Jump Start in Math

We just had our Michigan developmental education conference (“MDEC” see http://www.mdec.net/conference/2015/program.html for details).  One of our colleagues at Schoolcraft College did a presentation on their “Jump Start” program for math.

The Jump Start program has two components (each 2 hours long).  The first component is on math study skills; since the person conducting the workshop is a professional in the learning assistance center, she has a good background to provide clear direction to students on being successful in math.  Within this study skills component, she also deals directly with motivational issues — her goal is to provide HOPE for all students.

The second component is the content, where students choose the one that matches their course for the upcoming semester.  Since Jump Start is offered within a few days of the start of the semester, this part of the workshop reviews content needed to be successful in that course.  The college offers a Jump Start option for the first 4 or 5 courses.

You can get some information about their Jump Start program at http://www.schoolcraft.edu/a-z-index/learning-support-services/learning-assistance-center/student-success-seminars-and-workshops/jump-start#.VRKngeFuNyE  with the current schedule at http://www.schoolcraft.edu/docs/default-source/lss—jumpstart/jumpstart-winter.pdf?sfvrsn=0 .

Overall, the Schoolcraft math curriculum is quite traditional; they still offer a basic math class, and do not yet have a mathematical literacy course.  However, I like their Jump Start program; in particular, the 50% (2 hours) invested on study skills (and motivation) is very appropriate for most students.  The professional doing the workshops has a math degree; in fact, she was originally a developmental math student who had to work very hard … and became a math major because “math chose me” (as she says).

The 50% (2 hours) on content would not be sufficient to correct for basic gaps in understanding, and the content done focuses quite a bit on procedures.  However, even this part likely is a good thing for students — the workshop covers a half dozen topics with multiple examples in each, which might help students develop accurate expectations for college math classes, as our pace can be quite an adjustment from high school.

The Jump Start model might be a good alternative for many colleges who can not commit resources to week-long boot camps or similar programs.

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When Does Reform Succeed?

I have been thinking lately of a problem considered back when we started the AMATYC New Life project (about 2008).  The problem is not mathematical in nature, which perhaps explains why we have not solved it before.  Now, I am not saying that we found the only solution; I’m not even sure that our solution is sufficient.  I can report that our reform has grown way (way!) past any prior reform of developmental mathematics.

The problem is this:

What properties or methods enable a curriculum reform to succeed over a period of years and across regions?

Prior to our New Life work, many intelligent people had created valid reform ideas or models.  None of them survived time and space; they resulted in temporary changes (in general) and were limited to a few locations (at most).

For those interested in such things, here are my thoughts on strategies the result in successful reforms.

1. Professional organizations need to be deeply involved.

The New Life project was born in the AMATYC Developmental Mathematics committee, which had a large group of faculty willing to work on the project.  In addition, several members of the AMATYC Executive Board both supported and contributed to the work.  The involvement of the national leaders of a group enable that reform effort to connect with similar reform efforts by other groups (see below).

2. Content in the reform math curriculum created by faculty in a collaborative process, based on professional references.

If you look at the material over in the New Life ‘wiki’ (dm-live.wikispaces.com) you will notice that the learning outcomes were drawn from multiple professional sources (MAA, AMATYC, NADE, Numeracy Network, etc).  Both parts of the process were important — collaboration resulted in content that was widely accepted by math faculty, and professional resources helped create content that had external validity.

3. Avoid a focus on one issue.

In general, a reform effort built on one issue is very unlikely to succeed.  That one issue will not appeal to the general math faculty population.   For example, the NCAT redesign work tends to deal (in the curriculum) primarily with technology; as in prior calculator-based reforms, people find that this is a weak motivation for reform.   Addressing multiple issues in the reform means that most faculty will see something they like, which is a critical property for getting the reform adopted.  In the case of New Life, we addressed several content issues, classroom pedagogy, and professional development.

4. Plan for, and support, long-term conversations with faculty.

For some reform efforts, advocates did not sustain conversations with faculty over a period of time.  Only a few faculty will accept any reform when they first hear about it; one could argue that these faculty are actually not good test cases for a reform.  For the New Life project, we sustained conversations online (email, wiki) and at many conferences, for over 4 years now; in addition, we have had people travel to put on local workshops.  In our case, these conversations often result in faculty concluding that teaching our reform course is just more fun than what they have been doing; this is a powerful force for reform.

5. Create or support multiple solutions sharing basic properties.

No matter how good one particular reform model is, some faculty will not be comfortable with it; some institutions or states involve conditions that conflict with a given solution.  Our New Life project is one of three closely related solutions:  Carnegie Foundation Pathways (Statway, Quantway), Dana Center New Mathways, and AMATYC New Life.  The three projects have collaborated, shared resources and talent, and provide faculty & institutions with choices.    The New Life project itself supported multiple solutions — we depend upon commercial textbooks, and each major publisher is creating a solution.

