Two Final Problems

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.

Extrinsic Motivation

It's been a weird semester and this is going to be a weird post. I'm trying to work through how I made a mess of it, and doing that publicly is odd... but it fits with how this blog has helped me develop as a professional.  I shared my portfolio at the beginning of the semester, which I put together to make a case for promotion to full professor, and I've been turned down for that by the college personnel committee and our dean, after a positive but slightly contentious department vote.  The reason for for the no was no peer-reviewed publications.  Serves me right, many say, applying for full without any.

I became interested in the teaching of math in a serious way when I got the chance to make over the math for elementary teachers at Penn State. We changed texts, and my friend Sue Feeley gave me some excellent reading recommendations in response to 'what do these teachers need to know, anyway?'  I got more interested when I realized how amazing and challenging it was to think about, and just fun besides. I was going to quit graduate school and go teach high school, but my advisor correctly urged me to finish. The nail in the math research coffin for me was realizing just how few people would care about the research I was doing.

At Grand Valley they hired me to be a math educator, in what I still consider to be a minor miracle. What were they thinking? I didn't think too much of publishing then because I was really just learning the field, and then I just never got around to it. I was also changing (hopefully growing) so fast that it felt weird putting something into print - who knew if I was still going to be doing that in a year or two? Plus work in the schools with students and with teachers in professional development was so much more satisfying.  That led me to blogging, as a way to share resources and post materials for teachers, and blogging led me to writing. (Such as it is.) It was ephemeral enough that I didn't feel chained by it, and informal enough that I could share my process and stream of thought, which I value over product.

Then I started getting positive attention at work for the blog. I had long accepted that the way I went about my job meant never being a full professor and I didn't mind at all. Several friends convinced me to consider applying for promotion, and when my chair mentioned to my wife Karen that I should, it became a home discussion, too.  I decided to try; I could be a test case, since I didn't really care.  But a funny thing happened on the way, and as I put together materials and considered the college criteria, I really convinced myself that I did fit the criteria. The one thing missing: peer review. I decided that my department would be the peers, and made the process about asking them about the quality of my scholarship. They felt it met the requirements, though some felt like that was the wrong question, and the right question was publishing. But our criteria don't require publishing.

So when the negative decision came, it was totally depressing. The dean made it clear that it's a "technical requirement," and, I'm sure he thought kindly, "if you had one paper accepted..." Which to me sounded like you're right, your work is deserving, but sorry, you forgot to check a box.  My negative reaction to this makes me feel foolish beyond measure, because my life is a constant stream of blessings. This is so totally a first world problem. It made me feel unappreciated at work, despite the great support I received from many people. Karen suggested my reaction came from a lack of previous failures, and that is part of it, too, I think.  It really mired me in negativity.

I went from doing what I love because I loved it, to caring what someone else would say about it. And now I probably will try to submit for publication, though every obstinate bone in my body says to hell with it. Because it makes a financial difference for my family, though I hate that this matters. Which takes me back to having been so fortunate that I can be such an idealist at this advanced age.

Then it finally connected to me both how we do this to students all the time. Care about the grade! And how this is parallel to the new and developing teacher evaluation programs. With much higher stakes, where a no means you're out of school or out of a job. It's a nasty proposition, having to manage your professional life or academic life with someone else's criteria and interpretation of those criteria hanging over your head. 

Sympathy for the people really subjected to these extrinsic measures is helping me come out of my funk.  Plus to still be doing the job I love, with the constant amazing work that students do when genuinely learning.  Two #mathchats on math games this week! A new batch of student teachers to mentor next semester.  I want to re-evaluate how I'm trying to motivate students, and to be honest about it.

It's a Wonderful Life, when measured by what actually matters.

