Each time, I can hope to make this percent mess clearer to the students.  Wish me luck!

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