Saturday, January 30, 2016

Where was I using the Process Standards?

MTH 110 is an Intermediate Algebra course that explores topics typically found in Algebra I and II. Most of the students have seen the content before but for some reason it did not stick (nobody's fault, just a fact). Instead of simply re-teaching the topics, we are using the content to extend our understanding of what it means to do mathematics.

One of the ways we are attending to the mathematical process is by making our thinking visible is through Metacognitive Memoirs. Because this is a new approach to many of my students, I spend a good portion of Flight School (the first three weeks of the semester) developing the idea of how we can go beyond showing our work. This past week, I facilitated a workshop that demonstrated one way to share our thinking.

Schema Activation: Predict how many small cubes are in Step 43
from Visual Patterns
Focus: NCTM Process Standards
As I share my thinking, please keep track of where I am attending to the bullet points associated with these processes
Activity: Metacognitive Memoir Demonstration

Reflection: Where was I ...
  • Problem Solving
  • Reasoning
  • Communicating
  • Connecting
  • Representing
It might help to see my notes.
Please share your thinking (where I made the processes visible and opportunities I missed or messed up) in the comments.

Thursday, December 31, 2015

How did Bi-N-Bi go?

Awhile back, I promised to tell you how it went when I tried out a new game I developed while playing Bingo with Dad. Bi-N-Bi, or Decomposition Bingo, is an attempt to add a bit of strategy into the familiar game. Instead of finding a called number, you use a modified Bingo Board to try to find a pair of numbers that sum to the called number. At least that was the plan when I began testing it out with a couple of classes of sixth graders. But they had some other ideas. And some of their ideas were quite good.

The first game went pretty quickly. While the Bi-N-Bi rules allow for choice, a lot of the players used the same addends to make up the sums. I had given all of the sixth graders the same board, to test out my "choice" hypothesis (that the players would create different results) but they were unsatisfied with the ties, and many asked for a different board. Having anticipated this, I had several other boards available. There weren't enough for everyone to have a different board, but they seemed satisfied with the variety.

We played another game using the different boards and the players were happier with the results. Several sixth graders called "Bi-N-Bi" at the same time but they each had different numbers covered. After claiming their prize, a Bragging Rights Trophy, I asked if they had any suggestions for improving the game. They had a few:
  • Use two or three addends;
  • Use pairs (to stay with the "bi" theme) but allow for addition or subtraction; this would allow for using traditional Bingo Cards;
  • Make the "Free" space a "Wild Card" that can be used to make a pair; I might use X to reinforce the idea of variable; and
  • Perhaps the most ambitious idea was to use all four operations and parentheses to reinforce order of operations.
We tried the three addends version. For some of the sixth graders, this was a struggle. So I tried to model some strategies for them as I walked around. For example, I would say, "46. I could use 40+1+5 or 30+11+5 or 20+11+15." These weren't necessarily numbers on their cards but simply different ways they could think about decomposing 46.

The sixth graders also wanted to try using something more than addition, so we played a version that used any combination of addition, multiplication, or parentheses. But only if they wanted, because a few of the students seemed overwhelmed by this change. A couple of winners are shown below.

At the end of the last game, I handed out Bragging Rights Trophies to everyone as a way to thank them for their help testing Bi-N-Bi. I told them that each had demonstrated that they were mathematicians. And if they ever found themselves in one of my math education classes at GVSU, they could turn the trophy in for 1,000 Bonus Points. I do what I can to encourage the next generation of Lakers and possible MTBoS participants.

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 ...