Quality Instruction and Class Design

Last year, my college created a new structure for departments and programs.  Instead of a chairperson for each department within the 3 academic divisions, we got associate deans and ‘faculty program chairs’.  The associate deans are the administrative players ‘in charge’ of two or three of our old departments.  In my case, math and science share an associate dean.  We have 7 faculty program chairs for the two departments; I am in the role of faculty program chair for developmental mathematics.  [Not much time provided in the workload, but the work is rewarding.]

Currently, I am focusing on one key idea for our program:

How do we create quality experiences for our students?

We want higher pass rates and completion (of course).  However, our students need classes that serve a real purpose.  Designing a course so that grades and scores are consistently higher than a student’s learning does not help students.  Some people talk about this under the umbrella of ‘grade inflation’, though our interest is in the striving for quality in instruction and class design.

So, here are some issues I have been thinking about:

  • Should any ‘points’ be awarded for completing homework?
  • Should points be awarded based on the level of performance during homework?
  • Does “dropping a low test” support or hinder a high quality class?
  • If a student does not come close to passing the final exam, should they get a passing grade if their other work creates a high enough ‘average’?
  • Is it okay if students with a 2.0 or 2.5 grade are not ready for the next math course?
  • Do high grades (3.5 and 4.0) uniformly mean that the student is ready for the next math course?

When courses are sequential, the preparation for the next math course is a critical purpose of a math class.  Assigning a passing grade, therefore, is a definite message to the student that they are ready to take the next class.  In practice, we know that this progression is seldom perfect — we usually provide some review in the next class, even though students ‘should’ know that material.  At this point, our efforts are dealing with the existing course outcomes, which tend to be more procedural than we would like; eventually, we will raise the reasoning expectations in our courses (with a corresponding reduction in procedural content).

Of special  interest to me are the issues related to homework.  Some faculty assign up to 25% of the course grade based on homework.  Like many places, we are heavy users of online homework systems (My Labs Plus as well as Connect Math).  When those systems work well for students, they support the learning process; most students are able to achieve a high ‘score’ on a homework assignment.  Should this level of achievement balance out a lower level on a test and/or final exam?  Take the scenario like this:

Derick completes all homework with a friend; with a lot of effort, his homework is consistently 90% and above.  All of Derick’s tests are between 61% and 68%, and he gets a 66% on the final exam.  The high homework average raises his course grade to 71%, and he receives a 2.0 (C) grade in the algebra class.

This scenario is a little extreme (it’s only possible with a high weight on homework … >15%).  What is fairly common is a situation where homework is 10% of the course grade and the student passes 2 of the 5 tests; one of of the 3 not-passed tests is ‘dropped’, and the student easily qualifies for a 2.0 (C) grade.  One of the cases I saw this past semester involved this type of student achieving a 52% on the final exam.

In our case,we already have a common department final exam for the primary courses (pre-algebra up to pre-calculus).  In the case of developmental courses, we have a policy that requires 25% of the course grade to be based on that final exam.  This design for the final exam is a good step towards the quality we are striving for.  We are realizing that we can not stop there.

Like most community colleges, our courses are taught by both full-time and adjunct faculty; the last figures I saw showed about 40% by full-time and 60% by adjunct.  Because adjuncts are not consistently engaged with our conversations, adjuncts tend to have more variations than full-time faculty.  We will be looking for ways to help our large group of adjuncts become better integrated within the program, even in the face of definite budgetary constraints.  Fortunately, many of my full-time colleagues are committed to helping these efforts to improve the quality of our program.

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Students Don’t Do Optional … or Options

In the Achieving the Dream (AtD) ‘world’, the phrase “Students do not do optional” is used as a message to colleges that policy and program decisions need to reflect what we believe students ought to do — if it’s a helpful thing, making it optional often means that the students who need it the most will not do it.  I tried something in my class that suggests a slightly different idea.

For the past two years, I have ‘required’ (assigned points) students to connect with a help location at the college.  The idea was that students need to know — before they think they need it — where they can get help for their math class.  I allow days for this — usually, until the 4th class day.

Until this semester, I provided students with options for how to complete this required activity.

  • my office hours
  • the college’s “Learning Commons” (tutoring center)
  • the college’s library tutoring (also staffed by the tutoring center)
  • special programs tutoring (like TRIO)

Typically, I would have about 70% of students complete this ‘connect with help’ activity; most of the struggling students were in the 30% who did not.  Some of these students eventually found the help.

This semester, I tried a revision to this connect with help activity.  I provided students the following choice(s):

  1. the college’s “Learning Commons” (tutoring center)

The result?  I have 100% completion for this activity.  All active students have completed the activity, and most of these did it right away.

This is summer semester, and “summer is different” (though it’s difficult to quantify how different).  However, the results suggest that the existence of options creates barriers for some of our students.  We have evidence that this problem exists within the content of a mathematics class — when we tell students that we are covering multiple methods (or concepts) for the same type of problems, some students struggle due to the existence of a choice.  [For those who are curious, you may wonder if students are not coming to my office hour -- so far, I actually have more students coming to my office hours.  No apparent loss there.]

