The Reflective Educator

Education ∪ Math ∪ Technology

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Month: March 2014

Pause and rewind this until you get it

Watch this video. Every time you feel you are confused, just pause and rewind the video and rewatch it. Do this until the video makes sense.


Can’t do it can you? It doesn’t matter how much you pause and rewind a video, if it doesn’t make any sense to you, watching it again isn’t going to suddenly change the video so it makes sense.

One critical job of the teacher is to find out where students are, and help them at that stage. Explanations that don’t make any sense aren’t particularly helpful. In fact, since our goal is to help students learn the mathematics, and not just the words that represent the mathematics, explanations may not be helpful at all.


What is teacher inquiry?

The Teaching Channel recently published four videos which describe much of the work I do with teams of teachers in a project known as Accessing Algebra Through Inquiry (or a2i for short). One of the primary objects of the a2i project is to build the capacity of school-based teams of teachers working collaboratively to improve their practice and make evidence based choices about their instruction. In other words, a primary objective of our project is to change how teachers collaborate and increase their opportunities to learn about their practice.

One thing I do is help facilitate a meeting at a school, with the objective of working through an inquiry cycle where we look at student work, identify a common problem of practice, unpack what the probable models the students are using to think about this particular area of mathematics, and then decide together as a team on an intervention for the teacher’s group of students. Next the teacher implements the strategy, and we study the resulting student work to see if the intervention worked.

These two videos describe the inquiry process in more detail.

Another portion of my work is individual coaching of teachers. I work with teachers where I typically observe their classes, usually focusing on a particular aspect of the lesson. During this time, I gather information about what strategies the teacher employed, and how these strategies played out with the teacher. Sometimes I model a particular strategy with a teacher, and occasionally I rehearse a strategy with them, particularly if they have never used it before.

After the lesson is over, or sometimes before the lesson, I meet with the teacher and we discuss instructional strategies for their students. Sometimes we discuss classroom management strategies, sometimes we discuss the core mathematical content of a unit, sometimes we look at different instructional strategies, and sometimes I reflect back to the teacher the questions they asked, and we discuss the impact of these questions.

Here is a short clip from an individual coaching session between a former colleague of mine, Xiomara Gonzalez and one of the teachers in our project, Anna Tabor.

One of the most important benefit of our project is that we are providing a structure through which teachers get more feedback on their work when they collaborate with other teachers, and at the same time, get feedback from an instructional coach.

Here is a video that explains the impact of the feedback Anna receives from her colleagues, and from the work with her instructional coach, Xiomara.

It has become clear to me that large, impersonal conferences, workshops that are disconnected from teacher practice, courses which overly focus on the theoretical, and spaghetti-style professional development (throw the PD at the teachers and see what sticks) are not accomplishing their objective; getting teachers to reflect on their teaching right now, with this group of students. This process of inquiry that we are working as a team of instructional coaches to implement in our schools is designed to give teachers peer feedback on their teaching, and to help them grow and improve their skills while still connecting this learning directly to the work teachers do with their students.


Note: If you can’t see the videos above (because this post is in your email or your RSS feed reader), you can view them here.


PARCC sample assessment items for high school math

PARCC recently released some sample computer-based test items for ELA and high school mathematics, so I thought I would check them out since NY state is still officially planning (eventually?) to use the PARCC assessments.

First, some kudoes to the team that created these assessment questions themselves. In general I found that the questions were looking for evidence of mathematical reasoning, and would be difficult to game with classroom test-preparation. What I think is missing is an opportunity for students to demonstrate the complete range of what it means to do mathematics, including asking questions themselves that they answer, but for an assessment with this function, this seems much better than the current generation of standardized tests.

If you want to stop reading this and preview the questions yourself, feel free to do so (if you are only interested in looking at the sample math questions, you’ll have to skip through the sample ELA questions).

Here’s my preliminary thoughts from attempting the first few problems myself.

  1. Use of space is critical. The first assessment question does not do this very well. Look at the video below that explains my reasoning on this.


  2. The second question has two issues, one of which is really very minor, the other of which is something PARCC should make an effort to fix.

    I’m okay with questions that use approximate models for mathematics, but it might be at least worth noting to students that these models are approximations.

    Taking a test on a computer is already extremely distracting as compared to taking a test on pencil and paper. Given that there is research that shows that people generally read slower and with less understanding on a computer, PARCC should make an effort to mitigate the platform issues as much as possible. Imagine if your work flickered in and out of existence while you were writing it down on paper?