Holiday Game Design

#mathchat last night (twitter stream, wiki) was on "Games: Where's the math? How can we use games to teach mathematics?" One of my favorite topics, and a good discussion. There are so many things I like games for in mathematics: playing a game is quite like math, strategy is an excellent context for problem solving, engagement level for repeated exercises or tasks, etc. But one of the things I like best personally is making them. (That's definitely one of the appeals of collectible card games; building a deck is a lot like game design.)  The amount of math that goes into making a game can be quite a bit greater than playing it.

So today for the 5th graders I brought a half-formed game based on the Traveling Salesman problem. Georgia Tech has a nice Traveling Salesman Problem site, with a few games of their own, nice explanations and history of the problem. It was inspired by the ultimate Traveling Salesman: Santa Claus. Every home in a night? Mathematician Elves on the job.  I eventually changed the game to running Christmas errands in town here, and intentionally left it rather drab.

We played a few turns of the game to get the idea. Then I shared how I wanted them to be game designers today, and we discussed possible things to work on.


Game Design To Do:
1) Playtest

• Are the rules clear? Do they need to be changed? 
• Are the mechanics of the game okay? (Right number of destinations, how to move, placement of stop, dice to roll…) 
• Is it fun enough? How can you make it more fun? 

2) Develop

• Should there be obstacles on the map? 
• Decorate the board; add fun details or pictures. 
• Make nice game pieces. 

3) Create!

• Make your own map. 
• Change to the world map or the US map. 
• Change the story of the game. Santa, UPS, mail carrier, … 
• Completely new game idea: 12 days of Christmas, Christmas tree, Hanukkah Candles, Winter Break, … 

I also brought a blank grid, a polar map (to do a Santa Claus version) and a United States map. (Click for full size. PDF of the whole document on Google Docs.)


Nobody used the polar map - poor Santa!  The class had many people make improvements to Santa Haven, and several who made their own game.

Some of the improvements: new goals, like get all the presents to Grandma's house.  Board alterations, like road block, traffic jam, hazards, stop signs, school zones, etc. Some quite clever, like a gas station (you have to go in if you pass), or a muddy spot that divides your speed by 2 until you get to the car wash. Play alterations, like the bank after every present, or a specific chore list (home, school, presents, back home then school then home). One student made walking and driving rules; driving was double speed, but had school speed zones and roads they had to stick to. At the end we talked about how this was mathematical modeling, where they tried to take real life stuff and figure out what they would be like in the game.

The new games included 2 Risk variations on the US map, variations on the traveling salesman with all new maps, and a candy cane math game with problems on the spaces.










Quite a lot of creativity, so much enthusiasm. And, I think, a nice lead into designing some of their own games later.

Rigor and Relevance in Parallel

(The math is at the end of this one.)

Last week I had one of those teaching collisions where it felt like every idea was dovetailing.  First some twitterer retweeted Terie Engelbrecht's post on rigor and relevance in the context of motivating students with respect not points. Her post was a riff on this International Center for Leadership in Education chart. My preservice high school teachers had asked for parallel lines, circles and proof for our geometry topic.  Elissa Miller tweeted about parallel lines in a way that also brought up relevance.

Source at leadered.com

Fig 1.1 from this ASCD article

Some comments from my colleague Dave Coffey connected the Relevance framework to the Levels of Transfer, a professional development framework from Joyce and Showers. Before I switched to a communication framework, I tried to grade using that framework! Very ingenuous, as most students are not at an executive level of transfer, and it is not fair as an expectation for a large quantity of work. I did like the sense of ownership that it promoted, and the encouragement for students to show that they had made the learning their own. When it came time to convert to grades, Integrated Use was an A, and Executive an A+. Is that fair?

Terie's central question was: 'So why not change the "doing the work for points" idea into a "doing the work to improve my learning?" idea?' This has been an issue all semester with this particular group of students for me. Because this is a teaching class, we've even explicitly discussed from whence comes thier lack of responsibility. (As in the Condition of Learning, Responsibility; not as in a guilt trip from your parents. I hope.)

