SMWhat?

Educators seem to be a mixed bag of afraid of vs. jazzed about the CCSSM Standards for Mathematical Practice. At first read-through, they seem very sensible, and like things math students should be doing as a matter of course.

But if you think you really get them, try a little experiment: ask a small group of math teachers what they think "Attend to Precision" means. What does it look like if a classroom task requires it? What does it look like when a teacher is facilitating it? What does it look like when students are doing it? Here are some responses you might hear:

1. Rounding correctly according to the directions
2. Rounding sensibly based on the problem's context
3. Being careful when plotting points
4. Labeling axes and diagrams correctly
5. Drawing sketches and diagrams to scale
6. Using an appropriate number of sig figs based on the precision of the measuring device
7. Using precise mathematical terms in written and verbal communication
8. Defining variables and symbols

I've spoken to teachers who express their understanding with numbers 2, 6, and 7, but I've talked to teachers whose understanding hews closest to numbers 1 and 3. Which is not to pass judgment, but is to say: it might be wise to be aware that you and your colleagues could have different, and potentially incorrect, assumptions about the SMPs. 

And "Attend to Precision" seems like one of the more concrete ones. See what your colleagues have to say about "Look for and express regularity in repeated reasoning," and I bet the answers will be even more all over the place.

Another observation: it can be really hard to evaluate which SMPs are highlighted or emphasized in a classroom task. When I try, I tend to go "uummmm...all of them...?"

So what kind of task lends itself to "Modeling with Mathematics"? What does it look and sound like when teachers and kids "Look for and Make Use of Structure"? 

I'd like to point you to a recently published resource: A Rubric for Implementing Standards for Mathematical Practice. It was written in July of 2011 by Danielle Maletta, Mimi Yang, and Mariam Youssef as part of the Visualizing Functions working group at PCMI. It gives an observer specific items to look for in a task, as well as specific teacher behaviors, to help evaluate how faithfully a standard is being met in a particular lesson. The accompanying Resources document will also give you a deeper understanding of each standard.

Also, heads up that Illustrative Mathematics, in addition to the Herculean undertaking of trying to illustrate every K-12 content standard, has put a significant amount of effort into illustrating the Standards for Mathematical Practice using both sample videos and classroom tasks. 

Check them out. Share widely.

Pop Quiz

Was this published in 2001, or this morning?

New York State United Teachers, the state's largest teachers union, is urging members and parents to call on the state Education Department to stop implementation of this year's tests, which will be more challenging, because schools have not received all of the necessary curriculum. 

"If we want our children to be ready for college and meaningful careers, we need higher standards — and a way to measure whether those standards are being met — and we need them now," Education Department spokesman Dennis Tompkins said.

Give up?

The Tests Matter

Here is what is going on right now, in the time before the Common Core Standards have really hit high schools, and before a common assessment has been inflicted on any live children. The non-teachers in education are going: "Just start teaching the right way. Pay no attention to the tests. If you teach right, you don't have to worry about the tests. The tests will take care of themselves." The teachers are saying: "The way I teach is basically fine, anyway, so I'll make whatever adjustments I need to make once I see what they want kids to do on these new tests." I know there are probably some teachers changing their practice, and some non-teachers with half an eye on assessment. I'm painting with a broad brush. Go with it.

This is what I am afraid of: the thing that happened in New York State, starting in 1999. That's when NY changed from Course1/2/3: a decontextualized, integrated curriculum with very predictable though rigorous exams that were none of them a graduation requirement... to Math A/B, standards with more focus on applications and much less predictable tests -- also, kids had to pass the Math A exam to graduate. (This was a huge deal. Regents exams had traditionally been taken by your college-bound academically-oriented students, and suddenly everybody had to take one of them.) The new requirements were supposed to make things tougher, with all the rhetoric that comes with such changes. June 1998:

Yesterday, officials at New York City public schools welcomed the tougher tests, while some education advocates worried about the lack of resources to train teachers to teach for the higher standards.

If it sounds familiar, that's because it's straight from whatever school-reform-article-generating-machine the news has been using for thirty years. Moving on.

