Take Time to Save Time – Hall of Fame reference sheets

Inevitably, teachers get known for their mottos. Sam's mottos are justifiably world-famous. Personally, I love "Don't be a hero." Mine are known mostly around my school, but it is interesting to see how they trickle down into students' unconscious minds.

Color telling the story

Mottos pay off. My favorite is one I stole from my former colleague Alex Wilson: "Color tells the story." I don't understand how anybody can do math at a deep conceptual level without colored pencils. Color really does tell the story, especially in Geometry (see popular worked example at right).


One of my best math class mottos comes from published patterns for knitting. It is, "Take time to save time." In knitting, this means to make sure that the tension of your actual knitted work — your hands, your needles, your yarn — match the tension or gauge described in the knitting pattern. There are no shortcuts here. My knitting gauge tends to be extremely big or loose compared to most pattern-makers. I often have to use much smaller needles than specified in order to achieve a good match with the specified knitting gauge.

In my classroom, "Take time to save time" means, synthesize your learning into a reference sheet. For all tests but the final, I allow students to have and make a half-page reference sheet.  The first rule is, you can have anything you want except a photocopy of my work on your reference half-sheet. The second rule is, if you have more than a half sheet of 8.5 x 11 inch paper, then I get to tear it in half and choose which half you get. This rule gets tested even when I emphasize it. Every year somebody tests this rule. "But Dr. S! I only wrote a half-page worth of stuff on the paper!" It doesn't matter. I usually rip the whole thing lengthwise so they only get the right-hand half of the paper.

It makes its point.

In knitting, this point gets made by the scale and size of your finished object. If you insist on not checking your gauge, at some point, you will end up with a finger-puppet-sized sweater or a scarf the size of Lake Tahoe.

Clearly this student is going to ace the final.

In our classes, this point gets made by your performance on our common final exam. Students who have been practicing making clear, concise, summaries and examples of their work and key points tend to turn in consistently strong performances. So on the final, I allow a full-page reference sheet (both sides). I emphatically want students to consolidate their understanding and create their own examples. That is where the learning happens.

So I was thrilled today when I asked to see examples of in-progress reference sheets. Many of them made my Hall Of Fame request to scan for posterity. This Algebra 1 student has totally nailed her understanding of mixture problems. This is the best example I've seen of a student consolidating her understanding of these modeling challenges.

First thoughts on completing Exeter Math 1

I just finished doing the 2010 edition of Math 1 (91pages) today.  Now begins the synthesizing and summarizing, which I will put into blog posts.

Math 1 is an Algebra 1 course that includes an incredibly deep coverage of proportional reasoning, in addition to the usual linear, quadratic, and exponential function topics.

I did Math 1 because most of our incoming students are incredibly bright and hard-working but they were not the math monsters in their middle schools. They have many of the typical middle school gaps, but they are much more sophisticated than most 9th grade Algebra 1 students. So the fact that Math 1 is a REALLY TOUGH course that dives very deep into Algebra 1 material is a great thing because it will give my students the deep rich course they deserve, even though they are placed into Algebra 1 based on their current skill level.

My Algebra 1 learners find themselves stuck in a ZPD no-man's-land: their ZPD as math learners is nowhere near their ZPD as readers. 

This presents a huge problem in the classroom. The math in CPM Algebra 1, for example, is rich and interesting, but the text is written for reluctant readers, discouraged readers, and English Language Learners, which is a huge turn-off for the vast majority of my enthusiastic and highly capable readers.

They feel insulted by it, and they are not shy about expressing these feelings. So my student population tends to dismiss it and resist it, even if they really do need to learn the content. This raises the question of how best to serve a population of learners who need to be challenged with greater nuance in textual interpretation and presentation in an introductory high school math class.

For all of these reasons, Math 1 is going to form a terrific problem-based “spine” for my Algebra 1 classes. The problem sequences are rich and interesting and engaging with sophisticated contexts, though they start from first principles. They develop to a point where even a mathematically sophisticated adult will find them very challenging.

