Creative Pattern

So, like most semesters in most of my teacher prep classes, we started out by watching Sir Ken pose the question, "Do Schools Kill Creativity?" Especially for preservice elementary creatures, who often have trouble seeing themselves as math teachers, who often have had very negative math school experiences, and will even sometimes bust out with "I hate math" in front of their math teacher.

This semester's group got pretty into it: the story of Gillian Lynne was high impact, the idea that things need to change had traction, several recognized that they had been subject to this, and the desire to incorporate movement really resonated. (We have a drummer in class, so that might happen.) Some students wrote about creativity for their weekly work: Lauren and Kyrstin, for example.

One of the ways I'm trying to encourage creativity is a work structure (syllabus) like this:

Daily Work: I’m asking you for 1 hour per class. Document what you did somehow and keep in a binder. It is not evaluated on correctness, but on percent completed. Keep an index/table of contents for which days you have work for. This work should either be doing math or learning about the teaching of math.  It is okay to double dip - use daily time for Family Math or weekly work. Just keep track of getting in your hours. I will offer suggestions, but this is your responsibility. It’s a good opportunity to practice generating ways to meaningfully work, which will be an important part of your work as a teacher.

Creating: from our work each week I am asking you to put an additional hour or two into deeper work of your choice. Revise or extend a daily work, play or make a math game, make some math art, find and read something in an area of interest, work on a math problem of interest or create a mathematical task… there is so much different work that teachers do. If you can connect it to our course work, it’s probably okay. Each week’s work will get feedback in terms of our rubric and qualitative.  But those aren’t grades. At the end of the semester this weekly work will be evaluated ⅓ on completion (did you complete work for each week) and ⅔ on exemplars. You will pick two examplars of your doing math, and two examples of your preparing to teach math.

There's a urli.st of their weekly blogs The list helps me in finding them all for giving feedback, but I ask them to link posts to our Facebook group as well. That gets more readership amongst the class than I've ever had before. One of the purposes of blogging their work is to increase their sense of audience. So if you do take a peek, please comment!

The math content we paired with this is patterning. Our first activity (close to this previously blogged one) got us playing with the appropriately named pattern blocks, trying to get at the idea of what makes a pattern a pattern instead of a design. Our ultimate idea was that it needs to be extendable. Not necessarily predictable, but when you see what comes next it should make sense with what came before. They built and then we talked about repeating patterns and growing patterns and then sequential patterns. To emphasize the extendable idea, we built patterns, then rotated to have someone else add on. Clearly - time for pictures.

Clear to everyone

No discussion

People accepted extension,
but felt like 3rd red block
could go "anywhere"

Generated interest because
the start was in a line, and the
pattern was extended 2-dimensionally

Patter creator admitted they
didn't know what came next, but
liked the extension. Next: 3 blues
top and bottom.

Arguments! Pattern creator wanted the trapezoids
double each step, extender focused on blues
"adding one" each time.

Is this a pattern? Designer claimed it was just a design.
Extender felt like the red-blue-green were lines
extending out each direction. All agreed: lovely!

Here's the handout, if you're interested.

The next day I wanted to build on the idea of the sequential growing patterns with explicit connections to algebra. My colleague Pam Wells has the best activity I know for this, adapted from a Mathscape activity. Here's my version. (As a Word doc, if you want to edit. Wasn't displaying correctly...)


Everytime I've used it the lesson has been engaging, provoking discussion, and very supportive of symbolic representation with the visual. Students wanted to work through all the letters on the front, though I only asked them to pick a couple. Many wanted to jump to building their own pattern immediately. Most glossed over the verbal description, so I pushed for that. In general with our pattern work, visual to verbal has been uncomfortable. This is a good activity for the connection between rate of change and the symbolic rule, as several students made that jump. Some students went from data to rule, and some from the visual.

A couple students extended this for their weekly work. I based my sample weekly work on the letter patterns, so I expected more, actually; but that's why we give students choice. Brett extended the letter idea to his whole name, which is actually a pretty nice context for adding functions. (File that one away!) Emily did a really interesting project, making some mathart that  had layers of patterns.

