How can we pass the time?

For the past three year, we have spent New Year's Eve at the Wealthy Theater in Grand Rapids listening to Michigan supergroup, Starlight Six. They usually play three sets of music, with short intermissions between sets. During one of the breaks last year, I was looking for something to do (trying to find a problem to play with) when I noticed the light string at the back of the stage.

The string of 25 lights were hung in a way that I could see two groups of 13. 

This seemed quite appropriate given that it was 2013. And it got me wondering about what other groupings I might make with this string of lights.

I imagined using two interior anchor points (adding two more lights) in order to create three groups with nine in each group.

Making four groups meant adding three more lights. With 28 lights, each of these groups would have seven lights.

Five groups created a problem. When the four anchor lights, which were being double counted, were added to the original 25, I had a number that was not divisible by five. But six groups, with 25 (original) + 5 (anchor) lights, resulted in five lights per group. It had me wondering if other strings would be as "friendly" to various groupings or if there was something special about 25.

So I thought about a string of 26 lights. Two groups added one anchor resulting in 27 total, which is not divisible by two. Three groups added two anchors resulting in 28 total, which is not divisible by three. Four groups also didn't work. But five groups added four anchors for 30 total, and 30 is divisible by five - resulting in 6 bulbs per group.

This still left a lot of questions to explore. But the band was back on stage, so I filed this found problem away for another time.

Feel free to use it as a way to pass the time this coming year. Or better yet, find your own problem.

Why does this bother me?

First, because some of my students check in on this blog, I want to be clear that the thoughts presented here represent my own issues and are not meant to shame my students or dismiss their perspective. Second, for my 200th post, I thought I would share one of my issues with you. So, consider yourself warned.

Near the end of the semester, I often ask students to reflect on the course and write a letter advising future students on what to expect. (I wrote about it here.) This time, I asked them to narrow it down to three important points. One student wrote:

"This is an education class not a math class."

For some reason, this point really bothered me. I do not know how my student intended it (and it doesn't matter - this is about me), but I took it negatively. One of the ways I can tell I have had a button pushed is when I start getting defensive. So when I began to line up all my arguments, like soldiers on a battlefield, in order to attack this point, I knew that it was time to take a step back and consider why I was uncomfortable with this point.

It did not come to me immediately, but then yesterday I read these Tweets, 

and the linked articles (here and here, respectively). Both use reports from the National Council on Teacher Quality (NCTQ) to reinforce the narrative that U.S. efforts to train teachers are "abysmal." Yes, I know that the NCTQ teacher prep rankings are non-sense (none other than Linda Darling-Hammond makes the case) but I seem to have bought into the non-sense. Therefore, when the student wrote that my course was an "education class," I began to worry that I was part of the problem.

Most teachers can probably understand this line of thinking. Regardless of what we know about the realities of education, the message seems to be that teachers in the U.S. are to blame for all of our educational ills, whether real or imagined. Therefore, I want to do all I can to "make things better" - even if it is unwarranted.

A friend of mine once told me that if my buttons were being pushed, then I needed to move my buttons. In this case, I needed to shift my attention to aspects of the course that were preparing these future teachers for the realities of the profession; this was, after all the overarching goal of the course.

Please pardon me if this next piece seems self-serving, but I need to acknowledge some of the points other students shared:

• Keep up with the readings! They are very important towards your teaching future.
• As future teachers you understand that the main goal is helping the students learn. That is your professor's goal as well!
• You will work extensively in collaboration with your peers, as you will as future classroom teachers.
• Do not procrastinate! Otherwise you will have an overwhelming amount of work to do.
• Put a ton of effort into this class even though a lot of the things that you do won't be graded because you will get out what you put into the class.
• Take time to reflect on different aspects of class because there is a lot of useful information, and you want to be sure you have learned most, if not all, of it.

Reviewing these, there does seem to be a lack of attention to the mathematical content; this is something to consider for next time. However, as education classes go, I am satisfied that this course provided students with some of the tools necessary for success in teaching: reading research, focusing on learning, being prepared, giving less attention to grades and more on learning, and recognizing the importance of reflection. 

Finals! Huh! Yeah! What are they good for?

