The thing that’s difficult about graphing trig functions for the first time is that regular old graph paper just isn’t up to the job. When we have points like and to plot, you need something that is suited to the task, and the numbers involved. I have met teachers that say they just have students plot the multiples of , and then tell the students to trust them, it’s not really a zig-zag. And it seems to me that having them spend time plotting points in order to develop a *wrong* intuition is an amazingly bad use of time. You could just use the calculator and it would be better. But in order to make the time spent plotting points valuable you have to be able to see the shape of the curve from the points are plotted. So you need *y*-values of and and *x*-values that are multiples of and .

So behold:

Of course, you need a key for this:

You also need to call the students’ attention to how the *x*-gridlines aren’t evenly spaced, and that the unlabeled ones are multiples of and . But after that…

you have something that can allow student to connect their knowledge of the trig functions in the context of the unit circle to the traditional graphing context.

I don’t feel like wrestling with the code tags, so you can check out the latex source here, at sharelatex. (You don’t need a ShareLaTeX account to access that link, but if you’d like to sign up for one, please use this referral link, which lets me earn points toward referral bonuses. Thanks!) That document has two types of graphs– one for sine and cosine, and another for tangent and cotangent. There’s a second set with the functions graphed as well, which I don’t give to students, but it’s useful to have to point at in later classes.

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latex code on sharelatex.

latex code:

\documentclass[border=4pt]{standalone} \usepackage{amsmath,mathpazo,gensymb} \usepackage{tikz} \begin{document} \begin{tikzpicture} [>=stealth, scale=2.5, % Toggle commenting on the next four lines for the completed unit circle: angle/.style={draw,text=white,fill=white,minimum height=1cm, minimum width=1cm}, point/.style={white}, %angle/.style={fill=white}, %point/.style={}, ] \draw [white] (-3.6,-3.6) rectangle (3.6,3.6); \draw [thick,fill=white] (0,0) circle (3cm); \draw [thick, ] (-3.3,0) -- (3.3,0); \draw [thick, ] (0,-3.3) -- (0,3.3); \draw (0,0) -- node [angle] {$\frac{\pi}{2}$} (90:3) node[point, above right] {$ \left(0,1\right) $}; \draw (0,0) -- node [angle] {\footnotesize{$\pi$}} (180:3) node[point, above left] {$ \left(-1,0\right) $}; \draw (0,0) -- node [angle] {$\frac{3\pi}{2}$} (270:3) node[point, below right] {$ \left(0,-1\right) $}; \draw (0,0) -- node [angle] {\footnotesize{$2\pi$}} (0:3) node[point, above right] {$ \left(1,0\right) $}; \draw (0,0) -- node [angle] {$\frac{\pi}{4}$} (45:3) node [point, above right] {$ \left( \frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [angle] {$\frac{3\pi}{4}$} (135:3) node [point, above left] {$ \left( -\frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [angle] {$\frac{5\pi}{4}$} (225:3) node [point, below left] {$ \left( -\frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [angle] {$\frac{7\pi}{4}$} (315:3) node [point, below right] {$ \left( \frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{\pi}{6}$} (30:3) node [point, above right] {$ \left( \frac{\sqrt3}{2} , \frac{1}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{\pi}{3}$} (60:3) node [point, above right] {$ \left( \frac{1}{2} , \frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{2\pi}{3}$} (120:3) node [point, above left] {$ \left( -\frac{1}{2} , \frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{5\pi}{6}$} (150:3) node [point, above left] {$ \left(- \frac{\sqrt3}{2} , \frac{1}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{7\pi}{6}$} (210:3) node [point, below left] {$ \left(- \frac{\sqrt3}{2} , -\frac{1}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{4\pi}{3}$} (240:3) node [point, below left] {$ \left( -\frac{1}{2} , -\frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{5\pi}{3}$} (300:3) node [point, below right] {$ \left( \frac{1}{2} , -\frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{11\pi}{6}$} (330:3) node [point, below right] {$ \left( \frac{\sqrt3}{2} , -\frac{1}{2} \right) $}; \foreach \n in {1,2,3,4} \foreach \t in {0,30,45,60} \fill (\n*90+\t:3) circle (0.03cm); \foreach \t in {30,45,60, 120,135,150, 210,225,240, 300,315,330} \node [font=\tiny, fill=white,inner sep=1pt] at (\t:.75) {$ \t\degree $}; \end{tikzpicture} \end{document}

latex code on sharelatex

Get the latex code here.

