Wednesday, December 26, 2012

GeoGebra Animation

There was a challenge to make a GeoGebra Christmas card this year, through Instagram with a #ggbxmas tag, and there were a few really sterling entries. Like this one from Wengler, a lovely Pentomino puzzle from Renata, or milanoff's slick animation. I'm not an Instagrammer (yet)...

... but I tried to make a card too, and just wanted to share two bits that helped me make my mediocre effort. Making the card made me think that this would be an excellent class challenge, which could really help contextualize some algebraic topics.

The above image was made using GeoGebra's nice export to animated gif feature. The original sketch is at GeoGebraTube.

The first was to make a function that would be 0 for a while, then increase to 1, then stay at 1. There's smooth (mathematically speaking) ways to do that, but a piecewise linear function is good enough for most uses:
f(x) = If[x < 0, 0, If[x < 1, x, 1]]
Then it's a really nice use of graph transformations to adapt it. Want it to start increasing at x = 5? f(x-5). Want it to scale between 0 and 4? 4*f(x). Want it to take  6 units, starting at x=3, to increase from -1 to 8? That's a good problem! (Would 9*f((1/6)(x-3))-1 work?)

My first couple of pieces were animated by using the function directly on a point definition. Like Joseph's left foot,
A=(12f(1 / 3 d), 0)
and then I defined the rest of Joseph relative to that point. In GeoGebra you can do arithmetic with points like vectors and use the coordinate commands to reference part of a point.
A+1 = A+(1,1)
B=A + (0.5, 0)
C=0.5 (A + B) + (0, 1)
(x(A), y(A)+1)
W=(0, 1) + f(1 / 8 (d - 4)) ((12, 1) - (0, 1))
But then I realized that instead of animating each vertex, I could make the shape I wanted, and then use geometric transformations to move an image of it, controlled by a vector. So for the second time moving the Holy Family into the stable, I made vectors like:
Joe = 2.5(f(d - 7), 0)
Vector[(5f(d - 7), 0)]
And then moved shapes by translating by that vector.
poly7'=Translate[poly7, Mary]
I used the translate tool to do that, clicking on the names in the algebra view, but this is how it looks as an input bar command or in the algebra view. (Took me a while using GeoGebra before it occurred to me that using tools you can select from the algebra view, and that I didn't have to be able to click on it in the sometimes complicated graphing view.)  This technique is much more powerful, and lets you fine tune by changing the vectors instead of having to go change multiple vertices. Which is the kind of thing I have told students in linear algebra before, but never really had to do!

So Merry Christmas and Happy Holidays to all, and a GeoGebrafied New Year!

Wednesday, December 12, 2012


Michigan is just having a darling political month. First, becoming a Right to Work state, bitterly betraying our proud history of organized labor in as sleazy a manner possible. (Even if you supported Right to Work, hopefully you would be ashamed of how it was passed.) But now we're busy wrecking public education also.

Here's the gist of a letter I drafted about it:
To whom it may concern:

Please do not allow House Bill 6004 nor Senate Bill 1358 to become law.

These two bills will establish an Education Achievement Authority (EAA) operating outside the authority of the State Superintendent of Schools and the State Board of Education. Funding for the EAA would be without any oversight from the legislature. Any school in the state into the new EAA district could be taken in by this new government-run statewide district, and then these schools could then specify which students they would serve. The EAA can seize unused school buildings built and financed by local taxpayers and force sale or lease to charter, non-public or EAA schools. This completely bypasses our State Board of Education as well as local government, leaving the Education Achievement Authority with no elected oversight.

Furthermore, there are hugely negative impacts on the funding of existing local public school districts. Local districts are already financially compromised. And with this legislation moving so quickly, most citizens are not aware of it. They would not approve when they do find out. I urge representatives to be representative in this matter.  You can gather the necessary information to see the impending disaster. Vote no on these bills!

Doing all this for an experimental school model that has yet to be verified is nothing short of gambling with our children’s educational future. I am in favor of creating room for innovation and personalization, but this takes us farther and farther away from research verified successful educational models.

Another bill that seems disastorous, especially in combination, is House Bill 5923. It creates several new forms of charter and online schools with no limit on the number. Selective enrollment policies could lead to greater segregation and the bill creates new schools without changing the overall funding available, further diminishing and compromising resources for local public schools. This also creates unequal access to quality programming.

My spouse persisted in getting it through to the  Governor's office, and got this reply.

Thank you for recently contacting the Executive Office of Governor Rick Snyder regarding education in the state of Michigan. Governor Snyder appreciates your valuable input and has asked that I respond directly on his behalf.

If you're interested, we've just released a series of videos, where parents and educators speak out about the impact of the Education Achievement Authority in their school.  Take a moment to review the videos by clicking here:

To address the link between Michigan’s economy and the quality of Michigan’s public schools, Governor Snyder sought to address the state’s Persistently Lowest Achieving schools in his education agenda – currently designated as Priority Schools by Michigan Department of Education.  These are schools that are persistently low achieving and regularly in the bottom 5% of student achievement. Michigan started ranking schools under the School Improvement Grant three years ago, and then refined it for the state Persistently Lowest Achieving legislation and federal Elementary Secondary Education Act flexibility. For years, the lowest achieving schools have been identified without the tools to do anything about it.  The Education Achievement Authority is that tool.

The goal of the Education Achievement Authority of Michigan (EAA) is that all students should have access to a quality education and successfully complete their K-12 educational experience career and be college ready.  One of the principal objectives of the EAA is to transform public education from a one-size fits all model.