6. Do not depend upon “one good book”.

Prior reforms, at all levels, often involved the creation of one set of materials.  New books face several challenges both in publishing and in getting adoptions.  A single book is just not going to be good enough to result in reform long-term.  The current reform in developmental mathematics involves commercial texts, foundation developed materials, and self-published materials.

I think other areas of college mathematics need basic reform, some perhaps even more needed than developmental.  I want reforms to succeed in ‘college algebra’, pre-calculus, finite math & modeling, calculus, quantitative reasoning, and statistics.  These courses impact hundreds of thousands of students every year; the impact is not uniformly positive.

As  you look at the points above, I am hoping you reach the single biggest conclusion:  Reform is something we do together, with each other, over a period of time.

 
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College Algebra … an Archeological Study

As we make real progress on improving mathematics education in colleges, shown especially in developmental mathematics, our attention is going to focus on college algebra and the “STEM Path”.  Of course, the name “college algebra” is given to a variety of courses, some of which serve a pre-calculus purpose (and some do not).  For years, I have thought of the name (college algebra) as a statement of “not being remedial”.

Could be that I was wrong about that.  During some recent searching, I came across a paper that Jeff Suzuki gave a few years ago.  Most of that talk is available at https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxqZWZmc3V6dWtpcHJvamVjdHxneDo2MWI5YWE4YzU2MDM1MmY3; if you have trouble with that link, search for “Jeff Suzuki Project” to get a list of his presentations.

Assuming that the content of that history is essentially correct, here is a brief statement of what college algebra is today:

College algebra is a collection of mathematical topics for general education, taken in place of calculus.

Some of the information in the Suzuki paper is in the form of book references to the 19th century.  This led to a book, possibly the first, to use “College Algebra” in the title — George Wentworth’s “A College Algebra” (1888); a later edition (1902) is available at https://archive.org/stream/acollegealgebra07wentgoog#page/n12/mode/2up .  In the same period, Webster Wells authored “University Algebra” (1879) and “College Algebra” (1890); see the 1879 text at http://books.google.com/books?id=uKZXAAAAYAAJ&pg=PR7&source=gbs_selected_pages&cad=2#v=onepage&q&f=false

These courses were taught as universities (Harvard, Yale, Princeton, Bowdoin, etc) reduced their mathematics requirements.  The college algebra course was not designed to prepare students for calculus.

These early college algebra books did not contain some current topics (factoring and graphing, for example).  The addition of graphing (including properties of functions) is related to calculus preparation; factoring is generally not so related.  Overall, the current college algebra course is clearly a descendent of this earlier course.

One of my current projects is to study the math courses required before calculus in my state (Michigan); Michigan does not have a system for higher education, which results in diversity in mathematics — college algebra, precalculus, and other courses are used.  However, the overall approach (in Michigan and elsewhere) is to consider these as being an equivalence or subsets; either the college algebra course(s) equate to the pre-calculus course(s) OR the college algebra course is a prerequisite to pre-calculus (that is very rare in Michigan).

Therefore, I believe that this is our current method of preparing students for calculus:

After establishing that the student does not need further remediation on high school mathematics, the student enrolls in an antiquated general education math course with a few valid preparatory topics, with the unreasonable hope that this will prepare them for calculus.

Much of our apparent curricular dependency (stuff in college algebra that is needed for calculus) seems to be an artificially created dependency — we need this radical simplification because that technique is needed for a few problems in calculus, and those problems were created in calculus to show why we needed radical simplification; we need this multi-step factoring topic in college algebra because we have created a set of problems in calculus that require creative factoring.

I encourage us all to study the “Mathematical Sciences in 2025” (available at http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025 ).  Some parts of our curriculum are archeological artifacts from the 19th century, and some parts date from the mid-20th century.  Very little of our curriculum reflects either current needs of client disciplines; not much more of it reflects the needs of mathematical sciences.

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Just for Fun … Creative Factoring

I’m teaching a class focused on individualized learning and flexible pacing.  One student in that class took a test on factoring in our intermediate algebra course.  In the process, I experienced something very enjoyable — a creative way to factor a polynomial.

Here is the situation:

Problem:   Factor r^4 – 16

Student:   (r – 2)(r³ + 2r² + 4r + 8)

Initially, I found this a bit confusing; I was not expecting to see a proposed factor with 4 terms.  In the materials, we focus on patterns to factor binomials involving the difference of squares.  So, I asked the student why he did this; his answer was “it checks”.  [This is exactly what I tell students when they ask WHY we factor a polynomial in a specific manner.]

After a quick transition from confusion to mathematical thinking, I looked more closely at the cubic factor.  Sure enough, it factors to produce:

Correct answer:   (r – 2)(r +2)(r² + 4)

This particular student (planning to be an engineer of some sort) had a creativity I would like to see more of.  The only negative feedback I had to deliver was “Finish the factoring”.

I found this to be just a lot of fun (though I doubt this student enjoyed it as much as I did, though he did enjoy it).  Mathematical fun is meant to be shared.  In 40 years, I’ve not seen a student do this; it’s too good of a thing to keep to myself.

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