Thursday, November 12, 2015

How do you play Bi-N-Bi?

I spent a lot of last winter playing Bingo with Dad - sometimes, three days each week; it was a bit much. Don't get me wrong, Bingo is a fine game. However, it isn't very challenging. 

"Find this number. And, by the way, it's in this column."

I understand the point (and keeping track of multiple cards with an "auctioneer" calling the numbers can be a struggle) but I wanted more. So I began to wonder what it would be like if I could cover a pair of numbers that summed to the number that was called. For example, 46 is called and I cover 30 and 16 instead. I tried this during a few games and found that most of the times when I could decompose a number into two addends, I was using the B and I columns. This lead me to create my own card.

I liked the simplicity of this design. It would be easy to create, and players would have to decompose numbers greater than 45. Also, because the game included the element of choice, everyone didn't need a different card. Hilary could cover 43, John could cover 22 and 21, and Andrew could cover 13 and 30. (I planned to only call the number, not the accompanying letter; this would allow players to pick numbers in any column.) Finally, I liked the name, Bi-N-Bi, because it could reinforce decomposing numbers into two (bi) addends.

Today, I tested the game out in the classroom of a teacher I've been working with this semester. For the first game, I gave everyone a copy of the card shown to the right to see if the element of choice was enough to keep it interesting. The B and I columns were repeated to make it easier for players to know what numbers were available. I started out making sure that everyone was familiar with the goal of Bingo - getting five in a row or the four corners. The sixth-graders agreed that this wasn't very challenging, and they were excited to explore the changes I was suggesting.

The last issue to address was checking to see if a winning card is accurately covered. A player cannot simply call out the numbers, as happens in the original game, since many covered numbers are the result of decomposition and not because they are directly called. I toyed with idea of players marking the number called on the two chips used to cover the addends but I found that confusing when I tried it (and it meant cleaning the chips or throwing them away afterwards - not very sustainable). So I had players write their number sentences out on scrap paper. For example, if I called 34, 8, and 22, players might write:

  • 34 = 8 + 26
  • 8 = 3 + 5
  • 22 = 0 +22

And then they'd call, "Bi-N-Bi," provide each of the number sentences, and tell which of the addends they had use in their five in a row: "On the diagonal, I covered 8, 26, free space, 3, and 22."

With the instructions out of the way, I explained to the sixth-graders that I was looking for their feedback. I wanted to know what worked, what didn't, and what we might try differently. They were eager to be a part of the testing of this prototype and said so. 

A bit more nervous than I thought I'd be, I picked the first number. How'd it go? I'll tell you - in the next post.

Tuesday, October 20, 2015

What is the purpose of pre-assessments?

In our course, Probability and Statistics for K-8 Teachers, we are trying to apply Design Thinking to a project involving teaching in a local 6th-grade math class. We will be focusing on 6.SP from the Common Core State Standards. Before we begin planning our lessons, we want to know what students can already do in order to build on their strengths. Therefore, we decided to design a pre-assessment.

We recently went through the Design Thinking process and tested out our pre-assessment ideas with the 6th-grade teacher. Design Thinking is an iterative process, and the feedback we received from the teacher reinforced this idea. It was clear that we had not spent enough time defining the project, which resulted in a lot of disconnected pre-assessment ideas. So tomorrow we will return to the Define step using the Project Priority Puzzle shown below.

  • Select a phrase from each row (whatwhyhow, and when) in the table below that you feel ought to define our 6.SP pre-assessment. If you think a phrase is missing, write it in one of the blank spaces provided; 
  • Use scissors to cut out each of your selections, along with the top phrase; and 
  • Combine the phrases in order from top to bottom using tape to create your “Define” artifact.
Project Priority Puzzle

You could help us out by providing your definition of pre-assessment. Use the puzzle pieces above or create your own. Please add your definitions to the comments. Thank you in advance for your support of our future teachers.

Wednesday, September 23, 2015

How can we assess 3.MD.B.3?

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. [3.MD.B.3]
Teaching-Learning Cycle
Teachers use #MTBoS as a way to find interesting and effective math lessons. Recently, some of us have noticed that assessments are often lacking as this community shares in the work of teaching. So I am proposing the #MTBoSAP (Math Teacher Blog-o-Sphere Assessment Project) as a way to pass along our wisdom and experience assessing students in our mathematics classes. I figured I could start by sharing some work I am doing in one of my courses for preservice elementary teachers.

The teachers are currently researching assessment items related to the Measurement & Data Domain (focusing on Data) from the Common Core State Standards (CCSS) in Mathematics. One of the third-grade standards in that domain is written at the beginning of this post. Because we are partnering with a school district that uses EngageNY, we looked for assessment items that already existed within that curriculum. This is an exit ticket that we found in the Grade 3, Module 6, Lesson 4
We thought that this item did a fair job of assessing the last part of 3.MB.B.3 but completely missed the first part, "Draw a scaled picture graph and a scaled bar graph..." So we looked through the rest of the lesson and thought this item from the problem set looked promising.
This item has students "draw a scaled bar graph" (still no scaled picture graph) and asks that they solve "two-step 'how many more' and 'how many less' problems." As we thought about it further, however, we were concerned that it was not clear whether students would use the chart or the graph to answer the questions.

Therefore, I decided to try to modify the original exit ticket in order to assess more of the standard. I got rid of two of the bars and wrote questions intended to have them "draw (a part of) a scaled bar graph" and answer multi-step questions that require some "information presented in scaled bar graphs."

Please complete the bar graph using the following information. 
  • The number of books checked out on Thursday was 10 more than the number of books checked out on Tuesday. Draw the Thursday bar.
  • 1,480 books were checked out Monday through Friday. Draw the Friday bar.

What are your thoughts about using this modified assessment item to gather data on students' mathematical understanding related to 3.MD.B.3? Do you have a good item for this standard that you'd be willing to share? If so, please share the item or a link to the item in the comments. Also, If you have any other effective CCSS assessment items, please share them on Twitter using #MTBoSAP. Thank you in advance for all you do to advance the profession.

Monday, August 24, 2015

How can we support Math Talk in our classrooms?

For the past couple of years, I have been working with a colleague to support local districts interested in improving student-communication during K-8 mathematics lessons [Math Talk]. We do not have an agenda related to a particular curriculum. We are not pushing any method. Our goal is simple: provide assistance, whatever that may be, to elementary and middle school teachers trying to increase productive mathematical conversations in their classrooms.

After a recent professional development day, participants were asked for suggestions for future workshops. One theme that arose, was the need for support establishing classroom norms around Math Talk. For example:
  • Specific ways to implement the math talk norms in class.
  • Establishing safe atmosphere/learning environment
As university math educators, we have resources (research, time, technology ...) that K-8 teachers might lack. Given the teachers' request for support around Math Talk Norms, we went about trying to find and consolidate resources the teachers might find useful. This included putting a call out on Twitter. Below is the workshop that resulted from our efforts.

[The workshop focuses on the Thinking Together resources provided by the University of Cambridge Faculty of Education.]

How do we support Math Talk in our classrooms?

Schema Activation: Talking Points Process (from last time)
  • You are naturally good at talking, or not, and nothing can be done about it.
  • If you help people solve problems in class, it’s cheating.
  • Everyone can learn how to be part of a learning conversation.
Focus: B.R.I.C.K.

Because we are trying to avoid suggesting a particular set of norms, we focus instead on a process we and other teachers have found helpful. As you are thinking about developing Math Talk Norms (or any norms, for that matter), developing a solid foundation - a brick, as it were - can come in handy. These are in no particular order, except to make the acronym work, of course.
  • Keep your contributions brief: I told my middle school students that my classroom expectations were basically respect and responsibility. Respect in the way we communicated with each other. Responsibility for being prepared to participate.
  • Role-play how Math Talk does and does not look: We often think Math Talk is natural. It is not. Students will need examples and non-examples. I used scripts, like these, to support students' development as math talkers.
  • Incremental efforts: Learning takes time. Accept that changes to the way students communicate during math lessons requires a long term commitment and ongoing adjustments. You might need to revisit the norms after a break or when new issues arise.
  • Connected to other content areas: Teachers in other disciplines might already being using communication norms in their classrooms. Don't hesitate to build on their work. For example, elementary teachers often use  ideas from The 2 Sisters (ideas like role-playing) to foster productive literacy discussions.
  • Involve your kids in the development of the norms: "Buy-in" is an important part of the success of your norms. Students who believe that they have had a voice in the development of the norms find it easier to follow the expectations. This does not mean giving up complete control - your brief expectations ought to somehow be incorporated.

Activity: Do you need a Model, a Mentor, or a Monitor?
Monitor: Do you think you have an idea of what developing Math Talk Norms looks like? Then the resource we offer is time and a listening ear.

Mentor: Do you think you have some general ideas but need some support? Then one of us will collaborate with you to develop a plan.

Model: Are you stuck with what to do next? Do you need a demonstration? Then we will provide examples related to our focus. We have not used these example eourselves. We are hoping you'll help us to consider how they might work.
Reflection: Monitoring Sheet
Create a 5-by-5 grid modeled after this Talk Tally Sheet

Along the top, write the numbers 1 through 4 in the rightmost cells. On the side, identify four types of talk that you believe represent elements of productive Math Talk.

Extensions: Evaluate these other resources on Math Talk Norms shared via Twitter

Assessment for Inquiry by Darrin Burris

Tracy suggested Sheila Tobias' questionnaire about math myths

Teacher Talk Moves and Research Basis by Conceptua Math

Setting up Number Talks by Zones Math based on Sherry Parrish's book

Shared by Connie Hamilton during #TMChat

Shared by @EarlyMathTeach during #TMChat

    Saturday, July 18, 2015

    But what about the colleges?

    This question comes up a lot whenever people suggest making changes to K-12 education. I hear it at conferences and professional development sessions. Versions of this question even show up on Twitter:
    As the above example shows, the question is often in response to taking on some sacred cow in education, like eliminating homework.

    The question was asked multiple times this past week at the LMF4PD Conference. Rick Wormeli, the featured speaker the opening day, challenged many of our traditional grading practices (like averaging zeros and giving only partial credit for re-takes) and this made some of the teachers uncomfortable. I understood their concerns given the current emphasis on ensuring that students are "career and college ready," but I wanted to reassure them that colleges (and more importantly the teachers' kids) would be just fine if they transitioned from preparing students to empowering learners. So along with Dr. Clark Danderson from Aquinas College, we held an edcamp session on the third day of the conference to address what colleges and universities expect from learners.
    First, not all institutes of higher education are the same. I talked about how when my own kids were considering college, I discussed the difference between a university that focuses on research and one that views teaching as its primary purpose. High school seniors interested in attending University X need to know how to do research into what they can expect from a university's teachers and courses.

    Second, even within a university, different departments might have very different philosophies of education. For example, my department is committed to keeping class-sizes manageable in order to make lessons more interactive and alternative assessments, like portfolios, doable. Other departments at GVSU continue to to use large lectures and multiple-choice tests (no judgement - really). Again, it's up to the prospective learners to do the research.

    My last point was that even if learners find themselves in college classrooms using traditional methods of instruction or assessment, those that have learned to self-assess and adjust will find ways to be successful. On the other hand, those that have only been prepared for this "worst case scenario" (the traditional approach) will struggle at universities that expect more than "consume and regurgitate" from their scholars. Unfortunately, we see that happening a lot in our department. Students struggle in our courses and with our major because they are waiting to consume, and we want them to construct.

    Now, when it comes to being career ready ...

    Sunday, July 12, 2015

    How much do we owe Andrew?

    Andrew getting ready to add some doors
    to a utility room at our camp,
    Our son, Andrew, has been doing some work for us this summer. The other day was payday, and he let us know that he had put in 11 hours of work the past week. We are paying him $18.75 per hour. (He's 27 and has a degree in Building Technology from NMU, so these are not simple chores.)

    As we did the math to pay Andrew for his services, I was interested in the different approaches we picked to determine what we owed him. Kathy grabbed a pencil and paper to do the standard algorithm. Andrew looked at me and asked how I would do it. "Honestly," I said, "when there's money involved, I'd grab a calculator." Andrew proceed to talk through how he would calculate 11 x 18.75 mentally. (He has always had an affinity for numbers, though he struggled with school math that relied on "rules without reasons".)

    That same week, I participated with a group of about three dozen elementary teachers in training for Math Recovery. When it came to supporting students' multi-digit multiplication and division strategies, several of the teachers discussed how kids' mental math needed to lead to more efficient strategies. This seems reasonable; it's even in the Standards for Mathematical Practice(emphasis mine):
    ... procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), ... 
    But what does "efficiently" mean when it comes to multi-digit computation? Who calculated 11 x 18.75 efficiently: Kathy, Andrew, or me? What criteria are you using for efficiently? This is not a rhetorical question - I really want to know. 

    Wednesday, June 24, 2015

    Why do you want to learn to play the mandolin?

    It was time to learn something new. That was my mindset when I signed up for mandolin lessons at Earful of Fiddle Camp. Even though I have had some musical training (piano and flute), I knew learning to play a stringed instrument was going to be a challenge. So I spent last week swatting mosquitoes and strumming strings, all so I might someday call myself a mandolin player.

    Laurel and Michael
    of Red Tail Ring
    Oh, and as the sign says, we also got to listen to some old time music. On Tuesday, Red Tail Ring taught a workshop on harmony in the afternoon and put on a concert that night. Afterward, I asked Michael (he played the mandolin on a few songs) if he ever gave lessons. He said, "Yes." And then, perhaps noticing my age, he asked, "Why do you want to learn to play the mandolin? What are your goals?"

    I responded, "Don't worry, I'm not looking to replace you." He laughed, but I could see that he was serious about knowing my musical aspirations. Perhaps his instructional approach would depend upon what I wanted to accomplish.

    "I'm struggling with my fingering."

    He asked, "You mean your fingers are tender?"

    "They are," I said looking at the blisters on my fingertips, "but I mean creating a clean sound when I play."

    He explained that the two were related. Once calluses formed, I'd be able to press on the strings with more commitment. However, we still hadn't addressed the original questions. He tried again, "Who do you want to play with or for?"

    Now I got it. "Mostly with my family. My wife and son-in-law are here, too. You know, like around the campfire. Oh, and I'd like to be able to play for my grandson; songs like Puff the Magic Dragon and Itsy-Bitsy Spider." This made sense to him and we made arrangements to connect later in the summer to arrange some lessons.

    At the end of the week, campers perform for the rest of the camp, friends, and family in an event called, Earful of Idol. When the group I played with finished our song, I was reminded of my conversation with Michael. I had been miserable most of the day trying to memorize the song. During the performance, I only played about half of the notes, and only half of those at the right time. My "bandmates" picked up the slack and the audience was generous with their applause, but I was ready to quit mandolin. There was no point continuing if I wasn't enjoying it.

    That's when Michael's question hit me, "Why do you want to learn to play mandolin?" My purpose for picking up the mandolin was to have fun with my family. It wasn't to perform in front of strangers. Even the best audience couldn't hold a candle to this guy.

    Vance wants more cowbell!
    Maybe I'll be grateful I played at Earful of Idol someday - maybe not. (I do not subscribe to the "someday, you'll thank me" school of teaching.) At the very least, I gained empathy for students who "quit" math because the effort does not seem worth the experience. Whether it's music or math, learners need to find their own purpose in order to stay engaged in learning.