Multiple methods

Glenn Beck is criticizing the new Common Core standards. I have some interest in politics. But this is not about that. I have 25 years of my life invested into math education. So far what I have seen of Beck's criticism is wildly off base. Here is one correction. There may be more to follow.​

Beck states that addition with regrouping is a new-new-math that is dangerous and that dumbs down our children. I'll explain what regrouping is and why it is actually awesome.​

I'll use Beck's example: 29 + 17. How would you do this problem? (Without a calculator)​

Most of you would want to grab pencil and paper and commence the adding algorithm. This involves "carrying a 1", takes a few seconds and is reasonably accurate.​

However, do you know why you "carried a 1"? Do you realize that 9 + 7 = 16 and that 10 of the 16 can be transformed into a 6 in units column and a 1 in the tens column?​

Regrouping emphasizes student understanding of why "carrying the 1" makes sense.​

Most math folk don't do mental math from right to left. It doesn't work very well or is as quick as regrouping. If I did this problem I would think ​

20 + 10 is 30

9 + 7 = 16

30 and 16 make 46.​

Beck ridicules this very method. When you write it on paper, it looks longer. In your mind, it is remarkably nimble. You might not trust my mental tricks. But perhaps you would trust Arthur Benjamin's. ​I've heard Dr. Benjamin talk about mental arithmetic. Like most mental arithmetic experts, one of the first things he will do is explain to you that you don't work mentally like you do on paper. And that you often work with regroupings and from left to right.

But its not just mental arithmetic that motivates regrouping. It is an understanding of place value.​ Let me move to a second example. How do you multiply 33 x 21? If you follow the typical American procedure, you will multiply by 1 x 33 and write down 33. Then you write a zero in the next row. STOP! 

Why? ​Why a zero to start row two? Because you're not about to multiply by 2. You're about to multiply by 20. That's right, TWENTY. (I have a friend who is a heart surgeon. When I explained this to him he said he never understood why the zero is there. And he's a smart guy. But he had never been taught this fundamental fact with any understanding!)

So if we regroup, we get something like this:​

33 x 21 = (33 x 20) + (33 x 1)​

And 660 + 33 is pretty easy to do in your head. It's 693. ​

Mr Beck might complain that this is longer (and he's sure to write it on his nifty chalk board as if it is much worse. I wonder if he knows that white-boards have been invented?) but if he paused for a second, he might realize that it's looking kind of familiar.  It's looking like the distributive property! The same property that's so handy in Algebra 1! And pretty much the rest of advanced math. In fact, a strong multiplication unit (in the common core) will distribute the problem out all the way:​

33 x 21 = (30 + 3) x (20 + 1) = 30x20 + 30x1 + 3x20 + 3x1 = ...​

You might recognize this as FOIL'ing (I'll have to correct the atrocity of that mnemonic on another post) and rest of the world introduces the distributive property in this manner pretty early on.​ It builds algebraic thinking from a young age.

More to follow. Comments welcome.​

Late work

Every fall I get a list that tells me which of my students have medical conditions. It is clearly useful for me to know if I have students whose blood sugar may drop due to diabetes, may be prone to seizures, etc...  There is usually one student on my list, however that may surprise you. I invariably have one student listed who has urinary tract infections. My instructions printed on the sheet are to let her use the restroom whenever she requests. 

If this sounds odd to you, I'm right with you. Why wouldn't a reasonable person let a student use the restroom when they need to? If you think that a request like this is only created because some parents out there are crazy, you can just stop reading this post right now. I'm only going annoy you.

School culture is a strange thing, that's what I've been mulling over recently.  The four oddities that strike me are:

• Going to the bathroom 
• Turning in late work
• Cell phone use
• Tardies

As a general observation, I would say that many teachers spend a lot of time and energy around this issues while students feel that their teachers care more about these things than they do about anything else (including learning).

In fairness to teachers, managing students can drive you nuts.  Its a crazy task with too many bodies, too many pieces of paper and endless interruptions. And all of the things on the list above can create distractions from our primary goal: Learning.  However, you don't have to think too long before you realize that we're kind of nutty about these issues.  A few examples:

• On my campus, teachers regularly come late to school.  Some of them are the same teachers that never tolerate tardies with their students and will drop them from the class after their fifth tardy.
• At our staff meetings you can see teachers checking their phones constantly. I see this behavior everywhere I go. Schools are the only place I can think of with zero tolerance for phones.
• The IRS, the DMV and the credit card companies accept late payments, usually for a 10%penalty.  Yet at two back-to-school nights this fall, at least half of my own children's teachers made it clear that late would not be accepted, Ever.

