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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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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Theory of Everything … A Presentation on College Mathematics

Presentation done at the MichMATYC conference on October 13, 2018 at Kalamazoo Valley Community College … with a goal of understanding everything about college mathematics in the first two years.

Presentation:  Theory of Everything presentation Oct2018

References (handout): References Theory of Everything Oct2018

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AMATYC Standards … Where is the Math in IMPACT?

What is the role of professional standards?  Do they address instructional practices only?  Do we need or want guidance on curricular issues?

I am hoping that you consider these questions routinely.  Within college mathematics, both AMATYC and MAA have released standards over the years … with the MAA standards strongly tilted towards curricular concerns.  The AMATYC standards (in the original “Crossroads”), dealt with both instructional practices and curricular issues (see https://amatyc.site-ym.com/page/GuidelineCrossroads/Crossroads-in-Mathematics.htm), though the curricular guidance in that document was primarily a validation of the status quo.

Since that time, AMATYC has released two more standards documents — Beyond Crossroads (2006, https://amatyc.site-ym.com/page/GuidelineCrossroads/Crossroads-in-Mathematics.htm) and IMPACT (2018, https://amatyc.site-ym.com/mpage/IMPACT).  The 2006 standards focused on a process, the improvement cycle, which was to be applied to both instructional practices and the curriculum.

So, what happened with IMPACT?  Curricular issues (the mathematics) are not addressed at all.  The only math in the document arises coincidentally as instructional practices are described.  What does this exclusive focus on instruction mean?  Does it suggest that AMATYC does not see any need to update the curriculum?  No, I don’t believe so.  Is there a lack of consensus?  Very likely, but that does not prevent the development of standards on curriculum.

I think this missed opportunity was the result of a focus on perceptions of what members and faculty in general WANTED to see.  I see this as a confusion between want (a comfortable condition) and need (a challenged condition).  If we judge ‘need’ by what is popular at conferences, the need is certainly for instructional practices — especially those which can be implemented now, regardless of other instruction and regardless of curriculum.

However, leadership involves also judging what people need even when this need is not measured by session attendance.  In some ways, judging this need involves measuring need indirectly — such as the actions of state legislatures and policy makers to mandate curricular changes.  We certainly enjoy discussions about effective and cool teaching methods; it is not happy or comfortable to deal with curricular challenges.

By being even more silent on curricular issues, the newest AMATYC standards leave us (the professionals) with no guidance and no support when faced with external forces and directives.

Now, I need to disclose that I was deeply involved with this newest AMATYC standards project.  From when it started in 2014 as a vague ‘update Beyond Crossroads’ until 2017, I was on the planning team.   In late 2016, I even wrote a chapter for the new standards on faculty issues — combining curriculum and instruction.  You will not see a single word from this chapter in the final document, and that (by itself) does not bother me.  For personal reasons, I had to stop my involvement with the project last year so I had to ‘let go’ of the work.  Other people took the responsibility of creating the document, so they got the decisions.

I would not have minded if “IMPACT” had material having no connection to the chapter I wrote — as long as it supported faculty and departments in all ways.  Instead, we have a very nice catalog of good ideas for teaching … with no guidance on ‘what’ to teach.

The MAA has continued to issue guidance (sort of standards) on curricular issues; I use the CUPM 2015 material far more than I use the AMATYC standards — even though I had been a member of AMATYC for over 25 years and a member of MAA for one.

Our curriculum is under pressure to change, and our curriculum needs to change.  Much of what we teach has remained constant for over half a century, while the needs of our students and our client disciplines have changed dramatically.  If we do not have professional standards from AMATYC, we will have to update the curriculum based on MAA standards — which focus on calculus and ‘beyond’.  We are dealing with a period of change without any professional standards for the curriculum in the first two years of college mathematics.

The question becomes — how can we support each other?

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