Graphing Stories Meets Estimation 180: A Love Story

On Friday we started our Functions unit, and as always, my go-to is to start with Dan Meyer's Graphing Stories.

After they pick up a Graphing Stories worksheet, I give them a brief set-up, explain that the guy in the video is my friend Dr. Meyer (though now I have to explain that this was a long time ago, in a galaxy far, far away, when he was a young Jedi-in-training), and promise them that I will rewind the videos as many times as they want, to whatever point they want, for as long as they want, so we can figure out as closely as possible what their graphs ought to look like.

Portrait of the artist as a young Jedi warrior

Only this time, we encountered a spontaneous twist.

The first video went off without a hitch. Using QuickTime, I have an action-only version of the first 15-second video (Dan walks over a bridge in a Santa Cruz park), so that it doesn't accidentally reveal anything I'm not ready to reveal yet.

I explained that my friend Dr. Meyer is unusually tall (I gave his actual height, which can be found on his web site) and this gave students the idea to "measure" him onscreen so they could try to better estimate the rise of the bridge.

This gave me an idea.

The second video in the series is of Dan descending some exterior condo stairs, but after the first viewing, an argument broke out in the discussion as they tried to find some hook they could use to improve their estimations. "A car is, like 6 feet high!" "No, the stairs are about 5 feet in total!" Blah blah blah.

Enter Estimation 180 thinking.

My classroom is right next to the door to the stairs up to the small faculty parking lot. I pointed to two kids. "OK, you and you — take a yard stick, carefully cross the driveway, and go measure some average car heights in the teachers' parking lot."

The arguments continued so I pointed out another kid, one who was extremely interested in the stair height but who has never before piped up in class. "Pick a helper, grab a yard stick, and go out to the stairs and find an average for the stair height."

While our data gatherers performed their missions, the estimation arguments continued inside the classroom. "The car is only about 4 feet high!" "No, it's not!" "What about the lamp post—can we use that?"

Five minutes later, our intrepid explorers returned and we harvested our newfound information.

Then we continued working on the Graphings Stories task.

The funny thing was that our estimates led to wildly wrong answers, but that wasn't the most important thing. Instead, what was powerful was the level of engagement in discovery that electrified the room.

Instead of considering the need to quantify to be yet another tedious task that stood in the way of getting "the right answer," students started to lose themselves in the flow of the process of mathematizing their world.

And isn't that the whole point?

So thank you to Dan (@ddmeyer) and Andrew (@mr_stadel) for giving me the tools to slow my kids down and help them to find the wonder in everyday situations.

PS — I still haven't revealed the "answer" to Graphing Story #2 yet. But I'm curious to see what unfolds in class when we come back tomorrow. ;)

Scaffolding Proof to Cultivate Intellectual Need in Geometry

This year I'm teaching proof much more the way I have taught writing in previous years in English programs, and I have to say that the scaffolding and assessment techniques I learned as a part of a very high-performing ELA/Writing program are helping me (and benefiting my Geo students) a lot this year.

I should qualify that my school places an extremely high value on proof skills in our math sequence. Geometry is only the first place where our students are required to use the techniques of formal proof in our math courses. So I feel a strong duty to help my regular mortal Geometry students to leverage their strengths wherever possible in my classes. Since a huge number of my students are outstanding writers, it has made a world of difference to use techniques that they "get" about learning and growing as writers and apply them to learning and growing as mathematicians.

We are still in the very early stages of doing proofs, but the very first thing I have upped is the frequency of proof.  We now do at least one proof a day in my Geometry classes; however, because of the increased frequency, we are doing them in ways that are scaffolded to promote fluency.

Here's the thing: the hardest thing about teaching proof in Geometry, in my opinion, is to constantly make sure that it is the STUDENTS who are doing the proving of essential theorems.

Most of the textbooks I have seen tend to scaffold proof by giving students the sequence of "Statements" and asking them to provide the "Reasons."

While this seems necessary to me at times and for many students (especially during the early stages), it also seems dramatically insufficient because it removes the burden of sequencing and identifying logical dependencies and interdependencies between and among "Statements."

So my new daily scaffolding technique for October takes a page from Malcolm Swan and Guershon Harel (by way of Dan Meyer).

I give them the diagram, the Givens, the Prove statement, and a batch of unsorted, tiny Statement cards to cut out.

Every day they have to discuss and sequence the statements, and then justify each statement as a step in their proof.

This has led to some amazing discussions of argumentation and logical dependencies.

An example of what I give them (copied 2-UP to be chopped into two handouts, one per student) can be found here:

-Sample proof to be sequenced & justified

Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts! The conversations about intellectual need have been spectacular.

I am grateful to Dan Meyer for being so darned persistent and for pounding away on the notion of  developing intellectual need in his work!

Algebra 1 inequalities unit - notes on a conceptual and problem-based approach

My Algebra 1 inequalities unit is now the unit with which I am most satisfied, including a solid conceptual launch and framework, some serious problem-based learning, a deep treatment of quantitative reasoning and logic, and an excellent amount of discourse-rich practice activities designed to achieve both procedural and conceptual fluency, so I want to document here how it works. If there is useful stuff in it for you, then great! But I'm not going to be posting a lot of user-ready materials here to print off and give to students, so I want to be totally up front with you.

