Math Hombre

 I'm teaching fully remote this year and have been struggling with many aspects of it. Like one does.

My courses were scheduled online synchronous, hoping for ways to get these preservice teachers working with kids... which hasn't happened yet. We do an hour of synchronous time, then there's asynchronous time for the remaining, with a set assignment, and then homework for a time, usually with a fair amount of choice. Reading (NCTM journals, teacher blogs, ...) , video, problems, etc.

Today my class was pretty quiet. We do Zoom, using Jamboard for an interactive whiteboard. Often there are problems or discussions that start in breakout groups, randomized every few weeks, then come back for whole group discussion. Today, on our last day of trigonometry, we were tackling identities. I tried something new, which was going old school. We stayed whole group, I did an example, and then asked them to work individually and contribute. 

I usually start out with a Jamboard data prompt, sometimes my question and sometimes theirs. (I share these pretty frequently on Twitter.) Today was their question, and I tried having a parallel chat, where I asked them 'what's the scariest math problem or course?'

Not everybody, but good participation. And, whoa, integration. Would not have called that.

The visual prompt was my favorite trigonometry diagram.

We talked a bit about the history, the mistranslation of sine, and a recurring theme for this unit, the connections between geometry and algebra.

We then looked at just one section together to get at the similar triangles we would be using. 

We started with the triangle in the center. I shared how I learned from a student in the long ago to pull them apart and reorient them. We worked through three questions, where each time I left them more to fill in in the proportion. They had a minute to think by themselves, then we discussed. But only two students were willing to take the mic and share what they're thinking.

So I asked them in the chat publicly to share how comfortable they were with these kind of problems, 0 = don't know how to start, 5 = confident they could solve them. I also asked them to privately message me a word about their participation. We've been using WAIT - Why Am I Talking or Why Aren't I Talking (from Jennifer White's great Desmos norm activity) as a framework for this.

Sorted (public): 

1

3  3  3  3  3  3
3.5
4  4  4  4  4  4
4.5
5  5

Timeline (private):

• Wishing I was on fall break
• I think I am confused on the proportions. I'm not seeing where to get those. 
• I guess I’m not uncomfortable participating in discussion, but I didn’t, and haven’t been lately, because I am really burnt out and need to just focus on listening
• I am at home and can’t really turn on my mic
• I'm just confused about why we are talking about sin, cos, and tan as side lengths, and not ratios of the sides.
• honestly I am fine with talking sometimes I knew my answer was wrong however but after switching the triangles around I really understood it
• I found these problems kind of confusing. I was listening to the others to see if I could understand it from there and that did help.
• Trig for me has always been a hard subject, it took me a lot longer to realize how all 3 triangles are related and then create the ratios
• I feel ok with talking, but there are times where I would say something ang you're talking but I can't raise my hand, I'm not sure how. So I never get to say my thought.
• I was not completely understanding the triangles
• I felt like I could follow along well enough, but couldn't really piece it together. I'm still half asleep, I'm not a morning person
• Very busy week with other classes and I’m trying to keep up with them too
• I've always had a hard time with trigonometry so I don't have a lot of confidence in answering those types of questions in front of class
• I could follow what was happening and what was being discussed, but I was struggling to answer the questions and explain the thinking behind them. But after they were explained, they make sense
• I would have felt comfortable speaking aloud because I understand this material 
• I don’t like talking in front of the whole group because I’m afraid I’m going to be wrong. This stuff is sort of confusing, and I’m not terribly confident in my answer.
• I wanted to observe what other students come up with and have others the opportunity to do trig

(Lost the source of this!)

I got to give a whiz-bang 60 minute (with an option for 30 extra minutes) intro to GeoGebra at the New Tech network conference this week. 50 plus tech-savvy teachers... so it was good. I am always worried that people expect me to tell them about GeoGebra for an hour, when purpose is to get them started using it on the spot, in ways that make sense of their potential use. (Note that if you are in driving distance, I am more than happy to come do this at your school. No GeoGebra lectures, however.)

Purposes. So what are the ways that people make use of it? Oh, let me count them:

1. World's best graphing calculator. (A little weak on statistics and CAS, but that's improving quickly.) For you and your students. For algebra, calculus, or geometry.
2. Mathematical image editor. For uses in reports, papers, handouts or assessments.
3. Demonstration tool. Project a great visualization on your screen to show to or discuss with students.
4. Focused mathematical activity for students.
5. Open-ended inquiry tool. Pose a question and let students investigate.

Requirements. The AMAZING thing about this tool is that with version 4.0, all of these are accessible to teachers in that 60-90 minute start up.

1. Open the program, start typing equations on the input bar.
2. Needs some quick familiarity with the tool bar to make your image, then File > Export > Graphics View As A Picture.
3. GeoGebraTube. If you have not looked at this, you are missing out. 14,000 sketches and counting; free accounts, search, likes, tagging and you can collect them in teacher mode or show collections in student friendly mode. This is why you need minimal expertise to start using the program deeply. If you can run YouTube and you are a teacher, you can do this.
4. See #3. 
5. Students today are geared for this kind of tool. You give them access, they'll figure things out about it that I don't know.

