What Mathematicians Do in Collaboration

Okay, the subject for this post is just a hint about the content; don’t expect news of a large-scale project involving multiple R1 institutions.  I’m talking about a very specific collaboration in a small zone within our work.  Namely … dealing with frustrating flaws in student reasoning in mathematics.

So, here is the context … After a recent test, I felt a need for the stress relief (aka “therapy”) due to a specific flaw in reasoning which was seen from a number of students in the class.  The particular problem begins with a context dealing with the terminal velocity of a falling object.  The problem provides a formula, which (upon substitution of values) leads to this equation:

30 =  √(64h)        [30 equals the square root of 64h]

Our office area includes several collaboration spaces, and one of them is adjacent to my office.  In that space, I posted this (as part of my stress relief):

I was disturbed by the number of students who lost their algebraic reasoning when dealing with radical equations, and decided that eliminating a coefficient was best done by subtraction (perhaps just because the result is ‘nicer’ than the correct process).

A couple of days later, one of my colleagues (who has not yet confessed) posted this response:

Of course, my first reaction involved two components.  First, that’s creative … why didn’t I think of that?  Second, this is hilarious but probably not the reasoning my students had in mind.

Adjacent to this, the “perpetrator” posted this rationale for declaring “yes! … mod 73”.

I’m still bothered by the students’ use of the ’22’ step (and they heard about that in class the next day).  However, my colleagues contribution did help me by showing a role for creativity and humor among mathematicians.

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Trigonometry and the Phys Ed Teacher

Follows …  a story, perhaps an allegory.

Mr. Trubac is a physical education teacher at a local school; in fact, Mr. Trubac is a loved and respected teacher in middle school, a rare situation indeed.  Based on years of experience, Mr. Trubac has been able to devise content and pedagogy to help his students.

The core outcome for the physical education program at the school is:

Students will develop attitudes and knowledge to support life-long healthy living, based on experiential and active learning.

Being a ‘gym’ teacher, Mr. Trubac engages students in structured active learning every day using a diverse selection of such activities in support of this outcomoe.  Both team and individual work are used.  He is especially proud of basketball in the team dimension, and gymnastics in the individual dimension.  Being a professional, he uses experience to adjust the curriculum and pedagogy.

One of Mr. Trubac’s observations was that students tended to have a lot of trouble on the balance beam.  None of the falls were catastrophic (more of an embarrassment), and the resulting humor was appreciated by other students.  To help with this, Mr. Trubac began to include practice exercises for balancing on one foot and for walking along a painted side-line in they gym.  He noticed that this did, in fact, improve work on the balance beam; therefore, the current curriculum includes a significant amount of time on these balancing and walking skills — perhaps 20% of the course.

In basketball, Mr. Trubac observed right away that students had difficulty with the accuracy of their shooting.  Even from close to the basket, most students were getting more ‘rim’ than ‘net’.  The result here was also a matter of embarrassment and humor, though the students wanted to do better.  Since the simplest basketball shot is the free throw, Mr. Trubac started emphasizing practice from the free throw line.  He now uses about 10% of the course on free throws, and is pleased to note that students become quite proficient at that skill.  Sadly, there seems to be little transfer of this shooting skill to other attempts at getting a basket.

If we anticipate what Mr. Trubac will do in the future, he is very likely to emphasize practice on balance and free throws even more than currently.  They seem to work, so more is better.

Back to our world … we (mathematics educators and mathematicians) are “Mr. Trubac”.  The physical education class represents our curriculum, specifically the calculus and pre-calculus courses.  The balance exercises represent trigonometry … the free throw, algebraic manipulation.  Just like Mr. Trubac, we emphasize algebraic manipulation in pre-calculus and then are disappointed when students are not able to transfer those skills to calculus.  Just like Mr. Trubac, we notice that the work on trig does help students deal with trig functions in calculus, but that this seems to come at a high price in the prior work (30% to 40% of pre-calculus is ‘trig’).  [This is looking at “college algebra plus (trig or precalc)” as pre-calculus.]

The work we do in trigonometry is self-defeating.  First of all, we can’t decide “unit circle versus right triangle”, so our books have separate chapters on each (or we have separate books).  If we can’t integrate the two approaches in books and in our classes, this is a failure on our part.  Secondly, we expect students to practice and master ‘trig’ in very artificial ways — in other words, not connected to current usage of trig in our client disciplines.  We base our trig work on the ‘balance practice’ we have developed in calculus, where problems were created for the sole purpose of ‘showing’ why we need to know trig.  Think about all of memorizing many of us expect our students to do with trig … identities and formulas; only a few identities and formulas are critical.

We’ve been making curriculum changes based on anecdote — we notice that students struggle with something in calculus, so we say ‘more of that stuff in pre-calculus’.  This is a possibly reasonable approach.  An approach with at least equal validity: “We need to look at the nature of our work with this topic in pre-calculus in terms of effectiveness and quality of learning”.  Perhaps the problem we are observing is a reflection of the quality of our prior work with the student, and not an issue of needing to ‘cover more’.