The lesson after this was dominoes - but that's clearly a story for another day. Later in the semester we'll do more patterns using ideas of perimeter, area and volume.

Number Addict

I have an addictive personality, so why would I even start a game called this?

Number Addict is a free iOS or Android game. (Also has a Facebook page. The sequel is a 99¢ app.[App store link]) It's a Tetrisish puzzle, where you are trying to amass groups of equal numbers to score them. Two nice wrinkles: it takes a group of at least as many as the number to score them, and you can combine adjacent numbers by addition up to the max number tile available. (Which increases as you level up.) You can see upcoming tiles, so this and the adding makes for many strategic decisions. I think it's good for reasoning as well as mental compositon of whole numbers.

The scoring is completely non-obvious. Scores increase both for number scored and how many in a group. This post is really just about sharing my data for a possible lesson.

In an effort to continue to seek the most boring possible video, here is a screen capture of me playing the game. I used screenr.com and the Air Server app to make the video. 


(It has a fun song 'Pocket Calculator' that I muted for the video. Most Boring Ever.)

But, I spent some time (too much) gathering screen caps of scores... (google pdf)



So I've even collected the data for Act 2... (google doc)


That's editable if you gather more data...

Some questions are obvious. What are the missing data points? What will the level 9 scores be like? (I've never gotten there!) What's the pattern to the increase in scores? Is it determinable or were the programmers just having fun?

If you give it a try, share what you find in the comments!

EDIT: collected a bit more data... great patterns. If you want it to verify student work, here's the spreadsheet, and here's the evidence.

Patterning

I gave an algebra assessment this week (in SBG style) and a student really surprised me. So I thought I'd share.

The assessment:
Possibly relevant standards.
A. Algebra: representation, operations and modeling

1. Linear equations and functions 
2. Quadratic equations and functions 
3. Exponential and logarithmic equations and functions 
4. Higher polynomial and rational equations and functions 
5. Functional representation and operations. 

The problems: take 10 to 15 minutes to consider the following problems. You may do 1 a couple or all. Your SBG score will not depend on a correct answer, but rather on showing your understanding of the ideas involved. (Of course, good understanding helps find solutions, so it’s not totally unrelated; but it’s easy to give an answer and show no thinking.) Since the answers are not the central issue, be sure to communicate your thinking, process and understanding. If you read these instructions, clap your hands once.


1. Using the parabolas at right, estimate the equations of the curves.
2. Discuss: how do you recognize a quadratic pattern from a table, and what does that have to do with the familiar parabola shape of the graph?
3. Use the idea of parallel and perpendicular slopes to give the coordinates for vertices of a square that has no sides parallel to an axis or to y=x. What is the area of your square?
4. Find a symbolic rule for one of the patterns below and look for connections between the symbolic and the visual. Extend the pattern one step to check your rule.







Analysis:
Problem 1 gives lots of nice information. Whether the student uses standard (harder) or vertex form, what the coefficients mean to them, and if they are consistent in applying that understanding amongst the different parabolas. Almost no one uses the roots, and I've yet to see someone apply regression to points.

Problem 2 pointed out that students over-identify parabolas with quadratics, as most students who struggled talked about increasing slope and symmetry as opposed to first or second differences. (Also that there was a homework that they didn't get to!)

Problem 3 was interesting in that no one started with the points. Everyone who tried it gave linear equations, established parallel and perpendicular, and struggled with how to get the sides to be equal length.

Problem 4 was the surprise. Most people chose the pattern on the left, and used visual identification of pieces to generate a direct rule. Only one student tried the pattern on the right. But instead of finding the quadratic pattern I intended, she noticed that each element was as long as the three previous elements summed. Definitely true for the picture... and got me to wondering if it would be quadratic. Nope. but definitely non-quadratic. I made a Google spreadsheet to compare, and this was unusual that the quadratic matched the sum of the first three pattern so well. I got interested in the ratios as they seemed to be converging. (Here's the spreadsheet if you want to play around for yourself.)


Finally I gave in, and looked up the sequence in the On-line Encyclopedia of Integer Sequences, where it is known as the tribonacci series. (Cute, eh?) This limit ratio is the solution of x^3 - x^2 - x - 1 = 0. I think I'll call it the Golden Ratio.