\documentclass[border=4pt]{standalone} \usepackage{amsmath,mathpazo,gensymb} \usepackage{tikz} \begin{document} \begin{tikzpicture} [>=stealth, scale=2.5, % Toggle commenting on the next four lines for the completed unit circle: %angle/.style={draw,text=white,fill=white,minimum height=1cm, minimum width=1cm}, %point/.style={white}, angle/.style={fill=white}, point/.style={}, ] \draw [white] (-3.6,-3.6) rectangle (3.6,3.6); \draw [thick,fill=white] (0,0) circle (3cm); \draw [thick, ] (-3.3,0) -- (3.3,0); \draw [thick, ] (0,-3.3) -- (0,3.3); \draw (0,0) -- node [angle] {$\frac{\pi}{2}$} (90:3) node[point, above right] {$ \left(0,1\right) $}; \draw (0,0) -- node [angle] {\footnotesize{$\pi$}} (180:3) node[point, above left] {$ \left(-1,0\right) $}; \draw (0,0) -- node [angle] {$\frac{3\pi}{2}$} (270:3) node[point, below right] {$ \left(0,-1\right) $}; \draw (0,0) -- node [angle] {\footnotesize{$2\pi$}} (0:3) node[point, above right] {$ \left(1,0\right) $}; \draw (0,0) -- node [angle] {$\frac{\pi}{4}$} (45:3) node [point, above right] {$ \left( \frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [angle] {$\frac{3\pi}{4}$} (135:3) node [point, above left] {$ \left( -\frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [angle] {$\frac{5\pi}{4}$} (225:3) node [point, below left] {$ \left( -\frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [angle] {$\frac{7\pi}{4}$} (315:3) node [point, below right] {$ \left( \frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{\pi}{6}$} (30:3) node [point, above right] {$ \left( \frac{\sqrt3}{2} , \frac{1}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{\pi}{3}$} (60:3) node [point, above right] {$ \left( \frac{1}{2} , \frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{2\pi}{3}$} (120:3) node [point, above left] {$ \left( -\frac{1}{2} , \frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{5\pi}{6}$} (150:3) node [point, above left] {$ \left(- \frac{\sqrt3}{2} , \frac{1}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{7\pi}{6}$} (210:3) node [point, below left] {$ \left(- \frac{\sqrt3}{2} , -\frac{1}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{4\pi}{3}$} (240:3) node [point, below left] {$ \left( -\frac{1}{2} , -\frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{5\pi}{3}$} (300:3) node [point, below right] {$ \left( \frac{1}{2} , -\frac{\sqrt3}{2} \right) $}; \draw (0,0) -- node [near end, angle] {$\frac{11\pi}{6}$} (330:3) node [point, below right] {$ \left( \frac{\sqrt3}{2} , -\frac{1}{2} \right) $}; \foreach \n in {1,2,3,4} \foreach \t in {0,30,45,60} \fill (\n*90+\t:3) circle (0.03cm); % \foreach \t in {30,45,60, 120,135,150, 210,225,240, 300,315,330} % \node [font=\tiny, fill=white,inner sep=1pt] at (\t:.75) {$ \t\degree $}; \end{tikzpicture} \end{document}

As a sidenote, ~~I am really pleased that WordPress’s code tags seem to have improved greatly since the last time I tried to use them.~~ Never mind. It’s better than it was, but it’s changing < and > to `<`

and `>`

. And it took the away entirely. For the actual code, you can check out the versions on sharelatex, as well as this document, that has them both embedded in it. (You don’t need a ShareLaTeX account to access these links, but if you’d like to sign up for one, please use this referral link, which lets me earn points toward referral bonuses. Thanks!)

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I can’t believe that I didn’t even realize that I didn’t have that on there. Now, is it its own category, or does it fit under something else? I’m leaning towards its own category, but then I have to break it down. What are the qualities of perseverance?

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I had five students transfer into the class after they heard that there was going to be a huge project worth 50% of the course grade, and that exams were going to be worth a total of 35%. I spoke to each of them, and made sure they were aware that this didn’t mean the class would be easier, and they would probably have to do more work rather than less. They still joined our class. (We also had two students who transferred out, one because he didn’t have the prerequisite).