Some shit started hitting some fans. October 2000:

Mr. Mills said middle schools ''need to rethink what they are doing'' and quickly figure out how to teach students the skills they need to meet the new standards. He said he had no intention of backing down on the standards, which as of last June required every high school student to pass an English Regents exam to graduate, and by next June will require every high school student to pass a Regents math exam as well.

People started freaking out when they realized that requiring a passing score on an algebra test was going to be a graduation-rate debacle:

Students in the next class, which entered in fall 1997, will have to pass both the English and Math Regents to get their high school diplomas. If the results hold steady, about a quarter of this year's seniors will not be allowed to graduate.

There were protests (May 2001). There were districts trying to opt out (Nov 2001). 

I don't know what happened to all the kids in the early 2000's who were denied a diploma because they couldn't pass the Math A Exam. A bunch of heartbreaking shit, I'm sure. 

In June 2003, there was TESTMAGEDDON. The Math A Regents exam was the straw that broke New York's resolve. 

Though many districts have not finished tabulating their scores, superintendents, principals and math department heads are reporting preliminary results that some described yesterday as ''abysmal,'' ''disastrous'' and ''outrageous.''

It was not a good test. Post-Course 1/2/3 exams were not good tests, generally: problems that didn't make sense, weird, contrived contexts, a fetishization of goofy vocabulary and notation. Too much content was a huge problem. A test that didn't know whether it was an algebra or geometry test was a huge problem. A test that didn't know what it was measuring -- readiness for higher mathematics courses? Basic skills that should be expected of every graduate? -- was a huge problem. In the end, the test measured nothing but whether or not a kid had passed that test. The accountability movement compelled schools with lower scores to make their math courses all about passing the test. Math A became a de facto curriculum, and a horrible one. 

NY tried to raise the bar. Then, a whole mess of kids ran head-first into the bar and fell on their asses. Then, instead of re-evaluating any of their faulty premises, NY responded by lowering the bar.

On the June 2003 exam, they relented and lowered the cut score. 

Then, they eased up on subsequent tests. 

New York State's education commissioner, Richard P. Mills, said Wednesday that the state would loosen the demanding testing requirements it has imposed for high school graduation in recent years, including the standards used to judge math proficiency.

They made the tests easier. Lots easier. Also, the thing happened that took all the respectability out of the historically respected  regents exams: for the tests required for graduation, the score you needed to pass got dramatically lower. They said it was a 65, but after June 2003, you only needed a raw score of around 42% to pass the Math A with a scaled score of 65. (The raw scores in the linked table are not percentages -- they are out of 84 points.)

I wasn't around when this all happened. I didn't start teaching until 2005. And I don't think we're getting exactly a repeat with the Common Core. For one, there does seem to be a coordinated, genuine effort to support teachers in changing their practice, independent of testing. For two, there's a coherence and focus in the CCSS that New York was sorely lacking. But also, there's the whole added wrinkle that tests are trying to fulfill still another purpose: teacher evaluation. The disaster story might not be "so many kids can't graduate", it might be "so many teachers are being rated poorly, even good ones that kids, parents, colleagues respect."

But I still think it serves as a cautionary tale, and I'm still curious about how this is going to play out once the new tests hit a computer lab near you. If they really measure the stated goals of the new standards, they're going to be very different. Because of that, they're going to be perceived as too hard. How the test-writing consortia, DoE, states, districts, etc react to that is going to be really interesting.

On Writing Lessons for Others

(Cross-posted to Mathalicious Blog)

We sat down recently to rewrite the core Xbox Xponential lesson. In it, we tell students about Moore's Law. They use it to make predictions about how we would expect video game console processor speeds to increase over time. And then compare that prediction to how console processors have actually improved.

It took nearly three hours. For four people. To write five questions. Let that sink in.

Part of what is going on here, and the tension I want to think through here, is us trying to balance what is best for students' learning with what we can realistically expect of teachers. We want to give students the opportunity for as much inquiry as possible. But scaffold too little, and the lesson becomes too hard for too many teachers. Scaffold too much, and the lesson does too much of the cognitive work for the student, robbing them of the chance at the learning that comes from insight.

