The Festival of Reassessment (SBG)

December is when I am truly grateful for strong routines. They mean I can split the class up and still count on everything moving forward. I am running behind in my pacing, so I needed to set up a mass SBG reassessment opportunity for slope skills yesterday. I set up all the reassessors along the window side of the classroom and all of the non-reassessors on the hallway side of the room. The non-reassessors worked on linear systems skills and problem-based learning while the reassessors worked on demonstrating mastery of slope skills.

Whenever a reassessing student finished their work, they brought it up to be rechecked. I went through it right then and there while they watched. “Uh huh... uh huh... uh huh...” until “Oh noooooo! Why is this negative?!?!? It ruins everything! Go back and fix it!!!” And then the next student moved forward in the queue.

We went through this process of working and my checking and sending kids back for about 40 minutes. “Nooooo!!!” I would circle something and pretend to freak out, sending them back to fix stuff. It became a carnival. “AAACK!" I would say. "Go back and fix this!” Seeing this happen in real-time changed kids’ relationships with their misconceptions. It reminded me of piano practice as a child. Something would splatter in the midst of a phrase or figure and I would have to go back to the beginning, willing the figure to come out right through my fingers on the keyboard.

We kept going until all 12 students had triumphed over slope and point-slope form. What I loved about this process was how it transformed the social focus of the process of understanding to us (the whole class) against the skills. Kids were going wild and cheering when somebody finally triumphed over the "slope and point-slope” process, jumping up and down and hugging their newly non-reassessing classmates and friends.

The goal became one of getting everybody in the classroom over the finish line. From individual mastery to collective. Students stopped focusing on who had status in the classroom and who did not. They stopped thinking about their place in the social hierarchy and instead lost themselves in the flow of doing this f***ing slope and linear equation problem.

And then as each new classmate finally triumphed over the skill, the whole class felt truly victorious.

Everybody got the individualized attention they needed as they drove their skills forward, and we also bonded as a collective community, advancing our work as a group.

In these divisive times, I think this might be an important process to cultivate.

TMC #14 Group Work Working Group Morning Session – Annotated References & Framework

I'm having a lot of fun planning the Group Work Working Group morning session for Twitter Math Camp 2014, and it's time to start sharing.

Here is the background material I'm using for developing the group work morning sessions. Please note that this is NOT required reading!  Recreational reading only! So please don't freak out!  :)

I wanted to give people a sense of the framework and background I'd like us to start from so attendees can decide whether this morning session will be right for them. I also wanted to provide links and titles to valuable materials.

These are listed in order of relevance to the Group Work Working Group morning session — they are not in formal bibliographical form.

National Academies Press, How People Learn (downloadable PDF here)
This amazing free book provides the framework within which we'll consider the use of group work. I am especially keen for us to explore how we can develop and implement tasks that fit within their (approximately) four-stage cycle for optimizing learning with understanding while also fitting with our own individual school and district requirements. In a nutshell, the four stages are as follows:

STAGE 1 - a hands-on introductory task designed to uncover & organize prior knowledge (in which collaboration cultivates exploratory talk to uncover and organize existing knowledge)

STAGE 2 - initial provision of a new expert model (with scaffolding & metacognitive practices) to help students organize, scaffold, & develop new knowledge (in which collaboration provides a setting to externalize mental processes and to negotiate understanding)

STAGE 3 - what HPL refers to as "'deliberate practice' with metacognitive self-monitoring" (in which collaboration provides a context for advancing through the 3 stages of fluency with metacognitive practices)

STAGE 4 - transfer tasks to extend and apply this new knowledge & understanding in new and unfamiliar non-routine contexts

Malcolm Swan, "Collaborative Learning in Mathematics" (downloadable PDF here)
A short and highly readable summary of Swan's instructional design strategy for collaborative tasks, including notes on his five types of mathematical activities that constitute the bulk of the Shell Centre's formative assessment MAP tasks and lessons.