  3. I put the wording for the third question through a reading level estimation calculator, and it estimated that the reading level was grade 10. While it is reasonable to expect a certain amount of competence from students in this respect, we have to be careful that our assessments of the mathematical thinking of students aren’t actually measuring whether or not they can read the prompts in our assessment.
  4. Question 5 assumes a certain amount of cultural knowledge, specifically knowledge of playing golf. Having worked with students who do not have the this sport in their cultural background, I found assessment items like this frustrating. Usually, the questions are doable without the cultural knowledge, but imagine you are a student who comes across a question that contains an idea with which you know nothing. Regardless of whether or not the knowledge is required to do the mathematics of the problem, it impacts student confidence and therefore their performance.
  5. The sixth question assumes that students have some minor technical knowledge, which I would classify in the same genre as my fourth point; students with a minimal technical background may struggle with the mechanics of this task. This may not affect a huge number of students, but the assessment instrument should be as neutral as possible to allow the greatest number of students to interact with the mathematics of the task, not the mechanics of the task.
  6. The seventh question has a video. It’s probably between 4 and 10 megabytes in size. Can you imagine what this will do to your school’s bandwidth if every student in a particular grade is accessing the resource at the same time?


There are some things which I think are obvious to me about the computer based assessment that PARCC is working on.

The first is that many of these questions are still going to require actual math teachers, with some experience looking at student work, to look at. Most of these questions are not just reformated multiple choice questions (although some of them are). While this increases the per-student cost of the assessment, I do not think that there are computer programs (yet) that exist that can accurately capture and understand the full range of possible mathematical reasonings of students.

Next, some of the more adaptive and social aspects of the work Dan Meyer and company have put together, are not present in this work. This assessment is intended to capture what students think now, rather than what students are able to do once given more information. This is still an assessment of the content students have learned, and does not appear to do an ideal job of making sense of how students make sense of problems and persevere in solving them, attend to precision, or any of the other standards for math practice (SMP). While it is clear to me that students will have to use these standards when doing this assessment, I do not see how anyone looking at the resulting student work is going to be able to say with any accuracy what is evidence of each of the SMP.

Unfortunately, unless the standards for math practice get captured somehow by an assessment (a goal of ours during next year with our project is to make an attempt to do this systematically), it is unlikely that teachers will use them.



The Treachery of Words

This is not a pipe


Words are not ideas, anymore than the picture above is a pipe (it’s a picture of a pipe).

When we communicate about ideas we are forced to use words (or gestures or images, which are also not the ideas themselves), and so consequently we are never communicating ideas directly. We communicate about ideas through the medium of language.

It is possible for someone to learn the words that represent an idea without learning the idea themselves, even so much as to be able to mimic the output expected of someone who understands the idea.

This happened to me. I learned about linear functions in school. I learned about “constant rates of change” and “y = mx + b” and “find the rise over the run” and many other phrases which were connected to finding and subsequently graphing the equation of a line. I even later learned how to think of a specific line as a transformation through a translation and a rotation from any other line.

However, in a fourth year topology class, my professor used a completely different set of words. He said, “Let’s do an easy example. We have line segment AB and line segment CD, and our objective is to find the mapping function from AB to CD. Think about that problem and bring your work to next class.” Well, I spent a week thinking about that problem and could not do it. I spoke to my professor about it, and he gave me a look and told me that it was “not my fault, this is unfortunately due to how you were taught.” Note how I was earlier able to articulate “transform one line into another”, but not able to repurpose this information to create a function. It turns out, the solution is a linear function.

Here I am, many years later, with a much better understanding of the wide range of things in our world which are linear functions. In fact, in only a few minutes, I am now able to “prove” to myself (at least experimentally) a more general version of what my professor asked me to prove all those years ago. I have moved past a “just the words” understanding of linear functions into a more flexible and useful understanding.

This is the objective of good mathematics teaching, and the reason why I think that we need far fewer lists of things to achieve with our students, and far more time spent looking at very similar things but in a very wide range of contexts. We need to move past students learning about the words that represent mathematics and giving them enough time to actually learn about the corresponding ideas.



Keynote on Formative Assessment

I recorded some video and the audio from a keynote presentation I gave a couple of weeks ago. It turns out the video wasn’t all that useful, but I did a screencast of my presentation notes, and added the audio from my keynote to it.



Were I to do this again, I would definitely do a better job of summarizing my main points at the end, and I would probably explore more closely some of the different concrete methods through which one can do formative assessment.