Is THAT the Best You’ve Got??

A student comes to college, and needs to meet their general education requirement.  One of those is in mathematics, and this student actually has some options:

• College Algebra (called pre-calculus at their college)
• Introductory Statistics
• Quantitative Reasoning

Being a typical student, this student wants to avoid the college algebra course; they thought about being an engineer but are too frightened of mathematics.  The next choice would be statistics, because everybody seems to think it is the best choice.

In looking in to the course, the student discovers that the statistics course has some nice features.  Most of the material is taught by first looking at data from the world around us, and the description says that the quantitative work is somewhat limited.  The student becomes worried when they look at the content in the text materials used — it’s got words used in a weird way (normal, deviation, inference, significance); it’s like statistics is a foreign language without any visible culture, so the student feels like much of it is arbitrary.

So, the student tries to find out what “Quantitative Reasoning” means.  The course description talks about voting, networks & paths, logic, and ‘proportionality’ (whatever that is).  Like the statistics course, it looks like the material often involves data from the world around us; however, it’s not clear how much quantitative work is actually involved.  The student is not too worried about any particular topic or phrase in the content descriptions; however, the course does not seem to have any pattern to the topics … it looks like an author’s 15 favorite lessons.

The student thinks about the basic question:

Will any of these courses help me in college courses, in my work, or in my life in general?

Basically, this student will reach the conclusion that none of these three courses will be that helpful.  As a mathematician, I would summarize the basic problem this way:

• The college algebra course and the statistics course focus on a narrow range of mathematics.
• This quantitative reasoning course does not focus on any particular mathematics.

There is a mythology, a story repeated so often that we believe it, that statistics is a better pathway for most students.  The rationale is something like “our world is dense with data and decision making” or “making decisions in a world of uncertainty”.  I see a basic problem, that remains in spite of what has been written: statistics is an occupational science, with few broad properties or theories.  Statistics is about getting helpful results, and for statisticians, this is great.  How does it help students when we use “n”, “n – 1”, and “n + 4” for calculations involving sample sizes; the ‘plus 4 rule’ is a typical statistical method for producing the results we want — even when there is no mathematical property to justify the practice.  [In a field like topology, we don’t let inconsistent procedures survive.]  I think we also over-estimate the value of statistics in occupations; there are limited uses in  other college courses, and some nice uses for life in general (for those motivated).

The quantitative reasoning (QR) course has a different problem — we don’t have a shared idea of what this course should accomplish.  For some, it’s an update to a liberal arts course (like the example above).  For others, QR means applying proportionality and some statistics to life.  Still other examples exist.

Is that the best we’ve got?  We are giving students options now (a nice thing), but the options are really not that good for the student.  For the student above, they really should take the college algebra course — perhaps they will find that mathematics is not their enemy after all; they might become an engineer, an outcome not likely at all with the other two choices described.

As mathematicians, we need to claim the problem and be part of the solution.  That college algebra course?  Modernize the content and methods so that it actually helps students prepare for further mathematics without becoming a filter that stops students.  That QR course?  We need professional conversations around this course; MAA and AMATYC should jointly develop a curricular model of some kind.  In my view, the QR course is the ideal general education math course; we should include significant mathematics from multiple domains, done in a way that students can discover that they could consider further mathematics.  The statistics course?  Let’s keep a realistic view of the value of this course; it’s not for everybody, and we tend to think of statistics as the option for people who never need anything else.

No, THAT is NOT the best we have.  We have some basic curricular work to do; together we can create better ideas, and help our profession as well as millions of students.

 
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Never DO Mathematics in a Math Class?

Within our efforts to make major improvements for our students, both in developmental and gateway college math courses, we have been looking at our content and our methodologies.  I want to connect the over-used phrase “Do the math!” with our phrase “doing mathematics”, within this process of building a better future.

I’ve commented before that the phrase “do the math!” is frequently used as a propaganda technique, to imply that everybody would reach the same conclusion as the speaker (and often used when the ‘math’ in the statement involves a small set of numbers, far removed from meaningful evidence for any argument).  I wonder if our phrase ‘doing mathematics’ serves a similar purpose within the profession.