The EAA’s approach to teaching is to institute a system known as student centered learning.  In student centered learning, they test each student individually to determine their achievement level in each subject.  They then develop a unique education plan for each individual student, working with that student at their individual level.  In student centered learning, no longer are students moved on just because they have spent nine months sitting in a chair.  They move on when they master a particular subject.  Students no longer identify themselves in grades based on their age, but by their achievement level in various subjects.

All EAA direct-run schools use a cutting edge, digital learning platform that allows teachers to provide a personalized learning experience to each student. Each student is empowered to own and navigate her own learning path.  The EAA is dedicating both public and private resources to building a new learning platform to give kids who simply have not had a chance an opportunity for a bright future.

Again, thank you for contacting the Governor’s office. Should you have further questions or comments regarding this or any other state-related matter, please do not hesitate to contact this office if there is anything we can be doing for you.


Melanie Ellison
Constituent Services Division
Executive Office of the Governor, Rick Snyder
O: 517-335-7858
 How are we ever going to make progress in education if we are losing ground to this sort of unsubstantiated posturing for profit?

Tuesday, November 27, 2012

Workshop for a Living

I plan the vast majority of my lessons now with a workshop structure. It's become an integral part of how I think about lessons, much as Launch-Explore-Summarize used to. Or the "keep the students busy the whole time" strategy that came before that. (I can keep going here...)

What do you want preservice teachers to know about lesson planning, and when do you want them to know it?  The Math for Secondary course that I'm teaching this semester is the first math ed course that our students take. Then we get them for a Math for Middle School course, and then we got to observe them in the field and have a weekly seminar during the first semester of their internship. Just recently we've added more time with them during student teaching, their 2nd semester of internship. We're trying to get away from doing lots of lesson planning for imaginary students. But when they start their internship, lesson planning is what they feel most nervous about. (Survey says.)

Recently I had a class with the objective to get these ultra-novice teachers started on thinking about lesson structure. There's so many ways to consider...

Students were working from a section in Statistics in Action from Key Curriculum Press, and tried every model except 3 acts. (Makes sense given their resource.) Why ITIP...?

So why workshop for me? An article from a bunch of us that we've been working on summarizes the workshop with this table:

Despite the fact that we all implement the workshop differently, these are the commonalities.  So maybe just a word about the phases and why I feel they're necessary.

Objective: what? That's not in the table! But it's this structure that has helped me be clearer with myself about what it is I want students to get out of a lesson, and clearer in communicating it to students as well. I used to like lessons to be like a surprise, but that is a cheap way to get suspense. If you can tell the objective up front, and still generate the mystery... that's good teaching. I'm working on it.

Schema Activation: part for the students to give them something on which to build new understanding, part for me as pre-assessment, this has been very helpful.  One of the best things I have to share as a math teacher is my connected view of mathematics. How does this relate to what we've done before, what ideas do you want to refresh before tackling new material, or what questions do you have about previous times you've seen this that you might not even know you had. Why does that work? Why does it matter? Is there another way to do it?

Focus: this phase has helped let me back into the classroom. When I became convinced of the centrality of student activity, I went extreme. I tried to minimize what I shared to things that just got students started on their activity. It was an improvement, but it left a lot of my students without the support that they needed. I see this phase as part equipping for the activity, part selling of the purpose, and part testimonial as I share some of my experience or thinking.  Activities that students used to struggle with nonconstructively have become high impact learning opportunities with this phase, and frustration has reduced.

Activity: always the heart and soul of the lesson. Rich tasks on important mathematics. Often with some kind of choice built in. Usually cooperative, as I value the power of that mathematically and for learning. This phase can have none or many places where we come back together as a whole class. Sometimes to share what groups are doing, sometimes to address a common stumbling block, sometimes to just refocus attention on the task at hand.

Image by Duncan~ @ Flickr
Reflection: but activity is not enough. The research that showed differences in retention depended more on consolidation than on activity vs lecture really effected me. Whereas I used to just want to summarize - and that is still important to me - now I want the students to spend time thinking about what they did, what was important, what was new and to get time to record that. It has helped me with student retention, formative assessment, and re-emphasizing the objective. My most common form here is to have students write a bit about what they want to remember, what seemed important or what comes next, and then share with their group what they wrote. Excellent eavesdropping opportunity. It's been hard to cut off an engaging activity for this, but it is always worth it.

So that's why workshop is good for me. I shared with the novice teachers that I used to want them all to try to teach , but now I hope for two things: 
  1. that they will teach intentionally, purposefully choosing a structure rather than relying on what's always been done.
  2. that they will evaluate by meaningful assessment of student understanding.
That's going to lead to some good teaching and learning.

What instructional structure(s) do you use? Why?

Postscript: Dave, Esther and I are presenting on this tomorrow. We probably won't use the slides, but they make a good resource.

Pretty Petals

I was appreciating the geometrydaily:
#332 Totem – A new minimal geometric composition each day
And got to noticing the always neat equliateral triangle proportions in the circle where the side length is the radius. So I have to fire up GeoGebra and start playing.

I got fascinated by the lenses formed by the circular arcs trapped by the triangles, and started wondering about tangential lenses in osculating congruent circles. (Petals, I was thinking. Pretty petals.) This is the special case when the cutting chord is a radius. So I made a sketch to play with lenses of different length.

Now I'm quite curious about those special lengths of the petals. Here's the sketch if you'd like to play with it:  teacher's page (download) and student's page (applet).

Other GeoGebra inspired by Geometry Daily:

Thursday, November 22, 2012

Mind Filling, part I

Image from a
Joe Bower post on the book.
Ellen Langer pioneered what she calls mindfulness research – coined to deliberately contrast with mindless behavior and action. Her principal book on the subject is Mindfulness, but she has also written many follow up volumes. One of them that is focused primarily on applications in education is The Power of Mindful Learning, our department book group read this semester.  It was raised as a possibility by my colleague Esther Billings. (Here's the promo I put together for the dept.; it includes a couple videos.)