Malcolm Swan, Improving learning in mathematics: challenges and strategies (downloadable PDF here)
An in-depth introduction to Swan's approach to designing and using the kind of rich tasks offered by the Shell Centre and the MARS and MAP tasks.

Chris Bills, Liz Bills, Anne Watson, & John Mason, Thinkers (can be purchased from ATM here)
The richest source book imaginable for ideas for activities to stimulate mathematical thinking. Often credited by Malcolm Swan and Dylan Wiliam.

Anne Watson & John Mason, Questions and Prompts for Mathematical Thinking (can be purchased from ATM here)
The richest source book imaginable for variations on questioning and prompting strategies.

Dylan Wiliam, Embedded Formative Assessment
This book is a gold mine. Don't leave home without it.

Standards-Based Grading, or How Teaching For Mastery is Different

Teaching for mastery is different.

Teaching for mastery especially means giving up a lot of old and cherished assumptions about assessment. Anybody who has adopted SBG in any way can attest to this. But I am continually amazed at how unwilling many of us can be to letting go of old, ineffective methods, beliefs, and assumptions about assessment.

At its essence, valuing mastery means not only tracking relative mastery but also accepting mastery as the measure of student success in our classrooms. And that means letting go of the value we have always placed on the routinized behavior of the the dutiful student.

This is is probably the hardest shift of all.

As I shifted over to SBG, I noticed how much of our system of math teaching is organized around students being merely dutiful: sitting still, listening quietly, practicing silently, accepting information without challenge. It's a model of student passivity that places everybody into the known and accepted hierarchy. The "good" students land at the top. The "middle" students land in the middle. And the "weak" students land at the bottom.

But as we all know from having inherited, taught, and assessed these students, this schema does not measure mastery, skill, or comprehension. Dutiful students often lack conceptual understanding or procedural skills. They often have distorted memories of algorithms they heard about but never owned.

Changing over to an SBG system of teaching and assessment has meant that I have to create conditions under which any student — even ones with problem behavior or lack of "dutiful-ness" — can achieve mastery.

To me, this idea exposes the biggest flaw in the existing system. If a teacher or administrator decides from the outset that a given student is a "B-" student, then what reason does that student have to make the effort necessary for improvement?

This system also fails to allow for individual (or group) movement up the fixed staircase of the classroom hierarchy, except for improvements in "dutiful-ness." And it seems to me that if we want to improve access and equity to mathematics for all students, this is the single biggest obstacle we face.

It also seems to me that we need to consider the possibility that any hierarchical model might be transformed from a staircase to an escalator, in which all students can be expected to reach the target floor or level of skills and understanding. And that means we will have to allow for the possibility that all students in a class demonstrate the mastery that is asked in a way that permits them to receive a higher score than the "B-" or "B+" that they have always been pigeonholed into.

TMC 12 SESSION: Increasing intrinsic motivation using the ideas in Dan Pink's Drive

Dan Pink's bestselling book Drive: ___  has given the business world new ways to think about increasing intrinsic motivation in the workplace, but his ideas have resonance in math education too. The purpose of my Twitter Math Camp 12 session was to summarize the main ideas in Drive and to talk about how I have applied them to the specific situation of the math classroom.

In the model he sets out, Pink presents three fundamental pillars of intrinsic motivation:

• AUTONOMY, which he defines as "behaving with a full sense of volition and choice” as opposed to feeling pushed around by “external pressure toward specific outcomes” (Drive, p. 88). 
• MASTERY, which is a growth mindset in the model of Carol Dweck's work, a way of thinking about one's work that requires both effort and engagement. He also describes mastery as "an asymptote," an impulse that moves toward an ideal of perfect oneness without ever fully achieving it (Drive, pp. 118, 122, & 124).
• PURPOSE, a sense of being connected to the why of what one is doing (Drive, p. 233).