We just finished finals at Grand Valley. Coincidentally, this week of exams overlapped with the release of the second Hobbit movie; this might explain why these lines (modified from this poem) were going through my head Thursday as I walked around campus:

I am not trying to suggest that like the One Ring, final exams are evil and must be destroyed. However, I do wonder about the wisdom and unintended consequences from placing so much emphasis on a single assessment event.  Rarely do final exam scores make that much of a difference on final grades. Yet, there is so much stress associated with final exams, for both the students and the teachers, that it got me to wondering if these tests are worth it.

During the MTH 323 final event (we did not have a written exam), I asked the teachers to consider whether or not our school ought to have final exams and why. This is the course for preservice teachers that I designed to reflect what it looks like to be a math teacher in a K-8 school. So we created a "school" and went about developing a curriculum. Therefore, it seemed reasonable to discuss the final exam policy we would have at our school.

Grade-level groups talked about the question and then shared their thoughts and rationale. I added my two cents, but mostly these points represent the teachers' perspective.

• We need to have some sort of culminating experience as a way to identify what a student has mastered and possible weaknesses in the curriculum.
• The experience needs to be different than the traditional exams that represent a cram and purge approach that does not result in lasting learning.
• Like it or not, exams are a part of our culture and we would be doing our students a disservice if we fail to provide some experience with these types of assessments.
• Perhaps we could give the exams a couple of weeks before the end of the semester so those who need to can make revisions and the rest can extend their learning.
• Whatever we have them do needs to be meaningful.

It was the last point that resonated with me. How do we ensure that the culminating experience is meaningful - that it fosters the upper levels of the Engagement Continuum and not just compliance?

Developed by MTH 323 Teachers to monitor their engagement
Based on Engagement Taxonomy of Morgan and Saxton

At breakfast this morning, I asked a colleague what he thought final exams were good for. He said, "To see if students have synthesized the information addressed in the course." After a pause, he continued, "Of course, those are the questions they often struggle on most, which is discouraging."

"Why do we give all the other questions, then? The ones that focus on recall. Why not focus on the synthesis questions?" I asked.

"Probably because it's what we've always done," he responded. Neither of us were satisfied with this answer, but finals week was over (it was, after all, final) so there was nothing we could do about it now except think about what we might do different the next time around.

How does it fit?

At the beginning of the semester, I wrote a post about using the history of the LEGO company as a cautionary tale for innovation in education reform. The idea came from a Diane Rhem interview with the author of Brick by Brick. But there was another idea presented in book that also caught my attention - clutch power.

When a child snaps two bricks together, they stick with a satisfying click. And they stay stuck until the child uncouples them with a gratifying tug. And therein lies the LEGO brick's magic. Because bricks resists coming apart, kids could build from the bottom up, making their creations as simple or complicated as they wanted. … it is clutch power that makes LEGO such an endlessly expandable toy, one that lets kids build whatever they imagine. (page 20)

It was the last sentence that got me thinking. How might I design learning experiences that use clutch power - allowing learners to "build whatever they imagine?" In particular, I had one of our math courses for preservice teachers in mind. It explores a variety of mathematical topics that students often see as disconnected: algebra, geometry, measurement, and data.  Was there a way to treat concepts in that course as building blocks that learners could use across the domains to build new understandings instead of simply following the instructions I provide?

I am still working on that question, but I did start experimenting with simple problems that allow learners to add their own ideas while demonstrating their content competency. The following is one problem (modified) from the final I just gave. I would be interested in your feedback.

Plot the points (2, 2) and (2, 4) on the coordinate plane. Pick two other points that could combine with these first two to be used as the vertices of a trapezoid. Use this context to develop items that would align with the following targets (be sure to justify your alignment using the indicators):

• Graph points on the coordinate plane to solve real-world and mathematical problems (5GA1, 5GA2)
• Classify two-dimensional figures into categories based on their properties (5GB3, 5GB4)
• Solve real-world and mathematical problems involving area, surface area, and volume (6GA1-A4)
• Draw, construct, and describe geometrical figures and describe the relationships between them (7GA1, 7GA2, 7GA3)
• Solve real-life and mathematical problems involving angle measure, area, surface area, & volume (7GB4, 7GB5, 7GB6)

Because this was a new assessment for most of my students, this time around I offered some example items to pick from:

1. Define the term "trapezoid" and then use that definition to identify what other names apply to your shape.
2. Find the area and perimeter of your shape.
3. Extend each side (creating 16 angles) and find all the angle measures.
4. Write your own question for this context.