Let me be clear.  I want to teach students responsibility.  I want them to learn when and how is the appropriate time to use their phones. I like it when my co-workers are punctual in their duties. But consider this anecdote.

One Thursday our campus had a football game.  As I looked across my 5th period stats class, I noticed that I had several football players, a few cheerleaders and several band kids. These kids would pretty much be busy from 2pm to 11pm.  So at the end of class, I announced that the homework I was assigning that night could be a day late for all involved. To me this seemed reasonable. My students responded like I had given them a huge present. This surprised me. Was it so unusual for these great students to be given one night's grace on their work? I teach on a campus with GREAT faculty. They amaze me. Yet apparently this was an unusual offer.

In the end, I think this post is a plea. A request for reasonableness. I think as teachers we can surmount the challenge of dealing with the craziness of these headaches by embracing two concepts: compassion and creativity.

Compassion guides us as we realize that our students are kids. They are growing and learning and all too often not in control of all the variables in their life.  I have student with an iPhone. Her dad has yet to buy her a scientific calculator this year. So every day she borrows one of my mine. So be it.  I had a student fail to bring in his big polygon project. He tells me he was kicked out of dad's house in the afternoon and had to go to his mom's instead.  Where he had no supplies. So I took the project the next day and took off just a few points. Who knows if it was his fault or his parents or some mixture in-between? But I'm thrilled that he let me know what was going on. He was failing my class initially, but went to some of our extra Saturday study sessions and has rasied his grade to a C.

Creativity helps us get what we want out of students, and separate those who deserve mercy from those who don't. A 10 minute delay in granting a bathroom pass can help discern a biological need from a leisurely stroll across campus. I often use writing lines as a silly punishment for silly behavior. Writing "My future employer will appreciate my punctuality" 25 times usually convinces the tardy student that one last kiss just wasn't worth it. And his classmates usually notice (especially if the lines are taped to the window during the last week of the year when tardies grow with spring flowers) and join in the race up the stairs.

In the end I think a zero-tolerance policy or a rule that is 100% "consistent" (read: inflexible) is more comforting to many teachers. But it rarely actually results in justice. Our students are children. And humans. They deserve thoughtful and compassionate classroom policies that have the flexibility needed to bend with their growth and to maximize learning.

You're using it wrong!

​This problem, and ones like it, have become popular on Facebook.  People enjoy arguing about "the right way" to do these problems.  The discussions revolve around PEMDAS, or the mathematical order of operations.

While I would never want to deprive anyone of some good ol' math fun, these problems frustrate me.  The answer to the question is:  Use some parentheses please!  No one writes strings of arithmetic like this.  They use parentheses to make their meaning clear.​

People find these problems fun, at least in part, because 6th grade math books teach problems like these.  But this is an exercise in missing the point.  What is the point, you ask?  Let me tell you!​

The point of PEMDAS (parentheses, exponents, multiplication, division, addition and subtraction.  Or, Please Excuse My Dear Aunt Sally!) is to allow us to write algebraic expressions without parentheses!  A simple quadratic like 3x^2 + 5x - 6 would require three sets of parentheses without order of operations.

My son's recent school work actually finished strong.  ​The obligatory word problem at the bottom of the worksheet went like something like this:

Write an expression for buying x shirts for $10 each and y hats for $5 each.

So the answer, 10x + 5y is actually a great example of when and why we want to multiply before we add.​  Three cheers for context!  Boo and hiss for random strings of arithmetic!

PS:  The correctly written problem would read:​

6 - (1*0) +(2/2) = 6 - 0 + 1 = 7​

engagement

I've spent a lot of time thinking about improving student engagement.  It seems that many of the solutions are simple.  Here's one that worked well last week.​

A student asks for help on a homework problem.  You all know the drill.  You do the problem on the board.  Five people pay attention.  You figure that if one person asked, probably the whole class should be following along.  But most aren't.  ​

I simply decided to take their lack of interest off the table.  I asked the whole class to read the problem.  Then I randomly picked a student to start the problem.​

[sidebar:  I have been highlighting my seating chart.  Every time someone participates, whether voluntarily or by my choice, I highlight their name on my seating chart.  When everyone has a yellow highlight, I switch to green.  It gets a bit messy and in time it looks like it's filled with psychedelic asterisks.  It allows me to call choose carefully who I call on, while ensuring that no one can be ignored.]​

We scaffold through the problem with about 8 different students being called on to contribute to the solution.  The entire class is engaged and everyone has seen a correct solution.​

Now as to why most of class doesn't care about a solution to a challenging homework problem.  Well, if I had all the answers, the comments would be closed.​

procedures

This is probably more of rant than an enlightening observation, but...​

What's up with these rules we teach our kids about moving decimal points around for multiplication and division?  First off, these procedures are rarely used by anyone.  We use a calculator for goodness sake.  But more importantly, these rules become pure magic.  There is no number sense, no common sense and no math sense.  My son has no clue how many times $0.25 should go into $6.  There is no checking for the reasonableness of an answer.  There is just a procedure and a box around the answer.  One of his "word problems" (read: do the same thing you just did 12 times and put a word at the end of your answer) had four-digits divided into five-digits.  The answer was clearly a bit less than 10.  ​But he had no clue which way to divide, much less about where his answer might be.  I used a calculator to verify that the answer was 9 and saved him some pain.

