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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