Inequalities are still one of the most procedurally taught of all Algebra 1 topics (see Holt Algebra 1, CPM, Exeter, or anything else you can find, if you have any doubts about this), and yet, it has seemed to me for a long time that they offer the opportunity for students to ground their new understandings in one of their most accessible, intuitive existing mathematical understandings — namely, their sense of when a quantity is "more than" or "less than" some other quantity.

As Nunes and Bryant clearly show, children have a very deep understanding of comparing quantities from a very young age — including both discrete and continuous quantities. For this reason, it has long seemed essential to me to hook Algebra 1 students' work with understanding with their own authentic and meaningful prior knowledge.

So in my unit,we start by activating prior knowledge with Talking Points and problem-based learning from the Exeter Math 1 sequence. In the initial part of the unit, we investigate whether quantities are going to be more than or less than other quantities. We go from comparing known, computable quantities (such as one sum or difference to another sum or difference) to more abstract quantities (such as, If x is greater than -1, then x + 2 is greater than 1), always pressing them to rely on their own understanding or constructed understanding.

The conversations that were sparked during the initial phase — both this year and last year (with Patrick Callahan in my room) — were fascinating to all.

My rule for this unit is this: whenever I give students a "rule" (or whenever we develop a "rule" together),  it has to be a guideline that will help them to ground their new ideas in their old understanding of something. This has led me to come up with much more sensible and conceptually-based guidelines that students both remember and rely on correctly. For example, one of our "rules" is that when you have some kind of inequality, it makes sense to put it into "Number Line Order" — in other words, always organizing statements or representations about smaller quantities to the left of statements or representations about larger quantities. This has had the benefit of getting students to realize that the real number line is a tool that they can use for their own benefit. It is not just another arbitrary math teacher whim.

Another "rule" we co-developed as a community was the idea of the subjectivity of x — namely, that in mathematical sentences and statements, we should organize our symbolic representations so that x is the subject of our sentences. Another way that students expressed this idea was that x is the hero or heroine of our story, so x should be the subject of our sentences.

This too has had a profound effect on students' understanding of inequalities. Most textbooks emphasize equivalence of the statements 2 > x and x < 2, but from the perspective of student understanding, privileging both statements equally is just stupid. What we are hoping for students is that they will see that as they are trying to represent values of x, that makes x the hero of our story. So our mathematical statements should be organized to reflect the subjective status of x — not the "correct" or "incorrect" equivalence of the statements. This gave students the idea that they have permission to change statements around to suit their own  learning, and that they do not need to distort themselves to please whatever mathematical authority decided to place given statements in a form that is backwards for them.

We have also developed a "rule" for distinguishing absolute value expressions from other groupings in equations and inequalities. The idea we have developed is that absolute value expressions need to be "isolated" and "unpacked" into all possible cases of ordinary equations or inequalities (i.e., NON-absolute value statements) before you can apply any solving or simplifying techniques. Our shorthand for this method is: 1 - ISOLATE, 2 - UNPACK, 3 - SOLVE. Then once you have solved, you should put everything into Number Line Order so you can INTERPRET and GRAPH your findings.

This idea of unpacking absolute values first into all possible cases is a strong way of getting students to reason quantitatively and abstractly rather than blindly applying rules. Since they have some guidelines that are grounded in their own logical understanding and personal experience, my students have been much more thoughtful and intentional when they approach absolute value inequalities and equations. It has also led to a deepening of students' understanding of when the distributive property can and should be applied to a grouping and when it should not. This has greatly deepened the discussions of the functioning of groupings among students to the better.

Because we have build a solid conceptual foundation for our work with absolute values, practice activities have become opportunities for investigation and application of our conceptual understanding, rather than blind shooting at a list of targets from 1 to 35 odd.  At every step of the way, we have been using Speed Dating as a discourse-rich activity in which students can apply their understanding to a wide range of different problems of varying levels of complexity. It also demands that they share their understanding and speak about it with their classmates, sometimes giving and sometimes receiving assistance.

In this way, practice activities have become a new form of conceptual investigation themselves, and are not mere mental exercises designed to enhance procedural efficiency. As I am discovering through my work on National Board Certification, our profession has a major problem with privileging "discovery" activities as the only ones that occasion and showcase conceptual understanding. As How People Learn clearly lays out, it is often the case that students do NOT have a 'Eureka' moment during the initial 'discovery' segment but rather, like the young learners they are, they have a lightbulb moment once they have experienced a problem in multiple different contexts. This leads me to one of my pet peeves about why teacher certification and National Board Certification overvalue only the initial discovery activity as a venue for showcasing student insight, but that is a complaint for a different blog post.

Another benefit of treating practice as an opportunity for applied investigation is that it avoids the contrived, dog bandana pseudo-context problems that strive to provide a real-world context, but only end up twisting themselves into senseless knots. The contexts should not be harder for students to make sense of than the absolute value problems themselves! The most accessible real-world absolute value problem we have worked with is the question, When is water NOT a liquid? This makes sense to my 9th graders. Convoluted problems about the conditions under which a value will land within a range of possible values are not helping here, people. I am finding that this is a case where treating absolute value problems as a system of equations or inequalities to be unpacked and investigated as number sense problems has been far more worthwhile — and far more rewarding for students. As Deborah Ball has said, Sometimes mathematics is its own best context.