In PCMI's Reflecting on Practice class this year, we focused on making connections. Here are some things that made me think and that I would like to try.

### 1. Pre-Assessment

One place that it is important for kids to make connections
is when they are introduced to a new concept and they build their understanding
from their prior understandings. We talked about alternatives to a written,
individual pre-test that were primarily short ways to activate students’ prior
knowledge and get a sense of how comfortable they are with the pre-requisite
(or co-requisite) skills. For example, this is a WODB that could be done at the
start of a unit about linear functions with eighth graders, to see what they
remember about proportional relationships and to preview some of the new ideas.

In the past, I have generally not done this type of
pre-assessment. I like to start my units with a task that students can use
their intuition and prior knowledge to solve, but that introduces a new idea in
the unit (for example, solving a contextualized system of equations where the
difference between the two situations is very obvious). I like that because I
get a sense of the different ways students are approaching the new concept
before I have given them any sort of direction. I also like it because it is
something that we can repeatedly return to as students start to formalize the
new ideas. But I see the value in doing this type of pre-assessment as well,
particularly if it is pretty quick and low stress. This prompts a wider range
of vocabulary, procedures, and concepts than my one task might.

### 2. Contrasting Cases – from Star and Rittle-Johnson

The goal of this routine is to develop procedural fluency
that is flexible. This is a format that is pretty similar to things I have done
in my class before, but I like it because it formalizes the process a little
more and provides a bank of question options to choose from. Students are first
presented with two different methods for solving a problem, where the work is
accurately done out. There’s then a sequence of questions, first for
understanding, then comparing, then making connections. Finally, students
choose which method to use and why for each problem in a handful of similar
problems. Here’s an example that we did in class.

And here are the different question options:

This seems most valuable where one method is not always
better/more efficient than the other. I think that this could either be done
where the original problem is a toss up between the two methods, or the
original problem is obvious but the follow-up problems are split.

### 3. Examples and Non-Examples

Another routine I could see using in my class is having a
set of examples and non-examples. Having students individually choose one that
they know is an example, one that is a non-example and explain why. Then have
all of the options up on the board and have students place one color post-it on
which one they chose as an example, another color on which one they chose as a
non-example. Then discuss as a class
based on where there is the most conflict.

Here’s one we did where we had to determine whether there
was enough information to tell if the triangle was isosceles.

### 4. Different types of comparison

This slide, from Day 2 in the third module, summarizes the
different types of comparisons that we talked about.

For a lot of reasons, discussions in my class this past year
were often about what steps to take to solve a particular problem. I don’t
think that this is particularly valuable to students or a good use of
discussion time (particularly when it is the only type of discussion
happening). This is something that I want to work really hard to change this
upcoming year and I think it will be helpful to think about for what type of
learning goals might I use each of these types of comparisons.

### 5. Quiz Feedback through Highlighting Mistakes

We watched this teaching channel video where Leah Alcala
explains how she gives feedback on tests by highlighting anywhere a student
makes a mistake in their solving process. I like this in contrast to what I currently do (check or x
next to each problem, with a handful of random comments) because I think that
highlighting where the mistake(s) is gives an entry point into
revision/correction that many students need. Students who are overwhelmed or
stuck when they get back a quiz with just a check or x next to each problem
have a place that they can start. I still think, though, that you would need to
figure out how to support students who even when they know where the error is
don’t know how to fix it because they don’t understand the concept/know the
procedure. Having students work in groups to do the corrections is a step
toward this, but I think as I am highlighting I would also need to keep track
of what the misconceptions are, how many people have them, and whether or not
they need to be addressed in the whole class, small group, etc.

### 6. Thinking Classrooms

Peter Liljedahl video-conferenced in to talk to us about
thinking classrooms. In his research, he found student behavior depends much
more on what’s happening in the rest of the school than what the teacher is
doing. He defines what’s happening in the rest of the school as institutional
norms: normative behavior we take for granted in our institution’s classrooms.
And then he researched the best ways invert the institutional norms because the
more opposite teachers were to institutional norms, the more engaged kids were.
Here are what he found were the biggest disruptors, where the gear size
represents the relative impact.

I am really intrigued by his framework and definitely want
to read more of what he has to say here. My overall PCMI commitment to change in my
classroom next year is “Use the whiteboards [which I installed all around my
room last year] so that students learn from multiple strategies.”

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