Thursday, January 30, 2014

Don't Pay the Ferryman

I love story problems that look like they don't have enough information but they do.

The Futility Closet, an excellent source of puzzlers, had a beauty today. I wanted to dynamicize it to make a visual, and to allow for new numbers and a mechanism to provide for guess and check strategies. Here's the applet, which is also on GeoGebraTube.

What makes this solvable?

I have older sketches of two other problems, an escalator problem, via Bowman Dickson, and a burning candle problem, originally from NRICH.

Does the visualization add to the problems for you? Or does it not make a difference? Do these kind of problems engage you? I do worry about the frustration barrier with these, and am interested in how you might scaffold them.

The GeoGebra was fun. The most complicated piece was building a piecewise function that modeled the movement of the ferries. After the numbers are randomly generated, the sketch has the times for the ferries to cross, and then it's a point-to-point parameterization modified with graph transformations. It was fun to figure out as it took a lot of wee algebra bits to fit together.

The title is a nod to this Chris de Burgh song from the heyday of music videos.

Wednesday, January 29, 2014

Making Competition

One of the benefits of a college teaching job is the constant stream of high energy, bright, creative students with oodles of initiative. But it can keep you busy.

Math-Team-Matics is one of those things.

Andrew Otten and Sarah Jamison, two preservice teachers graduating next fall, came by with an idea for a math competition. They had seen the competition at Grand Rapids Community College for 9th and 10th graders. It's a little ironic as I am not competitive myself (though we had a big family discussion about this recently) and spend most of my time thinking about how to support struggling students. In addition to supporting Andrew and Sarah I thought it would be good for me to spend some time thinking about the other end of the academic success spectrum. Plus there's a bit of the game aspect to this - something I'd love to increase in future years of the competition.

In addition to Andrew and Sarah, who also provided most of our ideas on what the events should be, the key players on the Math-Team-Matics-Team were:
  • Chelsea Ridge, Regional Math and Science Center math coordinator. Great organizational skills and administrative support as well as event design.
  • Karen Novotny, full prof in the math department, amazing math knowledge, teaching and problem design.
  • Ruth Meyering, affiliate instructor in the math department, former high school teacher and coach of math competition teams. She was the last person to join of this group, but her experience with the competitions helped immeasurably.
All of us and several great student volunteers put in a long, hectic day on the competition to make it a reality. In designing the event, we put a lot of thought into making it an actual team event. Our lack of familiarity with the existing competitions led to some nice variation on how they usually proceed. The competition was open to 7th to 10th graders, but the content was open through some high school geometry.

Volunteer needed to try out this problem herself.

First Event: Team Challenge
Karen came up with this problem and next year's possible team challenge in 20 minutes of brainstorming. We knew this was pretty stiff, but wanted to come up with somethingthat they would need teamwork to do. Many students had not been exposed to isometric drawing and - as Ruth warned us - geometry was a challenge throughout the day. The staircase pattern can often be one of those problems that people either know because they have seen before or find it a stumper.

PDF directly

Second Event: Individual Test
How do you make this about a team? Andrew, Sarah and Ruth saw this as essential and a key part of other competitions. So we made the scoring a bit novel: drop the high and low score for the team. But to include the benefit of having an exceptional individual performer, also add a team score that awards a point for each problem that someone on the team got right.

PDF Directly

Third Event: Relay Race
Ruth runs a relay with coach cooperation.
I had never seen a structure like this before. One student does a problem, then hands back a numerical solution that is needed in the next problem. Ruth helped me understand these with a bunch of relays she uses for review in her algebra classes.  There's no need for the problems to be connected, but obviously it's more aesthetically pleasing if they are. I was particularly fond of the number-themed set here, as students think numbers are the simplest. However, these were the problems with which students struggled the most, and I learned that they should scale in design and complexity. Next year these will be easier overall, and gauged to make the most of the different amounts of time students on the team have to work.

