Applications — Starting on Correlation

In our Applications course, we develop some basic statistical reasoning.  I always start key topics (like what is statistics … what is correlation) by having students work in groups on discussion questions.  I recently changed the ‘correlation’ activity, and thought other people might be interested in seeing it.

My goal with statistics in this course is to develop statistical thinking (and caution!), in ways similar to the standards in GAISE (see http://www.amstat.org/education/gaise/GaiseCollege_Full.pdf)   We use real data whenever possible, and focus on reasoning first … computation later.

In the case of correlation, I originally used a variation of the “Cereal Plots” activity like that used in Statway™.  In the cereal plots, students are presented with various nutrition values plotted against the ranking of cereals by Consumer Reports.  Conceptually, this is really nice.  However, in practice, students have too much overhead — they don’t know about rankings in general, and certainly not cereal rankings by Consumer Reports.  We ended up spending about half of the class discussion time on secondary issues.

This semester, I created my own sample of graphs.  These scatterplots deal with contexts familiar to almost all students, such as cars (price, mileage, etc).  Here is my activity for this semester

So, this was the initial part of class yesterday.  Students were in groups of 3 to 5, and answered the 10 questions on the sheet.  The context for these graphs was much better — somebody in every group knew enough about cars to provide some additional wisdom, and everybody knew enough about cars to get started.  [The last graph, on grip strength versus arm strength, is accessible to all students.]

Overall, this new activity works much better.  All of the discussion was about the graphs and statistics.

This does not mean that students magically understood what correlation is … or how to judge it from a scatter plot.  We are still working on unlearning ideas about cause & effect.  However, we did make progress on judging a linear pattern in these graphs; when I say “negative correlation”, most students can connect this to a pattern like we visualize from the phrase.

In case you are curious, the application course is pretty limited in the statistical topics included.  We include some reasoning topics (samples, population, bias, correlation, and describing distributions); we also include some displays (frequency tables, bar graphs, etc) and a few calculations (mean – median – mode, standard deviation, rule of thumb for margin of error).  We do not calculate correlation coefficients, nor do we do regressions; we do not calculate actual margin of errors, though we do calculate 95% confidence intervals (using the 1/sqrt(n) rule of thumb).  Out of 16 weeks, we spend 3 weeks on statistics; 2 weeks are spent on basic probability.

By the way, all of the graphs on the correlation activity were taken from online searches.  I was honest with the class — we do not know how valid any of them are, though I believe that they are valid.

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Percents as Evil … Percents as Good (Applications of Math)

Given a percent and another number, do we multiply … divide … or something else?

A few years ago, I was at a presentation about a pre-algebra course where the presenter classified percent problems as either growth or decay.  My initial response was that these are concepts too advanced for that course; after a few minutes, I liked the idea, and my experiences since have strengthened that opinion.

Within a few days, I had a chance to work on percents in both a beginning algebra course and in our applications course.  In both courses, the percent problems are varied; one thing that was constant — students ‘wanted’ to multiply a percent and the other number in a problem, regardless of the context.  Sales tax rate and marked price … multiply and add.  Sales tax rate and final price … multiply and subtract (wrong).  Percent decrease and old amount … multiply and maybe add.  Percent decrease and new amount … multiply and add (wrong).

We seem to have reinforced overly simplistic rules about percents to the point where students are impervious to a need to change; 40% wrong answers is not enough (even if I asked ‘8 is 40% of what?’).  It’s really that 100% value that is the problem.  The connection between a growth rate of 3% and a multiplier of 1.03 is a challenge.

In the applications course, I had students work in small groups on a sequence of problems to make a transition from a simple percent value to a multiplier.  They worked hard, explained to each other, and seemed to do well.  The next day … a quiz on percents where they could use the multiplier; result — not so good.  In the applications course, we use this multiplier again — in our finance work (1 + APR) and in our exponential models [y = a(1+r)^x].  I suspect that a deliberate focus on the multiplier in 3 chapters might result in some improvement.

I actually fault our presentations on percents as the root of this ‘evil’.  We do “2 places to the left”, “is over of”, and mechanical use of “a is n% of b”; sure, we include problems where students need to find the base (divide), but the work is too superficial.  Students do not generally understand the contexts where percents are used.  An initial approach on growth or decay, which means seeing the multiplier, might just help.

The most common uses of percents in developmental courses is usually in that pre-algebra course.  Based on long-term goals of understanding, if we are not going to cover the whole story of percents (with the multiplier), we should omit percents entirely.  However, percents are one of the richest zones of overlap between math and the world students experience that we need to see percents as a good thing — and do it right.  Fewer tricks, a lot more understanding!

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Math — What is so good about THAT??

I was thinking that I could post a “What we see … What they see” item on math.  After all, we see many good things but our students see very little worth their time.  Then I thought some more about the ‘we see many good things’ part, and realized that this is not as obvious as we might think.

You’ve probably heard the phrases ‘intended curriculum’ and ‘received curriculum’ (or similar phrases).  When we advocate for a mathematics curriculum we are suggesting that the intended curriculum has sufficient value for students.  In mathematics, I think we confuse daily topics with the curriculum more than most disciplines do.  Instead of saying that a course deals with properties and relationships involving symbolic expressions of certain types, we say that the course covers factoring – rational expressions – radical expressions – and quadratic functions.  One problem is that these are internal code words that mean almost nothing outside of mathematics; the bigger problem is that these statements do not communicate any mathematics (in most cases).

