Mathematical Literacy Course WITHOUT a Math Prerequisite

We did a session (Mark Chapman and I) at the recent MichMATYC conference on our newest course … a second version of Mathematical Literacy with no math prerequisite.

Here are the materials from our session:

Presentation Slides:  math-literacy-without-a-math-prerequisite-for-web

Handout 1 … Information on the course:  math-literacy-without-a-math-prerequisite-handout

Handout 2 … Math Lit Goals and Outcomes: mlcs-goals-and-outcomes-oct2013-cross-referenced

We started offering the new course this semester, so these materials describe the course design.  Data will come later!

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The Right Answer is Not the Thing

This is not another post on assessment, though the content is related.  The central theme in this post is faculty being wise about the process of helping students navigate through mathematics in an efficient manner (something we might call “learning mathematics”) 🙂 .

As context, I want to share part of a lesson from our math literacy course.  Like many such courses, we both use accessible situations and recognizing patterns in the learning process.  This particular lesson uses interest (simple and compound) with these basic steps in the process.

1. We deposit $500 in an account that pays 6% interest each year, on that $500.  Find the interest earned in the first 4 years by competing the table.  [The table shows a row for each year.]
   Find the total money by adding the interest and the original $500.
2. We deposit $500 in an account that pays 6% interest each year, on the current balance (including prior interest).  Find the interest and current balance for the first 4 years by completing the table.  What is the total money for this account?
3. Which account results in “more money” for us?
4. We found the current balance by calculating “0.06 × 500 + 500”.  Is there a way to simplify this calculation so there is only one multiplication and no addition?

Of course, much time is used in the first two steps.  Students often have misunderstandings about percents, but these are motivational questions … as is #3.  However, the learning in the problem is all about the fourth step, which is looking for “1.06 × 500”.

Many teachers will present the 4th question in a manner that defeats the purpose of the question … “we added 6% to 100%; what do we get?”  This approach ‘works’ in that many students will see how we got the 1.06, and we feel good that they got the right answer.  Unfortunately, we just avoided all of the meaningful learning in this context.

First of all, students need to really know that percents do not have any meaning by themselves.  When we say “added 6% to 100%”, we have reinforced the basic misunderstanding that percents work like decimals in all situations.  It’s easy to determine if students have this misunderstanding by asking a variation of the classic question:

We had a 10% decrease in pay last year, and this year we got a 15% increase in pay.  Our current pay is what percent increase or decrease compared to the pay before the decrease?

This problem is tough for students because it does not explicitly state the core situation … that the base for each percent is the current pay … and we might think that this is the main reason we get the wrong answer “5% increase”. However, even when this fact about the base is pointed out, students continue to add the percents.

Secondly, the “we added 6% to 100%; what do we get?” question divorces the situation from the algebraic reasoning.  We’ve done adding of fractions, where a common base is required.  Somehow, with percents, we are comfortable leaving the base out of the problem when this produces more ‘right answers’.  Each of those percents has a base, which happens to be the same number in this ‘interest’ situation.  A more appropriate instructional move is to provide a little scaffolding:

Let’s write 0.06 × 500 + 500 this way:  (0.06 × 500) + (1.00 × 500)

Remember how we added 4x + 2x?  We got 6x.

Does that suggest how we might do the adding first?

Now, this instructional move will not make the problem easy.  The goal with this move is to connect the new problem to something fundamental in mathematics:  “like” things can sometimes be added.  Having the right answer without applying this concept is not learning any mathematics.

In our Math Lit course, this lesson introduces the concept of ‘growth factor’ which is then used as we identify sequences that are linear versus exponential.  That discrimination in sequences can get quite sophisticated, though we generally keep the level reasonable for the needs of the course and students.  The phrase ‘growth factor’ is used temporarily until we consider declining situations … however, this “adding to get one multiplication” is behind all exponential models.

Unrelated to the main point of this post, it’s interesting that many of us think of the number ‘e’ when exponential models are being discussed.  There are, of course, very good reasons why that is the most commonly used base in mathematics; unfortunately for the learning process, using base ‘e’ presents a disguise of the direct process involved in the situation … a multiplicative factor based on a percent increase or decrease.  I don’t see using ‘e’ prior to a pre-calculus course, in terms of helping students.

Back to the main point … whether you are teaching Math Literacy, Algebraic Literacy, or even the old-fashioned courses, “right answers” are a poor measure of the quality of learning.  The learning process itself needs to be richer and more valid than using a measure known to have limited validity.

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Scaling Mathematical Literacy Courses

My college (Lansing CC) has implemented a second version of Math Literacy, which allowed us to drop our pre-algebra course.  I posted previously on the ‘without a prerequisite (see https://www.devmathrevival.net/?p=2516).