It’s a short book with easy to read chapters, yet there’s much to discuss. We’re having two discussions, and this post is generated from notes from my reading and the first discussion. Each chapter is about an educational myth, introduced with a fairy-tale, and cross-referenced with particular bits of research. (In this summary I'll just pick one bit of research I found intriguing.)

The Book

Introduction: Little Red Riding Hood, overview of the 7 myths; pitches the book as a why-to rather than a how-to. Although she is a professor at Harvard, so is some mention of how she uses some of these ideas in her teaching. Mindfulness has three characteristics:
1.    continuous creation of new categories
2.    openness to new information
3.    implicit awareness of more than one perspective.
You can see how the alternative to these is also a good description of mindless behavior.

Chap 1: When Practice Makes Imperfect.
•    Story - the Little Prince and the lamplighter.
•    Myth - Basics must be learned so well they become second nature.
•    Research - same physics lesson for two groups, but one was told that this is one of several outlooks on physics, which may or may not be helpful. Students did equally well on direct testing, but students in the mindful group outperformed the others on extrapolation or creative use of the ideas.
Some of my connections were to student carrying out a skill regardless of whether appropriate (multiplying out factors when the objective is simplifying) or efficient (long-hand adding  1327 + 998).

Langer asks the costs for following rules without consideration of the context. In particular does drilling basic skills encourage carrying them out mindlessly? My questions: do we allow for individuation?  Do we give permission explicitly for different approaches? (Different than not forbidding them.) There’s explicit discussion of absolute and conditional approaches and how that relates to gender in K-12 math. Since we traditionally teach math so as to contradict earlier teaching. (You cannot subtract a larger number from a smaller; oops, yes, you can.)

Chap 2: Creative distraction
  • Story - the Three Languages (Bro.s Grimm) (new to me)
  • Myth - Paying attention means focusing on one thing at a time.
  • Research - students were asked to pay attention to a detail filled poster while sitting still, moving back and forth between marked spots, and sitting and shuffling their feet. The movement group outperformed the shuffling group which out performed the sitting still group. A related experiment in a Montessori (usually active) setting found a still group outperforming a movement group. Langer suggests novelty is the key.
I strongly connected this with the crushing repetition in many math classes. Homework! The idea here that distracted is distinct than differently attracted is powerful. Having students find and try different ways to approach work and obtain new information. Read for different comprehension processes, different objectives for doing homework, different choice schemes...  Feels like this might also help explain the myth of the great lesson. One of the effects of novelty is reducing the differences in achievement amongst diverse learners. To me that's evidence of the power of engagement, which means empowering learners to control their own engagement. (Course that could be spending sufficient time with Dave, read this.)

Chap 3: Delaying gratification is important.

Hold on a second. Scone break.  OK.

Chap 3: The Myth of Delayed Gratification
Elizabeth Bishop, Primer School
  • Story - bit from Elizabeth Bishop on starting school. 
  • Myth -  necessary work that is unnecessarily arduous can successfully be justified by future rewards.
  • Research - participants were asked to watch something they disliked. One group was asked to look for 3 or 6 distinctions, the other was not asked to do so. The groups that drew distinctions liked the activity more, and the more distinctions they found, the more they liked it. The other group saw no change in liking.

Interesting choice for the story. I wasn't familiar with Bishop,  and was disappointed this wasn't the fairy tale. Google books has quite a bit of her Collected Prose, including this whole story. Langer leaves out the math bit, which almost makes her point better than the part she does include.

This chapter felt like a stinging indictment of School Math As Usual, Grueling.  Why are we learning this? You'll need it for... Why do we have to memorize? You'll need it for...

This also leads to the next chapter.

Chap 4: The Hazards of Rote Memory

  • Story - Hansel and Gretel
  • Myth - memorization is way more useful than it is
  • Research - students were asked to read a high school literature essay, and either asked to "learn the material" or "make the material meaningful to themselves" with suggested ways of doing this.  Half of each group was told they would be tested. They were tested immediately, then given homework with the same instructions, and tested four days later. Students told to learn it for a test performed the worst. Testing made little difference among students told to make it meaningful.
It seems like the most frequent recipe given to struggling students is to do lots of practice with an emphasis on memorization and impending tests. 

Teaching Gap culture point of view.

The Discussion
One neat thing about the discussion was that Sadie Estrella joined our small group from Hawaii by Google Hangout.  I am definitely interested in exploring this tech more.

  • People noticed that up through this point, Langer had not addressed student interest and desire. How does desire and attitude effect us as teachers and students?
  • The rote memorization/drawing distinctions section really made us want to make the math more meaningful for students. 
  • The question of whether we make the material meaningful or teach students how to make the material meaningful. We can make ourselves into a pretzel, and it can affect achievement, but is it sustainable?
  • How much do students build the wall that stops them themselves?
  • Interesting the tension between the delaying gratification approach and noticing. 
  • How do we get students noticing? 
  • Representations
  • Choice:
    • Reduce the amount of: this is the way to do it
    • Choice results in more investment
    • What's good for me? Make students cognizant of what their own needs are.
  • Anticipate their strategies.
  • What is a good level of novelty?
  • Loved the idea in the book of being otherwise attracted vs distracted.
  • Very low effort shift: presenting things conditionally. "You could..." 
  • Interest in experimenting with Langer's notes on gender differences in math possibly being related to the absolute/conditional presentation.
  • What do multiple perspectives look like in math?
  • Someone connected these ideas with the book Math Exchanges by Kassia Wedekind. (Looks interesting.)
  • How could math practice look more like athletic practice where athletes put their own spin on skills and personalize their approach?
  • Connections must be important here:
    • connections from and to applications.
    • connections between different mathematical ideas.
  • How is this mindful learning the same as and different from critical thinking? Creative learning?
Have you read this book? Any thoughts on these ideas?