So I have a lovely class of 17 students, most of whom seem enthusiastic about the blogging project, and at least for now, none of whom are resisting the idea of this nontraditional math assessment.

The blog project will be graded three times, at week 5, week 10, and week 15. I’m calling these “checkpoints” for lack of a better name. For the first checkpoint, students have to schedule a meeting with me to go over their blog. They have to have six posts by then, and they’ll have until the Monday of week 6 to make changes to improve their grade. So next week, I’ve got a bunch of student meetings to look forward to.

The grading scheme for this class is:

Blog Project Checkpoint 1 10%

Blog Project Checkpoint 2 15%

Blog Project Checkpoint 3 25%

Daily Grade Average 15%

Exam Average 25%

Final Exam 10%

One of the things that surprised me with the first time I did the blog project was how much students objected to getting letter grades for the project. So numerical grades it is. Not that I feel obligated to give them everything they prefer, but when I’m asking them to come this far out of their comfort zone, I can keep the superficials familiar.

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The math department was in an especially desperate situation this semester– one week before the semester started, there was somehow a surge of unexpected new students, and 800 people were on a waiting list wanting to take college algebra. I really don’t understand how something like that happens, but then I don’t know much about how that sort of administrative thing works. The university was frantically trying to get us more classroom space, and the department was moving people around in an effort to create more College Algebra sections.

At the same time, for the first time since 2008, I had a class whose initial registration count was under 30. Way under. It was originally 14 students in one of my sections of Calculus for Life Science.

I cannot emphasize enough how bizarre that is. I haven’t had a class for the past two years that started with less than 40. Usually it’s 48, which is the fire-marshal limit for our classrooms. And since this happened at the same time that the department was scrambling to get more college algebra, I just kept my head down, and hoped that they wouldn’t notice me and kill my class.

And now I’ve got a class of 17, for the first time since I was a grad student teaching developmental math. And it’s glorious. It’s just… amazing. I’m doing the blog project in that class, and I can keep up with everything that everyone posts. When we do group activities, I can actually check in on each group.

I know I complain a lot about class sizes, but there’s been this little cynical voice in the back of my head saying that it’s probably not the class sizes, I’m just burning out on teaching. But no! It really is the class sizes. This class comes at the end of a very long day, and I have more energy for it than I do for my 10 AM precal that has 48 students in it.

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For years I’ve been telling myself that I needed to make my expectations for groupwork more explicit. I’ve told them what not to do: groupwork is not four people working silently side by side, it’s not dividing up the problems and sharing answers, it’s not one person doing everything and not taking input from their teammates. But that’s not the same as saying what it actually is.

But that sort of thing takes a lot of time to develop well, and being pulled in a hundred different directions, I just never took the time to sit down and do it. But when Cheesemonkey gave me such a great starting point, all it took was a little editing to get this:

## Mathematical Collaboration Expectations

## Active Inclusion

You helped the group to develop its shared mathematical thinking by:

- allowing others adequate time to express their own thinking
- demonstrating patience when other group members have difficulty putting their ideas into words
- making sure that everyone understands why or how a piece of shared thinking or reasoning is so
## Individual Participation

You made your own personal contributions to developing the group’s shared mathematical thinking by:

- developing your own unique insights
- sharing your thinking and ideas respectfully with the group
- encouraging and supporting others as they speak their ideas,

confusion, or questions- managing your desire to do more than your fair share of the talking
## Deep Listening

You developed your openness as a collaborator by:

- listening attentively to your teammates
- asking clarifying questions
- building on others’ ideas
## Exploratory Talk

You developed your voice as a math learner and as a member of a learning group by:

- noticing and wondering about a problem
- extracting information and forming questions
- trying a variety of approaches
## Reflective Talk

You developed a sense of self-awareness as a math learner by:

- noticing out loud other learners’ insights, strategies, or contributions that helped to move the group’s learning forward
- noticing what you personally did well as a member of the group
- noticing what you personally need to keep working on to become a more effective member of a mathematical learning group.

I didn’t want a rubric, so I stripped off all the points, and just made it a set of expectations. If you’re interested in the latex version, you can find it on my sharing handouts page (it’s at the bottom).

And this is my absolute last blog post until I get that stupid syllabus done.

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And so we try to cope.