I think the basic question is this:

Given that choices (options or optional) creates some risk for some students, WHEN are there sufficient advantages to justify this risk?

If dealing with a choice has the potential for improving mathematical understanding, I will continue to place choices in front of my students.  We should resist the temptation to provide simple answers when students struggle with mathematics; the process working (learning) depends upon the learner navigating through choices and dealing with some ambiguity. On the other hand, when the choices deal with something non-mathematical, we should be very careful before imposing the choice on students.

Some people might be thinking “So, it’s okay for us to be rigid and not-flexible” in dealing with students.  That is NOT what I am saying.  If one of my students gave me a valid rationale for why they could not do the ‘one option’, I would offer them an equivalent process.  Our rigidity needs to be invested in what is important to us; I would hope that the important stuff is something related to “understanding mathematics” (though we don’t all agree on what that means).

I would suggest that the AtD phrase be modified slightly:

Options will cause difficulties for some students.  Allow options when this provides enough advantages to students.

We usually try to be helpful to students, and part of this is a tendency to provide students with options. Putting choices in front of students is not always a good thing, so we need to be selective about when we put options in to our courses and procedures.

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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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Homework and Grades in a Math Class

I am trying to reconcile two recent conversations, and I suspect that most of us have had similar discussions.

First, a student (after missing a passing grade by 5%):

I have worked hard, and did extra homework.  The instructor set up the class to fail.  Very unreasonable.

Second, a potential faculty member in response to a question about classroom assessment techniques:

I am really in favor of homework.  I don’t grade it, but it shows what students do not know.

The potential faculty member was reacting to the practice of some colleagues who assign significant points in a class for the completion of homework.  Their institution also suggested that they use the publisher’s homework system for the tests in online sections (My Math Lab, Connect Math, or Web Asssign).  The sad part of this candidates response was that they never mentioned any other assessment technique.

In the student’s situation, she was mostly responded out of frustration.  She is trying to overcome a false start in college a few years ago resulting in a very low GPA, and was dealing with a family health situation; a failing grade in our math class meant that she would not be able to continue in college.

The question is this:  What is the primary purpose of homework?  We know that students do not learn if they do not apply significant effort.  This leads many of my colleagues to give credit for completing homework, sometimes up to 10% of the course grade.  They either give all points regardless of ‘score’ on the homework, or they prorate the points based on the performance level.  The ‘all points’ approach tells students that the main goal of homework is to complete the problems regardless of learning; the prorated method tells students that mistakes in homework can cost them points.

Doing assignments (reading, studying, practice, checking answers, etc) is the critical learning activity in any mathematics class.

With good intentions, we award points for homework; however, that’s pretty much a no-win situation:  Awarding points will encourage students to get it done.  However, getting it done does not mean that students are learning anything.  As a sports metaphor, points for homework is a bit like telling a batter in baseball to swing at anything close to the plate: you need to swing to hit the ball.  The  problem is that swinging at anything means hitting the ball is mostly a coincidence — just like learning when points are awarded for homework.  I would prefer to not settle for accidental learning.

My own conclusion is that doing homework should not be connected directly to a grade in a math class.  Without a learning attitude, the homework will not help; with a learning attitude, the work will get done.  If I can build a learning attitude with my students, they are better prepared for success in any math class they take.

The current media treatment of this concept uses words like ‘grit’ and ‘perseverance’; these phrases reflect the infatuation with educational outsiders creating solutions for educational problems.   Two weeks ago, I sat through a day-long professional development session featuring a psychologist who tried to tell us how to flip our classrooms.  This was a practicing therapist with great expertise in generational issues, but with no particular understanding of learning in a classroom.  [He suggested awarding 25% of the course grade for completing the preparation for class, and another 25% for the assessment activities in class dealing with that preparation.]

The problem I have with this media approach is that grit and perseverance have strong cultural components reflecting the student’s history (especially familial). I much prefer to focus on a learning attitude; this concept is accessible to students, and we have some tools to  build a learning attitude.

A quick list of ingredients for a learning attitude in a mathematics classroom:

  • All students involved in conversations about mathematics (I use directed small group work for this)
  • Cold-calling on students (expecting every student to be engaged, understanding that “I don’t get it yet” is an acceptable answer)
  • Encouraging discussion and reasonable disagreement in class (an initial step is to ask “is there a different way to do this?” … and then waiting 15 seconds or more)
  • Quizzes at the start of class (I do a quiz in about half of the non-test days)
  • Spending class time working with individual students on something they are struggling with (I do ‘test drives’ for students after doing examples, and worksheets with challenging problems)

This particular list is a practitioner’s list, not entirely founded in learning theory.  However, I can say that my reading of cognitive psychology leads me to believe that the critical necessary condition for learning is the brain engaged with significant and accessible material — this is reflected in most of the items above.

Homework is a poor assessment; homework points is a weak motivator for learning.  How do you build a learning attitude for your students?

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