Our discussion of Chapters 4, 5 and 6 is this Monday, so look for part 2 soon!

Sunday, November 4, 2012

Mashup Artists

As I'm here at #edcampgr, I am struck over and over today by the idea of teaching as a mashup art. People talk a lot about teachers 'stealing' from teachers or other equivalent expressions, but it's more creative than that. It is amazing listening to these edcamp teachers.  The kind of teachers that give up a Saturday for free professional development, are willing to present when they didn't come here to do that, are here explicitly to take control of their own PD - these are amazing people.

It's really cool how much these teachers have in common, and yet have so much of their own to add, also. They share so many values, yet are implementing them in different ways. Being such a twitter-savvy conference, you see ideas intermingling from concurrent sessions.  Over the course of several sessions, you see people applying immediately what they heard from multiple sources in the next conversation.

Not actually math-positive.
Someone was talking with Dave Coffey and I about Khan (that happens), and that essential nature of teaching was part of the conversation. Constructivist teachers get frustrated with students asking us to tell them what to do, but then those teachers can ask that same thing. Tell me what to teach or how to teach. Or, worse in my eyes, teacher educators tell novice teachers (or, arrogantly, inservice teachers) Teach Like This.

Not the teachers at edcamp. Even when they hear something amazing, they think 'how can I adapt?' or 'what will this look like in my classroom?' or 'how does that relate to what someone else was talking about?'

A persistent idea for me is the student/learner distinction (if you haven't, watch these videos, please. I went there, used caps), but I feel like we need a similar distinction for teacher of students vs teacher of learners. This relates to another word-confusion, the Skemp idea of instrumental vs relational. We need a way to talk about these different ideas in a way that supports all teachers in improving.

That happens at an edcamp.

Some of the resources from the day:
Sessions I was in:
@benrimes Tech Savvy Ed website
Video Story Problems vimeo channel
@kiemanha's Inquiry Google site
Project Zero's resources
Thinking Routines for Numeracy
Neato Comprehension Continuum framework (pdf)
Edward Burger, Williams prof, author of 5 Elements of Effective Thinking
Burley School's iPad blog
K-12 open source eTextbooks 
@anthonydilaura's Innovative iPad blog

Others: (twitter lets you eavesdrop)
@taramaynard shared @kfouss' GeoGebra and Google Forms
@delta_dc shared Project Based Learning explained (YouTube)
@sjunkins How Twitter is reinventing collaboration among teachers
@Packwoman208's Flipped resources

More? Check out the #edcampGR twitter stream.

Thursday, October 18, 2012

Exponential Potential

It really struck me listening to Shawn Cornally in this week's #globalmath session on SBG (click on the Recording tab) how his perspective as a physics teacher leads him to approach his math lessons as experiments. Starting with an experience that makes us want to model or makes modeling useful is definitely the start of some of my favorite math lessons. (While I'm writing this he tweets: "Leaf-Blower soccer went over *really* well today in physics. (vectors, f=ma)")

Starting exponential functions with my preservice teachers, I love to use this lesson adapted from a 5th grade Math in Art lesson. (From my pre-blog webpage.) The idea is the multiplicative patterns present in a Sierpinski Carpet.

(Also in Word format if you want to edit.)

One of the interesting discussions in the initial exploration is the 9 or 17 issue. 9 squares if we count the number of squares as distinct shapes, 17 if we unitize to the smallest level square. For algebra students there's some good opportunities for equivalent expressions, regression and even deduction of function rules. This is a good opportunity for sharing how recording how you're getting your answer can be more powerful than recording answers. The 17, for example, is 1·9+8, then the next level is 9·(17)+8·8. But later, most write it as \( 9^n–8^n \) - which can lead to a pretty neat binomial expansion. Maybe even more interesting and accessible is the 9=1+8, so the next step is 73=1+8+8·8, and gives them a way to generalize this pattern besides recursion.

Once we get to the design your own carpet, there are so many new patterns to find. Here are some samples from this week:

Note that these last two aren't really Sierpinski patterns - but they still raise interesting patterning questions that are extensions of what we already noticed.

Probably easy to see why this is one of my favorite lessons. I also have seen the power of adding in places where students who have not traditionally been strong in math class can do amazing work.

The next lesson to follow up this one has some other opportunities to gather/generate multiplicative data.

It always strikes me how even college math majors find things to be surprised about in this data. Especially the penny balancing one. This class made some neat displays of their data - but I haven't taken the pictures yet. (Didn't know I would be blogging this, as I thought I already had! Maybe I was thinking of the quadratic simulations?) I'll add them at first opportunity.

First Opportunity:

Next we'll look at modeling this data symbolically using technology, and asking questions that raise the need for logarithms. Since, of course, every exponential data set is logarithmic when seen through the looking glass.

Sunday, October 14, 2012

Angle Acquisition

Quick game idea. I've had a few bustling around, and I've got to get started writing them down.

Observing student teachers is a great job. I get to see and have in depth teaching discussions with lots of hard-working, talented teachers.  And see a broad range of content. We use a coaching model, which helps these be positive exchanges and ratchet up the interest level of the dialogue.

I was observing Terry Austen (the class he writes about here) and realized that once students knew the terminology well for parallel lines they were correctly identifying and applying the relevant properties.

