Monday, May 30, 2011

Was the scoring consistent?

The following is a statistics workshop I have used in a class for preservice K-8 math teachers. Because the topic being addressed, Mean Absolute Deviation (MAD), represents a specific procedure, parts of this workshop are more direct than what I typical do in a lesson. Still, I try to leave enough space for learners to explore the procedure and make sense of it.

The goal of this activity is to consider how to quantify consistency while addressing the idea of inter-rater reliability. It is based on the article Means and MADs and is intended to help introduce the concept of deviation using a measurement that is more accessible to middle grade learners. I updated the activity to reflect the upcoming Common Core State Standards (CCSS).

Grade 6 CCSS in Statistics and Probability [6.SP]
  • 6.SP.2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
  • 6.SP.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Schema Activation: Turn and Talk
With your partner, discuss which set represents the greatest consistency.

Focus: Quantifying Variation
Consistency is a major concern when it comes to grading.  We want the scores that different teachers assign a piece of work to be relatively close. While sometimes variation is obvious, quantifying it can be helpful in identifying inter-rater reliability. In this workshop we will learn about a particular measure of variation called the Mean Absolute Deviation that can be used to describe how values vary with a single number.

Activity: Small Group
The table and graphs below show the scores one group of teachers assigned Ross for his exploration project in each Math Thematic assessment category: Problem Solving (PS), Mathematical Language (ML), Representations (R), Connections (C), and Presentation (P).



  • In which category do you think the scores were most consistent?
  • In which category do you think the scores were least consistent?
  • Which representation (table or graph) was most useful in addressing these questions? Why?

Find the mean distance (direction does not matter – why?) each data point is from the mean of the data set and compare it with your predictions.  Which representation (table or graph) was most useful in finding this Mean Absolute Deviation? Why?

Reflection: What? So what? Now what?
  1. What were your results?
  2. So what do the results say about consistency?
  3. Now what are some ways to improve scorers’ inter-rater reliability?

Thursday, May 26, 2011

Why all the questions?

“The important thing is not to stop questioning.”

Albert Einstein

By now you may have noticed that each Delta Scape blog post title is in the form of a question. Why is that? I am glad you asked. Like most of what I do as an educator, it is intentional and built around ideas gleaned from multiple sources.

In the reading comprehension literature, asking questions is identified as one of the core strategies effective readers use. The questions readers ask serve two purposes: (1) Questions help readers to monitor if what they are reading makes sense; and (2) Questions propel readers deeper into the text. A person who reads a passage and asks, “What just happened?” or “What happens next?” is likely to be more highly engaged than someone who is just reading the words.

Questions are also an essential part of the Understanding by Design approach to unit planning. These essential questions take three forms: (1) big-idea questions; (2) key-content questions; and (3) making-sense questions. The goal is to design the unit by determining the essential questions that will frame an authentic and engaging learning experience.

In How to Solve It, Polya uses questions as a means to support problem solving in mathematics. Examples of questions a mathematician might consider during each phase of the problem solving process are provided in the book’s introduction. If you get a chance to watch Polya’s video, Let Us Teach Guessing, you will see him modeling the use of questions to work through a problem and make sense of it.

Asking Better Questions by Morgan and Saxton is another good resource. On page 27, they write: “Learning springs from curiosity, from the need to know.” The questions learners ask contribute to this need and their level of engagement. In order to support teachers and learners in increased involvement in any learning experience, the authors introduce the Taxonomy of Personal Engagement. I use a version of this taxonomy in the courses that I teach as a means of supporting my learners in monitoring their engagement and considering questions they could ask that would improve their involvement in the task at hand.

I hope this answers the question why all my posts have a question in the title. If not, feel free to ask your questions in the comments. Sorry, I couldn’t resist.

Tuesday, May 24, 2011

How do I plan for problem solving?