Here is another concept about curriculum:  A course should have a basic story to tell.  Think about asking students who have completed a math course with a good grade:  “What was that course about?”  or “What are some good things to come from that course?”  Given that we do so much of our work in a symbolic world without a strong narrative, students will have great difficulty answering these types of questions compared to non-math courses.

So, if we want students to see the ‘good stuff’ about mathematics, we better increase our use of narrative forms.  This is not easy for us, since mathematics encourages brevity and non-repetitiveness.  However, this lack of narrative is one reason students find math ‘different’ … and ‘difficult’.

Which brings up a related point:  Can a person reason using only symbols?

We have a reputation for focusing on procedures and answers.  We often justify what we do by saying that we are building the reasoning skills of students by requiring that solutions involve a specified level of detail in symbolic form.  Instead of thinking about the reasoning, many students respond by mimicking our solution steps — which is not what we want (for most of us).

My favorite course to teach avoids some of the problems I’m talking about.  The Math – Applications for Living describes the content in phrases that many students can understand — quantities, geometry (a tougher one), finance, etc.  We also employ more narrative in the course (which frustrates some students, but usually helps overall).  We talk about good steps and solutions, but we talk more about how to figure something out — reasoning.  It’s not that we ‘cover Polya’ or anything like that, but students know that we are learning how to solve problems.

I think the curricular problems in math are causally related to the calculus-fixation that still drives much of our work:  It’s like everything prior to Calculus I is considered remediation, so we don’t do any real problem solving before then.  In the old days, this remediation approach was stated explicitly at the larger universities; I have not heard this recently, but our courses before calculus have not changed very much overall.

Developmental mathematics is changing.  We are developing courses that tell a story, we emphasize reasoning, and use narrative more than previously done.  It’s time for courses after the developmental level to look at these issues; it’s time for US to look at these issues and what they mean for our courses and students.

What would you want students to say was “good about that math class”?

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Math Lit/Applications for Living: Seeing the Power

Both the Math Literacy course and the Applications for Living course deal with two common models — linear and exponential.  I’m finding it interesting to watch how different and similar the experience is.

For both students, they have not seen exponential models in their college (developmental) courses; none of the current Applications for Living students had the Math Lit course previously.  (That will change as some Math Lit students take Applications for Living.)  In both cases, we explore models from numeric and symbolic forms; the Applications for Living course includes more variety, and also requires active graphing of exponential models.

In both courses, students have a difficult time leaving the linear world of adding and subtracting.  There is confusion about the role of slope in an adding world; during the exploring process, we take the time to show repeated adding as a multiplying, and identify the number as the slope.  When we work in exponential situations, the linear view seems to dominate.  During the exploring process, we show repeated multiplying as an exponent and learn about the role of the multiplier.  The performance learning outcomes are not what we would want; there are some differences between numeric and symbolic problems.

For example, the final exam in the Math Lit course had a doubling problem for which students needed to write the model.  Something like:

At the start, 25 people knew about the latest i-product; this number is going to double every day.  Write the exponential model for N (the number  who know) based on t (days since the start).

Another problem for the Math Lit final was a growth pattern from a numeric standpoint:

The cost of a machine is $400, and this is expected to grow by 10% per year.  Complete the following table of values.  [The table shows years 1 to 5, where the value for each year needs to be completed.]

In Applications for Living, the corresponding problems were this symbolic one:

The value of an investment is expected to grow by 6% per year.  Write the exponential model for the value in terms of the number of years.

And, this numeric one:

At 3pm, 20 mg of a drug were in the body.  At 4pm, 15 mg were in the body.  Complete the following table of values.  [The table shows hours 1 to 5, where the amount of drug needs to be completed.]

Almost half of the Applications for Living students treated the last problem as a linear one: They showed values of 10, 5, 0 and 0 (sometimes with a puzzled comment about having zero as the amount).  In class, we had done drugs in both half-life and percent decrease models; we had calculated the multiplier as well.  They did a little better on the symbolic form; part of this is the fact that this course also does work with finance formula, and one of those formulae is basically the answer for this problem.

The Math Lit students did well on the numeric problem; part of that success was the remediation we did earlier when most students had difficulty on all things exponential.  Few of the Math Lit students wrote a correct exponential model, which is noteworthy since the problem is a slight variation of a situation we used to introduce exponential models.  Most of the incorrect answers were variations on y =mx + b.

Clearly, this assessment feedback is indicating a need for an adjustment to the instructional cycles.

However, I also think that the results reflect a math curriculum that tends to treat topics in isolation.  How often do students need to deal with both linear and exponential models in one assessment?  Also, do we use the word “always” with students?  As in: “Compare the y-values; the difference always tells you what the slope is.”  Or, “If you can see how to get the next value in a table, you can always use this to complete a table.”  Or, “In a function, you can always get the next function value by adding or subtracting.”

During the instructional cycles in both courses, I can see the resistance to leaving the linear model.  It’s a bit like distributing, where students become fixated on one process.  I want students to see the power of understanding exponential models; students want the comfort of one model for all situations.

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