That gave me the idea for a game. I thought it would be neat if students used the terminology to capture points. My first idea was to have a parallel line puzzle - I like those as practice, too - and have the players build it and then take the pieces. That's a lot of set up, though, so I decided on cards to make for less preparation. There's name cards to cut out, still.

Let me know what you think. I'll try it out with the PSTs this semester and post an update.

Friday, October 12, 2012

3rd Degree

I Love Charts, constant source of fun graphics, had a fun temperature comparison chart. temperature graph, complete with fun. I did share my graphs finally. I used GeoGebra to make them, of course, because that's the best accurate mathematical image maker. Easy enough to make all three variations. Which, of course, lead to wondering what features would you put in a dynamic sketch for it? How could you make the sketch switch between the three ranges?
But it wasn't to scale, so I started thinking about what that would look like. Just lengths? What's the norm? Then Shawn Cornally shared his oneupped

So I had to make it then. Obviously. It's on GeoGebraTube: for download or use as an applet.

Instead of checkboxes, I knew I wanted a slider to switch amongst Kelvin, Fahrenheit and Celsius as the input. I thought about adding an input box for conversion, but those still don't work on the iPad, and I'm trying to think about that more.

The idea came to make the segments for the 0 to 100 degree temperature range where the endpoints were a function of the slider. Then I just needed a particular value for three different inputs, so it would only have to be a quadratic. I started to go to Wolfram|Alpha to do the regression, when I realized that, of course, GeoGebra would do it more easily. So with the slider at values 0, 1 and 2, the range for Fahrenheit, for example, is just determined by
F0(x)=FitPoly[{(0, -459.67), (1, 0), (2, 32)}, 2]
F100(x)=FitPoly[{(0, -279.67), (1, 100), (2, 212)}, 2]
a=Segment[(3, F0(n)), (3, F100(n))]
I was struck again by how nice GeoGebra is as a source of activities for students, but also how rich the task of making things in GeoGebra is, and wonder: how do I encourage more sketch creation from students?

Having sunk the time into making the sketch, I did think about what activity could go with it, and whipped this up:

I'd be interested in feedback on the sketch. Does it have the features you'd want for your students to use? Should it have more cute doodads? What do you think about the design of such a sketch as a task? How far should the students be past the material before trying to design the sketch?

Tuesday, October 9, 2012

Why Questioning?

Found at From-Student-to-Teacher Tumblr
From Joy of Literacy

Last week became questioning week with student teachers. It came up in two action plans and we were able to have some really interesting discussions about it.

A lot of what I'm writing about here is from work with David Coffey, inspired by Kathy Coffey, and processed from Mosaic of Thought (link includes Chap.1 as a sample) among other books. Neither Dave nor I can remember the actual origin... which is sometimes symptomatic of having done it yourself.

In my own growth as a teacher questioning is definitely one of the places where effort and reflection have helped me improve. The first level was just asking better math problems. More open-ended, that required more problem solving. When the problems are better, there's more interesting things to ask the students about later. The next was to ask more appropriate questions. Some of those excellent problems I gave to students were too much for students. This is a zone of proximal development idea. When it was too much, I needed to scaffold. Later I learned about rephrasing the question instead of narrowing it. Later still I learned about demonstrations of how I think about a problem, sometimes more appropriate than guiding students through it. Then you ask 'what did you notice?', which is - of course - one of the all time great math questions.

One of the best things I ever learned was to stop being the authority. I don't say what is right or wrong. I ask the students if they 'agree or disagree?' That might start an actual conversation. 

One of the first things I really noticed about teaching was that students would tell me that they couldn't do it (whatever it was at the time) but when I asked them questions they could get from beginning to end with no difficulty. So, obviously, I needed to teach them to ask those questions of themselves.   It was difficult. Really difficult. Shifting from asking them 'how to do (next step)?' to 'now what?' and 'how do you know?'

Literacy learning experts are good about this idea of questioning as a process, with the idea that questions are how we move ourselves forward. I like this framework to help me think about the kinds of questions that I'm asking.

Obviously, we have had a tendency to ask too many literal and application questions in math class.  I think about inference questions being predictions, reading between the lines, hypothetical questions and the like. Analysis questions are reflections, synthesis, connections, recommendations and so on.

What I shared that seemed to tie it together for the student teachers is a simple idea: ask to find out what I want to know about. I don't need to ask for answers - I know those. I don't need to ask right or wrong. I need to ask about what they are thinking. Students know when a question is genuine, and this simple idea has improved my assessment more than anything else.  I'm more persistent in getting answers when I really want to know them, also.
From A Softer World
Some good student teacher writing on questioning:

Sunday, September 30, 2012

Greater Than

Previously: planning and coaching on inequalities.

The origin of this game was trying to think about a game that helped give students enough experience to intuit the rules for how operations with signed numbers affect inequalities. I think this is doubly hard for students because they don't have much intuition for signed numbers, and learn both integer operation and inequality rules by memorization rather than understanding.

I love playing cards as a material for signed numbers because the red and black make such a nice positive and negative visual. (For more integer games, see this collection elsewhere on the blog.) I thought about students somehow constructing expressions, but I couldn't think of anything not clunky. I knew we wanted comparison, and I like War as a comparison structure. That made me think of the exciting moment of war, playing down extra cards on a tie.

When I played around with the cards, though, it didn't get at inequalities because each player was changing the value of only one side of the comparison. A big no-no in the context we want. So the played cards had to effect both players' values. It seemed confusing to have both players reveal at the same time, so I decided that - at least to start - players should take turns, even though that makes the optimal strategy pretty determinable. (If it's not a word it should be.)

After students get the strategy, go to the variation where both players flip at the same time or the dealer has one less card. If playing with middle schoolers for integer operations practice, try the flipping off the top variation.