Almost a month ago, I wrote a post "Is direct instruction a better approach to teaching math?" that got a lot of attention (relatively speaking). My post was in response to an article which used one poorly constructed (my opinion) study to suggest that problem-solving or inquiry-based lessons were less effective than lecture-style instruction when it comes to standardized-test results. What seemed to get the most attention/ire was a comment by the article's author, Paul E. Peterson. 
"I, too, like those problem-solving classes. They require less preparation and are easier to teach."
This might be true for a tenured university professor who: (1) enjoys academic freedom; (2) has no "accountability" to a standardized, national test; and (3) does not believe in the problem-solving lesson as an instructional approach. But for the rest of us, a problem-solving lesson requires a great deal of effort. I want to share my process of preparation for such a lesson in this post.

Because I am also a university professor who enjoys those first two perks, I want to focus on a fraction lesson that I planned and taught a few years ago at a local elementary school. When the fifth grade teacher contacted me for help, she was very specific about the content that I needed to addressed in the unit. In Michigan, the driving force for most K-8 teachers is the Grade Level Content Expectations (GLCE) and for my series of lessons I needed to get at this standard: 
N.FL.05.14: Add and subtract fraction with unlike denominators of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 100, using the common denominator that is the product of the denominators of the 2 fractions.
My first step in planning was to gather data about my learners. Through a series of emails, I found out that this school grouped fifth graders by ability for math and that this teacher taught the lowest group. This included several learners with special needs supported by a special education teacher. While I am not a fan of ability grouping, this was not my fight. I was just grateful for the information.

Mathematics understanding is about experience not ability. It was up to me to plan a problem-solving lesson that offered learners an experience that would support their development of a relational understanding of fraction computation. Fortunately, I was familiar with an excellent resource that provides just such an experience. Planning a problem-solving lesson is not about developing activities from scratch (but if you have time and training, then this can work). Still, the planning does require effort in identifying appropriate resources and structuring them in such a way that they support learning.

Cathy Campbell wrote about this excellent resource on her blog. In particular, she discusses the clock model used for adding and subtracting fractions, which can be found in Minilesson for Operations with Fractions, Decimals, and Percents. Cathy does an great job describing this resource, so there is no reason for me to say much more except that I find its use of context and connections to prior successes very supportive for learners.

I am also fortunate to have the professional development packet that goes along with the series. This packet includes videos of teachers modeling some of the lessons. Before planning my lesson, I watched Joel teach the clock model, and it gave me some ideas of how to organize the lesson. In particular, it showed that he introduced the model in a whole-group setting.

Finally, I was ready to write out the plan. I decided to use a slight modification of a lesson planning framework Debbie Miller shared at the Michigan Reading Association Conference in 2008 and described in her excellent book, Teaching with Intention. You can view my plan here. As you can see, it is quite detailed, yet I do not consider it a script. I am a firm believer in Jon Stewart's approach to planning, "Creativity comes from limits not freedom ... When you have a structure, then you can improvise off of it..." (I still wish I had remembered to share that quote during my TEDx Talk). This detailed plan allowed me to make necessary adjustments as I taught the lesson, but that is for another time.

I hope this makes the point that planning for problem-solving is not easy. "Where's the problem-solving?" you ask. Let's compare the plan with the National Council of Teachers of Mathematics Process Standard for Problem Solving:

  • Build new mathematical knowledge through problem solving;
  • Solve problems that arise in mathematics and in other contexts;
  • Apply and adapt a variety of appropriate strategies to solve a problem; and
  • Monitor and reflect on the process of mathematical problem solving.

Please let me know in the comments if any of these are unclear in my planning.

Monday, May 23, 2011

What does the grade say?

The following workshop is based on an activity used in a class for preservice elementary math teachers. The original intent of the activity was to reinforce measures of centers while considering philosophical issues associated with grading. It has been updated to reflect my current understandings about teaching and learning and the Common Core State Standards (CCSS).

Grade 6 CCSS in Statistics and Probability [6.SP]
Develop understanding of statistical variability.
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

Schema Activation:  Journal Jot
What does an A grade communicate?
What does a B grade communicate?
What does a C grade communicate?
What does an F grade communicate?

Focus: Use the statistics that we have explored so far (mean, median, mode, and range) to describe the learners’ scores provided below.

