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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Inequity in the Math Classroom

I had an experience last week which I just need to share; I’ll try to explain why.

One of my classes this semester is a “Math Lab” class in which we have no large group lectures; in fact, the class has students in 3 different courses.  Students can work faster, and take tests when they are ready.  The basic methodology involves students working problems in the online home systems.

The classroom used for this Math Lab class is clearly going to be different; we provide 10 computers (desk tops) in individual work stations around the wall.  Students can also bring in their laptops and notebooks to use, at tables.

On this particular day, 16 students were in this class.  Six of the students were minorities (african american in this case), and the other 10 were majority (white).  That is not unusual.

What was unusual is the classroom geography.  Every one of the minority students was at one of the computer work stations; every one of the majority students was at a table using a laptop or notebook.

This separation speaks to inequities — the minority students lacked the resources of their own, so were using the provided computers.  For the work being done, the computers were adequate … but the difference bothers me quite a bit.

Students with a portable device can move with their computer; they can socialize in different ways, and they can bring their computer to me with a question.  Students using our computers do not have those choices.

You might be thinking … “so, provide laptops instead of the desk top computers”.  Sure, we could do that (we’ve been trying).  However, it still bothers me: when a category of student tends to lack a resource, students in that category face additional challenges to completion.

I understand that the causes for this inequity are complex (employment, wages, and financial aid … as starters).  I understand that the situation is “not my job” as a math teacher.  Those facts do not change the moral dilemma: a category of students face additional barriers.

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Conversation II: Herb & Jack on the Why — Practicality of Theory

In response to a post about STEM students and the traditional developmental mathematics curriculum, Herb Gross began with this quote (from a prior talk he gave):

The music is not in the guitar.

I think Herb is saying that mathematics is not in the visible tools used, whether these tools are procedures written down or technology used to answer questions.  This is a great point, and it suggests that we question any suggestion that we limit the content of mathematics courses to just those things seen as ‘practical’.  Seeing mathematics as being bound by the practical (for STEM or non-STEM) is a self-defeating behavior; a health profession is based on continuing growth, and growth depends upon research both applied and theoretical (the two work together in surprising ways).  Our students are future policy makers — do we want them to only value mathematics that is practical NOW?  (Think University of Wisconsin budget cuts.)

The music, and the mathematics, is based on connections among concepts.  This speaks to the growth of mathematical reasoning and critical thinking.  Herb adds this comment:

So I am not overly impressed with the pass rate improving as much as I am in seeing what the effect is further down the road.  In fact one of the reasons I don’t like non-algebra/calculus based courses is that even the students who are most successful in these courses tend to know how to crunch numbers into the calculator but have little feel as to what to do when the distribution is anything other than normal.

I think Herb is speaking to a basic goal of education — the improvements retained over a longer period of time, meaning improved capabilities.  The comment Herb makes is important, and I think it applies to most algebra based courses; I also wonder about calculus based courses.  Look at this re-phrasing of a critical part of Herb’s comment:

Students tend to know how to manipulate symbols or numbers often with the use of tools but have little understanding as to what to do with mathematical concepts applied to a new situation. (JR)

Creating scalable change within an individual involves some of the same work as creating scalable change in a profession.  A more complete view of learning is required, with less focus on ‘passing'; passing is a great thing, but it can not be the core measure of our success.  We seek to create mathematical abilities, including the willingness to apply existing knowledge to new situations where this knowledge is not sufficient.

Students in STEM programs need a broad foundation in mathematics, combining procedural and conceptual fluency.  To some of us, we follow that statement with “Non-STEM students to not”; this is where we can make large mistakes.  The mathematical needs of citizens and the mathematical needs of our partner disciplines are not different in a basic way — they need procedural and conceptual fluency as well.  The difference, overall, is a matter of degree and extent.  STEM students need MORE, not so much ‘different’.

Our work in the AMATYC New Life project supports this single-source approach to mathematics — the Mathematical Literacy course serves the needs of all students.  The initial uses of the course have often been for non-STEM students; however, the outcomes of the course were designed back from the needs of all students.  I agree with the design of the New Mathways Project (Dana Center), which has a similar course serving all students.

