I have been in Washington. DC for the past few days at the Teaching & Learning 2014 conference in our nation's Capitol.

When I stood at this podium, our meeting and session with policy makers had just ended. No audience occupied the opposite side of the podium. I began thinking, however, about the opportunity for positive change an educator could bring to a public office. Towards the end of the conference, we had the privilege to hear Angela McLean. Here is an article about Angela.

Below are some of the resources and knowledge from the conference sessions I attended.

My self-imposed holiday blogging break is over. <cracks knuckles>

The holiday break affords teachers the opportunity to reflect. One dimension I think many overlook in teacher reflection is to sit and think deeply about one's content area. I would even be bold enough to say the best math teachers are mathematicians at heart. When I sit down to think about mathematics, as a high school math teacher, I like to approach it like a person that can solve a Rubik's Cube. When you watch this feature below on a speed solver, Chester Liam makes the following claim about speed solving a Rubik's Cube:

As a speed solver? No, there is no math involved, no thinking involved. It's just finger dexterity and pattern recognition. There is nothing, no thinking involved in the entire solving process. [1:02 - 1:14]

We can unpackage Chester's thoughts about the Rubik's Cube and how he solves it so quickly by applying mathematical structure to the cube. However, we should ask: what is the objective? What are we trying to do? There's gobs and gobs of mathematics wrapped up in speed solving a Rubik's Cube, but if Chester were to pay attention to the procedures he is applying, it would inhibit his ability to solve the cube quickly.

But consider this: what if Chester attempts to teach how to solve the cube to another person? What would he have to do? What examples, explanations, and demonstrations would he utilize to teach his pupil speed solving? What implications does this thinking have on teaching mathematics? There would be many features of speed solving the learner may not perceive until Chester brings it to the learner's attention. And if Chester's choices are calculated, deliberate, purposeful... the learner may be helped or hindered dependent upon Chester's ability to communicate his thinking which leads to the automaticity of the procedures he applies to solve the cube. Just like learning how to read or learning how to drive a car, we want to teach learners how to do these tasks so well they become 'automated' at some level. To understand mathematics deeply, I believe it is often necessary to unpackage some of these automated tasks. I will share an example of such a mental exercise below.

Today I've been thinking about procedures we accept as true while doing math at the high school level, algebra in particular. As an example, suppose we want to determine the location of the x-intercept for the line 5x - 3y = 7. We might approach this 'task' in the following way:

And we might even graph the line to confirm our solution...

Yep. There it is. The x-intercept at (1.4, 0). As a student, we might simply yawn and move on to the next exercise. The student must recognize the y-coordinate of the line will be zero when the line crosses the x-axis. Yes, we have a solution, but I'm not so sure as a math teacher I'm satisfied to stop there. What if we take a different approach? Let's turn the Rubik's Cube and look at the problem another way. A 'typical' algebra student might subscribe to the church of y = mx + b and do the following...

Just another equally valid path to the value of the x-intercept. I'd like to focus on something in one of the above lines.

This statement says five-thirds of some mystery number is seven-thirds. So, what's the mystery number? I think if we asked a room full of high school math teachers to draw a diagram explaining why we multiply each side of this equation by the multiplicative inverse of 5/3, namely 3/5, we would get some really surprising results. It's not an indictment of teacher education. Rather, it's to say some teachers may not have considered the how's and why's of this procedure before for the same reason someone learning to speed solve a Rubik's Cube may miss a key structure. They may not have the experience of needing to know why it works. Rather, it was more important that they can find the x-intercept of the line; the multiplication by the multiplicative inverse was viewed as "below" the task or level at hand. Perhaps a student never asked "why?" at the critical moment to give the teacher pause.

I struggled to produce the corresponding fraction diagram without trying to reverse engineer the solution. When I write this blog, I often worry I might make a mistake that will be indelibly written into the electronic space of the Internet. But this worry violates the spirit of my blog's theme: that we need to make mistakes in a good direction to evolve our mathematical understanding. Here's the images of my failed by-hand attempts to generate this fraction diagram:

Chicken scratches, page 1

Chicken scratches, page 2

I had trouble thinking about making the diagram because 5/3 and 7/3 have the same denominator, so I wrote out some equivalent fractions. Then, I wanted to use "ninths" because that made sense to me in terms of the grid on the graph paper I had cut apart. But, I then realized I would need to cut fifths to find the mystery number, and I was REALLY struggling with trying to free-hand cut fifths with the grid in the background. That led me to coordinatize the points of the polygon. Then I had a problem with relationships between the linear units on the horizontal axis and the area (the fact the polygon is not "one" unit vertically... which is basically the notion of a unit fraction we see emphasized in CCSSM). So I abandoned the paper approach in favor of Geogebra because I could generate better precision and be more efficient with respect to time. <Sorry for the sloppiness of this paragraph, but it does describe my thinking and the mistakes I made.>

Below is an image of the fraction diagram I constructed using Geogebra.