Activity:  Small Group

Math Scores (Each project is out of 20)


Reflection: Think-Pair-Share
  • Based on your analysis of the scores and your responses to the Journal Jot, assign a math grade for each learner.
  • Please provide some insight into how you assigned these grades and what you hope they communicate.
  • Historically, teachers have used the mean of a set of scores to assign grades. Do you think this is an appropriate strategy in this case? Why or why not?
  • In general, why do you think teachers typically average scores to determine grades?

Exit Ticket: Now what grade do you think each learner earned and why?

Thursday, May 19, 2011

What is an EdCamp? Part III

Before this year, I had never heard of EdCamp, and then I got onto Twitter and began seeing tweets about it all the time. People raved about the experience and what they were learning at these unconferences. Was this a fad or would this be a transformative moment in education? I had to see for myself. Fortunately, Wayne State was scheduled to host an EdCamp - not exactly close, but close enough for me to check it out. Reflecting back over the EdCamp Detroit experience (the sessions I attended and the session I facilitated) I can see how it has helped me to refine my vision of 21st century education.

The two main points I wanted to address in my session were: (1) how isolation is destroying our education system; and (2) how technology can help us to break this isolation and connect. These were based on my view of education occurring in silos. Teachers isolated in their classrooms. Schools unaware of what was happening across their district. District disconnected from one another. Universities and K12 schools fighting over curriculum, methods, and resources. I thought that the answer was a lab school. In particular, a virtual lab school that would make use of the available technology to connect teacher educators, inservice teachers, preservice teachers, and K12 learners.

What I learned at EdCamp Detroit was that my vision was too limited. While a lab school would be nice, it did not go far enough.  My vision changed because of what I saw there. I smiled as other session facilitators discussed web resources I had been using myself. Hundreds of miles apart and we had shared an educational experience. I watched as one of my sessions Skyped in with another session at EdCamp Boston. When Nick Provenzano (AKA @TheNerdyTeacher) began to describe an internet project he had used, the teachers in Boston screamed (actually screamed like Nick was some rock star) because they were familiar with his work. Mostly, I was amazed as these teachers referencing the online work of people I respect and follow on Twitter. People like Dan MeyerEric MarcosAlan November, and Derek Muller.

Reflecting on all this has helped me to realize that a lab school might be thinking too small. What I really want is a virtual one-room schoolhouse. A place where it is safe to explore ideas with other like-mined learners. A system that expects success through collaboration and support. A learning environment where the more experienced mentor those with less experience. A school where we share common knowledge instead of isolating it based on subject or grade-level.

So, are EdCamps a fad or a transformative moment in education? I think they are an opportunity. We have worked in silos for far too long. It is time to breakdown the barriers that separate us and work together to educate all learners. EdCamp has the potential to move us toward this goal. 

See for yourself at the nearest EdCamp. Or you can start your own. I know that I will be at EdCamp Grand Rapids on November 5, 2011. I hope to see you there - maybe over Skype.

Wednesday, May 18, 2011

What is an EdCamp? Part II

In my previous post, I offered an overview of my experience at EdCamp Detroit. Well, it did not provide the entire experience. I still need to share my presentation and the wrap-up of the unconference - that is the purpose of this post.

I called my presentation, Collaboration between College and K12 Classrooms: Win-Win-Win-Win. It was essentially based on two of my prior blog posts: one dealing with using technology to support distance learning and my dream to reclaim assessment from politicians and testing companies. Earlier EdCamp Detroit sessions on technology and collaboration also gave me some resources. Mostly, though, I wanted to listen to the participants' experiences and ideas about ways to consolidate our efforts to improve education.

Consequently, my PowerPoint was fairly bland. The version provided here includes backchannel comments from Kristen Fontichiaro (@activelearning) who tweeted much of the session. I am grateful for her contribution.

After my initial remarks, I opened up the conversation to the participants. Some of the College-K12 collaboration experiences included: Preservice teachers acting as substitute teachers and an Autism Center in Novi partnering with a local university. Kristen also made us aware that the University of Michigan is already exploring a lab school.

The new ideas generated were wide ranging. A member of the local college faculty suggested that her students could hold extracurricular classes to address K12 students' interests. A high school social studies teacher thought that it would be helpful for college students to describe the process for entering and succeeding in college to potential first-generation college attendees. Participants also suggested using Skype as a way to provide homework support, using college students to staff existing projects like Destination Imagination, and teaming college students with National Honor Society students in mentoring younger students.