I think this is a good educational game, but only close to a good strategy game. I can't quite figure out what's missing, so if you have a variation or adaptation to try, please let me know.

Evaluating this on my game design framework:
  1. Goal(s). Gain experience with effect of integer operations on inequalities. Works well. Also good for gaining experience with integer computation.
  2. Structure. The game generates a lot of these situations and the cards guarantee a mix of operations and values.
  3. Strategy. Pretty simple. On variation becomes a pretty nice bluffing game, but not intensely strategic.
  4. Interaction. What you do completely depends on opponent.
  5. Surprise. Hidden information and opponent plus randomization of cards helps here.
  6. Catch-Up. Victory is almost always possible.
  7. Inertia. Might be too simple for loads of play, but good enough for the objective.
  8. Rules. The idea of applying your card to both is non-intuitive, and remembering suits-operations connections is hard. You might want specialty cards
  9. Context. No context, but all the variations generated engagement from the preservice teachers.
One thing I want from this experience is a deck with operations for suits. Not sure if it should have all 4 operations, or multiplication and addition with positive and negative numbers or some other variation. What do you think?

Photo credit: Abulic Monkey @ Flickr

Friday, September 28, 2012

Put Me in Coach

I took my new inequality game (that was the fruit of my planning last post) to Dave Coffey's and Hope Gerson's student teacher assistant seminar. The idea was to teach a lesson, and then get coached as a model for the TAs.  GVSU has a great education program where our student teachers have a full semester of being in the schools all morning before a more traditional student teaching semester. They get supervisors from the College of Education and, for secondary, from their major content area for both semesters.

Inspired by the Learning Network and the Cognitive Coaching model, as well as the Instructional Coaching model from Jim Knight et al, we've moved away from an evaluative assessment model for observations to a collaborative improvement model.  Dave has written a few excellent posts on coaching. One of the things we try to do for the TAs is a coaching demonstration, where they can see what this process is like. Very understandably, they are nervous about being observed. This semester, I've came in and taught a lesson as if to my preservice secondary teachers, and then we debriefed for the demonstration.

We ask the students for whatever they're using for teaching notes (as opposed to expecting a lesson plan) and for them to fill out an action plan. My lesson plan is pretty bare bones... many student teachers wanted to know if that was okay. Yup. The principle is do what is helpful to you. What I brought:
Greater Than Lesson Plan
Math 229, 9/26/12

Objective: understand nature of inequalities and their interaction with operations; apply to measurement

5 Start Up, DOS, share HW
45 Greater Than Game
SA: Fill in blanks: 3 ___ 10; -3 ___ 2; -1 ____ -3;
F: explain game rules, play a round vs whole class
A: they play; pose question “what effect did operations have on inequalities?”
R: (10 min) discuss
•    what effect did operations have on inequalities?
•    strategies in game
•    suggestions for improving the game

50 Error analysis investigation
SA: Measuring a line with +/-
F: overview of measuring stations; introduce contest
-they measure volumes and areas
-class-wide table of results
R: explain how did they compute errors for their measurements
5 Contest reveal; debate winning conditions if there’s an opportunity
5 Concerns/ Wrap Up

To Do:
Observation Journal WS how many/how much investigation

playing cards
rulers/measuring tape
contest jar
I realized I omitted the lesson objective of analyzing an activity as a teacher, and put that in for the teaching of the lesson 'for real.'  We only did a representation of the first part of the lesson for the coaching. Here's the action plan:

Dave took notes while I was teaching. (He may produce a sharper version of this.) When we're taking notes we often make note of other things to talk or think about aside from the action plan, but don't necessarily bring them up unless it is something that could greatly help the observed.

The dialogues following an observation are some of my favorite times at work. Discussing teaching with another professional based on something that we both just saw happen is amazing. Exciting as a teacher and satisfying and growing as a colleague.  We've tried to think of a way to encourage this kind of teacher to teacher interaction at the university, but it hasn't happened yet.

The video of this coaching isn't exactly like what it is when not for demonstration, but what is here is pretty authentic. (For one thing, the discussion is usually 30 min to an hour.) I did make some of the changes that were suggested here or in the discussion afterward with TAs participating in the coaching. They recognized the value of positive feedback, seemed less intimidated by the prospect, and were interested in how the person being coached does more of the talking.

What: the game does work for generating experiences to consider inequalities. I think it also works on the level of an activity for novice teachers to evaluate for use in class. It raises the issues of materials, effort to implement, and when a game might be good in class.  (Some of this is based on my use of the game with my PSTs.) I did hear a couple things that made me think about how do I make sure more of the connections that I'm thinking of get shared in class, for example the connection to War.  I do like the structure of the game.

So what: I want to consider the idea of generating a good demonstration game vs authentically playing a game to model.  I feel hesitant to stage a game, and I'm not sure why.  I want to think more about when should a game be well-determined and cleanly set, and when should a game be something that you kind of unroll over a few days.
        The idea of being clearer with preservice teachers about the difference between how things work in our college classroom and how it would be with K-12 students is definitely a valuable one. 

Now what: I did clean up the instructions as they suggested. I think the game is good as is for high school. The choice of cards vs flipping off the top helps emphasize that the only thing that flips the inequality is multiplying by a negative. The flipping is a nice variation for younger students who are focused more on operations with signed number. As a game I have to think more about the full strategy version. Is there a way to make it a real game?
        It's a constant danger that PSTs will decide our experiences are irrelevant when confronted with the schools as they are. So thinking about this transition and connection piece from teacher education to teacher reality should always be considered.  I also want to continue to be sensitive to the idea of big shifts and subtle shifts. Subtle shift - questioning, medium - finding a game online, big - designing your own game or writing your own curriculum.

