Wednesday, November 20, 2013

How Might I Orchestrate the Discussion?

In the last post, I shared how I used ideas from Orchestrating Discussions to:
  • Set goals using established standards for a Family Math Night (FMN) activity;
  • Anticipate learners' responses to certain tasks and questions aligned with those goals; and
  • Create a monitoring sheet to collect information on learners' actions and answers.
When the time came to test out the activity during a mock FMN, I asked preservice teachers from each group to observe their peers carrying out the tasks and ask question that would make their peers' thinking visible. The observers recorded this information using the monitoring sheet, which I gathered onto a single form (provided below).

Because my principle goal was associated with the target, Represent and Interpret Data, I concentrated on selecting work associated with the second task on the monitoring sheet. (I took pictures of the participants' efforts in order to share them here.) While some participants laid out the Pattern Blocks they grabbed into something resembling a real graph and others translated their handful to Unifix Cubes, everyone went on to create more permanent graphs. Therefore, besides acknowledging this step, I would not want to spend class time asking participants to share these real graphs.

Instead, I would focus on the permanent representations participants created and how these representations might be interpreted. In particular, the work selected would relate to the Common Core indicators 2.MD.10
Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories
and 3.MD.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories.
With this in mind, I selected L1, T1, A1, and J2 to share their work.

L1's work reflects the representation closest to the actual activity, so I would ask her to share her work first. Then we could discuss ways it might be interpreted. If it is not raised by the class, I would point how the graph might be misinterpreted: "It seems like the graph shows there is more yellow hexagons than red trapezoids. Is that truly the case?"

Next, I would have T1 share his graph. He drew the shapes inside squares on a grid; this seems to address the possible misinterpretation that the more direct representation might foster. A person reading this graph can see immediately which shape occurred the most in T1's handful. But as the number of red trapezoids is so close to the top, it raises another question: "What would we do if there were more of a shape than squares in a column on the grid?" Two participants encountered this problem and I would ask them to follow T1.

As you can see by her graph, A1 dealt with the problem of having more blocks than squares by amending the bar graph. She added ellipses to the top of the graph to indicate that some items from the data set were omitted. Also, the number of total items in each of the "over-abundant" categories were written directly on the graph. While this graph conveys all the necessary information, it can impede quick interpretation of the data because some of the visual characteristics are lacking.
J2 addressed the issue of having too many blue rhombi for the column by splitting the squares in two. Each half represents one block. She standardized the unit for the other shapes, thereby making it easy to compare the quantities. This approach begins to get at the idea in 3.MD.3 of a "scaled bar graph." The sharing would end with this example and lead to a whole class discussion meant to bring the ideas associated with these models together.

Looking Back
I would want the participants to notice the multiple approaches that can be used to represent a data set and how the representations are related. For example, most of the graphs reflect a one-to-one correspondence. Discussing the limitations of this approach seems a natural next step as we could revisit the possible misinterpretations (L1's work) and the need for creative modifications (A1's work).

Looking Forward
In order to move toward 3.MD.3, I could ask them to consider which approach they might use if they were going to graph the "handful" results from their group or the entire class. Hopefully, increasing the size of the data set will push them to considering a scaled approach similar to J2's work. This would lead to the next task associated with this activity

Next Time
If I were to do this activity again, I would continue to provide participants with a variety of ways to represent their data sets. This results in a rich source of approaches to choose from when engaging in the Selecting phase. 

Furthermore, because I am not asking for the "best way," participant are free to select methods like the picture graph because they want to be creative rather than feeling the need to be efficient. Hopefully, this lessens the likelihood that participants will somehow link approaches shared earlier in the Sequencing phase with the creator's ability - it was simply a choice.

Next time, I want to make sure that the Pattern Blocks are equal in thickness. While having a combination of Pattern Block sets mixed together made for some interesting approaches, it could potentially be a distraction, as this stacking shows. It can create extra categories that are not always consistent across the buckets from which the participants grab a handful. The task is rich enough as it is without adding this extra variable.

Also, after graphing one handful, I would ask students to predict how many would be in two handfuls. This would hopefully, create interest in conducting another experiment. If they do grab two handfuls, this increases the likelihood that students creating a bar graph would encounter the issue A1 had with not enough grid space.

Friday, November 15, 2013

How Many in a Handful?

One of the projects in #MTH221 involves hosting activity stations during a Family Math Night at local schools. Our preservice elementary teachers are assigned a topic (patterning, measurement, data, or probability) and asked to find an activity related to that topic that might interest K-5 students. As a group, they decide on two that they want to run and begin gathering/developing the resource they need to make the activity work.