All three of these elements support the development of FLOW — a profound human state of "optimal experience" which was first studied in depth by the renowned psychologist Mihalyi Csikszentmihalyi (pronounced "chick-sent-me-high").

Flow is what many of us who teach math feel when we lose ourselves in doing mathematics, and my argument in this presentation is that helping our students to experience the flow state while they're doing math should be our top priority when thinking about motivation.

We can help students tap into the flow state by using Pink's three elements of intrinsic motivation to create "on ramps" for students to the flow experience.

PURPOSE is a terrific building block for many of our most capable students, but for the discouraged or disengaged student, it is necessary but not sufficient. What Can You Do With This?, Three-Act Digital Problems, and AnyQs? activities can be helpful in cultivating a sense of purpose in students, but it is important to keep in mind that there are other factors — including social, emotional, and psychological factors — at work with our most discouraged students.

Using a Standards-Based Grading framework helps students understand talk about MASTERY by clarifying expectations and improving communication between and among students, teachers, and parents.

AUTONOMY is the hardest of the three elements to encourage, so I spent most of my talk about ways to develop a sense of autonomy with math students.

There are two parts to autonomy: (1) an outer component and (2) an inner component. The EXTERNAL part can be built up by disrupting student expectations through alternative  activity structures. Games, game-like activity structures, treasure hunts, creating foldables, making up dances or songs, creating and performing skits or puppet shows that demonstrate definitions or processes, and other such reframing activities redirect student attention away from what causes them anxiety or trauma and toward something that allows them to relax and let doing mathematics be simply a means to an end. REFRAMING can be a crucial part of helping students find themselves in flow while doing mathematics.

The INTERNAL component of boosting autonomy has to do with helping students to NOTICE their fears or reactive responses and ALLOWING there to be space for their authentic feelings and conditioned reactions. We can support students by not taking their reactions/reflexes personally and by noticing our own reactions/reflexive responses to different kinds of disengagement we experience from students. Encouraging a posture of noticing and allowing enables us to help students loosen their identification with past negative experiences and open up space for newer, positive experiences to overwrite those in their minds and bodies.

By honoring and encouraging the flow state in our students while they are engaged in mathematics, we can help them to renegotiate their relationship with math class. And that creates space for the positive and self-reinforcing intrinsic motivation that will help them get out of their own way and find lifelong success with mathematics.

Renegade Math Teacher Brings SBG into the English Classroom: Film at 11

SBG works so well in my Algebra classroom I've been looking for ways it could transfer into my eighth-grade English classroom -- especially with regard to writing. One of the hardest things about teaching persuasive or expository writing is that each "skill" is composed of multiple, discrete sub-skills, each of which could itself be broken down further into tinier and tinier (i.e., more refined) sub-skills. In many ways, it's an M.C. Escher-like process -- a mise en abîme of nested skills.

BACKGROUND

Our school and district use our own combination of two methods that work for us, adapted through our own collaborative practice, reflection, and research over many years to fit our district goals and population. As a starting point and foundation, we use the Jane Schaffer method for teaching the actual composition process and a slightly modified version of 6 Traits program for assessing the finished product (or the work-in-progress). 

What I like about the 6 Traits assessment system is that it has a strong SBG orientation. It is a rubric-based system that assesses idea development, organization, voice, word choice, sentence fluency, and conventions -- the main basic categories that young writers need to master to produce both competent and coherent arguments, paragraphs, and essays. In our district, we have whittled it down to a four-point scale, which gives most middle schoolers a fighting chance of making sense of their scores.

What I like less about the 6 Traits assessment system is the complexity and denseness of its rubric. As Edward Tufte, the infographics pioneer might say, its information density approaches near-total opacity.

While I appreciate that its authors are trying to be comprehensive, my experience is that, for a middle school or high school student trying to juggle the many skills that come into play in writing a persuasive paragraph, it's just too damned complicated.

