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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Avoiding Problems and Disasters in the Learning of Mathematics

In some implementations of reformed mathematics courses, there is a strong emphasis on particular ‘learning’ procedures in our effort to improve student outcomes.  For developmental mathematics, the learning procedures du jour are problem-based learning and discovery learning (both heavily influenced by constructivist viewpoints).  These methods have some basis in research and theory, but are easier to implement badly than well (as is true for most procedures).  The purpose of this post is to suggest some guidelines that can avoid issues with these methodologies.

First, I recommend people read a summary of such methods by Krischner, Sweller, and Clark called “Why Minimal Guidance During Instruction Does Not Work” available at http://www.cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf .  Here is their conclusion:

After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of
research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports
direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to
intermediate learners. Even for students with considerable prior knowledge, strong guidance while learning is most often
found to be equally effective as unguided approaches. Not only is unguided instruction normally less effective; there is
also evidence that it may have negative results when students acquire misconceptions or incomplete or disorganized knowledge.  [pgs 83 and 84]

Second, here is my own summary of other research:

Lectures are a poor method of providing direct, strong instructional guidance.

Our problem, and confusion, stems from a reaction to the lecture methodologies.  Students tend to be passive and not engaged, so our reaction looks for methods that make activity visible.  However, visible activity may be worse than a passive student (that’s what the research summary above says).  We can not settle for what is easy to see; we must go beyond the ideology of ‘learner centered’ and focus on LEARNING.

How people learn is not that much of a mystery.  For people with a brain functioning in the normal ranges, here is the recipe:

  • New information that does not fit existing knowledge
  • Effort applied to reconcile this gap or conflict
  • Access to information related to this gap
  • Validation of the resulting new knowledge

Of course, this is overly simplified to be prescriptive for use in a classroom.  However, we need to keep our minds on these ingredients, not on visible activity.  When discovery learning fails, it is often due to a design where the focus is on the first two ingredients; the mythical lecture mode involves a focus on the third ingredient.

I received a message from somebody teaching in a Math Lit course, who was frustrated by the difficulties students were encountering.  To me, those difficulties originated from an almost exclusive focus on new information and effort; the message spoke to the missing information (step 3), though step 4 is an issue in some courses as well (practice and assessment).  We can’t expect students — at any level of mathematics — to spontaneously create good mathematics that is integrated in their brain.

Yesterday, I had a brief conversation with my Provost after a professional development session emphasizing the advantages of a flipped classroom (which is a different issue than those above).  When I told the Provost that I was not impressed with the presentation, the Provost responded with something like “That’s okay; we mostly wanted to get people moving away from lecture.”  Yes, we need to move away from lectures; most of my colleagues have already done that.  We also need to move away from the antithesis of lectures to models where students are active but not productive.

So, provide summaries and mathematical statements to your students.  I don’t care whether this occurs before or after they attempt problems, as long as you design the experience for learning.  And, provide instruction to your students; you are an expert, and they will tend to remain novices when they do not see how experts do the same work.  Keep a large emphasis on validation and assessment; however students learn, they tend to store information either partially correct or completely incorrect, and it is our job to provide a rich diet of feedback on this learning.  Remember that learning is never visible; what happens in a brain is hidden from most of us (who lack fMRI machines in our classrooms).  And, be sure that you don’t confuse activity with learning.

Learning is not easy; designing a math class for learning is not easy.  The ideas of learning are simple, but the application to a classroom is nuanced, requiring attention to all components of learning.  Any instructional design that emphasizes a subset of ingredients for learning is going to fail students.  In our rush away from ‘lecturing’, we need to avoid jumping off the cliff in to the lake labeled ‘active students’; this lake might look attractive, but being in the water does not mean there is learning about the water.

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Outcomes for a Quantitative Reasoning Course

When we look at reports summarizing enrollment trends in college mathematics (like CBMS; see http://www.ams.org/profession/data/cbms-survey/cbms2010-Report.pdf) the entry “Quantitative Reasoning” (QR) does not appear — which will likely change, given the increase in the number of colleges offering QR.  As a profession, we have not settled on the general nature of the learning outcomes for a college level QR course.  As a supplement to the entry of Principles for a QR course, I will list our QR outcomes; these are from our Math119 Math – Applications for Living course.

So, here is the list of outcomes:

  1. Use mathematical principles, concepts, processes, and rules to investigate, formulate, and solve problems in disciplinary and career contexts.
  2. Work with others in teamed situations using mathematical principles, concepts, processes, and rules to investigate, formulate, and solve problems in disciplinary and career contexts.
  3. Use appropriate tools and equipment, including graphing calculators, in investigating, and solving problems in disciplinary and career contexts.
  4. Use standard references and resources, both print and electronic, from disciplinary and career areas as resources in investigation, formulating, and solving problems in disciplinary and career contexts.
  5. Use measurable attributes of objects and the units, systems, and processes of measurement in disciplinary and career contexts.
  6. Apply appropriate techniques, tools, and formulas to determine measurements in disciplinary and career contexts.
  7. Use and develop formulas for applied situations in disciplinary and career contexts.
  8. Use proportions, ratios, and percents in disciplinary and career contexts.
  9. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships as they apply in disciplinary and career contexts.
  10. Specify locations and describe spatial relationships using coordinate geometry and other representational systems in disciplinary and career contexts.
  11. Apply transformations and use symmetry to analyze situations in disciplinary and career contexts.
  12. Formulate questions in disciplinary and career contexts that can be addressed with data and collect, organize, and display relevant data to answer them.
  13. Select and use appropriate statistical methods to analyze data in disciplinary and career contexts.
  14. Develop and evaluate inferences and predictions that are based on data in disciplinary and career contexts.
  15. Understand and apply basic concepts of probability in disciplinary and career contexts.

We blend occupational and academic contexts in this class, as you can see from these outcomes.  As you would expect, some outcomes are emphasized more than others.  Proportionality and percents are very important in the class; functions are emphasized using a variety of representations.

When I teach this course, I organize the content in these units:

  1. Quantities and Geometry
    Converting units (linear, area and volume) and dimensional analysis; significant digits; scientific notation; geometry (2D and 3D) applied to objects, including compound objects (2D).
  2. Percents and Finance
    Growth and decay to algebraic statements; relative change; interest; savings plan balance; savings plan payment; loan payment.
  3. Statistics
    Concepts (population, sample, bias, hypotheses, significance); confidence interval; measures of center; distributions (concepts — symmetry, variation); communicating statistical information (frequency tables, bar graphs, histograms, line charts, 5-number summary).
  4. Probability
    Calculating outcomes; basic probability; sequences of events (independent and dependent); at least once probability; counting formulas (sequences, permutations, combinations)
  5. Functions and Models
    Linear and exponential models; writing models from verbal statements; solving for parameters (finding slope and y-intercept in context, finding multiplier and starting value in context); doubling time and half life; logistic growth; solving exponential equations numerically; graphing linear and exponential functions (including creating scales for axes).

This is not an easy class.  Regardless of background, many students have difficulty with the transitions from verbal information to mathematical symbolism.  We blend presentations and workshop activities in class, and — due to student effort — usually get a pass rate about 70%.

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