For example, I found I Have a Handful in the November 1999 issue of Teaching Children Mathematics in the Math by the Month department. I made a poster and began identifying Common Core State Standards in Mathematics (CCSSM) that this activity might address. We encourage our teachers to connect at least two Standards for Mathematical Practice and two content standards. For I Have a Handful, I decided to focus on:
Standard for Mathematical Practice:
  • Model with Mathematics [SMP 4]
  • Use Appropriate Tools Strategically [SMP 5]
Content Standards:

  • Kindergarten: Classify objects and count the number of objects in each category [K.MD.3]
  • Grades 1, 2, and 3: Represent and interpret data [1.MD.4, 2.MD.10, 3.MD.3]
  • Grade 6: Develop understanding of statistical variability [6.SP.1]
  • Grade 6: Summarize and describe distribution [6.SP.4, 6.SP.5a]
Next, I identified some questions I might ask to help make students' thinking visible during the activity and aligned them with the CCSSM.
  1. Which shape shows up most often and how do you know? [K.MD.3]
  2. How could you record your result on a graph? [SMP 4, 1.MD.4, 2.MD.10, 3.MD.3, 6.SP.4]
  3. Why did you use this type of graph to represent your results? [SMP 5]
  4. Which shape occurred the most? The least? How many more? [SMP 4, 1.MD4, 2.MD.10, 3.MD.3, 6.SP.5a]
  5. What would happen if you grabbed another handful? Why? [6.SP.1]
I decided to concentrate on the first three questions, and using the framework from Orchestrating Discussion (5 Practices), I began to anticipate possible student responses (Practice 1). This lead to the monitoring sheet (Practice 2) that is provided below.

On the second page, I tried to arrange the responses in such a way that they represent movement from a concrete approach to an abstract one. In essence, a rubric reflecting various levels of comprehension related to the idea of creating a display of the results from the activity. It was not as evident, to me, how to break up the questions on pages 1 and 3 using this approach. Perhaps seeing people engage with the activity will make these responses easier to arrange.

In the next post, I will share the results of carrying out the activity and address the remaining Practices (Selecting, Sequencing, and Connecting).

Thursday, November 7, 2013

What did/does the 'A' say to me?

In the previous post, I used the song What Does the Fox Say? to frame a discussion about what grades communicate to various stakeholders in education - in particular students and teachers. It is my view that we do not share a common understanding of what a grade means and this impacts learning. I suggested some of the different ways we interpret an A by modifying the song's lyrics. Readers responded in the comments with what an A grade meant to them as a student and what they hope it says to their students as a teacher. Now is the time for me to share my perspectives about what did/does the A say to me.

When I got an A, I thought that I had pleased my teacher. I distinctly remember having a conversation with a friend who was struggling in school about how I achieved success. I tried to find out what the teacher wanted and then went about meeting that vision. The grade I got would tell me how close I came to giving the teacher what he/she had in mind. It did not matter the subject, the teacher was the all-knowing arbiter of my work. For me, school was not so much about learning as it was mind-reading.

As a new teacher, I simply flipped this perspective. An A meant that the student had done at least 90% of what I expected (i.e. what I would do). To my credit, I did not want my students to read my mind so I made my expectations very clear. I got a lot of student-work that looked just like my work. Instead of mind-readers I was fostering mimics.

Now, I see an A as representing what Joyce and Showers (2002) called Executive Use. The student has demonstrated complete content competency (I was uncomfortable with idea that there might be a 10% gap in a teacher's knowledge) and an ability to analyze under what circumstances the learning could be applied appropriately (phronesis) or how to adapt it to new situations. Granted, because I mostly teach teachers, this standard might be easier to implement now than when I taught middle school math. Still, I have applied a similar idea in a College Algebra course with some success - it was a tough sell.
Basically, I want an A to say to students that they have achieved sustainability in the topic being graded. They can apply what they have learned beyond what we talked about in class, and they can learn more on their own if needed. The teacher (me) has become obsolete.

Friday, November 1, 2013

What does the 'A' say?

I was introduced to this little ditty for the first time last week. 
[Warning:  possible earworm]

Dog goes woof
Cat goes meow
Bird goes tweet
and mouse goes squeek

Cow goes moo
Frog goes croak
and the elephant goes toot

Ducks says quack
and fish go blub
and the seal goes ow ow ow

But there's one sound
That no one knows

What does the 'A' say?

What does the 'A' say?

You got 90% or better!
You got 90% or better!
You got 90% or better!
What does the 'A' say?

What does the 'A' say?

What does the 'A' say?

Big bold type
Pointy head
Causing joy
and spreading dread

So that's enough of that. But in all seriousness, the song reminded me of a project I have been interested in for some time: what does a grade of 'A' communicate to students, to their parents, to other educators, to the community? I have often wondered if we are all on the same page when it comes to grades and what they say. At one point, I thought about putting a questionnaire in the mailboxes of my colleagues asking this question, but for some reason I never got around to it.

Until now.

In the comments, please share what an 'A' said to you as a student and what you hope it communicates to your students as a teacher. I'll share what I really think the 'A' says after you all get the ball rolling. Thank you in advance for your participation.

UpdateThanks everyone. My response got a bit long for a comment - so, for what it's worth, you can find it in the next post.