I hope the common core will solve things like this.  I don't claim to know all the answers.  But there's got to be a better way.​

NOTE:  I'm quite impressed with my son's teacher.  I'm sure she is forced by California's calculator-free assessments to teach much of what she does.​

In the zone

I've spent some time this summer reading Colvin's Talent is Overrated.  I have read a lot of it.  But am trying to figure out more about how to apply its principles to my teaching life.  Specifically, I want to delibrately practice my teaching craft in such a way that I am improving.

But I realized that the book's principle of delibirate practice cuts two ways for a teacher:  we need to think about how we practice, but we also need to think about how our students practice.  

I had two students come after school for tutoring in AP Stats a lot this last year.  They both earned a 2 (not passing) on the AP test.  Heart-breaking 2's.  Every AP teacher knows what this feels like.  Hard working students who you like and they don't quite make it.

But it makes me wonder how I had them practice.  It was probably too routine.  Too familiar.  Learn the basics.  "Here, do another one" (just like the one you just did).  

I could be too hard on myself.  Maybe without the tutoring they would have earned 1's.

But the idea that ignites my imagination is the Learning Zone.  Colvin talks about the comfort zone, learning zone and panic zone.  Students usually come to tutoring because they feel panicked.  But do I let them slide all the way to comfort?  Or do I keep them learning?

For that matter, how do I keep my class in the Learning Zone?  I've been trying to lecture less and let students work more.  But Hunter's "Guided practice" may not be learning.  At least not enough learning.  It may just be comfortable practice that looks a lot like the lecture.

Much to learn!

Just stop

If we're talking about technology in a math classroom.  You know, calculators, computers, Wolfram Alpha, all that cool stuff.  And you say "Well if your calculator runs out of batteries..."  or "If you don't have an internet connection..." or something in that vein.  Then I would consider our conversation to be over.  And I would safely say that you're not thinking very carefully about the issues surrounding technology and learning.

It might be that you just need more time.  And that you haven't thought carefully enough yet.  Or you live in California and you've given in to the demands of the CST's.

But I can't believe that you've really thought carefully about your own life and how you do math.  Or watched someone under the age of 25 and how they interact with technology.  

More on what careful thinking on these topics looks like later...

I like...

I've worked with a few long term substitute teachers recently.  They have both commented at the end of the day "I like working with the AP classes!"  The problem is, the students didn't seem to share their enthusiasm.

I don't say this to be mean.  I say this because it made me pause as I looked in the mirror.  I have classes I like to teach (AP Stats!).  And my students seem to like taking the class with me.  But all that is really beside the point.  The essential question is simple, yet forgotten--Do they learn in my class?  

It's about the learning.  Period.

It's not about me liking them or me liking them.  It's not about having fun.  It's about students learning to think.  Don't get me wrong, I don't think animosity makes for a learning state of mind.  But at the end of the day, I hope my students learn.  

Ignite

I did a 5 minute Ignite talk for Key Curriculum Press at the CMC Palm Springs conference.  Here's the link.

Homework

I stopped grading homework this year.  It isn't worth anything.  Not "points", not a percent.  I also noticed that I was assigning less and less.  I have a mixed feelings about this.  Reading Dan Myer about this made me feel better.  I wonder if he still feels the same as he did four years ago.

Not giving out points for papers filled with work of dubious value was definitely a good decision.  I don't think I'll ever go back.  Now how much to assign and how to maximize class time.  That's still the battle.

Deadhorse #1

It's hard to let things go.  But if we're ever going to teach deeper and better, we've got teach fewer topics.  So here's the beginning of a list:

Rationalizing the denominator.  Time to kill it.  Call it dead.  

A bit of history:  this topic only exists because armed with a square root table and a piece of paper, it is easier to scratch out root-2 divided by 2 than 1 divided by root-2.  In other words, this topic has the same shelf life as interpolating on log tables. (Which I learned.  Despite the TI SR-52 sitting on my desk.)

You say some nice things about this topic.  You can connect it to other ideas.  But it doesn't have a point and its time to kill it.