And then what did you do?

In my experience, middle school students are often reluctant to try a problem they don't already know how to solve and uninterested in using alternative solution methods once they have one that works. Some middle school math teachers I worked with this week have found the same thing. So we considered how we might use techniques from improv to help students get started on a problem and possibly explore an approach they might not normally use. Improv techniques like the "Yes, And..." activity.

We decided to play it using this situation from Context for Learning Mathematics [CFLM] and a couple of starting choices.

Starting Steps:

1) First, I split up the dollar and the 25 cents and multiplied the dollar by 24.

2) First I doubled $1.25 and got $2.50.

We would make teams of three. Two people would engage in the "Yes, And..." activity while the third kept "numeric notes" that could be referred to after the activity. 

Here's how it might work:

Numeric Notes

First, I split up the dollar and the 25 cents and multiplied the dollar by 24.

Yes, and then I added back the 25 cents and got $24.25.

Yes, and then I remembered there were 23 more quarters that I need add.

Yes, and I don't know what 23x.25 is so I did a quarter plus a quarter is a half dollar instead.

Yes, and that reminded me that there are four quarters in a dollar.

Yes, and that made me wonder about how many dollars there are in 24 quarters.

Yes, and I divided 24 by 4 and got 6.

Yes, and that means 24 times .25 is 6 dollars.

Yes, and then I added 6 dollars to the original 24 dollars.

Yes, and that means the total cost of the turkey is $30.

Afterwards, the students reflect on the activity. This is an opportunity to observe their own thinking: what they did; why they did it; and how else they might have navigated the situation. I have done something similar with my college students, who finish up by writing a Metacognive Memoir about their experience.

What is Fluency in Math?

When I was a junior in college, several of my friends spent a semester studying in Heidelberg, Germany. I decided against going because knowing German or being willing to learn it was a prerequisite for the program. One of my friends, Scott, thought this was funny because the only German he knew/learned was, "Bitte, haben Sie einen Plattenspieler?" which translates to "Please, do you have a turntable?" I thought this was pretty funny, too, and I liked the way it sounded, so I also memorized the question. Clearly, this didn't make me or Scott fluent in German. In fact, unless I'm talking to some hipster familiar with record players, my question probably doesn't even make any sense to him or her anymore. I often share this memory when a discussion about fluency in math turns toward the necessity to be fast and accurate with facts or some procedure. While speed and accuracy might result from fluency, speed is neither a requirement of nor a support for fluency. 

Consider the actors playing Klingons on Star Trek: Discovery: they correctly deliver their lines without any pause. Are they fluent in the Klingon language?

Perhaps (some Trekkies are), but more likely they are professionals that have memorized a script. They are not responding to something said by another Klingon actor, and I doubt many of them would be able to improvise new dialog. 

Is Siri fluent in mathematics? She is certainly fast and accurate.

But ask her how she arrived at her answer, and she is at a lost.

What does it mean to be fluent? Fluent speakers demonstrate flexibility in their language usage - the ability to play with words and use figurative phrases. Certainly, there is also a need to know the words and grammar associated with the language. However, this alone isn't enough to be fluent in the language.

And, in my opinion, knowing alone is not enough to be fluent in math, either - no matter how quickly you can arrive at the answer.

So, how am I wrong? I'm looking for pushback before I tell this story again. Make some comments and make me smart.

Should there be guns in schools?

I intend to share this letter with my governor and as a letter to the editor. I would appreciate your feedback in the comments. Thank you in advance.

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Dear Governor Snyder,

On November 8th, the Michigan Senate passed legislation (584, 585, and 586) allowing individuals with concealed weapons permits to request permission to carry guns in schools (and other previously gun-free zones) with extra training. “[A gun-free zone is] a target-rich environment for people that don’t abide by the law, and people should have the ability to protect themselves, wherever they are,” explained State Senate Majority Leader (and my senator) Arlan Meekhof. This follows the narrative often accompanying such legislation: the only thing that stops a bad guy with a gun is a good guy with a gun. 

I understand why this is a compelling story; it makes us feel safe, empowered, and may even touch on our dreams of being a hero. There's even a recent example of a good guy with a gun helping to stop a bad guy. But law enforcement officials, like the Houston Police Department, want you to consider a different story when involved in an active shooter event: run first; hide if you can't run; and fight only as a last resort.

Why would the police discourage you from being a hero? A couple of storylines come to mind:

• Good guy with a gun shoots an innocent student or teacher while trying to stop a bad guy (with or without a gun); and
• Good guy with a gun shoots some other good guy with a gun after misidentifying him as a bad guy with a gun.