Part of our problem is that we assume that there is a single meaning for ‘doing mathematics’.  Historically, the phrase seems to have grown out of the view of mathematics as seeking patterns, often in a constructivist approach; this ‘doing mathematics’ just means to immerse people in a situation involving quantities with a goal to establish patterns and statements that make sense to the learners.  Within professional mathematicians, we have two contrasting meanings — occupational (actuaries, for example) using systems of mathematical knowledge, and researchers using a variety of tools to establish new knowledge or applications.

As a learning tool, the “doing mathematics” has very limited benefits for the student.  Most research I have read suggests that learners need a very directive structure for the learning to occur by discovery; this guidance takes the process out of the original meaning of ‘doing mathematics’ into the more appropriate ‘learning mathematics’.  Doing mathematics, with the goal of learning mathematics, is a very advanced process — it is what some experts can do; expecting novices to engage in this process is a bit like expecting novices to become good piano players by having them sit at the piano (without any technique, without any theory).  Doing mathematics to learn mathematics does happen, often not by design, frequently with great excitement by those involved.

Perhaps the question is:

If students can experience what we experience when we ‘do mathematics’, they will be motivated to learn more mathematics.

Now, motivating students is one of the central roles in our classrooms.  Sometimes we focus so much on content and skills that we provide no information on our world of mathematics.  If we are the ones doing mathematics, presented in a way that novices can follow, then I can see some real benefits.  I have tried to do this in all of my classes; with colleagues, I use the phrase:

Students will see and perhaps do some beautiful and useless mathematics in every math class.

I include ‘useless’ in the description, and actually focus on that.  Why?  Because it is not reasonable to expect students to understand the eventual usefulness of the mathematics which they can appreciate; to them, it will likely seem “useless” even though it is not to a mathematician.  When I do this, I am walking a little beyond the limits of what students currently understand; I am in a beautiful field on the other side of a path, and want to share this perspective with my students.  My honest answer for ‘why I do this’ is simple:  It is fun!  I also have pedagogical reasons; walking a little beyond the current level helps to create an atmosphere of respect and one where learning for learning’s sake is appreciated.

So, my advice is: never have students to mathematics in a math class.  Help them learn mathematics, and you should do some mathematics in class so they know why we are mathematicians.  We don’t do mathematicians just for the money, or the fame.  We do mathematics because we want to.

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Curriculum and Instructional Improvement

We are doing some things at my College that might be of interest to others — not the type of thing we do a presentation on, though the information might be helpful.

Like many community colleges, our developmental mathematics courses have some of the highest enrollments on campus.  Therefore, these courses have large number of sections and are taught by a wide variety of faculty — full time/part time, new / experienced, rigid / flexible, etc.  Like many colleges, we follow student progression in the courses as closely as we can.  In the traditional courses (pre algebra, beginning algebra, intermediate algebra) the primary goal of each course has been to prepare students for the next course.  This progression data is not as good as we would like; nothing new there!

So, here is one thing we are doing about the problem.  We wrote a survey to be taken by instructors in a subsequent course.  In this survey, we listed the course outcomes for the prior course.  The survey asked the instructor to rank how important that outcome is, in preparing students for success in the subsequent course.  [The survey itself is being delivered through “Lime Survey”, a nice platform for surveys.]

The first survey asked intermediate algebra instructors about what students needed from beginning algebra.  We are currently working on the results (we had 16 surveys returned, from a pool of 33).    We are looking at the survey results as part of a process involving much discussion, rather than saying “this topic has got to be deleted because nobody needed it …”.  Our content in these courses is a little unusual in that few topics are covered in both courses — systems of equations is only in beginning algebra here, for example, as is most graphing concepts like slope.  Factoring polynomials is one of the few overlapping topics, which is likely why those outcomes were highly rated by instructors for intermediate algebra.