When stating the equation , we should think of it as a verbal statement: "Seven-thirds is five-thirds of what mystery number?" Well, if the green polygon represents a whole, then the orange polygon is one-and-two-thirds of that whole. Then the green area of 1.4, which equals 7/5, corresponds to the solution. Mentally, how in the world did I end up with fifths, then? How did I know to cut the horizontal into fifths using vectors and vertical lines in the coordinate plane?

We can think of a fraction in the most basic way. Consider 5/3. If the denominator indicates the number of pieces we partition from a whole, and the numerator indicates how many pieces we possess, then I knew we needed to cut the orange rectangle in a way that would make five pieces. It gets to the root cause of WHY we invert and multiply. The numerator 5 becomes the desired number of pieces, considered in a denominator.

For the sake of time, I will stop my mental exercise there. Between working the problem, generating the fraction diagram by hand and on Geogebra, and typing this up, I've spent about two hours roughly on this article. This professional development is incredibly powerful for me as a teacher, and it's absolutely free (well, not quite free, I do pay for the website hosting, but you get the idea).

Let's end by stirring the discussion pot. Consider our understanding of how to find the x-intercept of the given line. Does our understanding, or lack of understanding, of the fraction diagram and how to construct the fraction diagram (essentially "invert and multiply" in many high school classrooms) inhibit our ability to solve the original problem? Is it still possible to understand the original solution without knowing all the nuts and bolts of the fraction procedure? Stephen Wolfram argues in favor of using computers to automate trivial computation procedures to help us access problems in the world outside school. How will our teaching of mathematics change as computers continue to become faster and more powerful?

December is a busy time of the school year. Well, every month is a busy time of year, but at the high school level, as the end of the semester approaches, teachers have a lot on their plate. Administrative edicts, standardized testing demands, semester final exams, and kids stressing about their grades consume a good deal of the high school teacher's time. One thing I would like to share about how I approach teaching: always protect time to reflect on your practice.

Becoming a better teacher is a never-ending process. Growth can be accidental or growth can be purposeful. One thing I try to do each day, whether during a planning period at school, or a brief few minutes in the morning when students aren't around, or after school when I've finishing helping students for the day, or even while surfing professional articles on the Web on a school night, is protect a few minutes to think about what I can do to become a better teacher.

One thing I like to do when I visit teachers I admire is to take a picture of their bookshelf. As Peter Knox says, "Sharing your shelf is sharing yourself." In fact, here's a photo of one of my bookshelves at school.

The top shelf is an annotated version of my professional library. A great way to steal the knowledge of great teachers - a really quick and efficient way - is to take photos of their bookshelves. What books do they have that you don't? Which books are most important to them? I have been spending some of my recent reflection time looking at books from my undergraduate teacher education and my Master's work.

I used my time today to look at a phenomenal book I read cover to cover during my undergraduate work years ago. I would recommend it to any teacher at any level. The title is The Skillful Teacher: Building Your Teaching Skills. I feel like I entered a cheat code as a first year teacher in that I read this book cover to cover during undergrad, not just the assigned parts.

Below is an image of one of the many helpful research based practices and insights from the book. Think carefully on this table below. What do you believe? What do your students believe? What do your teacher colleagues believe? What do your administrators believe?

If that's fuzzy or tough to read, a PDF of the table is below.

Think about our school system and what gets rewarded. Ken Robinson points to this feature of schooling - the fact we are educated in a way that penalizes and stigmatizes mistakes - as educating students out of their creativity. The text excerpt above would point to this aversion to making mistakes as an unwillingness to take risks in the classroom.

This risk-taking dimension of climate has to do with the amount of confidence a student has and the amount of social and academic risk taking the student will do. If it is well developed, a student might be able to say, 'It's safe to take a risk here. If I try hard, learn from errors, and persist, I can succeed.'

I am constantly searching for ways to empower my students to be better risk-takers in the mathematics classroom. If students adopt the healthy attributions from the left-hand side of the chart above, we will see increases in tenacity, perseverance, and patient problem solving. How do we establish a classroom climate to empower our students to be better risk takers? And to respond with greater effort after making mistakes?

This was great to wake up to on a Sunday morning... I am officially a National Board Certified Teacher!

The application process was time consuming but rewarding. I hear that NBPTS is making some changes to streamline the process. I would encourage any teacher interested in developing their craft to pursue National Board Certification.