We must also consider that confronting an active shooter isn't the only story that can be told around a good guy with a gun in a school. In fact, the good guy with a gun is unlikely to actually encounter an active shooter in a Michigan school given current statistics. So what other stories might we tell that involve a good guy, a gun, and a school - minus a bad guy?

• Good guy with a gun gets angry about something and shoots the student, teacher, administrator, or parent that made him angry;
• Child finds good guy's gun carelessly left laying around and shoots self or another child; and
• Child finds good guy's gun carefully hidden away and shoots self or another child.

I get why we would rather not consider these tragic storylines. Most everyone would prefer the happy ending associated with "good guy with a gun stops bad guy with a gun." But just because you like one ending better than the other doesn't make it more likely to happen - especially when you compare the hundreds of children accidentally shot each year with the handful of heroes.

We ought not be making laws based on Hollywood happy endings. Instead, we need to consider the reality and the possible unintended consequences that occur when a gun is added to any situation. If we truly want to use "common sense" when addressing gun violence, just remember that no one ever got shot in a place without guns.

Should these bills reach your desk, please exercise your veto. Do not put our children at risk just so a few people can dream of being a hero.

Sincerely,

David Coffey

What does this mean?

The other day I was working with some high school students on problems like this one:

I asked what I thought was a pretty straightforward question, "What does this mean?" [SPOILER: it wasn't as straightforward as I thought.]

One student said/asked something like, "Split the square root? Make it the square root of one over the square root of 64?" (Seriously, these were implicit questions asking, "Is this where I start?") 

STRIKE ONE!

I responded, pointing emphatically at the problem, "But what does this mean? How would you say it in words?"

The other student interjected, "The square root of 1 over 64." And then asked, "We should rationalize the denominator, right?" 

STRIKE TWO!

I tried again, "You're focusing on what you would do. I want to know what you understand about square root problems. What if it was the square root of sixty-four, then what would the problem be asking you to find?"

The first student, clearly getting frustrated, "I'd take a half of 64." 

STRIKE THREE! YOU'RE OUT, COFFEY!

I didn't see any point to continuing to frustrate them or me with more questions. So instead, I tried to model my thinking about solving problems involving radicals.

When I was a kid, I used to bring a four-function calculator with me on long road trips (like the one shown here). There was no square root button on the calculator, so I would play What's the Square Root by typing in a guess, pressing the multiplication button and then equals, and use the result to see how I might need to adjust my guess. It was hours of fun! 

If my question was, "What's the square root of two?" then I was essentially asking, "What number multiplied by itself is two?" So when I ask what this (again, emphatically point at the original problem) means, I'm thinking about playing the game and trying to determine what factor squared gives this (more pointing) product, the number inside the radical.

I went back to our simpler problem, "So this is asking me, 'What factor squared equals sixty-four." They both answered eight.

Then we went back to the original. "What does this mean?"

They said, "What number times itself gives one over sixty-four?"

"Yes, What factor squared is one-sixty-fourth? Now that we understand," I said, "we can begin thinking about what to do."

It's troubling to me that so many years after Pólya wrote How To Solve It, that students race past the first phase of the problem solving process - Understanding the Problem.

Why is that? It can't be that we are focusing on the procedures and not the concepts. No, it can't be that, can it?

What's your vision?

From Black Elk Speaks

After the long winter of waiting, it was my first duty to go out lamenting. So after the first rain storm I began to get ready.

I recently heard Kent Dobson talk about this idea of "lamenting" - what is often thought of as a "vision quest." [I apologize in advance if I get some of the details wrong in this post. A lot of what I'm writing is a combination of my memory of and the connections made during his talk. Please let me know in the comments if anything needs correcting.] 

In the Lakota tradition, when a seeker comes of age, he or she goes off with an elder to "cry for a vision." After some preparation, the seeker is left alone to lament the current state of the community and seek answers in the form of a vision. From time to time, the elder looks in on the lamenting to advise and support the seeker. When the lamenting is complete, the seeker returns to the community and shares the vision. This is an important rite for the community because without new ideas, a community withers and dies.

As I begin a new year of supervising student teaching, I want this ritual to inform my work. I hope to play the role of the elder supporting coming-of-age teachers as they experience (and lament) the current circumstances in math education and seek new answers. I will listen to those answers and take them seriously because in many ways our profession is withering. And I will help the student teachers to share their visions with the larger math education community, so that these seekers might contribute to the development of our profession.

Thank you in advance for welcoming these seekers and helping to interpret and implement their visions.

What needs changing?

The Michigan Department of Education is considering changes to Michigan teacher certification and asking for feedback. Here is more information about their proposed certification structure.