Another area we are looking at is instructional quality.  We have had a common departmental final exam for these courses for many years; we all use the same exams, and grade them with a common rubric.  However, much remains for each instructor to determine — points for attendance?  points for homework?  drop one low test?  making up tests?  We are working on providing instructors with feedback about how their choices impact the student’s probability of success in the next course.  One tool we are starting to use is an easy data-reporting tool that each instructor completes for each course:

The goal here is not to identify individual student issues; we are looking for patterns.  Does a given instructor have a large difference between the pre-final average and the final exam score?  Does an instructor have a large number of students who fail the final exam but pass the course?  [The final exam is required, but passing it is not required.]

We’ve also begun doing a “lesson study” method.  In our modified process, a group of instructors decides on a small topic to focus on, such as integer exponents.  The group talks about the topic — which is usually part of one class day: what makes this difficult?  what do students miss?  what shows understanding in students?  The group then creates a plan for the lesson, and some faculty use this in class while other faculty observe; this happens in 2 to 4 classes.  After these observations, the group meets to debrief … it’s about the lesson, not about students or instructors directly: what went well?  did students understand?  do some parts of the lesson need improvement?  Ideally, the class lessons are video taped for use in this debriefing, though we have not done that yet.  The debriefing itself is very educational, and we would like to record this so other instructors can experience the conversation.  The lesson study process is methodical and focused on the long term, one piece at a time; after a year, we have finished one lesson.

We are finding that a search for curricular and instructional quality is a long road; no maps are available so we are not sure that any particular action will lead to good results.  We do know that the process will lead to improvements if the conversation is centered in the hands of faculty.  None of this work is for administrative purposes.  Our goal is to help every instructor become better over time, and we see the administrative actions as issues of last resort.  We share ownership of the courses we teach, so this is not an issue of “I have the answers … not pay ATTENTION!”; it is more of an issue of professionalism for all instructors.

Hopefully, you found something of interest!

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STEM Prep … Make That Path Straight!

We’ve been dealing with two basic issues in our work:  First, helping students succeed in mathematics as a service (general education especially).  Second, helping students realize dreams of being in a STEM program.

Since the current reform efforts started in 2009, most of the focus has been on the first problem (geared toward general education math courses); this is the problem that Statway™ and Quantway™ provide a solution for, and this is also the connection between the New Life course “Mathematical Literacy” and general education (like the Michigan Transfer Agreement).

However, the professionals involved with the New Life project have … from the start … kept the second issue in mind.  We saw a need to provide better mathematics for those in STEM fields, as well as a new model that enabled more students to reach their goals.  This is why we designed the second New Life course “Algebraic Literacy” the way we did; the content is based on professional standards, with special focus on STEM boosting learning outcomes.  I often refer to this Algebraic Literacy course as the “Missing Link” because it seeks to connect more students to the STEM programs with better employment and quality of life.

This week, the Dana Center (University of Texas – Austin) announced that they have launched their “STEM Prep Path”.  You can see some details at http://us2.campaign-archive1.com/?u=f75754127932b3bd8ffbea25c&id=3f956c1b40&e=23e0b88a42#STEMPREPTeams

Throughout our efforts, the Dana Center work on New Mathways has been consistent with our New Life work; in fact, the Dana Center has involved AMATYC members of New Life in all stages of their work.  In the case of STEM Prep Path, they will design a course serving the same purpose as Algebraic Literacy; I’m sure that they will differ in some basic ways, but am also sure that the content will be similar in basic ways.  One thing that is different — the STEM Prep Path for the Dana Center includes work in the domain of College Algebra & Precalculus.  This is very exciting work, and offers the promise of cleaning up the swamp our students face in those courses.

The STEM Prep Path is an effort to make the path straight — in other words, design the curriculum to serve the purpose and combine this with instructional methods and support that allow all students a high probability of success.  Currently, our path to STEM is not good for equity; developmental math classes can be high minority classes (that’s not necessarily a problem) while STEM math classes are very low minority (and that is definitely a problem).  The New Life work in Algebraic Literacy is part of this same effort to support a broader spectrum of students a path to STEM fields.  For information on the Algebraic Literacy course, see the presentation on the “Missing Link” at https://www.devmathrevival.net/?page_id=1807  .

I hope that you will take a look at STEM Prep Path, and a look at the Algebraic Literacy course, so that we make a straight path for our students with a goal of a STEM program.

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