Mathalicious lessons might feel very unscaffolded at first glance. But much thought goes into setting up students for success.

• We carefully choose given information. For example, since Moore's Law refers to "every two years" and the Atari came out in 1977, we found consoles released only in odd-numbered years. We chose two years later later than 1977, then four years after that, then ten years after that, to both gently ramp up the difficulty and implicitly suggest searching for shortcuts. 
• We are also very deliberate about the wording of questions. For example, the third question could merely consist of the last sentence. 
• We pay close attention to formatting, for example, since question one asks students to generate several numbers, that they will then use to look for and make use of structure, we organized them in a table.

With the understanding that lots of teachers, at least at first, are going to be uncomfortable with our lessons. With the understanding that right now, it's maybe not for everybody. With the understanding that many teachers are going to have to get better, if they want the deep learning and the rich conversations and the joy that comes out of real learning and teaching. With the understanding that the part of the solution that we are offering - obsessively thought-out lessons, with engaging contexts and rigorous mathematics, that promote effective teaching and enduring learning - is not an easy or quick fix for a teacher who truly wants to improve her practice.

We come to work and we have the same discussion. I hangout online with teacher friends and we have the same discussion. I go to conferences and we have the same discussion. The discussion is all about how do we help teachers be better. The "we" can be we at Mathalicious writing curricular materials, it can be we practicing teachers who want to help our colleagues or improve our own practice, we who believe the Common Core is a valuable framework, we teacher educators who work with pre-service teachers and reach out to inservice teachers.

The discussion goes like this: We know what it takes for real learning to happen. There is tons of research to support it. We want students to explore rich problems, in contexts that are meaningful, be critical, make generalizations and abstractions, try things that don't work and try something else. We want teachers to give kids time to struggle, to help them gain comfort and perseverance in uncertainty, explore incorrect reasoning and channel it into better understanding, ask them better questions and give them fewer answers, and not rob them of the thrill of making insights.

But, the discussion continues, this is going to take a huge shift. This is not the way we were taught. This makes many of us extremely uncomfortable. This is not what kids are used to. It can make them uncomfortable. It's also hamstrung by high-stakes skills testing that understandably makes teachers feel like they must employ direct instruction and lots of procedural practice. It can all be very frustrating if you let it.

So what can be done? What can I do from where I'm at right now?

If we consider all US math teachers as a population, some are further along than others. Some teachers are only comfortable with direct instruction of procedures. Some are comfortable with open-ended exploration, provided a high degree of structure and scaffolding. Some are comfortable with a minimum amount of structure. And there is at least one Shawn Cornally.

While it's tempting to believe the distribution of US math teachers on this continuum looks balanced (red), it seems more likely the actual distribution is skewed (green.) Just so we're clear about my own level of hubris, I would put myself as a classroom teacher to the right of that maximum, but not too far.

In writing lessons there are a million decisions to be made, and one of the biggest that guides all other decisions is, Who Am I Aiming At?

Aim at The Middle?

We could aim right at the fat part of that distribution. Our materials would be used by tons of people. They would also be pretty indistinguishable from all the crappy stuff that is already out there. They also wouldn't help teachers get better. Using our recent edit of Xbox Xponential, this is my interpretation of what that might look like, if it were reimagined by a standard American textbook company. I include it here even though it might give Karim a heart attack:

Aim at the Right?
There is a temptation to write ideal lessons for an ideal classroom - to aim way at the inquiry-based side of the tail. At those teachers you run across, online or in real life, that make you say "whoa." But there aren't that many of them, and they don't really need our help, besides. That lesson might look something like this. And, perhaps, we would not even provide a student handout.

Where We Are Aiming

Realistically, we figure, our lessons are appropriate for a range of teachers.