I still expected them to align the item with the targets, justify their alignment, and come up with a correct response.

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).

Selecting

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.

Sequencing

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.

Connections
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
Glows
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.

Grows
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.

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).

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.

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?

Well-done-well-done-well-donedonedadone!

Well-done-well-done-well-donedonedadone!

Well-done-well-done-well-donedonedadone!

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?

De-de-de-de-de-distinguished!

De-de-de-de-de-distinguished!

De-de-de-de-de-distinguished!

What does the 'A' say?

Ah-ah-ah-ah-ah-ah-ah-ah!

Ah-ah-ah-ah-ah-ah-ah-ah!

Ah-ah-ah-ah-ah-ah-ah-ah!

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.

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

What's wrong?

• Make sense of problems and persevere in solving them
• Construct viable arguments and critique the reasoning of others
• Attend to precision

These are three of the eight Practice Standards associated with the process of doing math that will now be assessed in states that have adopted the CCSS. Some teachers I have talked to are concerned about how they will incorporate the Practices in their math classes. I decided to share an activity that addresses these three Standards and can be adapted to nearly any content - a Crit Session.

I first heard of Crit Sessions while reading Jonah Lehrer's Imagine. Despite the controversy surrounding this book, I found his description of the Crit Sessions held at Pixar to be particularly interesting and began to consider how to apply this idea in my classes. So I developed the following workshop.

+++++

Crit Session for Presentations of Mathematical Thinking

This can be used for any content where students, individually or in groups, are presenting mathematics. The work does not need to be completely thought out. In fact, it helps if the work is "in progress" as the feedback provided by peers can help the students to move forward.

Schema Activation [5 minutes]: Review the information you want to share.

Focus [5 minutes]: What is a Crit Session?

Read the following edited extract from Imagine:

Concentrate on the second paragraph:

• Learning from other people's mistakes
• Distributed responsibility for everyone's success

Activity [40 minutes - assuming 4 group presentations]: Crit Sessions

• 5 minute presentation
• 5 minutes for feedback
• Critical groups (audience) take a minute to organize critiques
• Members of the presentation group splits up to meet with critical groups - this makes the critiques more manageable and seem less harsh than in the whole class setting

Reflection [10 minutes]: What; So What; and Now What?

Back in presentation groups, after everyone has presented, the members share:

• What feedback did we hear?
• So what made the feedback important?
• Now what should we work on?

The information does not necessarily need to focus on their own presentation since we sometime learn from other people's mistakes.

+++++

The students I have tried these Crit Sessions with have been positive about the approach. They appreciate the permission to be critical (critique the reasoning of others). Most times they feel as though they have to be nice, non-critical, in order to have a safe classroom environment. Yet, no one felt attacked or belittled and the classroom community seemed even stronger after the workshop. After all, they were working toward a common goal - success.

I am also impressed with how these Crit Sessions have supported a growth mindset in regards to doing math. The students seem more attuned to the need to put effort into working toward precision in their work and the work of their peers. Also, there is a built in expectation that they will need to make improvements (persevere) to their work. No one expects to get it entirely right the first time.

If you try this approach in your class, please share your efforts and any adaptions you needed to make in the comments. Thanks!

What makes someone an expert in education?

The book is I Got Schooled. M. Night Shyamalan, director and author and education, now, expert.

The above quote is from the interview provided below. To his credit, Mr. Shyamalan immediately corrects the interviewer and says that he considers himself a novice; this seems appropriate given his limited time looking into the educational research.

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Still, earlier in the interview he seems comfortable correcting Diane Ravitch. Can we be certain that in five years, Mr. Shyamalan was able to synthesize ALL of the research related to class size that Dr. Ravitch has considered given her over 30 years examining the history of education? Has Mr. Shyamalan considered how increased class sizes might impact the retention of good teachers (one of his 5 keys to closing the achievement gap)? These are issues we might expect in statements made by a novice but these issues were not raised.

To be fair, I can see why an information entertainers might be intimidated by Mr. Shyamalan's work on this book. For someone who has spent no time looking into educational research, five years must seem like an eternity. But for those of us who take education seriously and have been examining the research for a much longer time ... (with apologies to McKayla)

Image from Columbia 250