To get started, I printed all pages of the problem sets, answer keys, and commentaries and created a binder with the following sections:

1 - Problem Sets plus glossary at the end

2 - Commentaries

3 - My Worked Solutions (for each page of problems, I have one stapled cluster of my worked solution pages)

4 - Answer Keys

I did all of my work on three-hole binder paper, with each new page from the problem set being its own stapled packet (or "blob") in the Worked Solutions section. 

Whatever problem set I was working on I would take out of the binder along with the relevant answer key page. That way I could work on binder paper without having to carry the whole damn binder around all the time. Much of this work was done on a lap desk with my iPhone/Desmos for graphing, my TI-83-plus (sorry, Eli) for computation, and my monkey pencil case including my mechanical pencil, my ProRadian protractor, and my colored pencils.

A lot of people have asked me why I started at the end and worked from the end forwards, about 10 pages at a time. The answer has two parts: (1) whenever I started from the beginning, I bogged down or got sidetracked; and (2)  it enabled me to see where we were going and where students would end up. By seeing where they would land at the end of the course, I could better understand how things worked from the beginning.

More thoughts coming soon, but I wanted to capture these ideas right away. If you have specific questions you'd like to discuss, please put them into the comments section below.

Algebra 1 Systems of Inequalities - Dan Wekselgreene's Ohio Jones & the Templo de los Dulces treasure map

Some of Dan Wekselgreene's early puzzles, lessons, and projects are truly love poems for Algebra 1 students. And I have loved his Ohio Jones and the Templo de los Dulces systems of inequalities puzzle since the first time I read about it, did it, and used it.

So there was never any question that I would use it with my Algebra 1 students this year. The only question was, how would I make it accessible to my blind student?

Susan Osterhaus of the Texas School for the Blind and Visually Impaired has been generous beyond words with her ideas for teaching math to blind students. Her web site is filled with ideas, best practices, and links to resources for making mathematics accessible to blind students. I cannot recommend it strongly enough.

Here is how I adapted this activity:

FRONT PAGE OF DAN'S WORKSHEET

I typed out each of the three clues as a quote by a statue — i.e., Statue 1 says...

Then I plugged each quote into an online Braille Translator (I like http://brailletranslator.org) and downloaded the Braille text file as an image. I copied and pasted each image file onto an Omni Graffle document (though you could also use Word or Pages) next to the regular text quote. That way the student and the paraprofessional aide could easily collaborate and share information.

This took three pages, but it worked.

I traced the basic map at the bottom of the page but without all the grid lines. This became the "map" for this part of the puzzle. 

Then I copied these pages onto capsule paper and ran them through the PIAF (Pictures In A Flash) machine to create a tactile worksheet with Braille and a raised map. The PIAF machine (affectionately known around the math office as "the toaster") takes the capsule paper with all its delicious black carbon-heavy areas and raises them to create a tactile graphic that can be read by a Braille-literate blind person.

After solving the system and figuring out the target region, my student used Wikki Stix on the map to make a graph.

BACK PAGE — THE MAP

I traced the "big picture" outline of the map to remove as much visual noise and clutter as possible from the main image.  I added Braille labels to indicate the start and the hint at the end of the map:

I scanned this file, printed it on capsule paper, and ran it through the PIAF machine. Again, the student worked on Braille graph paper, then transferred her results to her tactile treasure map using Wikki Stix.

For each "sector" of the map, my student used Braille graph paper and Wikki Stix while her classmates used pencil and the grid on the worksheet.

It was such a joy to see her as just another team member at her table, doing mathematics and solving a puzzle. It was even more exciting to see how her table mates appreciated her mathematical skills.

All in all, a successful experience in creating an inclusive classroom!

My reduced version of the Teacher Packet (including the worksheet and instructions) plus the Braille-ready package are all on the Math Teacher's Wiki.