Those living at the arrow and to the right will value the quality of our materials and comfortably incorporate them into instruction. Those in the range, but to the left of the arrow, will find it challenging, at least at first, but we suspect that one important factor in excellent teaching is access to excellent curricular materials. Just like if you want to become a great novelist, part of your education is reading great stories. And if you want to become a great programmer, part of the process is understanding and using great code. I believe, due simply to my own experience using other people's stuff, that teaching a great lesson (even if someone else wrote it) helps a teacher learn what great teaching looks like. While Mathalicious lessons might be used mostly by the more risk-tolerant, skillful teachers in a building, we would love for their success to have enough gravity to pull their colleagues to the right. (Indeed, we are making plans for supporting them in doing just that. It's going to be so cool, you guys.) And that over time, that bubble will shift.

Obsessions

Happy spring forward day! In Syracuse, they don't get sun for another two months or so. Or birds. And in Buenos Aires, it's the oppressive mosquito-thick shank of late summer. It was 65 degrees in Charlottesville today, the sun is out, and birds are singing. I'm 10% of the way through a 30-day Bikram challenge, and all I can move right now is my fingers. Things are good.

Maybe you all have heard of all of these things already, maybe not. Here you go:

Jason Dyer is on a tear lately.

1ucasvb makes pretty and interesting things out of code.

Daily Desmos has given me a right headache a number of times already.

Michael and Tina are taking the best kind of teacher blog post and making it into another blog.

Ben has reminded us why we do this, and what makes it fun.

We're getting a podcast!

Collaborative Mathematics is so great.

ICYMI : Math Teachers Get Down With Their Bad Selves

This happened.



Sometimes I forget that things happen not-on-Twitter. It didn't occur to me to post here until Sam posted it. Which I only noticed because I opened my Reader for the first time all week.

Fun facts:

• The dog's name is Hershey.
• There are two Rubik's cubes.
• The equation on the board is a nod to the Simpsons.
• Greg is actually doing the Harlem Shake.
• Timon has some seriously underrated breakdancing skills. (Until yesterday, I suppose.)
• Julie did not have to dress up special. She happened to be wearing a cowgirl outfit that evening.
• The Matt in the first half on the monitors is the same Matt in the second half in person. We had him on the G+ Hangout for the first half because that's normally what he looks like in our office.
• You might have to look hard for Sam, since he is wearing a disguise.
• At the very end, Christopher is getting ready to no-kidding launch Tabitha across the room. That part got accidentally cut as a result of the slow-mo.
• The math twitterblogosphere is the best twitterblogosphere.

Two Tens for a Five

Thanks for all the ideas about how to talk to eighth graders about irrational numbers. Here is my stab at a question progression.

I don't know how to credit people who shared ideas that made it in here - they are so overlapping. Also, several people didn't provide their names.

I do want to give a shout-out specifically to Justin Lanier, as I copied his even/odd irrationality of √2 proof basically verbatim.

Thoughts appreciated.

----------------------------------------

Calculators away!

Let’s try to figure out exactly where √10 is!

What two integers is √10 between? Label them on the points plotted below.

Which numbers is √10 between, rounded to the nearest tenth? Find these by hand. Place them CAREFULLY on the number line.

Which numbers is √10 between, rounded to the nearest hundredth? Find these by hand. Place them on the number line.

How much more precise can you get?

You may have learned that when you turn a fraction into a decimal, the decimal eventually either ends altogether, or ends in a chunk that repeats over and over forever. 

For example, 3/8 = 0.375 and 1/7 = 0.142857142857142857… However, the square root of ten never ends or makes a repeating pattern! You can compute its value as precisely as you want, but there is no way to write it exactly as a decimal. (If you think about it, a decimal is really a bunch of fractions: tenths and hundredths and thousandths, added all up.)

This may seem too weird to be believed. However, we can come up with possible, theoretical non-repeating decimals. 

For example, can you spot a rule suggested by the start of this number, and write more digits?   0.13113111311113__________________

Can you come up with your own rule to create a decimal that will never be just a repeating chunk of numbers?

Many calculators claim that √10 = 3.16227766 (maybe even yours!) Explain how you can tell, beyond any doubt, that this can’t POSSIBLY be true.

Recall that the algorithm for multiplying fractions is stupid-easy. For example, 2/3∙4/5=8/15 and 8/7∙8/7=64/49

Let’s try and pinpoint √2

Explain how you know for sure that 3/2 is too big.