So I brought the framework into class and asked them to classify the kind of lessons they saw in observation... 10 to 15 lessons over the course of the semester.  I was seriously surprised how well distributed the X's were on the relevance framework, but not surprised by the low range of rigor. The students discussed how the ABCD was confusing, as D was the "best." They had quite an interesting discussion about whether math should be 'real world' or not. You all know the discussion: why do they ask us (math teachers)? Do they ask their history teacher? What about learning the math for math's sake? Sometimes the applications are so artificial.  The students hate the applications so much, why make them do them?

I think, from their discussions as well as their observation journals, I would have put most of their X's in A.  The application in the rigor framework led to a lot of the marks being along that line, as we haven't studied Bloom's, nor have they in other classes.  Personally, I do wonder about what the classroom should look like in terms of this scale. I like the idea of flexible access, and I want students to transition to analysis, synthesis and evaluation. The application framework seems like you do want a diversity of those problem types. Regardless, I'm really interested in what other teachers think about the framework.

To support the students in thinking about implementing this, I brought what I had thought of after Elissa's prompt. Their task:

Our question is: how can we plan a lesson or lessons that will support our students in moving towards being able to make proofs, understanding the required angle content, and engaging them all the while?

Consider the following and please add your own:
a) a puzzle made from parallels and transversals
b) a map of city streets
c) a classic Japanese problem using these ideas

Where do these ideas fall on the rigor/relevance chart? How should we sequence them or structure class to get to our objective?

Explore the activities in your groups, or design your own, and then we’ll come back together to discuss the ideas.

The puzzle: (Actually revised a bit from their use, based on play; changes were made to make it more accessible and focus on the mathematical properties. Click on the image for full size.)

The map: (Grand Rapids)

Tasks suggested for this were identifying parallel and perpendicular lines, vertical and transversal pairs of angles, combining measurement of street intersections, make informal arguments for congruent pairs, etc.

The Japanese angle problem:

(The top angle is 50 degrees, the bottom 30 degrees.)


Find the angle x. “Please solve the problem and if you can make an explanation that is amazing.” From the TIMSS video at http://timssvideo.com/66 (Love this problem; thanks to Rebecca Walker who first found it.)

Students tried all three, and they typically wanted a combination of them in their lessons for their hypothetical students.  The discussion of rigor in these problems gave us a lot of material (by which I guess I mean gave me a lot of informal assessment data) for a discussion of answer, solution, argument, proof in a later class.

I think these problems are pretty good samples of A, B and C lessons; I confess to being a bit stumped as to what a D lesson/problem/activity might be for this topic. The nature of abstraction in mathematics will lead to a lot of good lessons proceeding from B to C. Opportunities to venture onto D might be more rare. I'm very curious to hear what other people think.

SBG Resources

From Rainbowcatz @ Flickr

I was gathering Standards Based Grading (SBG) resources for a colleague and thought that would be worth sharing.  Maybe this should be a LiveBinder? There's a definite math focus to my selections below, those it's not strict. Many people refer to SBG as Standards Based Assessment and Reporting (SBAR), which gets the whole 'grade' idea right out.

People: (Name links to Twitter)

• Jason Buell - SBG resources that includes the SBG galas.
• Shawn Cornally - collected resources at his blog
• Riley Lark - SBG blogposts
• Frank Noschese - SBG blogposts

Fundamentals and Further:

• Ed Leadership article: Seven Reasons for Standards-Based Grading by Patricia Scriffiny
• SBG Beginners Wiki, which also has math skills lists , started by Elissa Miller
• SBG page at the MSmathwiki.
• Blue Harvest - Shawn's SBG software
• ActiveGrade - Riley's SBG software

Twitter discussion

• #sbar - find more SBG folk, or people trying it in your discipline, by a Twitter search.
• #sbarbook - book group that 'meets' weekly for discussion about a particular book on assessment. Doesn't look like this semester's book is very engaging, though.

For completeness sake, here's my 2 (so far) SBG posts. Hey, this makes 3! If there are more examples of SBG in college, especially college math, please help me find them.