The education world says instruction should be 'data driven,' but unfortunately, many administrators and teachers have little experience with conducting an exam autopsy analysis. Below is a copy of the form I made and use to analyze data from assessments for unit exams. I shared this form with my department last year. We used it to analyze midterm and final exam data. Feel free to share this form with others.

Here are some things I think about when I use the form above.

Numerical summary: I type the student's grades sorted in ascending order (without identifiers, of course) in Geogebra. I prefer to make a dotplot and a boxplot to display for students on the Promethean board. We use the quartiles to make verbal statements about performance: "Looks like the first quartile is 71, so 75% of the class scored a 71 or better." I can have powerful conversations with my students about analyzing data given their emotional investment in their own performance. I can ask questions like: "Which is preferable? An exam with a middle of the road mean and low standard deviation, or an exam with a decent mean but high standard deviation?"

Brief verbal description of exam format: I prefer a balanced approach between different question types, such as multiple choice, short answer, true/false, matching, and sequencing questions when possible. If I see poor performance for a whole section, I might reflect on whether to change the question format in hopes to capture more information to assist my diagnosis of student misconceptions or misinterpretations.

Five most frequently missed test items: To conserve class time, I prefer to spend time answering only the most frequently missed test items. If a student has another question, I encourage them to visit with me outside class. If I see a recurrent theme in the content of missed items, I know I probably need to revisit my lesson plans and unit activities to try and isolate the cause of the misconceptions.

Five items most frequently answered correct: We all need to try and catch kids in the act of doing things right. However, if everyone is getting an item correct, I might need to ratchet up the item difficulty or ask a deeper question.

Verbal description of content assessed: I have this information for my own reference. I use it to aid my construction of learning guides (some refer to these as curriculum maps or pacing guides).

Qualitative data regarding instructional methodology & delivery: I make notes of the instructional techniques I used during the unit. These may include guided lecture, partner work, small group work, cooperative formative assessments, writing tasks, and other instructional strategies. If students struggle mightily on an assessment, I may need to examine my delivery of the unit and make modifications to the activities to increase the probability of student retention.

Comments: This is the field I reserve for logistic issues going on at school. Perhaps an unannounced fire drill takes place 20 minutes into the exam. Maybe we are on shortened schedule for a Homecoming assembly. Perhaps the students mentioned having tests in almost all their other classes on that given day. All of this information is useful for the planning and preparation phase of teaching.

Our school district is involved with a statewide pilot program to revise teacher and principal evaluation. Here are some of the thoughts and resources I shared at yesterday's school district level meeting.

We started today's meeting with this presentation by Bill Gates on the Measures of Effective Teaching (MET) study and how our country needs to provide meaningful feedback opportunities to teachers. The talk features a segment on a highly effective English teacher using video of her own classroom for reflective purposes - what a phenomenal idea! Normally, when Teacher A asks Teacher B to observe his or her teaching, Teacher A asks Teacher B to watch for specific things. The limiting factor is the required time commitment for both parties. Filming one's own teaching and then performing self analysis and reflection on the video eliminates this potential time issue. For more high quality videos of highly effective teachers, visit the Teaching Channel site. Below is an image of the video topics available at Teaching Channel and an actual screen shot of the site.

Topics listed on the Teaching Channel site

Screen shot of videos on mathematics from Teaching Channel

Much of our discussion focused on transitioning teachers from the perception evaluations are for firing to the perception that evaluations are for growth. Both Charlotte Danielson and Robert Marzano have great frameworks for teaching to help move instructors away from the limited feedback the word "satisfactory" provides and towards more effective instructional growth practices.

Seeing mathematics in the world around you is a trainable skill. This presentation will focus on participants rather than the instructor. I am simply sharing the tremendous ideas of others and how I use them to improve my own teaching.

My Rough Guidelines for Incorporating Digital Media

Encounter an interesting image or video.

Formulate questions associated with the image or video.

Reflect on your own practice. Which questions are worth student investigation?

Structure tasks with you, the instructor, as facilitator. You should fade into the background. Students should be doing the thinking and the doing.

Determine where in the curriculum the activity fits and align it to standards.

Determine how you will craft assessment tasks associated with the activity.

Good Places to Start

Math Class Needs a Makeover Watch Dan Meyer's talk. Mr. Meyer points out some glaring issues math teachers face in classrooms and provides a rationale for incorporating digital media in the classroom.

101 Questions
Create an account on this site. Start formulating questions for images and videos. If you do not find an image or video interesting, skip it and go on. If you find an image or video interesting, be sure to use the links in the top right to find the URL or to download (pictured above).

3-Act Lessons
Dan Meyer's Google Doc on 3-Act Lessons. This is a powerful place to start. A great example of a low threshold high ceiling task is the Sunglasses Problem.

Twitter
If you are not yet on Twitter, you are missing out on a powerful tool for streamlining professional development.