Explain how you know for sure that 5/4 is too small.

Carefully plot 3/2 and 5/4 on the number line below. The points are plotted exactly at 0, 1 and 2.

Do any of these fractions exactly equal √2 ?   7/5, 11/8, 10/7

Plot them as precisely as possible on the number line.

Can you find any fractions that are even closer to √2 ?

As you may have guessed, there is no fraction, that when you square it, equals 2 exactly. √2 can not be expressed as a fraction – a ratio of numbers. That is why it is known as irrational.

But how do we know? Maybe we just haven’t looked hard enough for the fraction. Maybe if we could look nonstop for a week, we would find it! How can we know for sure that it doesn't exist?

Any fraction has to be one of only four kinds: odd/odd, odd/even, even/odd, and even/even. What can you say for sure about any even/even fraction?

Of course, even/even can be reduced to one of the other three kinds, so we only need to consider these. We’re going to show that none of these kinds of fractions could be √2—that is, that none of them squared is 2.

One example of a fraction that equals 2 is 18/9. Can you think of three more examples of fractions that equal 2? How can you describe them in general?

We’re just going to look at three cases of candidates. Odd/odd, odd/even, and even/odd.

Well, when you square odd/odd, what do you always get? Could one of these possibly equal 2?

When you square odd/even, what do you always get? Could any of these possibly equal 2?

So the remaining case is even/odd. When this is squared, we get even/odd—so it looks like it might be possible for the top to be the double of the bottom. But consider this: when an even number is squared, the result is a multiple of 4. (Pause a moment and convince yourself this is true.) And a multiple of 4 is never the double of an odd number.

So √2 can’t be a fraction that’s even/odd.

But then there’s no option left! So √2 is irrational.

On Not Being Irrational

From your friendly neighborhood Common Core eighth grade standards:

I am particularly intrigued by what students in eighth grade are meant to understand about what it means for a number to be irrational.

Okay hypothetical eighth grader, come with me down this road. As you work through some classroom tasks, this is what you will discover:

1. If you build a square with 3 things on a side, the square will have 9 things in it. 4 to a side, 16 things in it. A shortcut to how many things in the square is the side times itself. Notation for something times itself is something².
2. If you try to arrange a certain number of things into a square, you can't do it with any old number of things. only numbers like 9 and 16 and 25 will work. We call these numbers of things "perfect squares". To decide if a number is a perfect square, see if you can find something times itself that equals it. We call this function square root and use a funky symbol √ which is really a stylized r because it's a root.
3. There's no reason to restrict our side lengths to discrete values. If I can transition you to thinking about area, you can see that if I build a square on a grid with a side that's 2.5, there is an area of 6.25 square units inside the square. The 2.5² shortcut still works.
4. Likewise, if I tell you a square has an area of say 20.25, you can find the length of a side of that square. The square root thing again. Keep trying to square numbers until you hit on the one that gives you 20.25.
5. Now you will look for the square root of two. Sure you can use your calculator. Only use the multiplication function, please. I know there's the funky symbol. Just ignore it for now please. (Or maybe I didn't tell you about the funky symbol. But someone is heard about it, or will find it, and spill the beans. (Intentional nod to the Pythagoreans.))
6. No matter what, the class will quickly discover that they can ask their calculator for the square root of two. The calculator will give them a nine- or ten-digit number. If they think to square that number, the calculator will say 2. They will think they have found it.
7. Nothing I do will convince you that irrational numbers are a really different kind of number. 

So I try to get around this, the most extreme version of that goes like this, picking up at 3:

3. No calculators. We build a square on a grid with a side that's 2 and 1/2, which I will try to give as 5/2. There is an area of 25/4 square units inside the square. You will probably write this as 6 and 1/4. Maybe you will see that (5/2)² still works, if I can convince you to just work with improper fractions.
4. I tell you a square has an area of 81/4, and you can easily find the root.
5. Now you will look for a square root of two. Still no calculators. We guess 3/2, but (3/2)² is 9/4, and that's too big. Maybe you reason that 3/2 is the same as 6/4, so 5/4 is a little bit smaller. but (5/4)² is 25/16, and that's too small. Okay let's try (11/8)². Still too small.
6. You give up after a while. I tell you that, surprise, there is no fraction whose square root is two. The square root of two can not be expressed as a ratio. We call numbers like that irrational. You know how when we divided out fractions to express them as a decimal, and the decimals always ended up ending or repeating a pattern? Irrational numbers don't do that.
7. Just trust me, kid.

