Looking at student work

This week provided a great opportunity to apply ideas into practice centered around co-constructing criteria and looking at student work. The context comes from two fronts:

At school we spent a few days with Sandra Herbst and Anne Davies (see previous blog post) and have been putting into practice their work on co-constructing criteria with students, both as a means to clarify student’s work towards a higher level of quality, as well as create a sense of agency in students over their work, as they adopt a growth mindset about how to improve their work.

The Deeper Learning MOOC (http://dlmooc.deeper-learning.org) had a focus this week on Examining Student work as well. The conversation on Monday (video archive here: http://dlmooc.deeper-learning.org/live/archives/#012714) was a panel discussion (Joe McDonald, Ron Berger, Rob Riordan, Steve Seidel, Carissa Romero amongst others) about how and why we examine student work. In quick summary, we look at student work for at least 2 important reasons:

– To help us better understand what students are taking away from our instructional implementation. What skills and knowledge have they acquired?

– To help improve our professional practice. In what ways can we use this work to help future planning and revising our work for a better effectiveness?

During the conversation that happened, Carissa Romero from Stanford talked about her work with helping math teachers develop better strategies and mindsets in students. Since we were in the midst of some challenging conversation around approaches to solving systems of equations, I thought I would apply that to our work this Wednesday. Students were assigned a problem after having watched a couple of Khan Academy videos (a modified ‘flipped lesson’) and used to their tables to map out solutions as a group. We then did a gallery walk of their work table by table. Some examples of their work here below:

Instead of just looking at the work, one of the things we talked about as a class for each solution was “what are some examples of good mathematical process we see here? What are some things that could have made this work stronger?” This line of questioning and recording it helped identify elements of good mathematical work, and created a sense of ownership (agency) around those criteria that determine what a good mathematical solution look like.

In another example of co-constructing criteria, we had students finalizing their schematic diagrams that they created for their display projects. Students are locating a place on campus and designing an exhibit space for student work. Before they can create their prototypes to show our admin team, they were required to turn their conceptual ideas into schematic drawings represented as orthographic projections. We just finished doing this work two weeks ago on miniature catapults, so it was the perfect time to reach consensus around what elements we should see in a good schematic drawing. We put up examples of student work, and asked the students to consider what elements they think were strong, and what things they want to make sure these drawings had for them to be useful and appropriate. Their list included criteria like:

– use of pencil and protractor to draw all lines and curves
– scale included as a legend
– three views minimum
– attention to detail
– neatness of sketches and text that describes
– measurements given along major directions

We then agreed that these are the ways by which we should judge the quality of their work, and they set to rework their diagrams to be more aligned with these criteria

The level of student work in both their writing and their drawings went up dramatically. We just collected these documents on Friday, so I make sure to include some images from this work next week.

Continuing to Rethink Math Design on light and on motion

Over the past two weeks, a couple of different opportunities arose to rethink the way to approach the understanding and application of modeling mathematical functions like inverse relations and quadratic relations. One of my goals is to tie these inquiries to work we are doing that is authentic, but when that is not possible, to at least root it in a real world situation or phenomena.

On Light

Over the past 2 months, our MPX 10 students have been learning the skills to complete successful energy audits (more on that in another post) and one of the pieces for that is examining lighting in spaces and judging not just the cost (determined by wattage of bulbs and frequency of use) but also whether there are options to “de-lamp” in situations where there are too many bulbs causing spaces to be overly bright. Students learned how to use a light meter, terms like luminosity, intensity, units like lumens, lux and footcandles and what appropriate light levels should be.

In the midst of this, it became clear that we could examine the relationship between light intensity (in lux) and distance which was a natural fit for our work. The activity sheet below was designed to have students investigate this phenomena. I have converted it into a jpg so it is readable through this web page:

This investigation was real world, because there were all kinds of design issues – how far and what intervals to measure, working between scale levels in the meter, effects of ambient light and reflection from nearby surfaces. In the end there was a clear inverse relationship detected, but 4 different math models arose:

I = a/d — an inverse relationship
I = a/d^2 — an inverse square relationship
I = ad^n — a power relationship where n was a negative number between -6 and -2
I = a/d^n + b — a nth inverse relationship – most students got n=-2 – an inverse square relation

This really gave us two excellent opportunities for discussion that I have not anticipated!