PDF Directly

Fourth Event: Quiz Bowl
Waiting to steal.
(Psssst, it's Archimedes.)
There was a bit of a panic with these, as the slides were made up to use Classroom Response System as buzzers, which then did not work with the classroom computers. But here is where the coaches and teams proved their kindness and good spirit: they helped us determine a system of hand thumping to determine who had responded. Though I had arranged a Swiss structure (like chess or Magic tournaments) I had neglected to include tiebreakers! The points for this round were too swingy also, so we're modifying for next year. Winning three quiz bowls was pretty much a guarantee of making the finals. Since that was also quiz bowl - quite an advantage. Andrew got some old fashioned proper buzzers working for the finals, and that added a lot. As well as having all the players and parents to watch.

Top two teams in the younger Division 1 finals.
Since we originally didn't know if students would be age-mixed, as we didn't know how many teams to expect, we designed these to be mostly asked in turn, with a chance to steal if the team answers incorrectly. That worked pretty well, and will probably be a keeper. (Also be careful with these questions - there were some incorrect answers on the key (!) and I'm not sure they all got fixed.) People were doubtful about the Sometimes/Always/Never questions, but they inspired a lot of student reasoning in practice. There's also some math history here. These were tough, even though I tried to write those Jeopardy style where the actual question was distinct from the historical tidbit. We have a culture, people!

PDF directly

We're definitely interested in feedback, but since Math-Team-Matics 2 will be November 8th, 2014, it's even more beneficial than usually. Suggestions and questions, please!
And if you're anywhere nearby, we'd love to see you there! 

Division 2 Camps and Runners Up from the same school?
People will be gunning for Coopersville next November.

Friday, January 17, 2014


On Tumblr today, geometric aesthetic had this innoculous little image...

Wow. These are so elegant.  But there were no notes! What are these from? Google image search found this composite image all over, but also tipped me off that this was Josef Albers, a (literally) Bauhaus artist who moved to the US in 1933 (when the Nazis closed the Bauhaus) and became one of the 20th century's most influential artists (geometric abstract) and educators. Check out his quotations, frothing over with wisdom.  As much as I love art, there are these huge, gaping holes in my art education... I was even mooning over some of his paintings in the last year at the Hirschorn Museum in Washington! (By the way, the Hirschorn had a great retrosprective exhibition on Albers, and still has a lot online.)

The piece in the bottom left, ‘Structural Constellation, Transformation of a Scheme No.12′, particularly got my GeoGebra-juices flowing. (I found a nice image of it in a nice ArtBlart blogpost covering the Hirschorn retrospective.)  There is so much geometry to notice. But then to start to think about how to dynamicize it was the real mathart fun.

My first idea was to just make two overlapping rectangles, and see which of these proportions are consequences of that. It was much harder to make than I thought at first! This version is pretty robust; robust to me is about durability, keeping the constructed properties regardless of how you move the free points and whether you can make all the possible varieties. (For example, if you make a chevron, is it always concave, always a kite, and can it make any chevron?)

Play with the blue points to change the image. (This is an example of GeoGebra's new HTML5 embedding. Easy from GeoGebraTube on any platform that lets you edit the HTML. [I.e. not, unfortunately.])

Here's the direct link to GeoGebraTube. I made the overlap by starting with one rectangle, then determining the angle of intersection for the second rectangle by specifying the intersection points. Hmm, now reflecting, I should have continued that! But I wanted the rectangle controls to be at the vertices, so it's probably unnecessarily complicated. I do like how it lets you play to find some of Albers proportions. In particular I got an appreciation for how the vertices of the colored parallelograms come in collinear sets of three, and the symmetries that makes.

My next effort was making a version that built outward from the innermost parallelogram. I thought that would still allow for some dynamic variation, but capture more of the symmetry that makes Albers picture so gorgeous.

I used the center of the parallelogram as a symmetry point for the corners of the rectangles. This one is at GeoGebraTube, too.

Because of adding in the rotational symmetry, only two extra points are needed to determine the rest of the figure. But the control over that inner parallelogram gives a large amount of variety still. (Especially if you turn it inside out.)

Lastly I wanted to figure out what exactly were the proportions that Albers used. As I looked, I realized that you could either build it from two squares divided into thirds, or build the whole thing outward from a single isosceles triangle. I also noticed his exquisite framing... just a flawless image in design and proportion. This one is also on GeoGebraTube, though it's not very dynamic.