MY IDEA

One of the things I've noticed early on in this school year is that even our strongest writing students tend to have only a tenuous grasp of what makes an effective topic sentence. And having taught literature at the university level, I have seen how this confusion tends to persist and worsen over time.

So my goal was to come up with an activity that integrated two tools I've found useful in SBG in the math classroom: (1) a clear, simple, compelling four-point rubric for judging the effectiveness of a topic sentence, and (2) an activity to give students practice in judging a wide range of topic sentences, along with practice in using the rubric as a basic for analyzing, debating, and justifying their assessments of each one.

With that in mind, I created the two tools which are attached here: a Topic Sentence Rubric and a "Judging Topic Sentences" activity for use in pairs or small groups. The "Judging Topic Sentences" activity sheet includes twenty topic sentences I wrote based on a recent writing prompt for that staple of the eighth-grade English curriculum, "Flowers for Algernon." The writing prompt (which was deliberately broadly written) asked the student to compose a persuasive paragraph regarding the author's message in the story about cruelty toward people with mental disabilities. 

I gave them 30 minutes in class to work together on the assessing activity before we came back together as a whole class to discuss and give closure to the process.

RESULTS

What was fascinating as I circulated among the groups was how quickly everybody grabbed hold of the idea of using criteria from the rubric as the basis of their judgments. Suddenly I was hearing arguments about how, yes, a certain claim was definitely true and supportable but was basically pretty trivial! I was also hearing students argue that another example made an "original and juicy claim," but that it was awfully long and wordy and could easily be improved with better word choice and sentence construction.

When we came back together as a class, I asked for examples of the worst topic sentence on the list and the best. The discussion was productive in that it brought students to an understanding that an "OK" topic sentence could kick off a really great paragraph if the writer used all the tools at his or her disposal. It also made them realize that a truly outstanding topic sentence could launch a truly mediocre paragraph if it was followed by weak use of evidence from the text and lame or badly written analysis and interpretation.

My fellow eighth-grade English teachers used this activity in their own way over the next few days and found it to be very helpful in getting students to think about what makes a strong and effective topic sentence.

So now it looks as though it will become a regular part of our writing curriculum. 

Another triumph for SBG!

Something I really did right: Day 1 lesson on really working the rubric

I know I said I was going to go to bed at 8:07, but I wanted to capture and share the essence of something I really did right today in my (mixed 7th and 8th grade) Algebra 1 class.

It was an actual mini-lesson on interpreting -- and more importantly, on using -- the five-point rubric my teammates and I adopted.

Here is my version of the rubric (as a PDF): Rubric handout - Algebra.pdf

WHAT I DID
After the syllabus and the course requirements, I handed out the rubric and had the kids read the bulleted description of each element of a full-credit five-point answer.

We discussed each element, most of which were familiar to all students, but then I really stayed with the topic of "presenting one's work in a neat, organized, and logical manner according to the standards recognized by mathematicians around the world."

My kids are from very varied backgrounds, so we talked about how mathematics has become a lingua franca in our modern world -- so much so that a math teacher in Russia or in Egypt or in India or in Italy could easily read and assess the essence of a math problem written out by a well-trained student who does not speak the same language they do. We marveled at this for a bit (they're very smart and curious middle schoolers!).

Then this was the piece that really pinned the whole thing home.

HAVE THE STUDENTS ASSESS A WORKED PROBLEM USING THE FIVE-POINT RUBRIC
I wrote a very elementary prealgebra problem on the board and told them that the solution I was about to show them was presented on a math assignment turned in by one Hermione Granger. I told them that they were going to have to assess Hermione's work on the problem and justify their score using the rubric we had just reviewed.

Now, of course, being Hermione, she was very meticulous in solving problem #4, showing every step of her work, spacing her work out on the page so that it was easily readable, lining up the equals signs so that equivalence was as visually obvious as it was algebraically.

Then I asked them to give this work a score.

