Math – Applications for Living VII

Our class, Math119 (Math – Applications for Living) is in the middle of our work on statistics.  The last class included finding a margin of error and a confidence interval for a poll … like those pesky political polls we are constantly hearing about. 

So, here is the situation.  This month’s poll showed 63% of respondents supported one candidate, based on results from 384 people; last month, the same poll reported 58% supported that candidate.  The article stated that the candidate is enjoying the increased support … is that a valid conclusion?

As you know, this relates to two issues.  First, the standard error for a proportion like this is found with the statistical formula:  

Tests of significance are then based on z values for a normal distribution; the most common reference is z = 1.96 … creating a margin of error representing a 95% confidence interval.

In our class (Math119), we use a quick rule of thumb to combine these two ideas into one statement which just uses the sample size — and this rule of thumb works pretty good for the types of proportions normally seen in polls (p values between 10% and 90%).  The rule of thumb for the margin of error is just the reciprocal of the square root of the sample size     

For the poll data, the sample sizes are both about 400.  The rule of thumb gives an estimate of 5%, which is very close to the actual value (approximately 4.7%.  In our class, we make a reference to the presence of the more accurate formula, but we use only this rule of thumb.

In this poll example, we create the confidence interval … and conclude that there is no significant difference between the polls.  The confidence intervals overlap; even though the new poll has a larger number, it is not enough of an increase to be significant (with this sample size).

We also have talked about selection bias and other potential problems with polls, and have begun the process of thinking about the impact of sample sizes on things being ‘significant’ (whether they are meaningful or not).

 
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FOIL in a Box (algebra!)

Some of us have a ‘thing’ about FOIL as a topic in an algebra class; there are concerns about emphasizing the FOIL process as it can submerge the real algebra going on.  Some (perhaps the majority) are not significantly handicapped by being “FOILed”.  This post is not about FOIL itself … it’s “FOIL in a Box”.

Okay, so this is what I am talking about.  The problem given to the student is to multiply two binomials, such as (2x – 3) and (3x +4).  Here is the “FOIL in a Box”:

Some students like this approach, and I think this is because the box lets them focus on one small part of the problem.  The overall process is submerged, and the format does all of the work.  Of course, this is exactly what many procedures in arithmetic do.  The FOIL in a Box method is much like column multiplication, where partial products are arranged in a mechanical way to produce the correct place value.  If correct answers to multiplying were the primary goal, there would be nothing wrong with either FOIL in a Box or partial products in arithmetic.

My observation has been that almost all students who use FOIL in a Box are handicapped in working with polynomials.  Students have trouble integrating the Box into longer problems.  And, though they may have some ‘right answers’ for factoring trinomials, the transition to other types is more difficult. 

What should we do instead?  My own conclusion is that we need to keep emphasizing the entire idea involved.  FOIL is used for “distributing when both factors have two terms”, and “distributing is used to multiply when one factor has two or more terms”.  We too often assume that students will keep information connected to the correct context … they don’t automatically know that distributing does not apply to 3 monomial factors [3(2y)4z ≠72yz], nor to a power of a binomial  [(x + 3)² ≠x² + 9]. 

The achievement of correct answers in the short-term should not come at the price of handicapping the student’s future learning.  All learning should be connected to good prior learning, and imbedded within the basic ideas of the discipline.  We need to be comfortable articulating the full name of what we are doing (multiplying two factors each with two terms), and not use a mnemonic such as FOIL as a container for knowledge of mathematics.

 
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New Life vs Emporium Models

I am currently at the AMATYC conference in Austin — very good conference.

Earlier today, I had a session entitled “New Life Takes on the Emporium Model for Redesign”.  My intention was to provide a viewpoint on these alternatives, both of which are currently popular. 

Here is the file

Here is the handout from the session (1 page summary): https://www.devmathrevival.net/wp-content/uploads/New-Life-takes-on-the-Emporium-Model-for-Redesign-HANDOUT-final.pdf

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Pathways Blog Available

Our friends at the Carnegie Foundation for the Advancement of Teaching have launched a blog for their ‘pathways’ work.

The blog is at Math Pathways, and I encourage you to take a look; you can also join the community on that page to receive updates.

I like the fact that the current post at the Pathways blog deals with quantitative literacy; we would be better off if we focused on quantitative literacy instead of ‘developmental mathematics’.  Of course, I would rather it be called ‘mathematical literacy’ (to focus on the scientific aspects of mathematics, not just the tools) — but that is a relatively minor point.

I hope you take a look!

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