Art is a such a good entry into some wonderful mathematics. Look at all the angles, shapes, similarity, proportions and symmetry in this Albers design. I hope you take the time to explore more of Albers work - you won't regret it!

Thursday, January 16, 2014

What is Math?

As a preassessment for my capstone course in mathematics, I asked these soon to graduate math majors what is math, and what are the big developments in math. This goes along with both my belief in preassessment, and wanting to ask questions about what I really want to know. Here are some of their responses. (See all their blogs in a urlist.)
Roz Chast

What is math?
  • Lots of "math is more than numbers" or "it is not about computation"
  • Relationships
    • Sara: "math is patterns."
    • Jennifer: "Math is more of a way of thinking.  It is logical reasoning.  It is looking at patterns and relationships.  It is problem solving and explaining phenomena that seem unexplainable."
    • Kate: "I think math is thinking logically, understanding facts and finding relationships between different mathematical concepts."
    • Alex: "Math is the study of relationships that many people take for granted."
    • Annette: "math is an explanation or an attempt to explain relationships we find in nature or ones that we create."
    • Kristine: "math is how we can apply numerical values to how the world works."
  • Basis of science
    • Becky: "Math can be used to problem solve, find patterns, make predictions, provide reasoning, and much more which is all in the tool box of math – the resource for information."
    • Bryce: "Math is the science of solving problems."
    • Josh: "Math is the one science that every other subject has in common."
    • Kenton: "The reason that I call math a "science" is because without math, no scientific studies would be able to be quantified, all studies would have to be qualitative and therefore much less precise."
  • Logical system
    • Kerry: "Mathematics is a fantastically broad, beautifully intricate, complexly connected concept of numbers and symbols."
    • Emily: "math is thinking critically about different kinds of systems. These systems can range from the number system in algebra and calculus, to a system of shapes, such as geometry, to a system of rings as we saw in modern algebra."
    • Andy: "Mathematics is a method for conveying logical principles. A proven mathematical theorem is a reality about logic; it is some organizing principle inherent in the human mind."
  • Language. 
    • Duncan: "mathematics is the most beautiful language in all of the universe."
  • Final answers:
    • Biz: "For me, math has been a source of intrigue, education, and frustration."
    • Matt: "To ask what is mathematics is like asking what is life. There is no definitive answer."
Top 5
I also asked what are the biggest moments/discoveries in history of math. (Top 5 or milestones, etc.)
  • Famous Stuff
    • Pythagorean theorem 9
    • Fibonacci Sequence 9
    • Pi 4
    • Fundamental Theorem of Calculus 3
    • Natural logarithms and e 2
    • Kepler’s laws of planetary motions
    • Any of the Fundamental Theorems 
    • Law of Sines
    • Quadratic Formula
    • I Ching
    • Fractals
    • ϕ
  • Culture
    • Numbers 9
    • Calculators/Computers 8
    • Abacus 3
    • Discovery of zero 3
    • Understanding of different formulas
    • Development of measuring units
    • Ancient Geometry
    • Math in astronomy.
  • Concepts
    • Unit circle
    • Function
    • Pattern
    • Combinations (in counting)
    • Infinity
    • Axioms
    • Parallel postulate
    • the need for complex numbers
    • prime numbers
  • Fields
    • Calculus 7
    • Algebra 4
    • Calculus 3
    • Non-Euclidean Geometry 3
    • Geometry
    • Differential Equations
  • People
    • Euclid 7
    • Euler
    • Newton
    • Archimedes
    • Gauss
    • Karl Pearson (new to me!)
Not too different than I might expect, although the diversity of responses was pretty interesting here.

One more interesting comment:
  • Danielle: "The founders of mathematics struggled and devoted their whole lives to the theorems we now take for granted when studying in our classes. I can’t imagine devoting my whole life to proving what we consider now a simple concept." 
If you have a moment, I would love to hear your responses to these prompts in the comments!

Image: I saw this at Peter Liljedahl's very worthwhile site. Even Google couldn't help me find the original. Doesn't it look like it's from the New Yorker? EDIT: Tweeps nailed this one: Roz Chast, a frequent New Yorker contributor.

Sunday, January 12, 2014

Spin Square

Brief post to test GeoGebra embedding.