I had the students whiteboard in their groups their data sets so that we could have a discussion as assigned to the community about our findings. Some of that work here:

First of all, in considering experimental design, even though in general we agreed that light intensity decreased as distance increased we needed to consider what it meant that 4 different solutions arose. Error from ambient light greatly affected the results and was difficult to control for.
More importantly, the general idea in science that we should look for simple relationships needed to be addressed when some groups were stating relationships like I = a/d^.621

Secondly, the opportunity to discuss the series of curves generated by the functions list above let us discuss why I = a/d^n + b is a more general form of the other functions. Even though we didn’t go into the complete depth I would have liked on this, it will lead to a better understanding of how to find form and function in real data that we will continue to loop back to all year.

Here are a few summary documents (pdf format) by students in the class:

kai light lab

ethan light lab

isaiah light

On Motion

Although we will continue to work on our energy auditing with community members, we are moving on to understanding motion which serves two purposes – kinematics, the study of how things move and dynamics, the study of why things move. These core physics topics have been shown for many years to be the fundamental gateway for students entering college that need to take physical science whether it be engineering, medical school, architecture for example. To that end, our next round of activities will focus on having students understand motion, with the goal of designing a project that we choose as a class that will help us explain motion.

Beautifully tied to this, is the mathematics of kinematics, which involves multiple representations of polynomials and quadratic’s which are significant part of algebra 2.

To get us started in examining these mathematical underpinnings, we conducted an experiment last week in which students measured distance and time for an object rolling down an inclined plane. The activity sheet, which is not as well developed that some of the others I’ve done is below:

One of the things that excited me about conducting this experiment, was rethinking the design of this traditional kinematics investigation. Usually, timing the motion requires using computer aided systems like a photo gate, or a tickertape. As I was thinking of ways to quickly collect accurate data, I was wishing that the camera on the iPad inserted timecode, and then realized that if students filmed the phenomena within iPad timer in view, they could slow down the motion and measure the times accurately by reading off clock measures as the object slid down the inclined plane.
Here are a few photos of the experiment in action:

keeping in mind that the goal of experiment was really twofold:
begin an understanding of accelerated motion which will tie to our working kinematics
examine relationships that pose interesting quadratic challenges to investigate mathematically
the data lends itself beautifully to both of these goals, and we will unpack their findings this week – but here is a preview of what a one of first draft writeups looked like already:

bobby motion lab

onward and upward!

On the PSAT, looking at Energy and Art in Math

The past week and a half have seen us focus as a class on three main threads.

The PSAT
Every one of our sophomores takes the PSAT as a means to practice and acclimate that kind of testing environment so that when they take it again as it Junior they are more accustomed to it. In our program, that means an additional layer of testing with multiple choices, since we spend more of our time immersed in deeper thinking about what we know of math and less on practice and rote problem-solving. As a result, the students and I agreed we should spend time just about every day just doing some practice problems which will serve a couple of purposes: give us a better review of the types of math that they are expected to know, cover distinct strategies about doing well on the test, fill in the gaps of areas that they think would help them do better on this first pass at the test. So that end, we’re using resources like the practice tests on majortests.com, the College Board website, and resources that we’ve accumulated to support the math side of the PSAT. We typically start by taking a small number of questions, and then debriefing them either as a class or in groups to better deconstruct the mathematical thinking that underpins those questions.

Energy Auditing
While we are waiting for our full set of equipment to arrive, we have been doing group work tied to understanding how to evaluate a space for electrical energy footprint. Perhaps a example will help in that regard:
Today, their challenge was to take the data from an air conditioner that had been running for 27 minutes, then look at its kilowatt hour usage, and estimate how much it costs to run that device for a month (current Hawaiian Electric cost: $0.34 per kWHr). Once they had that determined, they were then challenged with going to a building, counting how many air-conditioners are in that building, and estimating what the monthly electrical charge is to run the air conditioners in that building. I’ve attached Josh and Kris’s sheet that they gave me as an example, there is also a bonus questions that they were given to determine what the cost should be to run a fan instead – you’ll see that on the bottom:

The goal of all this initial work, is to create a more careful eye on their part, not just in being able to calculate correctly the energy usage of the device, but to also think about where these things are and how they might take a position with the client in helping them consider ways to save money and energy.