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STEM Students and Developmental Mathematics

Pathways … Mathways … creating alternatives for non-STEM students.

The changes in pre-college mathematics are significant, and I am incredibly pleased with the work of my colleagues in dozens of institutions across the United States.  The modified curriculum for these students has dramatically increased the proportion who achieve their goals, with a significant increase in the number passing their required college-level math course (statistics or quantitative reasoning).  These gains have been achieved by putting thousands of students into a different path, wherein they avoid beginning and intermediate algebra.

We need to get over a myth about other students — the students who need at least a college algebra course, often because they are pursuing a STEM or STEM related field.  [STEM refers to Science, Technology, Engineering and Mathematics.]

Myth: STEM students are well served by the traditional developmental mathematics curriculum.

Alternate hypotheses: STEM students are ill-served by the traditional developmental mathematics curriculum.

As you may know, I have been in this work for over 40 years.  Much has changed.  However, developmental mathematics is currently bound by these constraints (which has been true for over 40 years):

Constraint 1: The pedagogy and content is limited by the preponderance of adjunct faculty assigned to developmental math courses.

Constraint 2: The content is based on textbooks reflecting a set of topics which were copied from a typical high school curriculum of 1965.

These constraints interact within our curriculum, including at my own college.  The rigor of a developmental algebra course is most often established by the complexity of the procedure students would use to solve problems; these ‘problems’ are copies or slight variations of exercises seen in the homework.  These exercises, in turn focus on the achievement of a correct answer to a well-defined problem either stated symbolically or in the disguise of a verbal puzzle (where such puzzles lack both value in the real world and value in their structure).  We have a sense of pride if OUR algebra course includes conic sections or inverse functions, based on knowing that these topics await students in their college algebra course.

Some people might wonder if I think the presence of adjunct faculty in a classroom results in lower quality; definitely not — some of my adjunct colleagues are better instructors than I am.  The constraint is based on the fact that these courses need to be ‘teachable’ by the pool of adjuncts available; the issues deal with the expectations that are reasonable for a group, rather than individuals.  Full-time faculty may, in some cases, face similar limitations in the knowledge and skills they bring to a developmental algebra course; the difference is that full-time faculty have greatly enhanced access to professional development and networking.

In terms of data, the pathways work is fueled by the low pass rates in traditional courses (50 to 55%) compared to the typical 65% to 70% seen in the reform models (Mathematical Literacy, Fundamentals of Mathematical Reasoning, Quantway I).  By saying that STEM students are well-served by traditional developmental mathematics:

We are apparently comfortable with 25% (or less) of students completing two semesters of developmental algebra.

The improved outcomes for the reform models is likely due to the fact that all 3 address both constraints — professional development for faculty AND improved content.  By saying that STEM students are well-served by traditional developmental mathematics:

We are apparently comfortable with STEM students having to survive lower quality pedagogy and outdated content.

I see other issues, as well — such as the relative lack of technology in developmental algebra courses as a basic part of the content; calculators are banned … we avoid numerical methods … and remain out-of-touch with the world around us.

Again, I say that STEM students are ill-served by the traditional developmental mathematics course.  The content is inappropriate, pedagogy is not supportive, and little inspiration is ever seen for why a student would persevere in their STEM field.  STEM students need a reformed curriculum just as much as non-STEM students; the needs of society would suggest, in fact, that STEM students have a greater need for a reformed curriculum.

Take a look at the reform curriculum; it’s actually not that complicated.  Instead of beginning algebra, use the SAME reform course as non-STEM (Mathematical Literacy, Foundations of Mathematical Reasoning, or Quantway I).  Then … replace your intermediate algebra course with a reform course.  In the New Life work, that reform course is called Algebraic Reasoning; you can see some information at http://www.devmathrevival.net/?page_id=1807 , or head over to the wiki http://dm-live.wikispaces.com/

The New Mathways project is starting their work on STEM path — take a look at this post http://www.devmathrevival.net/?p=1935

I hope to do a presentation on the Algebraic Literacy course at this fall’s AMATYC conference … as a ‘bridge to somewhere’!  I believe that the Dana Center will also be there.  I encourage you to learn more about the reform curriculum for STEM students.  The work is important, students need it, and we will find it very rewarding.

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