Every time I do a group quiz in class, it’s in the context of a “Participation Quiz.” For a description of a participation quiz, I’ll give a pointer to the excellent description and the discussion going on in the comments at Continuous Everywhere but Differentiable Nowhere. At some point, I’ll follow-up with a post here with some ideas I’m going to try out this coming year with regard to participation quizzes.
Some advice that was given to me when I first started teaching regular PreCalculus was to skip or deemphasize completing the square. However, I didn’t shy away from the topic because I felt it could be approached in a way that appealed to the fact that the average person enjoys solving puzzles. I plant the seeds by introducing “Avoid the Freshman Mistake Puzzles” as warmups or puzzles for those last few minutes of class, for about a week before we dive into the topic. This handout that I created is the most formal thing I have written up (we normally do this more informally in class).
I’ve also covered this topic in my Intermediate Algebra class (a post-Algebra 2 or Pre-PreCalculus class) with a population of students that have generally struggled with math in previous years. One of the struggles that we often have in teaching high school math is the need to teach or review fractions and how to do that with a population that is very fraction-phobic. This year, “completing the square” turned into an accidental ah-ha moment for many in terms of understanding fractions. (Working on my National Boards this year, I wish I would have had the camera rolling for this whole-group discussion…) Using words here, I know I’m not doing a good job of capturing the discussion, but basically the problem the class was wrestling with was how to find out what half of 3/8 was. Various students described their thinking and we ended up looking at a visual representation of 3/8 in a rectangle and cutting each shaded part in half and seeing that half of 3/8 was 3/16. Then the excellent generalization question came from the audience: “So can we always just double the denominator to cut a number in half?” Those are the types of questions that make a teacher proud of their students :). From that came the observation that that’s what happens when we multiply horizontally using fractions. Other ideas that came up in the discussion included the way we found half of 4/9 in a previous problem and the connection between thinking of it as four “ninths” and then half of it was just two “ninths,” versus going to 4/18 and then reducing, and why the denominator changes when multiplying but not adding fractions. And all this on a 5th period on a Friday afternoon, right after lunch and about an hour before the weekend.
Just thinking about how far this class has come from September to now makes me so proud of them! This is precisely the sort of thing I was refering to in my previous post about beginning-of-the-year review. Review in context, in this case not even a “real-world” context but a context of an “advanced” algebra move, is much more meaningful rather than just drilling a bunch of fraction addition and multiplication problems.
Right now, the impetus for me cleaning out my drafts folder of posts I’ve begun is sitting in a West Seattle coffee shop procrastinating from doing more National Boards writing… I was genuinely shocked to see the last time I’d posted anything was mid-August. I’m humbled by the fact that so many other math blogger folks turn out quality, thought-provoking posts on a regular basis amidst the activity and bustle of being a teacher. So along with the “how do they do it?”, this post is also a “thank you!” to all the other bloggers I read on a regular basis for taking the time to share your ideas and your thinking, and giving the rest of us food for thought.
In elementary school, I used to get excited right around when we reached p.100 of our math books, because that’s when we would really start doing stuff. Those triple digits meant that lessons would no longer be about practicing addition and subtraction, or rehashing things we were supposed to have learned the year before. Maybe that first month or so of the school year taught me patience, otherwise, I would have gone mad. Unfortunately, as a teacher, I’d fallen into the same trap of needing to do that all-important review as the first unit of the school year. What better way to kill the potential energy built up over the summer and bring whatever momentum had built up to a grinding halt? In retrospect, it seems that “review” seems to make students who already got it last year either check out or get lulled into a sense of complacency. For students who didn’t “get it” last year, instead of starting with a clean slate, it instead drives home the message that this year will be more of the same.