Note: I'll shortly have a post up that's just about the game with the current version.

Image credit: By Sgt. Robert Adams [Public domain], via Wikimedia Commons

Tuesday, September 25, 2012

Think Aloud: Planning Inequality

From the always fun Zappa Blamma
In my preassessment for preservice teacher (PST) high school mathematics course, I had a new topic grab their attention - inequalities.  In previous times teaching this course, that hasn't been an issue to which we've given much attention. My bad, as I've seen plenty of struggles with it in secondary classrooms. It's clearly an error where we retreat to instrumental understanding (using Skemp's terminology); rules, rules and more rules.

So what are the key ideas? It's an extension of the algebraic methods we're already using. So I want to focus more are the conceptual components of inequalities... which I haven't ever really deeply considered before.

The heart of inequalities is comparison, which is a key component of number sense. A key and often omitted component. Usually we introduce numbers (multi-digit, fractions, decimals, signed) and then are lucky if we even get representation work before we get to computation rules and rote. So that makes me think of a couple of my favorite games - Fraction Catch and Decimal Pickle. Probably not appropriate for a high school class, though I use them in the preservice middle school class to good effect. What would a high school version look like?

Hmm. It's the rules for changing the sign that get people all bothered. What if the players/teams started with a number, then multiplied or added to both players numbers... the goal could be to get the larger quantity in the end.  You could make special cards... or (I love being able to use regular dice, cards, dominoes, etc.) have the suits imply the operations. I'd have to think of how many turns, how many cards to have, some of the game mechanics. But that idea might be worth the work of developing something new.

Before being in math ed, I worked in index theory, that includes plenty of analysis. Analysis, as a mathematical field, involves a lot of estimates. Think nitty-gritty epsilon-delta proofs. This is less that that, which is less than or equal to the other, which means the whole thing is less than... I loved that stuff. So is there a way to get students making those kind of estimates?

That makes me think about experimental error. Error analysis is nice because of the connections to measurement. That gets us into volume formulas. Could even do a guess how many kind of competition. Hmm. I took a picture of a neat set up like that at the ND-U of M football game over the weekend, but it came out too blurry. And I don't know the result! (I guessed 1337. [LEET.] The bottom pyramid was 10 balls wide and about 20 balls tall.) Lots of good images for that on Could use in class or make a good HW assignment out of that, and get the PSTs looking at that great resource. It would also let us talk a bit about measuring technique - important to get in somewhere. Volume ties into some of the higher degree polynomial stuff we've been doing...

I think that would make for good foundational experiences, and then we could do a follow up day to see how the ideas of inequalities we see apply in traditional algebra and algebra 2 problems.

Now to get working on that game...

ps. What I didn't do here that I usually do when planning, is look around what other people have about inequalities. That usually involves a visit to Sam's virtual cabinet, searching reader, raiding Kate's archive, etc. Where do you look for inspiration, adaptation and out and out robbery? (Accredited, of course.)

This story continues in the post on coaching. With video of an actual dialogue!

Saturday, September 8, 2012

Slant Wise

I caught a great talk by Eugene Peterson this week. He's a pastor and spiritual writer who gave a talk at Valparaiso University in honor of Walter Wangerin, Jr. (who was also there); the talk was "What are writers good for?" (I found a pdf of a previous iteration of the talk.) There'll be mention of God below, but really, these are my connections from his talk to math teaching. I've written one other post inspired by Peterson, Jonah the Math Teacher.

Peterson's bottom line for writers is that they can reclaim language from debasing use. For religous writers this is particularly important, because we as a society have turned our spiritual words into godtalk that is easy to ignore and not worth our time.  In other words,

Knowledge of speech, but not of silence;
Knowledge of words, and ignorance of the Word...
Where is the Life we have lost in living?
Where is the wisdom we have lost in knowledge?
Where is the knowledge we have lost in information?

-TS Eliot, Choruses from The Rock
Poems and Plays, 96

Important ideas, I think. And amazingly close to the challenges we face in mathematics teaching.

"So what are writers good for? It is our vocation to maintain and practice this core, basic, revelational, personal nature of language, living speech.    In a world in which language has been uprooted from its originating God soil and put to the use of information or propaganda, it is the vocation of writers to represent and practice language as revelation, to re-orient language into the personal world in which men and women actually live—in their families, and neighborhoods and workplaces," says Peterson.

What are math teachers called to do? To recover the debased math, practiced in schools for years, to bring it into the world in which the students live, to share math as it's truly done, to share learning that will make a difference in our students' life.

So how does a writer do it?  For Peterson, the illustrating example is the middle parable section of the gospel of Luke. A lot of Jesus most powerful teaching. Stories with no explicit mention of God, no direct lesson. A story about fertilizing a tree instead of cutting it down. Told in Samaria to people who are not interested in his religion, and not fond of his people. He might as well have been a math teacher.

Not to double up on poetry, but Peterson also went to Dickinson.

Tell all the Truth but tell it slant –
Success in Circuit lies
Too bright for our infirm Delight
The Truth’s superb surprise
As Lightning to the Children eased
With explanation kind
The Truth must dazzle gradually
Or every man be blind
-Emily Dickinson

(Fantastic art by  David Clemsha. Shows up in my Reader the next morning as a wonderful coincidence.)

How well does that capture the essence of good teaching?  Peterson notes, "A parable keeps the message at a distance, in the shadows, slows down comprehension, blocks automatic prejudicial reactions, dismantles stereotypes. A parable comes up on listener obliquely, on the slant." A writer does this by having the reader come to them, going slow, countering the impatience of the age.