I was really excited by how engaged they were in assessing Hermione's work. One student gave it a five, and then another agreed, citing each of the criteria in the rubric in turn. Then one of the eighth-graders said, in summary, that this made sense because she had shown both that she understood what she was doing AND that she understood that how she presented her work was important in communicating to others that she understood the value in communicating clearly, logically, and neatly.

Needless to say, I was dumbfounded.

I then showed how Ron Weasley had approached the same problem. Unfortunately, Ron started by labeling his paper from 1 to 22, and squashed in problem #4 to fit the meager space he had left on the page, writing smaller and smaller until he had to have his solution creep up the right-hand margin of the page toward where the paper was dated in handwriting so cramped it verged on mental-patient scribblings. He circled his answer near the top of the page.

Then I asked them to assess his work using the rubric.

The students had a hilarious time criticizing the insane-looking presentation of the work. Much hilarity was had at poor Ron's expense, which was fine, since he is a fictional character without actual human emotions.

CONSOLIDATION OF LEARNING
To drive my point home, I took Dreikurs' strategy of using natural and logical consequences as a framing device and asked the class about the different choices Hermione and Ron had made in how they presented their work and their answers.

Everybody agreed that Ron had made lazy, poor choices because even though he had arrived at the "correct" answer, he had done so in a way that sabotaged his own efforts to do well in the class.

There was a lot of repetitive, reinforcing discussion of this point and we concluded that in a situation where you have control over choices (such as how to present your work), it would be foolish and short-sighted to sabotage your chance to have your learning be recognized by a scorer just because you're feeling a little inconvenienced.

And THAT was exactly the point I was hoping they would discover for themselves. Low-cost, high-impact point made.

And now I'm really going to bed.

UPDATE: 14-Sep-11
   I created a completely generic downloadable version of my 5-point rubric as a PDF which you are welcome to use / steal. It's on the Math Teacher's Wiki at:  Rubric-5-point-generic.pdf

Mathematical Language Manifesto — including mathematical language skills in SBG

Yeah, I'm talking to you. But mostly I'm really writing this for myself.

One of the biggest holes I see in our SBG skills lists is the skill of using correct mathematical language. Not just the names of things, but the verbs.

Oh, the verbs, they are killing me.

Here's an example of the toxicity of mathematical babytalk.

In the math lab at lunch time, I listen in on the students trying to answer tutors' or teachers' questions. "Um, you times it by two...?" or "You minus one." or "You put three."

Which makes me say to myself, You WHAT?

What bothers me even more is hearing how other teachers respond to this. Most of them don't. They simply cringe and try to ignore it, as if overlooking the use of babytalk will improve the doing of the mathematics on paper and in the mind.

And I have not noticed magical thinking to be an especially effective intervention in the classroom.

I started listening in with regularity and I noticed two related correlations:

• When students sound ignorant, they tend to be treated with subtle (or not-so-subtle) contempt.
• When students sound knowledgeable, they tend to be treated treated with respect.

This gave me pause. It also made me wonder if I am doing this too. Which naturally gives rise to worry.

So since the only way I know to address this kind of blind spot is head-on, I've decided to address it head-on. No shame, no blame. Just another couple skills for students to master on the skills checklist.

I want to set my students up to sound knowledgeable and be taken seriously as math learners. And that means I have to encourage their courage by means equipping them with the language skills (both oral and written) to get themselves taken seriously in the math classroom as well as in the world outside.

So I'm adding two mathematical language skills to the skills checklist this year -- both for Algebra 1 and for Algebra 2.

1. Use the appropriate mathematical names for arithmetic operations (noun forms) when answering short-response questions
2. Use the appropriate verb forms of basic arithmetic operations when answering short-response questions

Same SBG rules as ever: Just show me you can do it twice perfectly and you'll be off the hook for those questions in the future — forever.

But secretly, in my heart of hearts, I'll be hoping that by that point, they will already have internalized a better set of language habits.