Here is a summary of where we ended up in the first semester of having both Math Lit courses.

• Math105 (Math Lit [with math prereq] has 8 sections with about 165 students.
• Math106 (Math Lit with Review [no math prereq] has 9 sections with about 225 students

With these 17 sections of Mathematical Literacy, we have quite a few instructors teaching the course for the first time.  Most of the instructors new to this teaching have been involved with the development of the course and policies, where we discussed text coverage and technology expectations for students.

As part of our collaboration, we are having bi-weekly meetings with as many of the instructors as can manage the ‘best time’.  The leading issue being dealt with is the textbook purchase; we’re helping as best we can with that, but buying the textbook is outside of our control.

We are talking about learning and teaching issues.  For example … how to balance an emphasis on concepts to enable reasoning with an emphasis on procedures so that students can actually ‘do something’ with the math (like have an answer to communicate).  We are talking about which small-group structures seem to work well in this course.

Our approach to scaling up Math Literacy is based on a long-term professional development approach.  Our bi-weekly meetings will continue as long as there seems to be a need (one semester, one year, or longer).  We are looking in to setting up a shared collaboration space for the instructors, which will enable those not able to attend to be involved.

In our structure, students who do not place at beginning algebra (or higher) are required to start with Math Literacy; those at the beginning algebra have the option to use the Math Literacy course.  After the Math Literacy course, students have 3 options:

1. Take our Quantitative Reasoning course (Math119) … required for most health careers
2. Take out Intro Statistics course (Stat170) … required for some other programs
3. Take our Fast-Track Algebra (Math109) which allows progression to pre-calculus

Unlike some implementations, our vision of Math Literacy includes all students … even “STEM-bound”.  Faculty teaching our STEM math courses are pleased with the strong reasoning component of Math Literacy.  We will be collecting data on how the various progressions work for students, and can implement needed adjustments to make improvements.

For those near Michigan, we will be making a presentation at our affiliate (MichMATYC) conference next month (Oct 15 at Delta College).  See http://websites.delta.edu/math/michmatyc2016/ for details.

 
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The Difference Between Mathematical Literacy & Algebraic Literacy

As more colleges implement Mathematical Literacy courses, we are running in to a point of confusion:  what is the difference between Algebraic Literacy and Mathematical Literacy?  The easy reference is problematic … comparing these courses to the traditional beginning & intermediate algebra courses; those traditional courses are at the ‘same level’ in a general way, but this fact does not help us deal with the details of new courses.

I’ve written previously on the comparison of the new courses to the old, especially Algebraic Literacy compared to the traditional course (https://www.devmathrevival.net/?p=2347 and https://www.devmathrevival.net/?p=2331 ).  However, I’ve not talked that much about the difference between these new courses that share a word in the title (“Literacy”).  That’s the goal of this post.

First, the course titles are not perfect … the word ‘literacy’ was meant to imply that the courses deal with pre-college material; ‘mathematical’ was meant to suggest that we did not start with algebra directly … while ‘algebraic’ was meant to suggest some directionality (headed towards STEM and STEM-like courses).  We have focused on the goals and outcomes documents for the new courses as a way to clarify what the courses are designed to deliver.

MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2

Algebraic Literacy Goals and Outcomes Oct2013 cross referenced

Since these courses diverge from the traditional curriculum, these documents were not sufficient to clarify “what belongs in each course” for shared topics (especially algebra).

So, here is a side by side chart meant to provide some additional clarification.

The intent is not to avoid any overlap between the courses, though there is less overlap than the traditional courses (in general).  As an example, many Math Lit courses introduce systems of linear equations; the solution methods are usually limited to numeric (graphing & intersect) and some substitution.  In an algebraic literacy course, the problems would be more diverse and so would the solution methods presented.

Another example is factoring polynomials.  The classic Math Literacy course might cover “GCF” factoring only (pardon the redundancy) … though that is not assumed.  The intent is that Math Literacy avoid most factoring beyond that which is a direct application of the distributive property; Algebraic Literacy picks up most of the factoring concepts necessary.  We note that most ‘needs’ to include factoring are contrived; a deep understanding of functions (the core goal of pre-calculus) does not depend upon all the typical methods presented in the albatross “Intermediate Algebra”.

A solid Mathematical Literacy course will involve some algebraic manipulation (limited in types as well as in complexity), and these procedures would be further enhanced Algebraic Literacy.  Therefore, the distinction between Math Lit & Algebraic Literacy can not be reduced to a particular ‘problem’ being present in one course but not the other.  We really want to keep the focus on the purposes of each course; see the ‘goals’ part of the course documents listed above.

If you have questions about the distinction between the two new courses, I would be glad to provide any information I have.

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