This is not to say that review isn’t important or necessary, but what makes the difference is the way that it is done. This year, a colleague and I rearranged the sequence of topics in the PreCalculus courses that we taught (different schools in the same district). Instead of first semester being largely a rehash of Algebra 2, we started with Trigonometry first. This is the fifth year that I’ve been fortunate enough to teach PreCalculus, and I’m kicking myself for not coming to this realization sooner. Even though we’re starting with a topic that’s new for most students, we’re still able to secretly review solving equations, working with exponents, and working with fractions, all in the context of doing trig. Instead of two separate groups (those who got it last year and those who didn’t), everyone was learning something new, on equal footing, and with a clean slate. I don’t think that there’s anything intrinsically different about the makeup of students this year, but with the new approach, we’ve been able to keep the momentum going from the beginning of the school year when students come back from summer, eager to learn something new. We’ve been able to cover more material, go into more depth, and maintain a healthy pace, especially compared to previous years.
I’m slated to teach Algebra 1 next year and am hoping that this idea is something the team will be willing to run with, or at least consider.
I have a terrible memory and so acronyms are something that I rely on… “Every Good Boy Does Fine”, “FACE”, “Great Big Dogs Fight Alligators”, and “All Cows Eat Grass” are some that I live by. Even “SOH CAH TOA” is something that’s helped me for nearly 20 years. Acronyms and shortcuts are great for brute-force memorization. However, I’m putting this post out there to maybe plead for help in getting my anger management in check. I hate, really really hate, perhaps even despise the acronym “All Seniors Take Calculus.” I’m not sure why it prokes such a strong negative reaction within me – I think it’s because it really strips away meaning at the cost of coming up with a cute saying. After all, if one understands that (and why) the x-coordinate on the unit circle is the cosine of the angle, the y-coordinate on the unit circle is the sine of the angle, and y/x is the tangent of the angle, one doesn’t need the overhead of “All Seniors Take Calculus.” Plus, how does that saying help deal with figuring out that the cosine of 180 degrees is negative, for example? Help me not be so hateful…
I came across the webcasts for AP Computer Science 10 at UC Berkeley, which seems to be a pilot course for Computer Science Principles (http://csprinciples.org/). Althought it’s a non-major course, I think it’s an excellent idea. I suspect that many students end up majoring in the subject because of courses like this. Speaking from personal experience, I did end up majoring in computer science after taking the non-major introductory course (CS 3 at the time). I think it would be wonderful for mathematics to be able to better publicize and offer a similar course for math. Many colleges offer such a course under titles such as “Math for Liberal Arts” or “Math in Society.” Different books aimed at such courses are also published, such as Key Curriculum Press’ “The Heart of Mathematics”, or COMAP’s “For All Practical Purposes” (http://www.comap.com/product/textbooks/index.html). A course like this could provide a broader perspecive of applications of math and at least for me, what’s applealing about this approach is the logic and problem solving involved with more rigorous mathematics than just arithmetic.
I really can’t think of a good reason why synthetic division is still in the Algebra 2/PreCalculus curriculum. When I was in high school (at my HS, I was in the last class not to use graphing calculators), synthetic division was a handy shortcut because we had to do tons (often enough for a whole page for one problem) of divisions to test potential zeros for higher-order polynomials. However, with technology as it is, I can think of better ways to spend the 2-3 days it would take to cover synthetic division and finding rational zeros by hand.
I do still teach polynomial division, because it is useful for finding polynomial asymptotes of “improper” rational functions. But for the synthetic division shortcut, unless I’m doing tons and tons of polynomial division problems (and only the case where we’re dividing by linear factors at that), I really can’t think of a good reason to take the time to do it. Plus, I have a really bad memory and always have trouble remembering the proper sign for the number on the outside of the synthetic division box (r when the divisor is x-r…).
At my school, synthetic division is taught in Algebra 2, so I’ve been able to use that reason to skip it in PreCalculus. However, our local community college lists it as a topic that is taught and covered on the final in PreCalculus. So, if I have students who opt to do the college in the high school program, I do need to cover it in PreCalculus…
Please let me know if there’s something I’m missing, or why you do or don’t do synthetic division.