Leinwand Would Be Proud

Steve Leinwand wrote "Accessible Mathematics."  To quote the description on Amazon, "...Leinwand shows how small shifts in the good teaching you already do can make a big difference in student learning."  I read the book, loved it, and summarized and shared the ten shifts for the math teachers at our site.  Today during classroom visits I saw one of our teachers knocking it out of the park.  Read on...

This was an Algebra II class and the second half of the period was going to consist of an assessment on parent functions, transformations, and domain and range.  I walked in during the first half and the teacher was reviewing a few homework problems.  The problems had been done up on the board by students.

3.  Use multiple representations of mathematical entities.

Leinwand encourages frequent use of the number line.  The students had graphed and our teacher was meticulous about the accuracy of their graphing.  She touched on scale, the need to put labels when a point was graphed so that we could understand where that point was in relation to the other parts of the graph.  

Leinwand encourages frequent use for students to draw or show and then describe what is drawn or shown.  As she went through each problem she dialogued with the student about what the graph looked like, why a point was here, why it curved like that.  Again, the students had drawn the graphs.

Leinwand encourages teachers to use pictorial representations to help students visualize the mathematics they are learning.  Our teacher used her arm to represent this:

Of course she had the kids stand up and do it themselves.  She didn't stop there.  She offered two other way for the kids to quickly remember the shape of this graph, one of them being, "Think of an eyebrow."  I bet the kids remembered the shape of this graph on the assessment!

4.  Create language-rich classroom routines.  

Leinwand emphasizes that students and teacher explanations should make frequent and precise use of mathematics terms, vocabulary, and notation.  As our teacher reviewed the domain and range of the parent functions she reiterated that the kids chose what notation to use.  She made sure to dialogue with the students about the names of the notation (set-builder notation versus interval notation).  "What does a bracket represent versus a parentheses?  What is another method we could use to represent this range?"

5.  Take every available opportunity to support the development of number sense.  

This was SO COOL!  Leinwand calls for frequent discussion and modeling about how to use number sense to "outsmart" the problem.  A student had graphed the function:

This is simply a linear graph shifted square root of 2 down on the y-axis.  Square root of 2 is approximately 1.4 (which our teacher allowed the student to use as the estimation, another small shift Leinwand encourages).  The student had used a table to create values for her graph:

This created a bit of a messy graph, though not inaccurate.  Our teacher emphasized that students could choose ANY values they want for x.  Our student chose those five values for x.  Our teacher then asked the class what would happen if we chose these two values:

Which of course leads us to this graph:

Math problem, consider yourself outsmarted.

She used several of the other small shifts throughout the time I was in the room.  And this was only homework review/assessment prep!  This was teaching at its finest and I am so proud of our teacher for taking these small shifts and intentionally putting them into her daily practice.  I know the kids are benefitting because of it.

Why Testing Could Ruin The Common Core

A statement from this article describes the intention of the Common Core Standards:

"The standards, which were written by a panel of experts convened by governors and state superintendents, focus on critical thinking and analysis rather than memorization and formulas."

This is good.  Reflecting on the notion that anyone was at anytime focusing on memorization and formulas when teaching our kids is scary.  But why was the focus on memorization and formulas?  Because that's what the tests valued.  That's what the tests measured.  So what else would the teachers be compelled to focus on?

The thing is, you can't test critical thinking.  You can't test analysis.  At least not by a standardized test.  Because thinking critically and analyzing are not standard.

What will end up happening is teachers will begin to learn what the new standardized tests measure.  Will these new tests be better than the current iteration?  Probably.  But they won't measure what they are intended to measure.  So teachers will adapt.  They'll teach their students to recognize what the tests are asking for.  They'll develop testing strategies for a different kind of test, but still a standardized test that is scored by artificial intelligence.  And that kind of test will never truly measure what our kids can and should learn.

This is a shame, because the intention of the Common Core is legit.  Of course we should focus on critical thinking and analysis rather than memorization and formulas.  But quit worrying about how to measure it and how we compare to other countries that are worrying about how to measure it.  I think Goodhart's Law is appropriate:

"When a measure becomes a target, it ceases to be a good measure."

Dark, Massive Asteroid...

So I see this headline online this morning:

"Dark, massive asteroid to fly by Earth on May 31st."

And immediately I'm asking questions:

How close will it be?  Will I be able to see it?  How do they know when it's going to fly by Earth? How big is the asteroid?  Is Bruce Willis going to have to save us?

At this point I haven't even read the article yet.  I click on it and read, and now I've got more questions:

How did they figure out that the asteroid is 1.7 miles long?  How do they know it's orbital path?  If one theory is that it flew close enough to the sun to torch it but not destroy it, how close was that?  How can they tell it will be 3.6 million miles away from us?  Radio antennas are used to view the asteroid; how do those work?