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`\documentclass[12pt]{article}`

\usepackage[margin=.75in]{geometry}

\usepackage{exsheets,mathpazo,titlesec}

\usepackage{amsmath}

\setlength\parindent{0in}

\titleformat{\section}[block]{\large\bfseries\filcenter}{}{1em}{}

\setcounter{secnumdepth}{0}

\newlength{\spacing}

\setlength{\spacing}{1in plus 1fil}

\settasks{

counter-format = tsk. ,

item-indent = 2em ,

label-width = 2 em,

resume = true,

after-item-skip = \spacing,

after-skip = \spacing

}

\newcommand{\resetspacing}[1]{

\setlength{\spacing}{#1 plus 1fil}

\settasks{after-item-skip = \spacing,after-skip=\spacing}

}

\begin{document}

\section{PracticeHandoutTitle}

\begin{tasks}(2)

\task

\end{tasks}

\end{document}

This is going to make it a lot easier for me to get things to look like I wanted them to. Tables never really lined up right if the cells had text of different heights. And plus, using tables is just a pain. If you’re interested, here’s how my Pre-Cal Review of Prerequisite Knowledge looks now. (ShareLaTeX doesn’t have the exsheets package yet, so if you really want to see it, you’ll have to download it and compile it yourself. I put in a request, though!)

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At the time, I didn’t have a schedule yet, but all of the sections of the course I was designing it for were assigned to other faculty members. At first (before the workshop), I was really disappointed about having to wait to do the project I was re-designing, but during the workshop I realized I had a lot of work to do, and maybe having a few months to do things slowly was not a bad idea. I planned on asking if I could teach one section of Calculus I for Life Sciences in the Spring semester. (Spring is usually significantly less crazy than Fall, and they will grant a request like that if they can.) But then the schedule got rearranged, and I ended up with two sections of the course.

I decided very definitively last time around that there was no way that I could assess the writing of 90 students, do everything else that is required to teach mathematics well, and maintain my sanity, so only one section will be doing the blog project. Which is the perfect opportunity to evaluate whether writing about math really does help their conceptual understanding. I’m still looking for the best way to do that.

Last time, I told students they had to post twice a week, and they could post anything related to math or learning math, although I wanted them to focus on the material we are learning in class. That will still be the case, but I am defining several types of posts that they can write, and I may require that at least one of certain kinds is required. So many students had trouble just getting started last time. The type of post that I got help with in the workshop is one I’m calling an “exercise report”. This is basically making up a problem similar to a homework exercise and writing about the strategy (and why that particular strategy was chosen) and showing how to solve the problem. I am still more interested in seeing students reflections on the concepts of calculus then their exercise reports, but I think that having a nice well-defined type of post with examples and a rubric will give students with less confidence an entry point. And writing mathematically is definitely one of the standards I want to assess my students on.

I’m planning on writing up rubrics and examples for “application reports” as well, but that probably won’t happen before the semester begins. The other types of posts will just get descriptions for now.

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I got my MS in Math from Texas State in 2007. I also got married right at the same time. I was offered the lecturer job, and originally planned to teach for one year while I figured out which PhD program (in math ed) I wanted to apply to. At that time, my husband was getting several calls from headhunters every week, so we felt confident in our ability to move anywhere in the country. But then 2008 came, and the economy broke, and we couldn’t really justify moving and leaving his really excellent job just then. And now it’s 2013, and we have a baby (a delightful one!) and I’m starting to think about my career in a wistful kind of way.

I have some serious frustrations with my current working situation. Class sizes and lack of advance notice of what classes I’ll be teaching are the main problems. Of course, since I only have a master’s degree, I’m very limited as to where I can teach in higher ed, and those problems will probably exist anyplace that would hire me without a doctorate. And so I’m seriously considering several questions:

- Do I want to continue teaching in higher ed?
- Do I want to put in the time and energy for a PhD? The money?
- Do I want to to do research? What kind and how much?
- What is my ideal working environment? What would it take to get that?

I’m gearing up for the semester now, but these thoughts have been bouncing around my head all summer. For most of the summer (and previously in the spring), I’ve been shifting around on the pessimistic end of the scale on how I feel about the future of higher education in this county (de-funding, hyper-focus on job preparation, framing education as simply the acquisition of facts), but the technology integration workshop I attended really put me in a more optimistic place.

More on that later…

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