Really, any training beyond that first 90 min. is about if you want to become proficient in number 5, or if you want to be designing your own activities. Some teachers are doing that anyway by the end of an hour, most by the end of a half day.  Once you start using it, there's a big danger of being sucked in by the possibilities of what you can make. The power of dynamic examples is as much greater than static electronic images as static electronic images were than hand drawn. (My opinion. No research. Actually yes research, but they would never quantify so crazily.)

After my session, I got to go to Geoff Krall's (@emergentmath) session on formative assessment. He was using the MARS  MAP (Mathematics Assessment Resource Service - Mathematics Assessment Project) materials.  In particular, he used the Ferris Wheel lesson to get us collaborating and specific in discussion.

As we discussed, it really got me thinking about how I would use the task early on. It would make a good project or assessment, I think, but what about as an inquiry? The basic problem was to make a symbolic model [find a, b and c for a+b*cos(ct)] for the height of a car on a specific Ferris wheel. Then there was a card sort which got students comparing context, equation and graphs.

I've given many explorations before that got students experimenting with parameters to see the effect on graphs, but I love the idea of tying it to a context. That doubles up on the intuition they can apply - physical and visual. If the students had access to that, they might be able to do enough trials to start to generalize. Even without much trigonometry understanding, it's a nice context for graph transformations. For me, these kind of thoughts now lead to GeoGebra. I made a quick sketch, with the Ferris wheel in a 2nd graphics window, and was delighted to find that even the 2nd window worked on GeoGebraTube.

But since then I thought it would be worthwhile to develop a bit more. Both to familiarize myself with using the 2nd graphics window and to make the single model into a reusable activity. I knew I wanted to have either a customizable or random Ferris wheel, some animation of the situation and a way for the students to enter the equation.

That bore some thought: sliders, input boxes for parameters or an input box for the function?  Sliders are best for seeing continuously linked examples, but can make a problem like this too easy! The input boxes for the parameters helped support the idea of structure, require some thinking before making a new guess, and don't require as much typing as entering the whole function. Plus you can isolate one parameter and just adjust that. That might be a positive or negative.  It feels like a support for learners early in this, by encouraging them to focus on one parameter at a time.

The trick to working on two graphics views is the advanced tab of object properties. You can use any tools from the main window. Just select the tool and then use it in the 2nd graphics window. The objects you make there show up in the algebra view. But when you edit things, or create them in the input bar, they migrate or appear in the first graphics window. The solution is in the object properties, advanced tab; just check the box you need. Note that you can have something appear in both ... there just has to be a cool use of that.

I don't think there's anything else too tricky about the sketch. I used the Function[ , , ] to get the modeled equation to move with the tracing point, the ZoomIn[1] command on the button to clear traces, and the UpdateConstruction[] command to reset the Ferris wheel dimensions. (I had slick graphics window dimensions based on the Ferris wheel, but then the ZoomIn[1] command doesn't work. Ultimately I thought it was better to see the Ferris wheel changing sizes anyway.)




 The sketch is on GeoGebraTube: Teacher page for download or Student worksheet for in browser use. You have to click in the main window to get the animation button to show.

Trig Problem 2

For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.



The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.




Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

Have to love easy-to-draw memes.

Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.

Rebecca Walker and I modeled a lesson for our secondary student teachers on trigonometric equations, based on the first chapter of the Precalculus book from the very interesting CME Project curriculum.  While it has some interesting applications, this curriculum really does a good job of letting the mathematics be the context and addressing mathematical habits of mind.  The lead developer is Al Cuoco, who has a great history of interesting math and math ed work.

The lesson is a bit of a stretch, because we're just touching on one section, using a bit of information from three or four.  We did unit planning one week, lesson planning the next week, and finally the lesson.  The TAs read The Teaching Gap, so then we connected it to the idea of lesson study, and a discussion both about how to revise this lesson, and why lesson study might work as professional development.

We have two Geogebra sketches to help with visualization.

 As a sketch or a webpage.  This sketch supports visualizing sine and cosine with unit circle connections.
As a sketch or a webpage.  This sketch lets you invert trig functions using the Unit Circle representation.













This is my first attempt at a WCYDWT.  When I was making these sketches (don't worry, I disinfected them before posting) I had a bad cold, so was constantly reheating my tea.  Watching it go round and round.  Thinking, "so when do we know a position and want to know the angle, with possible multiplicities...hey, wait a second."  If I was using this, I think I would start with the video, and use that to motivate the idea of solving for information based on the circle position, as well as how periodicity relates to multiple solutions.

This has to be the world's most boring video.  Enjoy!


Here's a slightly more polished version of the handout we used with the sketches.  There was some discusssion with the student teachers as to whether the inverse trig or the algebraic solutions part should come first.  I think they could be switched, depending on what you wanted to emphasize with the students and how strong their trig background is.  Also, the handout is written as if the teacher is demonstrating with the computer, which is what we wanted to model for them, (no lab is no reason to no have technology) but the ideal would be to have the students have access to the sketches.


Solving Trig Equations