Wikki Stix are available in a big box on Amazon or any kids' art supply store.

Betweenness and non-betweenness: absolute value inequalities and Patrick Callahan

I felt a little nervous about having Patrick Callahan come to observe my classroom yesterday, but in the end, it was fun. I had asked one of our security guards, to bring him down to my room when he arrived at our school. He walked in as he always does, all mathematical open-mindedness and pedagogical curiosity.

And we got started.

I felt anxious about having him observe my conceptual lessons about betweenness and non-betweenness. I have never seen anything even close to how I understand and talk about absolute value and inequalities. I talk about boundary points and betweenness and I have students hold up their fists and point their thumbs to show me their understanding. “Is this a situation of betweenness — or NON-betweenness?” I demonstrate with my own fists, swinging my thumbs inward or outward. “Your fists are the boundary points and your thumbs are how you shade your graph on the number line. So is this a situation of betweenness... or NON-betweenness?”

If it is a situation of "betweenness," then students point their thumbs inward towards each other, touching the tips together. If it is NON-betweenness, then they point their thumbs outward in either direction, like a group of indecisive hitchhikers. And once we have done this analysis, then we can do whatever calculations we may need to find our boundary points.

So much of advanced algebra and precalculus depends on having this kind of deep conceptual understanding and thinking. Am I looking for quantities that are GREATER than...? or LESS than? Is this quantity going to be positive? or negative?

For me, the whole thing is intimately hooked together with the real number line. And with number sense. 

When we started last week, we began with an inquiry into “more than” and “less than” and widened our thinking outward from there.We connected more than and less than to number line thinking. I always emphasize Number-Line-Order and Number-Line-Thinking in my Algebra 1 classes. If they think about the number line, then they can anchor their thoughts in their bodies. LHS (or Left-Hand Side) and RHS (Right-Hand-Side) are fundamental ways of thinking in algebra. These ideas are eternal and unchanging. The number line is the foundation of everything. It gives you the “true north” of the real number system.

So we always ground our thinking in our bodies. I ask, “Left Hand Side or Right Hand Side?” “Is this a situation of betweenness or non-betweenness?” “OK, now that we know that, now what?”

I also anchor this unit in what they know about logical reasoning. They have an intuitive sense of how many possible cases a situation may present. I've been a huge Yogi Berra philosophy fan all my life, so I believe that when you come to a fork in the road, you should take it. When you come to a fork in the road, you can go left or you can go right. Or you can stay right where you are. Three possible cases. Over and over I ask them, “What’s going on here? How do you know?”

Absolute value inequalities are either situations of betweenness or situations of non-betweenness. Figure that out and then everything else will run smoothly. Then all you have to do is to use what you already know.

Once students have gotten that figured out, it’s just one more small step to combining their new knowledge with their existing knowledge. Follow the order of operations and common sense. Plus everything you know about the real number line and multiple representations. Then things can naturally unfold the right way.

But I always come back to number sense to what we know about the real number line. Numbers are the ground, the foundation.

So when Patrick walked in yesterday — this world-class mathematician and math education expert — what he encountered was my bootcamp in algebraic thinking. “Hold up your fists! Is this a situtation of betweenness or non-betweenness?”  "How do you know?" And then my waiting until everybody’s thumbs are pointing in the same direction.

It is Logic 101 and numbers and anchoring our thoughts about numbers in our bodies. Like the ancient Greeks and Babylonians and Egyptians before us.

Our next step is to solidify our thinking through what How People Learn calls “deliberate practice with metacognitive awareness.” We are going to do two days of Speed Dating. Now I have to make up Speed Dating cards and a test to use on Thursday. 

And then to document my thinking.

When the class ended, Patrick came up to my tech podium and was excited. He grabbled a whiteboard marker and started sketching and pouring out ideas.

For me, that was the best possible review I could have gotten on this lesson. A five-unicorn review. A direct hit. :)