Prior authors have used the phrase “lean and lively” to describe the goal of a curriculum focused on the core content with high effectiveness.  I see this is a good goal; in fact, this should be the emphasis of our profession including every one of us along with the organizations working with us (professional associations, publishers, etc).

We’ve been adding so many trivial and irrelevant ‘topics’ to our STEM curriculum in mathematics that the core content is submerged and partially hidden.  Some of that core content is so distorted by ‘extra stuff’ that few students discover what they are supposed to be learning the most.

Yes, we need significant trigonometry in calculus and most STEM fields.  What we are doing now?  Way too much, and not appropriate.

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Theory of Everything Presentation Page

Sharing is good, right?  Well, over the past 18 months, I have been working on a summative presentation on where college mathematics (first two years) is now and where we are heading.  The result of this long term project is available, sort-of like a webinar, on the page Theory of Everything.

I hope that you will take a look … perhaps there will be a thing or two in the presentation that will help you understand our work a little bit better.

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Modern Pre-Calculus Course

Good questions are very helpful.  At a presentation recently on the Theory of Everything (Theory of Everything presentation Oct2018) one of the participants in the session asked:

What do you mean by a ‘modern math course’?  What would be in it? How would it be different?

Being a good question, I could not give that good of an answer at the time.  However, it seemed like such a good question that I should make an attempt to provide a good answer.  The initial domain for the answer is “Pre-calculus”

In order to understand how inadequate the current courses are, we need to understand ‘modern’ thinking about learning and learning mathematics in particular.  Much of the traditional college algebra and pre-calculus experience is based on the presumption that working 10000 problems with some success will prepare you for a course with a higher conceptual basis and greater cognitive demands.  One reference for modern thinking about this comes from the book “Adding it Up”, which focuses on K-12 mathematics (https://www.nap.edu/login.php?record_id=9822) which provides an image to help us visualize the learning of mathematics:

We generally understand the names for these 5 strands, and we often talk about them with our colleagues.  However, the courses currently only address two strands directly (procedural fluency and (to a lesser extent) strategic competence).  Some of us use active learning strategies which (coincidentally) provide some support for the other strands.  A basic premise of a scientific approach is that “things we want do not happen when we want if we do not plan and act intentionally”.

A core problem in college mathematics is our separation of classroom practices from content decisions.  If your instructional practices encourage conceptual understanding within a course which does not directly state ‘conceptual understanding’ as a goal, there is a mis-match between instruction and content … and this will always result in reduced student outcomes because the assessments are likely driven by the learning outcomes.  So, here is the first standard for a modern pre-calculus course:

The learning outcomes in pre-calculus represent all 5 strands of proficiency, and instructional practices support the success of students in all 5 strands throughout the course.

In general, the learning outcomes for the Dana Center “Reasoning With Functions” reflect this type of approach.  Here are the outcomes:

• DCMP_Reasoning with Functions I_Overview and Learning Outcomes
• DCMP_Reasoning with Functions II_Overview and Learning Outcomes

As an example, a traditional pre-calculus course might list this learning outcome:

Represent and recognize functions

A modern course might list this learning outcome:

Create, use and interpret functions and use them to solve meaningful problems

Hopefully, this example of outcomes is helpful in understanding what I mean by a ‘modern mathematics course’ in pre-calculus.  There is a key feature not well represented by the outcomes above — the role of numeric methods within the course.  A modern mathematics course needs to provide a balance of symbolic and numeric methods for students, whether the course is calculus I or pre-calculus.  Some of this is addressed by the ‘overview’ portion of the documents above, though I would look for an explicit statement that the course will embed technology tools for both graphing (TI, Desmos, etc) and modeling (Mathematica, etc).

In a modern mathematics course, we would see evidence of all 5 strands of proficiency on each major assessment as well as the final exam.  A modern mathematics course removes the significant amounts of current courses that fail to meet professional standards for preparation … in this case, for calculus (where we have a sound basis for identifying the nature of the preparation (MAA Calculus Readiness test https://www.maa.org/press/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness and Characteristics of Successful Programs in College Calculus https://www.maa.org/programs-and-communities/curriculum%20resources/progress-through-calculus/cspcc-publications).  In other words, we make room in the pre-calculus course(s) by dropping the unnecessary topics and problems so we can focus on the goal … helping students get prepared for calculus.

The other aspect of a modern mathematics course deals with design principles.  It is generally not wise, and often is dangerous, to create a design with implementation not including a process to collect data on the effectiveness of the design.  Conferences and journals are often well stocked with reports of the first semester or year of a ‘new’ thing; that is not what I am talking about.  I am referring to a regular process of collecting data (aggregated and disaggregated) that will show meaningful trends in a process allowing for the assessment of corrections and modifications in the ‘treatment’.  This type of work is seldom ‘fun’ in the same way that a conference presentation is.  However, serving all of our students depends upon this continual examination of the basic question:

So, how are we doing NOW?

Moving from traditional to modern mathematics courses provides an opportunity to have all students experience good mathematics reflecting current tools and applications, and we might therefore conjecture that there will be an increase in majoring in the mathematical sciences.

Hopefully, this first approximation to a good answer to that good question was helpful.  I’d like to hear from you on that!

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