Inspired by a typically great Daniel Mentrard sketch.  I wanted to play around with some of the ideas. In this you can make a random or ordered mosaic of a tile you design yourself, that is spun through a number of possibilities which you can also set. Hope you make something neat!

Thursday, January 2, 2014

Smith and Stein Demand

(Note: crazy semester and I didn't finish a lot of started blogposts. But this one felt important to finish. Great experience in teacher ed, and I'd love some feedback on how to make an activity for these ideas for teacher prep.)

I was kvetching on Twitter - I mean, consulting my PLN - about preservice teachers whose desire for fun, shiny lessons can distract them from finding lessons of substance. Nicora Placa, a math ed prof and teacher educator (with a fabulous blog),  suggested Smith and Stein's framework for cognitive demand.

That's Margaret Smith, an all-star researcher at University of Pittsburgh, and Mary Kay Stein, who seems like her partner in crime at Pitt. Their paper(s) and book on the 5 Practices for Orchestrating Discussions has been very influential here at GVSU this fall. Several profs are using them to organize our first math for elementary teachers course. [Lengthy aside: book, NCTM's number one seller; MTMS article, very readable for preservice and inservice teachers; Christopher Danielson's 5 practices posts, start here; it synthesizes a lot of great research into worthwhile mathematical tasks and instructional practice that they have done with other partners.]

The cognitive demand article (full issue MTMS Feb 98; also available on JSTOR; Reflections on Practice: Selecting and Creating Mathematical Tasks: from Research to Practice, Smith and Stein, Mathematics Teaching in the Middle School, February 1998, v.3, n.5.) is great for a professional development discussion. They have several highlighted discussion questions that get to the heart of the matter. The article boils it down to four levels of tasks, with detailed breakout at each level.

The reflection questions are very theoretical for preservice teachers, though. (The article relates a professional development discussion with teachers, though, which is excellent reading for them.) But it also includes tasks to analyze. So I had my students try them for a while and discuss them at the table. Without exposure to the article or to Smith and Stein's cognitive demand framework.

(Click to see full size)

Luckily, there's a lot of overlap with out content. The fractions, not so much, but there's a lot of fraction language in the common core geometry and statistics, so it's been discussed.

Immediately, they went into test mode. Which is especially interesting as we don't take tests as such. (SBG, rather.) It took several urgings to get them to try the problems cooperatively, ask each other questions and share methods. I noted that they got very engaged - maybe I need more individual problem solving or even practice in class. (I'm reminded of our pre-assessment and how that's a tension for me.)

Afterwards, we talked about what did they notice about these problems. They focused on the subject of the content and the story problem aspect. Except they were impressed by the realism of these story problems. Many of them spent a significant amount of time on task C, and it was the most discussed at the tables.

Next I asked them to, in their groups, rank the problems on the amount of thinking required by students working these tasks.
We talked about the tasks that had significant agreement or disagreement or were at the least or most thinking ranks.

  • Task A was easy. The visual made it easy to extend the pattern. (Many skipped over the description.)
  • The group that rated Task B the hardest had trouble making sense of the text.
  • Task C had so much computation, calculators were required and the problem had extra unneeded numbers in it.
  • Task E started to bring out some task analysis. It was easy with the picture, but the content was hard. People who had difficulty understanding the picture shared that, too.
  • Task F was ranked easy because it was an easy pattern compared to the ones they had been doing in class.
  • Task H was "just a rule"
Now we looked at the Smith & Stein exposition:

It was astounding to me how ready they were for this discussion. They read for themselves, discussed in their groups, then rated the tasks by this framework within their groups.  In particular I was pleased at how they discussed non-routine vs a difficult-to-execute algorithm in terms of difficulty, and the value they placed on connections and representation. They were able to make connections from their own experience to challenges students might face. For example, the symbolic rule for the train pattern (Task F) and going from the observation of 2 more each time to what that means in the rule. Unfortunately, some of them feel the solution to this will be in what to tell the students, but the majority saw the value in presenting cognitively demanding tasks.

The activity is obviously raw. I'd like to formulate some questions to support them finding some of these task characteristics themselves - but that's challenging without students to observe. 

How might you teach a lesson on these ideas? Surely that's a teacher task with a high-level demand!