Math in Art

In my last blog post I talked a bit about the project we are undertaking to work with math and art. I just want to share a couple of artifacts that are still in their early stages, but they give a hint of the work that is to come. The pictures below is a shot of both a student hand-drawn art, and the mathematical models using Desmos that give an indication of how they can translate art into clear mathematical function and form.

***the skill and process of critiquing***

One of the things we believe fundamentally is the importance of critiquing and feedback in improving student work and developing a greater sense of autonomy/agency in student work. Once the students completed their first sketch, they used post it note pads to give each other feedback following our agreements development from Ron Berger:

Be Helpful
Be Kind
Be Specific

Examples of the process :

the students did blog posts last week in which they talked about their first round of drawings, the feedback that they received, and their directions. Here are two nice examples from? And? That show where they’re at and give some hint of where they’re going.

More updates to come in another week…

Mathematical Modeling, Authentic Math Activities and Standards

One of my goals for this year is to create (or borrow liberally or adapt) hands-on, minds-on engaging and authentic activities to teach and practice math concepts – both those in our Core Content (in my case Algebra II) and from the Common Core (which our Math Scope and Sequence aligns with).

For this past week I was designing around the following MPI core concepts:

Relation Properties,
Domain and Range,
Functions Properties,
Function Notation Direct Variation,
Slope,
Slope-Intercept form,
Graphing Lines

Which loosely corresponded to the following Common Core Standards:

Understand the concept of a function and use function notation (F-IF 1,2)
Create equations that describe numbers or relationships (A-CED 2)
Interpret functions that arise in applications in terms of context (F-IF 4, 5, 6)
Analyze functions using different representations. (F-IF 7, 8, 9)
Build a function that models a relationship between two quantities (F-BF 1)

My goal was to construct an experience that students would design and implement an experiment that would make them collect, analyze and model a behavior in the real world. This is what is looked like:

Mathematical Models and Modeling Functions
Goal: To design and practice the common core standards around linear model functions
Step 1. Define and design an experiment that you can conduct with existing classroom equipment that will generate a set of at at least 6 data pairs to examine the relationship between two variables of your choosing.
Your goal should be for this data to test a hypothesis that the relation is linear.
You must get permission for the teacher to move forward to step 2. You must define:
• your purpose
• your independent and dependent variables
• your anticipated domain and range
• your rationale for why you expect a linear relationship • your apparatus and procedure
Step 2: Conduct the experiment. Make sure to run AT LEAST 3 trials to minimize error.
Step 3: For your submitted report, you must complete the EVALUATION section of our standard lab report:
a table of calculated values
a graph that includes appropriate axes and labels as well as attempts to linearize your 
data
a statement of the relationship
a mathematical model of the data, including slope and intercept with correct units. 
(This should be in slope intercept form)
a brief discussion of the results and any divergent issues

The students proposed a variety of very interesting experiments (see below pics) – and although not all would actually lead to direct relationships, part of the ‘game’ was not pre-judging for them what might occur. That in itself led to some interesting conversations about best fit lines, the shape of data, and how we know what kinds of relationships are happening between variables in our experiments.

So how did it go? I think the notion of creating real opportunities to practice and implement our mathematical understanding are the places where real mathematicians and the habits of minds of mathematics at her. Certainly, it exposed areas where students were still unsure about something as simple as slope intercept form, or what kind of relationship a scatterplot of data that is not quite linear means. All in all, it lends itself to working and thinking as real mathematicians and scientists, which is the goal of a good STEM education. Here are two examples of submitted work from the activity (in pdf format):

slinky

Bounce Height of a Tennis Ball copy
So what’s next? The work that’s coming up next deals with inequalities, and so I’ve been inspired by some work that Alfred Solis did when he was at high Tech high on the connection between art and mathematical form. More to come on this in the next week… A little hint of where were going below: