Friday, September 23, 2016

Should I keep these math flashcards?

I am working on a project with Alyssa Boike called MacGyver Math. Before I impose my kind of crazy on that site, however, I want to write a prototype post. Here it goes...


Dear MacGyver Math,

I am a new second-grade teacher. Over the summer I went through some of the boxes the previous teacher left in my classroom and found a bunch of math flashcards. It's my understanding that addition, subtraction, multiplication, and division flashcards are bad because they reinforce the belief that being good at math means being fast. And who can forget those awful Around the World games in math class where one person dominated while the rest of us just sat there?

So, should I keep the flashcards or throw them away?

Enlightened Elementary Educator in Elmira


Dear Enlightened,

Thanks for your letter. It reminded me of one of my favorite Macgyver quotes. 

While you might not want to use the flashcards to reinforce false beliefs about doing math, they might serve other purposes.

For example:

We could ask our students to sort the flashcards; "Which facts do you know and which facts are you still learning?" Then we could see if there are any known facts that they could use to help them learn the unknown facts.

Students could use an interesting/appropriate subset of flashcards to create a real graph of the answers. They could analyze the graph looking for patterns, making conjectures, and testing the conjectures. Like, "I think that if we keep sorting out our flashcards, 27 will show up the most because that's the largest product in our set of cards."

We could use the flashcards to create Which One Doesn't Belong [WODB] scenarios. Or have students create their own WODB to challenge their peers. (See the WODB blog and the book by Christopher Danielson for more information about this instructional approach.)

Finally, if we want to play a game that's not Around the World, maybe we play Go Fish. (Make up your own rules depending on what you want student to experience. Or better yet, maybe have them make the rules if you want to encourage them to play with math.)

What do you think, regular (irregular, and new) MacGyver Math readers? How might Enlightened Elementary Educator in Elmira step back and take a look at what she's got (flashcards) in a totally different way? As always, leave your suggestions in the comments.

MacGyver Math


If you have suggestions for Enlightened in Elmira regarding using flashcards in novel ways, please address your comments to her. If you want to comment on this MacGyver Math format, please direct your comments to Dave. Either way, thanks for your participation.

Thursday, May 5, 2016

How does a mathematician see the world?

And how does a math teacher help learners to see the world through a mathematician's eyes?
This was a major challenge during the past semester in my Intermediate Algebra sections. Many of the students came to class expecting me to show them a procedure that they would practice until test time - when they would reproduce the procedure and promptly forget it. Nearly all of the students had seen the Intermediate Algebra content in high school (linear, quadratic, and exponential functions) but it hadn't stuck.

This was the problem. They had seen the content. Now it was time for them to use it to see the world. So I shared pictures and videos and asked them to look at them through a mathematical lens.


At one point, a student said, "Just once, I wish I could see the world through your eyes." Exactly! Unfortunately, they were often so afraid of making a mistake, of breaking the mathematical glasses, that they were not willing to even put them on. They did not know how to be playful with the math we were exploring.

So I introduced them to Yes, And ...; this is a problem finding activity that I developed using a well-know improv game where participants accept and build on the ideas of their partners. Here are the instructions for my version:
  • Provide a mathematical context (often pictures or videos) but without any identified problem;
  • Pair up the students and find a fun way to identify Student A;
  • Student A picks one of the contexts and finds a problem to solve;
  • For one minute (this is usually enough time to get started without completely solving the problem), Student A talks through his or her thinking while Student B writes as much as possible down on a piece of paper;
  • After one minute, the students switch roles. I say, "Yes, and ...," to signal the switch and to emphasize that Student B ought to build on the work that was already done.
  • Student B thinks out loud for one minute while Student A records the thinking on the same paper.
  • After one minute, I say, "Yes, and ...," and the roles reverse again.
Yes, And ... can go on as long as the teacher wants. I found six minutes (three rounds) to be about right the first time we played the game. One student submitted this to demonstrate her engagement with the task.

I cannot claim for certain that Yes, And ... changed my students' view of mathematics or helped them to see the world through a mathematical lens. What I know is that for six minutes they played with math. It's a start.

Tuesday, May 3, 2016

Isn't it great to be a teacher?

Yes, it is great to be a teacher. However, unless you are a teacher or know one very well, it's probably not for the reasons you think. The one example that I am constantly having to explain to people is the whole misunderstanding about the summer pay thing.

"Isn't it great to be a teacher and have a paid summer vacation every year?" asks a person I just met when she/he finds out I am a teacher. 

I explain that the money I get paid is deferred from what I earn during the school year. Much like a pension or Social Security, it helps me make it through a time I would not normally be earning a salary. Also, it helps most districts to spread payments out over the year. Win-win, right?

But what happens if the district is in financial trouble and runs out of money before paying the deferred salary to teachers? This is not a hypothetical question. It is happening right now in Detroit. And state politicians are having a hard time understanding why the teachers are worried about their summer pay.

Let's do the math. Imagine a Detroit teacher works for 9 months, September to May, for $36,000 (These are rough estimates. I am simplifying the problem to make it easier to follow.) The teacher can take the payment across 9 months, $4,000 per month, or 12 months, $3,000 per month. If a teacher in Detroit chose to spread her pay over 12 months and the district runs out of money at the end of June, then she will receive 10 months of pay or $30,000 of the $36,000 she earned. This translates to 7 1/2 months of teaching or through the middle of last month (April). In other words, if the legislature does not come through with the necessary funds, Detroit teachers have already worked for roughly 10 days for free. 

The governor's office tells teachers not to worry - that a $715 million aid package is moving through the legislation. Unfortunately, according to the New York Times
A year ago, Mr. Snyder, a Republican, proposed a $715 million aid package for the Detroit district, but it has been bogged down in the Republican-controlled Legislature.
No wonder Detroit teachers feel too sick to go to school.

Oh, Happy Teacher Appreciation Day!

Update [5/4/2016]: I was a bit off with my estimation of connivence, but it was close.

Update [5/5/2016]: Michigan House approves $500 million funding plan for Detroit Public Schools - but there are concerns.

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.