May your quizzes ever surprise you!

Pattern Generators

I like the theme of the first week of my Math for High School course to be teaching for creativity.

This semester's group did a nice job with Dan Meyer's Toast video. I had them ask questions and record what they noticed. There was a neat dichotomy: the WCYDWT responses were all pretty traditional math textbook questions. The what they noticed branched far afield, wondering what would make a good soundtrack, wondering about darkness of toast and toaster design.

We followed that with a workshop (I think based on an Esther Billings and Pam Wells workshop) based on verbalizing and algebrafying number patterns.

Is this a pattern?

The last workshop of the day was planned to be this, which in the past has been a pretty good activity:

Objective: TLW develop mathematical patterns from the teacher’s perspective.

Schema Activation: when (if you do) do you notice patterns in real life?

Focus: What we talked about in Class 01 was really curriculum. What are we going to teach? Here’s the NCTM’s take:

The Curriculum Principle
A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.  A school mathematics curriculum is a strong determinant of what students have an opportunity to learn and what they do learn. In a coherent curriculum, mathematical ideas are linked to and build on one another so that students' understanding and knowledge deepens and their ability to apply mathematics expands. An effective mathematics curriculum focuses on important mathematics—mathematics that will prepare students for continued study and for solving problems in a variety of school, home, and work settings. A well-articulated curriculum challenges students to learn increasingly more sophisticated mathematical ideas as they continue their studies. (From the Principles and Standards for School Mathematics (PSSM for short), NCTM.  All these principles have expanded information and explanation at the NCTM website.)

As we discussed, there are small shifts and large shifts. This activity applies to both: it’s an easy activity that allows for creativity (subtle shift) but may lead to you designing your own problems and questions for students (big shift).


Activity: we have blocks. Play with them!
1. Make a pattern of images with the blocks. A successful pattern for this task is one in which the next shape is determined, or is reasonable.
2. Describe your pattern in words. What’s happening, what’s changing? What can you say about the next step compared to the previous? What will the 10th step be like? A general step?
3. Describe your pattern mathematically. What can you say about the next step compared to the previous? What will the 10th step be like? The Nth step?
4. What connections do you see amongst 1 (the visual), 2 (the verbal), and 3 (the symbolic or mathematical)?
5. Repeat as time allows.

Reflection: choose 2
• Was this task too open-ended? Does it need more structure?
• Was this task engaging? Was it mathematically worthwhile?
• What were the strongest or most interesting connections you saw in a step 4?

But we didn't have time for that! (Not to butter them up, but they were jamming on the toast and I didn't want to cut that short.)

So what we did was:

Schema Activation: whole class discussion - what makes a pattern a pattern?

Focus: Pattern blocks, make what you consider a pattern.

Activity:
1. Make patterns.
2. Put a piece of scrap paper by your pattern with a Y and an N.
3. Gallery walk. Put a hashmark by yes or no if you consider that a pattern or not.
4. Stop by someone else's pattern, add three blocks.

Reflection: whole class discussion about what happened.Record your personal definition of pattern as it is now.

I'm kicking myself now that I didn't take more pictures. Most students made repeating linear patterns, some tessellation patterns, one person made a circular pattern, and then there was...

This was the only pattern that students did not see as a pattern. It was described as just random fitting together, and someone asked about the yellow.  "They were supposed to be orange." And then the author shared how they were lines of blocks fit together. Another student shared how to her it was a skewed checkerboard. Then the class unanimously agreed it was a pattern. I thought that was really interesting and asked how they would verbalize it. Good descriptions followed. Could they capture it symbolically? No, not that I expected it. So I shared a bit about the wallpaper groups, and how mathematicians seek the power of good notation so they can symbolically manipulate. I also shared how I had thought the yellow was a pattern in a pattern.

The characteristics of patterns they thought most important was that it can be explained (there is an idea or structure) and someone who understands it can extend it or fill in a missing piece.

If one week determines a pattern, it's going to be a good semester!

Photo credit: The tremendous image up top is from Tanya Khovanova, in this post. The original question was rather brilliant: "Which one of these things does not belong?" Many thanks to Sue, who pointed out the author below.