There are in-between methods, like working with decimals but not calculators. It seems to me that no matter what, we are going to run into the same problem. We'll be looking for something that is not there, and I'll have to just tell you it doesn't exist. CCSS doesn't expect us to prove it, and that seems too hard for eighth grade.

8.NS.1 says "Know that numbers that are not rational..." hold it right there. Is it even possible for an eighth grader to grok that there are numbers that are not rational? For that to mean anything and not just be a memorized definition? What definition would they be able to hold onto?

Potential Definition of Irrational Number	Potential Misconception
non-repeating decimal displayed by calculator	1/19 is irrational
anything with a √ in it	√2.25 is irrational
weird looking numbers like π and √2	π and √2 are the only example of irrational numbers I know

This is something that has been breaking my brain for a while, it's just freshly breaking it this week. I know lots of really smart people, and there doesn't seem to be a right answer. But, you know, it's okay. Questions are cool, too.

If You Are Confused, You Have So Much Going For You Already

Oooh, new careers, you are kind of scary. You find things I am bad at that I didn't even know were skills. 

I notice how very often I am asking people to restate things, or saying "Wait, I'm confused" or "Hold on, I am not following." I mean, it's kind of hard to miss. It's happening with a frequency that can't be ignored.

A little bit ago I was having a post-holiday lets-celebrate-that-we-live-in-the-same-time-zone chat with Ashli, and I said something offhanded like, "I'm sure I'm going to be bad at everything. I hate being bad at things." And she said, "Kate, that is not true. I think you like being bad at things, because you like learning."

Adventuretime

It's okay to be confused. (Elizabeth Statmore)

We should not be too quick to dismiss confusion or try to resolve it or spackle over it.  I would even argue we need to consider it a badge of honor and an activity worthy of our time, consideration, and cultivation. The only way to cultivate curiosity is to cultivate an environment that is supportive of wallowing — active engagement and presence in the process of being confused.

It's okay to be bad at something. (Josh Giesbrecht )

It’s not just that we tell kids, “You can’t do this.”  It’s that we tell them, “If you can’t do this now, then you can’t do this ever.”

I'm not missing class yet, but, oh, your basic freak out over that, it's coming. I am really grateful this exists.

It melted my heart.  It was so genuine.  It was exactly what I needed to hear at that moment. And it came from the mouth of a seventeen-year-old. (Rebecka Peterson)

But I can't deny, I am enjoying all the day-to-day differentness of not being a teacher. You know, not having to drop everything because a bell rang.  That sort of thing. And that I get to work with my friends, who are not just breathtakingly smart, and have really good hearts, but are also fun to be around.

Sandol Stoddard Warburg

I appreciate that I get to do something different every day. And that I have the time (at work! on the clock!) to explore things that I find really interesting. 

So let's all take a moment and appreciate all that being bad at something has to offer. Your assignment for this week is to notice a moment when you are confused, or you don't get something, and not try to hide it. Especially if you are around young humans. At a minimum say, "I am confused!" At a maximum, be really, flamboyantly bad at it, and celebrate how you found a problem worthy of you.

Repatriating Ahead of Schedule

Hello, Internet! Just a quick note to announce I've pulled up stakes yet again to go write amazing lessons and projects at Mathalicious. I imagine it's indicative of how excited I am about the work we're going to be doing, that it prompted me to move away from Buenos Aires, after one term, at a great school in a great city. My reasons for leaving the classroom now are as complicated and personal as they are boring. Also, I'm a little busy with my second international move in six months, which I do NOT recommend for a peaceful path in life. So I'm not going to explain myself, at least not right now. I am not sure what is going to happen in this space, but I doubt I'm going to run out of things to say any time soon.