8-10 questions, just like that.  What if we asked our students to read the title of the article and list all the questions that come to mind?  We could then have them read the body of the article and list all the questions that come to mind.  Then spend the next month at school utilizing all your "instructional time" learning the answers to those questions.  Use the Internet.  Use your teachers.  Use the library.  Use each other.  Call the Jet Propulsion Lab and ask them questions.  Call my buddy who is an Assistant Principal at La Canada High School where students have parents who work at JPL and ask him if he knows anyone who can help.  Just ask.  And find out.

What if they blogged about the process along the way?  Captured video?  Created a documentary?  What if they presented the math they learned to their classmates?  What if they summarized the process and submitted that summary as an assessment?

What if school were like that?

Instancy and Critical Thinking

"When I was your age I walked 5 miles to school, uphill both ways, in the snow..."

I think most of us heard some form of this when we were kids, even if it was just in jest.  I write this post with that in mind, amazed (and a bit embarrassed) by the fact that I'm at a point in my life where I'm going to talk about how different it was when I was a kid.

I love technology and all its benefits.  I utilize it non-stop.  But there's something to be said about the instancy of our world and how it has negatively impacted critical thinking.

I can vividly remember one of the first times I realized I was thinking critically.  Maybe it wasn't critical thinking, it might have just been simple problem solving.  And I'm sure I had problem solved many times before this one instance, but in this particular case I consciously realized I had solved a problem on my own.

I had a walkman.  It played tapes.  I don't know how old I was at the time, maybe 7 or 8.  It had three buttons: play, stop, and fast forward.  I wanted to hear a song again and I didn't want to listen to the entire rest of the side of the tape, and the entirety of the other side, before hearing it.  Short of me taking the tape out and twirling the rotor with my finger, I had to find a more efficient way to do this.  Somehow it hit me...if I switch the tape to the other side and press fast forward, that would be the same as rewinding it on this side.  I was proud of myself.

Technology changed that.  A more expensive walkman had a rewind button.  Then cds came out.  Instead of rewinding to a song I could just hit one button and it would immediately go back to the beginning of the song.  Amazing!  But with the advancements in technology, leading to instantly rewinding my song, there was a loss of problem solving.  A loss of the need to think critically.

I think this has happened in a lot of areas in our lives.  I used to have my best friends' phone numbers memorized.  I'm sure that was a small exercise of the brain.  My phone does that for me now.  Going on a trip I'd do the navigation myself.  An app does that for me now.  Things that took some time are now done instantly.  And more often than not, done for us.  In many ways this is a great thing, but not in all ways.

I wouldn't trade the advances in technology for the times of yesteryear.  But as an educator, and as a dad, it's crucially important that we allow our kids opportunities to exercise their intrinsic ability to decipher, think, wonder, and solve.  You'd think that because the world is moving so much faster we'd have more time to think, but the opposite is true.  The instancy has taken thinking away from us.  We've got to be aware of this and intentionally put it back into the lives of our kids.

Kobe Sucks?!?!

There's a group of freshmen boys that I try to check in with everyday.  They all sit together for lunch at the same bench.  They're sweet, funny, and curious.  They sort of remind me of me as a freshman, though that's not why I check in with them.  They're on the right path, and I guess me checking in daily is my way of ensuring that they stay on that path.  We talk about all kinds of things, but the other day I walked over and one of them hit me with this:

"Kobe sucks."

I guess I should have inquired more, because for all I know he had a piece of overcooked meat the night before and was forever tainted.  But I had a feeling what he meant, so I immediately engaged.  "Define sucks."  The others saw me jumping into the ring and gathered around to watch.  Slowly I took the youngster's argument from, "Kobe sucks," to something more substantial, "Look at his shooting percentage, it's terrible."  Okay then, "Define terrible."

We went back and forth for a while, with me not defending Kobe so much as I was working to get the young debater to base his arguments on facts and even validated opinion versus simple hyperbole.  And he was PASSIONATE.  I mean, we could have gone on for days.  He got into why he believes LeBron is better than Kobe (I didn't let on that I actually agree) and that Jordan couldn't stop LeBron (I agreed here, pointing out that Jordan is 50 and definitely couldn't stop LeBron).  I was a freshman in high school almost 25 years ago and I was having the same argument with my friends about Magic and Jordan.  I was PASSIONATE!  However, I was quite uninterested in school.  Not that these boys are uninterested in school.  Most of them are far better students than I was.  But I don't think they're as passionate about school as they (or at least this one in particular) are about debating LeBron versus Kobe.  Can we change that?

Not if we don't give them the chance to have their own ideas.  Not if we don't allow them to experience the curriculum on their terms.  Not if we talk at them instead of to them.  Or even better we could talk very little at all and ask them questions that would get them doing the talking.  Or even better we could teach them to ask the questions so that they'd have a genuine desire to go find the answers.

I came upon this website a while back and enjoyed one entry without looking around much more.  Then I came upon it again recently and have been unable to pull myself away.  What if school was more like this? What if we allowed kids to ask 'what if' and then dove in headfirst trying to figure out the answer?  What if for homework we asked every kid to tweet a 'what if' question and then chose one and it became the focus for the week?  How cool would that be?