Feedback is due today. (Sorry, I'm still a bit of a procrastinator.) However, if you consider yourself as having any stake in education, I encourage you to complete the survey, as soon as possible. It'll take maybe 45 minutes to watch the video and answer some questions.

For what it's worth, here's my response. (And, yes, I know I am biased.)

My issue with the structure being proposed is that it seems to presume that new teachers ought to be completely prepared upon certification. Even if we focused on giving specific grade-level certification, there is no way to address all the issues (textbooks, culture, community, resources, …) new teachers might encounter across the state (or country). Furthermore, this structure will limit the options available to teachers and potentially restrict administrators' ability to fill positions. I can imagine school districts asking for waivers to put PK-3 certified teachers in 4th grade because of a lack of options - a teacher that may have no training at that level.

I encourage that the state look at the Grand Valley State University teacher certification program as a model. We support the development of teacher-leaders in content areas by requiring even pre-service elementary teachers to get a subject major. Local districts regularly call us looking for recent majors because they know they will be content experts who can also integrate other subjects into their practice.

Also, the MDE needs to rethink professional development for inservice teachers. Because no teacher is truly finished learning, we must return to a robust professional development process that helps teachers build on current education experiences to deepen their pedagogical content knowledge in all discipline they might be teaching. Such a system is especially important now because we have so many new, inexperienced teachers entering the profession.

This professional development ought to be in partnership with accredited bodies responsible for initial teacher certification. These institutions are aware of the training new teachers bring to the profession and can provide meaningful experiences that can expand teachers’ current abilities. The experiences would be the result of the institutions and inservice teachers collaborating on areas needing improvement.

In my discipline, we often hold up Dr. Deborah Ball (former dean of the University of Michigan’s School of Education) as the model elementary math teacher. However, we fail to recognize that Dr. Ball rose to this level after identifying this as an area of weakness in her teaching – after being a certified, inservice teacher (see p. 9). She decided to do something about it by going back to school and learning more about the subject and its teaching. We need to allow all inservice teachers to follow the path from awareness to adjustment without fear of being labeled unprepared or ineffective. 

While teacher preparation institution must make changes to better prepare teacher-leaders, teacher certification is not the primary problem. A lack of opportunities for meaningful professional development is the issue. Changing teacher certification to try to address a broken inservice support system focuses our efforts in the wrong direction and may do further damage to education in Michigan.

Thank you, in advance, for your attention to this issue.

Wanna Play?

Grand Valley's College of Liberal Arts and Sciences has a policy that all courses must meet and have a culminating experience during finals week. Typically, I try to do something other than a traditional exam during these meetings. Most of my students are preservice teachers, and I want to offer them an alternative way to "show what they know" and reflect on the semester; this often entails presentations. But this year we decided to throw a party.

I got the idea when I attended a session at the NCTM 2017 Annual Meeting and Exposition that was led by Kassia Omohundro Wedekind and Mary Beth Dillane - "We are Mathematicians": Building Mathematical Communities Based in Sense Making, Agency, and Joy. During the session, they talked about kids at their school hosting a math party for their peer, teachers, and parents. At the party, the kids shared some of their favorite math activities. I decided this would be a great way to wrap up the semester.

We started the celebration with a popular party game: The Marshmallow Challenge.

Then the pre-service elementary teachers started developing their Math Teaching Vision Boards. I got the idea from this podcast that discusses the role instructional vision has on math teachers' instructional practice.

After about 20 minutes, we were ready to share our visions. We didn't have peers, teachers, or parents at our party, so we set up a series of viewing venues arranged by group.

The sharing looked a like this:

At the end of the party, the preservice teachers gave me some feedback. One said, "That was 2 hours of reflection disguised as fun."

Another complained that they didn't get to go to the viewing parties of their table-mates. I told them that was too bad but maybe they could connect with those peers after class to talk about their visions. Then I wondered out loud, "Do you think I did that on purpose?"

What's your next move?

Our presentation from Math in Action 2017

Description: 

Games are an effective way to engage students in learning. Participants will experience how to support the development of pre-adolescent mathematicians through purposeful play. [Grades 3-5]

Consider what you think it means to effectively teach mathematics. Now take the Simile Survey provided below. What are the characteristics of your simile selection that relate to good mathematics teaching?

A while back, Dr. Doug Fisher introduced me to another teaching simile: Teaching is like being an expert commentator. During the lesson, the teacher highlights important aspects of the "routine" that the student might otherwise overlook. In cases where the action moves too quick, the teacher might need to "rewind and show it in slow motion" in order to clarify some move. Here is an example from the 2016 U.S. Olympic Trials that demonstrates these characteristics. So what does this look like in math class?

Imagine we are in a 3rd-grade class playing BINGO. If the students are fluent in reading number symbols, there's not much to the game. So let's break it - add another dimension by allowing players to decompose the number that's called.