I am sure there were other ideas that I missed (always the problem with making a list). Several times I found myself so engaged with the conversation that I forgot to take notes. I would ask that you please add any other suggestions in the comments.

While these ideas were amazing, it was the connections that I made that I found most meaningful. I now have contacts across the state. Tech-savy contacts that I will tap this coming fall when I have 60 preservice teachers in need of experience working with real K12 learners. I went from my session to wrap-up filled with ideas and hope.

EdCamp Detroit ended with a summary of the day, another brief Skype session with Dan Callahan at EdCamp Boston, and door prizes. What sticks with me most is a question asked by Nick Provenzano: "What if your next Professional Development day was an EdCamp?" Well? What if...

Tuesday, May 17, 2011

What is an EdCamp? Part I

EdCamp Detroit was an eye-opening experience. A perk of being a university professor is attending a lot of conferences (NCTM and TEDxGrandRapids to name two recent ones). I had never been to an unconference, however, and was looking forward to checking out this new approach to improving teaching.

The first thing to know about EdCamps is that there are no prearranged speakers. Participants arrive and encounter instructions like those shown below. EdCamp Detroit had 32 slots available spread equally between four one-hour sessions. To my amazement (since we always struggle to find enough K12 teachers to present at Math in Action) these slots began to fill rather quickly.

I have given plenty of talks, but I will admit that this format intimidated me a little - maybe because I had not done my typical preparation. After overhearing one of the organizers encouraging another participant to put an idea up on the board, I decided to take a risk. I selected a time slot at the end of the day and a topic that I thought would allow participants to be actively engaged. Most people feel like their brains are full at the end of a conference and I wanted to provide an opportunity for them to release that pressure by reflecting on what they learned. There will be more on my presentation later, but first I want to provide an overview of the EdCamp Detroit experience.

I chose sections that I thought I could incorporate into my final session. The first was called "The Flipped Classroom and Screencasting." Dan Spencer shared his ideas for using available technology to enhance learning and referenced examples like Khan Academy and MathTrain TV. Steve Dickie led the next session on "Pseudoteaching." He referenced blogs by Frank Noschese and Derek Muller (which actually questions some of the assumptions made by Khan Academy). In the third session, Mike Kaechele led a discussion about "Collaboration Across Classrooms." This included a Skype session with a group at EdCamp Boston (led by Marialice Curran) which allowed participants to share their experiences and see the power of this resource in action.

@TheNerdyTeacher shares with participants at EdCamp Boston
Very little of this was new to me. Again, I have more time than your typical K12 teacher to explore emerging educational approaches, ideas, and resources. Still, I was energized by these sessions because of the involvement of the participants. EdCamp presenters expect everyone to get involved and in most cases participants oblige. I left Mike's session excited to see what would develop in mine.

To be continued...

Saturday, May 14, 2011

Which problem is "just right" for you?

Today on my Twitter-stream there was a great deal of discussion about how we use homework in math class. Being a university math educator, I have more freedom than most K12 teachers when it comes to assigning homework but that doesn't mean that I am any less concerned about this topic. Homework is a important memory for many math learners and we teachers need to consider carefully how we use it.
If our schools require us to assign homework from the text but we have some freedom in what it looks like, it is time for us to work our magic as problem solvers. John Golden and I team-taught a course for preservice teachers a few years ago and modeled some ideas of adding to pre-existing items from a text. We called them "just right" problems in reference to the NCTM article Vygotsky and the Three Bears. Here's what the original items looked like:
from Scott Foresman – Addison Wesley Math [5th Grade]
And here is how we tried to make them more thinker friendly:

Using these existing textbook items to demonstrate the processes associated with doing math. Pick one or two of the following to explore:
  • Do either the odds or the evens – your choice. Why did you pick the evens (odds) to work on?
  • Look over all the items. Which five do you consider the easiest? What makes them easy? Which five do you consider the hardest? What makes them hard?
  • Pick an addition item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
  • Pick a subtraction item that is just right (not too hard and not too soft). Solve the item using two of these three methods: using manipulatives, drawing a picture, or developing a real world context. Compare the two methods you selected.
  • Pick one item to solve and write a metacognitive memoir that describes your cognitive efforts.
  • The answer to one of the items is nineteen-twenty-fourths; which item is it? (Be sure to keep a record of your thinking)
  • Which answer is closest to one? How can you be sure? (Be sure to keep a record of your thinking)
  • Put the items in order based on their answers from least to greatest. (Be sure to keep a record of your thinking)
  • Write your own ‘just right’ problem related to this review.

Which problem is "just right" for you?

Is there another problem we could add to this list?

Friday, May 13, 2011

How might this apply to education?

Yesterday, I attended the TEDxGrandRapids Conference. This was my second TEDx conference. I attended and presented at the TEDxGrandValley Conference last year, but I was unable to stay the entire day because of a teaching conflict. Fortunately, I was able to stay for the whole conference yesterday. 

At TEDxGrandValley, I was able to stay engaged because I was the last speaker of the first session. The expectation is that TED Talkers will try to make connections with the content of previous speakers. Consequently, I kept asking myself, "How does this relate to what I want to share?" I was not presenting yesterday and I was unfamiliar with any of the speakers (besides what I read online), which meant that I needed to come up with another question to provide me with focus: "How might this apply to education?"

Because of a lack of internet and poor cell reception, I was unable to Tweet much during TEDxGrandRapids. Instead, I decided to share the notes I kept on Evernote on my blog (keeping them short a la Twitter). Please keep in mind that I was looking for connections to education. The fact that I was unable to do this in every case is not a criticism of the speaker. They had no idea what I was focusing on. Here are my notes, with some connections detailed in parentheses:

Session 1:
Sheryl Connelly
Innovate Uncertainty
• Uncertainty cannot always be predicted but we can be prepared (Planning)

• Alternative scenarios - writing more than one story (Planning)
• "The future belongs to those who prepare for it today."' Malcolm X

Innovate Ultra-Rapid Renewables
• Sustainability: consider impact, energy, and time. (Styrofoam or teaching)

Innovate Innovation
• Reflect on how far we have come (Reflective practitioner)
• Path to meaningful innovation [slingshot framework]: striving for more, enhancing seamlessly, unconventional thinking, audience [narrow to broad], & systematic implementation (Educational reform) - world awaits your invented game

Innovate Education
• What is the right thing to make? (Self-directed learning)
• LUMA Institute - teaching creativity through design (Education reform)

Session 2
Jeffery Kimpton
Innovation Art
• Innovation takes imitation and intention (Instruction)

Innovation Curation
• Gather material to tell a story (Planning)
• Your choices matter (Decision making)
• Be clear in your purpose (Instruction)

Innovation Entrepreneurism
• What are learners' pain point? (Assessment and evaluation)

Innovation Justice
• Birth pains - closer together and more intense (Education reform)

Session 3
Steve Frazee
Innovate TEDxGrandRapids

Innovate Cyborg Musically

Innovative Your City
• Vibrancy - stumble onto the fun (Planning and Instruction allowing for collaboration)

Innovate Philanthropy
• Corporations are good at creating powerful solutions to things that aren't really problems (Authentic teaching and learning)
• We don't think about success factors we've received (Reflective practitioner)

Session 4

Amy Davidsen
Innovate Clean Revolution

• We undercut efforts to innovate because we are afraid (Education reform)

Innovate Wonder
• Developing a sense of wonder (Planning and instruction)

Innovate Experience
• What do we want for the world? (Planning)
• Flow and Meaning - ways to happiness (Conditions of Learning)
• What sucks and how do I fix it? (Action Plan)

Musical Guest
• Check out his TEDx freestyle when you get a chance

I hope that these brief notes do justice to the TEDxGrandRapids speakers. I know that this post does not capture the whole experience, which was also about the people I met and the conversations I had around innovating education. Perhaps once I've had more time to process the day, or if you have any questions about the connections I made, I can post more about what I learned and how I plan to apply it to my teaching practice.