This leaves teachers the challenge of knowing what is best, in a culture that wants what is lesser. When we propose that there is better, they want solutions that are immediate and rushed. How can we convince them that the answer is slow? Real stories, finding the way themselves, experiencing the superb surprise. I think we have to just persist. Write our real lessons. Participate in the community. Share our success stories that will buoy us through a dozen bad days.

"The Truth must dazzle gradually."

It's our vocation to tell it slant.

Friday, August 31, 2012


By Samuel Hansen from Flickr
I've been asked multiple times this week about good iPad apps for math... so time to put together a list. I have not had an opportunity to use most of these with students yet, so I'm going on my impressions and recommendations. Many of these do have android versions. I'd love for you to add your experiences or recommendations or opinions in the comments. I have a previous post about iPad for teachers, too. My first response to people was to send this list from mathxtc, which has a lot of goodness on it. I'm particularly interested in  Solids Elementary HD (explores nets and solid geometry) and the Mathination touch solver or Algebra Touch.  Also note the official Apple collection of iPad math apps. (ITunes link. Most of the links in this post are web links.)

The big question to start with is "wifi or not wifi?" On the web, start with Wolfram|Alpha and Desmos.

I also have to give a shout out to GeoGebraTube which on a mobile device automatically optimizes sketches for mobile use. Input boxes and some other features don't work. It's worth noting here that there's a Kickstarter for a free GeoGebra for iPad app. I think they're having difficulty because it's hard to perk something which is going to be free. Very worth of our support as a community.

There's a lot of video lesson archives, such as ... you know, but I'm not going to cover them here for fear of terminal irony.  Vijay recommended Fraction Basics which does interactives plus video and has some free samples. (There are a few with the model of free video app, buy the content.)

The Calculator
If you have an iOS device, you should not need a calculator anymore. However, I have yet to see a calculator that combines regression with the nice graphing and computation that's available. Who will finally save us from TI's decades of benevolent tyranny?
  1. Wolfram|Alpha - the reasonably prided app. I also like the course focused variations that make the interface more direct for students. This I have used with students and it is powerful.
  2. Quickgraph - free, also Quickgraph+ for 1.99.  Good computation, nice interface, good touch interaction. (Worked with one of the authors, Alejo Montoya, on ParabolaX when he was at GVSU.)
  3. Free graphing calculator - upgrades to Scientific graphing calculator. Both very serviceable.
  4. Calculator Pro -free scientific calculator. No graphing but faster for computation than the others. Simple and clear.
Note that there are games here, but they are pseudo-games. The game structure is gamified content, as opposed to be a game that also addresses content. I obviously don't have a good way to describe this yet. I'm not against gamification, but think if it's overused it will have a short shelf-life.

  • Dragon Box - winning awards and notice left and right. $3 or $6. Content: solution of linear  equations. Should help students learn which manipulations are permissible, and maybe even choose which manipulation is desired, but I have qualms because there is no algebraic reasoning as to why we can do these things. My 8th grade daughter even noticed that she could figure out how to solve anything, but didn't know why anything worked. Transitions kids from pictures to regular algebraic notation. Clever and really well executed, but possibly dangerous as instruction. I'd use it as an alternate representation after some inquiry. Christian Bokhove is a big fan and pointed out that it's multi-platform.
  • ParabolaX - our GVSU contribution to algebra games. Content: factorable quadratics. Designed by Alejo with Char Beckmann. The idea of the game is to use game conditions to gain students experience that supports later symbolic understanding.
  • Sketchpad Explorer - free. Let's you explore Geometer's Sketchpad files on the iPad. Key Curriculum was nice enough to send along the link to their sketch sharing site: Sketch Exchange

  • MathEvolve - Lite and $2. The Dragonbox of elementary number operations. I wrote a post about it already.
  • Everyday Math Games like Top It. Usually $2. Love it or hate the curriculum, they have some excellent number games for computation practice. They were an early leader in putting them online, but that was only accessible to schools using the curriculum. Now they're apps, so everyone can have access. Follow the iTunes links from Top It to see the full selection. (No easy way to link to that.
  • Protractor 1st - free. A super-protractor.
  • Geoboard - free. Infinite rubber bands. Finally.
  • Number rack - free. Rekenrek with clicking beads, even. Great number representation.
  • Motion Math - $2 each. Really a stand out in this category. Get the feeling of a game, use the full range of input of a mobile device and they're making several games that are new and great. Try starting with Motion Math fraction game.
  • 24 game - $1. Great IRL and just as good as an app.
  • Set - $5. Remember that's cheaper than an actual set of Set.
  • Entanglement - $2. Graph theoretic reasoning that's really absorbing.
  • Pick-a-path - free. Interesting computation and estimation game from the NCTM.
  • Concentration - free. Early number representation matching game from the NCTM.

  • Soulver - $3. Interesting effort to blend verbal and symbolic representations ... not sure if it's there yet, but I like where it is going.
  • Mobimath - Lite or $1. Needs cellular to do distance, but still does angle of elevation. This kind of real-time data collection will be awesome once it's smoothed out. Even just the angle of elevation bit is neat, though.
  • Loopy - free. Trying to push the boundary of an interactive virtual manipulative, but it doesn't quite work for me yet. They're onto something neat, though. Free, so give it a try.
In addition to all this, there are a lot of excellent puzzles that echo both classical math problems (like tangrams) and newer archetypes (rush hour).  You might start with some of the neat Think Fun games, like Chocolate Fix.

This also doesn't get into the tools that students can use to record or create their own content. Paul Macneil recommended Explain Everything ($3).

Hopefully this is a good starter selection. What's missing? For what objective or classroom activity would you like to see an app? Do you know somewhere where the various instructional materials are written up?