If you were in a 5th-grade class, they might ask why they can't decompose the called number into more than two addends ... or use operations other than addition. Then the challenge might be, "Can I get a BINGO with just one number called?"

After (or during the game), what sorts of things would you want the students to notice? What would you highlight and maybe have to slow down? It depends on the game and our players.

• If I was playing the regular game of BINGO with young kids still struggling with number recognition, I might be sure to call "thirteen" and highlight ways to tell the difference between 13 and 31.
• If we are decomposing, I might want kids to recognize that 38 can be decomposed into 30+8 or 31+7 and highlight the concept of compensation.
• For 5th graders, I might show how "thirteen" can be written as 13+8/(9-7)-4 and highlight an important property of zero in our number system. [To demonstrate another important property of zero, ask students if they could cover the entire board if "thirteen" was called.]

It is important that teachers have the opportunity to play games before using them with their students. That way the teachers can consider possible modifications (ways to "break" the game) that would meet their students' needs. It also gives them experience playing the games that can lead to insights into important mathematical aspects encountered while playing that the teachers might want to highlight for their students.

Game Centers 

Number and Operations - Fractions 

Grades 3-5 

• Fraction Go Fish 
• Fraction Memory [pdf] 
• Cookie Fractions [page36]
• DecimalPickle 
• Pattern Block Fraction Game
• Fraction Track [app]

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Other Game Resources

• Adventures with Mathematics [MCTM]
• Gameswith Fraction Strips and Fraction Cards
• Rational Number Resources [NRICH]
• Investigations Games

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After playing the games, we reflect on our experiences using Math Teacher Chair:

• What games did you play?
• So what mathematical ideas would you want to highlight?
• Now what would you do to break the game or slow down the play so students would benefit mathematically from playing?

Thanks for your participation. You can reach us using the following contact information. 

"Rocket science is child's play compared to understanding child's play."

~ Unknown

If you are attending the upcoming 2017 NCTM Annual Meeting and Exposition in San Antonio, we will be presenting this session again. 

We promise it will be better next time thanks to the feedback you've provided on your session evaluations (or in the comments below).

Can we have five more minutes?

Excuse the pun, but it was like clockwork. I would assign a group project (something like making a concept map), give them 20 minutes, and set the classroom timer. The timer would go off and nearly all the groups would ask for more time - usually about five minutes. It got to the point where I would just add five minutes into the plan but then they'd still want more time. 

It wasn't as if they hadn't been working the entire time; they were just really invested in getting it perfect. Even the smallest detail, like the use of colors, had to be debated. I explained that these details didn't matter as much as the connections they were making, but somewhere they got the notion that the presentation of their ideas was paramount.

Then I was introduced to Design Thinking and the principles of bias to action and prototyping to a solution. I decided to apply these principles to the problem of students attempting to create the perfect poster. I went back to giving groups only twenty minutes but I broke it up into smaller intervals - each with a defined purpose.

First three minutes: organize the concepts in a way that reflects how you see them related and glue them on the paper.

I knew if I simply moved on to the next phase nothing would change; they would get stuck in the same old debates. Therefore, I had the groups rotate clockwise around the room. They were now looking at another group's vision of how the concepts might be arranged. 

Next three minutes: consider the previous group's arrangement, talk through what the arrangement might represent, and add connections (nothing more).

During this time, I kept reminding them that the previous group had only spent three minutes coming up with the configuration of concepts upon which they were working. There was nothing they could do to ruin it. This was simply a prototype and they had limited time to add their contributions. After three minutes, they rotated clockwise to a new group's poster.

Third interval of three minutes: consider the work of the previous groups, talk through what the work represents, and add descriptions to the connections.

I reminded them that only six minutes of work had gone into poster. The previous groups didn't really have anything invested in what was already done. So they shouldn't worry about doing anything that might change the poster. I even encouraged them to add new connections if they thought it made sense. After three minutes, they rotated clockwise to another new poster.

Fourth, fifth, and sixth intervals of three minutes: repeated the previous intervals - add more concepts, add more connections, and add more descriptions.

Each time, I repeated the mantra: "The other groups only spent three minutes on the poster. You can't ruin it. Just get to work." 

After 18 minutes, each group returned to their original poster. There were some audible gasps and laughs. Rarely had the poster turned out as expected but each group could infer the intent behind the decisions other groups had made. They spent the last two minutes creating an artist statement for their concept map - something they thought an observer ought to notice.

None of the groups asked for more time. They were satisfied that the posters were prototypes - works in progress that allowed viewers to add their own perspective. We used the "extra" five minutes to do a gallery walk and see how our work turned out.