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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Algebraic Literacy: Finding a Textbook

One of the new courses coming to a college near you is “Algebraic Literacy”, a modern course that prepares students for a STEM path (and related work).  This position in the traditional curriculum is held by ‘intermediate algebra’.

For a brief comparison of these courses, see the chart below:

In this chart, “MLCS” refers to the Mathematical Literacy for College Students course (also known as Math Lit, and similar to Quantway I).

One of the issues with the Algebraic Literacy course is finding textbook materials.  Books being written for this course are not available yet.  However, there are materials available which have enough similarity to be used.

One book I have learned about recently is “Algebra: Form and Function” (Wiley publishing, 2010).  This book was written by a team connected with the calculus reform efforts, and is designated as a ‘college algebra’ textbook.  However, the book does not assume that students have the higher background; it’s quite accessible by students in an Algebraic Literacy course.  For a quick look, see this link to the Course Smart page:  http://instructors.coursesmart.com/9780471707080

You can also find more information on this text at the Wiley page http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000346.html

 
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Product As Sum: The Language of Algebra

I’ve been puzzling over some types of errors that seem both common and resistant to correction.  Essentially, the errors involve a disconnect between meaning and symbols especially in the two basic structures of quantities — adding and multiplying.

Here is a brief catalog of the errors:

• 3x²+5x² = 8x^4
• 4a(2b) = 8b + 4ab  (or some other ‘distributing’)
• (5y²)^3=15y^6  or 125y^8
• (3n +2) + (5n + 4) = 15n² +22n + 10
• sqrt(4x^9) = 2x^3
• sqrt(-50) = 5i + sqrt(2)

I’ve been seeing these types of errors for many years; however, it seems like the first 4 are becoming more common.  The radical context is not that important by itself for most of my students — except as a window into the same fragile knowledge about mathematical notation and meaning.  The errors appear with both new-to-college students and students who have ‘passed’ an algebra course.

In talking to students about these patterns, I’ve concluded that quite a bit of the problem is based on procedures removed from meaning.  Students usually know the phrase “like terms”, but seldom talk about counting when we have them; they know to combine the numbers in front but are often unsure about the exponents.  A focus on the meaning of the expression would make it clear what should be done.

The fourth error (‘foiling a sum’ or ‘distributing when adding’) is triggered by the “distributing is great” attitude; students really like to distribute, and we talk about distributing all the time.  In exploring this error (which shows temporary improvement) students say that they did not “see” the operation between the parentheses; what they mean is that they thought that parentheses means a product.

It’s likely that experienced teachers are not surprised by any item on the list above.  The issue for us is this: If these are important enough, how do we change our curriculum to decrease the frequency of such errors of meaning?  My own view is that the basic errors (the first 4) are very important, and I want to address them in all courses (whether traditional algebra or a math literacy course).

One strategy that I plan to use is more “unblocked practice and assessment”.  Much of a traditional developmental math course is severely blocked: the problems deal with a small set of procedures, separated from other types that might trigger an error.  We need to provide opportunities for these errors to be shown during the learning process.  Instead of trying to include quite so many types of each procedure, I will include some competing types from earlier work.  A student who can complete 50 ‘foil’ problems with 90% accuracy may not understand much at all, and may mis-apply the procedure … if we’ve never given them a chance to develop skills in discriminating types of problems.  This unblocked approach needs to be in all stages of learning (initial, practice, assessment, cumulative, etc).

Another method I use in my beginning algebra course is based on language learning concepts.  The idea is not complicated: Present students with either the symbolic statement or a verbal equivalent and ask them to identify the other.  Usually, this is done in a ‘multiple-select’ format: more than one correct choice is possible.  Students need to know that there is more than one verbal statement for a symbolic statement, and that there are sometimes equivalent symbolic statements.

For years, I have included some vocabulary or concept questions on daily quizzes.  I am concluding that I need to expand this to other assessments including tests, and to include perhaps more types.  Some of the online homework systems we use have these types of items, and the students who need them the most tend to skip  them … putting more emphasis on these in assessments will encourage students to take them more seriously in the homework.

I called this post “product as sum” because I am seeing students not being able to consistently treat them accurately.  This is such a fundamental concept that such errors bother me, especially when they occur in students who have passed an algebra course last semester.  Perhaps this is more evidence that:

1. We are trying to ‘cover’ too much (not enough time to understand and connect knowledge)
2. We focus on procedure too much (removes meaning as a critical feature to deal with)
3. We compartmentalize content too much (problems tend to be blocked, sometimes severely)

Meaning, connections, and concepts are important.  Procedures by themselves?  Not so much!

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Quantway(tm) Materials, available — the Dana Center “FMR” course

The Dana Center (University of Texas – Austin) has launched an updated web site.  As part of this, the original Quantway™ materials (version 1.0) are available along with other documents — see http://www.utdanacenter.org/higher-education/new-mathways-project/new-mathways-project-curricular-materials/foundations-of-mathematical-reasoning-course/  

To get the Quantway information, you need to scroll down to the end of that page.  If you want general information on the New Mathways project, go to http://www.utdanacenter.org/higher-education/new-mathways-project/

This site is part of the New Mathways Project.  I’ve talked before about their project; the first course in their paths has now been named “Foundations of Mathematical Reasoning” (FMR).  It looks like this course will be shared by all 3 paths that will be developed — statistics, quantitative reasoning, and STEM.   The learning outcomes for FMR will be adapted from the Quantway outcomes, which were adapted from the New Life MLCS course.  The position of FMR in a math curriculum is very similar to that of MLCS; the Carnegie Foundation Quantway program is different in that students are tracked from the start … both New Mathways and New Life provide a flexible structure at this level.

The fact that 3 reform efforts share a curricular element (the first course) is part of my optimism that we can create basic change in developmental mathematics over the next several years.

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