Pedagogy

How it Works: Integer Multiplication

This past week, Josh Zagorski forwarded a tweet from John Fritzky, a middle school principal looking for an explanation of how integer rules work for multiplication and division without using rules:

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Graham Fletcher folded me in on the conversation and I haven’t been able to stop thinking about it. I haven’t been able to think of a quick way to sum it all up, either.  So, here we are.

A few people commented on this with some good ideas and I’ll highlight those here.

As I mentioned, I haven’t been able to stop thinking about this.  The more I thought of it, the deeper I dove into it.  I decided to focus on multiplication first and after the past few days of thinking (this is just the beginning), I think I this is involves 2 big ideas:

  1. What are integers and how can I use what I know about them (absolute value, other operations, and the negative symbol “-“, etc.) to make sense of multiplying these numbers?
  2. What does multiplication really mean and how can we use what we know about it to apply it to a new system of numbers?

When I’ve thought about this in the past with my own students, I focused only on the first idea.  But the idea of what multiplication really means plays a huge role in making sense of integer multiplication (and later division).

Integers

The whole numbers and their opposites have several ideas associated with them and all of them really need to be developed deeply and conceptually in order to get to the point of making sense of operating with them.  That said, here are what I see as the big ideas  that stand out for the sake of this discussion (blue bold highlights):

6.NS.5 – Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.6 – Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

7.NS.1A – Describe situations in which opposite quantities combine to make 0For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

7.NS.2A – Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

Multiplication

As briefly as possible, here’s what multiplication really means (I’m taking this straight from the standards, to be as transparent as possible here:

3.OA.1 – Interpret products of whole numbers
e.g. Interpret 5×7 as the total number of objects in 5 groups of 7 objects each

So, when reading a multiplication problem out of context:

____ x ____ = ____

really means:

____ groups of ____ things in each group = ____ total things

This idea is important for three reasons:

  1. In the absence of context, students contextualize the “naked math.” Giving meaning to numbers in order to work with them is one piece of the number sense puzzle.
  2. This builds on an idea of grouping and sharing from Kindergarten, so it is accessible to all students.  Building connections!
  3. Representing multiplication with pictures, arrays, and/or number line diagrams makes more sense when students think of multiplication in this way.

Making Sense of Integer Multiplication

 

 

 

 

 

Revisiting 3-Act Tasks

My lessons never stay the same.  They’re always evolving.  Recently, I’ve taken a look at some 3-Act Tasks I created and I noticed:

  • Some of the tasks are lacking an act.
  • Others have resources that no students ask for (at least students that I’ve worked with).
  • The quality is low (shaky camera, point of changes, etc.)

So, I finally had a minute (read 2 days) and revisited each.  Below, you’ll see the tasks I’ve chosen to revisit.  An explanation of the original, what I changed, and why I changed it follows.  If you’d like to skip this and get to the revisited tasks, click here.

Revisited #1 – The Candy Bowl

My very first attempt at a 3-act task was the Candy Bowl task.  I was working in an elementary school at the time and Graham Fletcher had created problem to get 2nd and 3rd grade students reasoning about subtraction by removing the numbers from the problem context.  His context involved the lunchroom and numbers of students in three classes.  We talked on the phone about this for a while and though I liked the problem, I wasn’t crazy about the context.  I sat in my room trying to think of a context that would be a bit more engaging for students to think about.  And the Candy Bowl was created.

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It was a good problem, but it really lacked one of the most basic parts of a 3-Act Task… The third act.  The reveal was weak, because it relied on the teacher to give students validation.  The updated version, which had to be done from scratch (apparently whoppers candies are no where to be found anywhere near Valentine’s day), can be found here with all new updated resources for Act 2 and new video including two reveals, depending on which question students decide to tackle.

Revisited #2 – Sweet Tart Hearts

Another one of my early tasks was Sweet Tart Hearts.  I really liked this one from the beginning. There is a huge focus on estimation which allows for students to obtain solutions that are close, but not exact in most cases.  This also allows for the teacher to facilitate a discussion about why answers may not be exact for a variety of reasons.  But again, it really lacked that third act.  The task was good, but the closing of the lesson was weak due to the fact that the students were relying on the “all knowing” teacher to give them affirmation.

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Apparently Sweet Tart Hearts are a hot commodity a few days before Valentine’s day.  I went out the other day for a quick run to pick up a bag.  I had to go to 4 stores and finally found a bag (the last one).  I thought it would take about 10 minutes to do this revisit.  Surely the  numbers for the colors would be similar to the last time.  Not only was that not true, but Sweet Tarts changed the orange hearts to yellow!  But, the revisit is all done and I’m very pleased with the new reveal which allows the video to reveal the answer and the teacher to focus students on the reasonableness of their solutions.

Revisited #3 – The Penny Cube

My final revisit is the Penny Cube.  It is probably my favorite task.  I’ve certainly heard more from teachers about this task than any of the others.  I think I got the reveal right on this one.  The problem I found with this task was that I thought students would ask for things that I would want.  The first time I did this task with students, I guided them to the information I had ready for them.  They didn’t care anything about the dimensions of a penny.  They just wanted some pennies and a ruler.  It’s amazing what you learn when you listen to students, rather than try to tell them everything you think they need to know. So, to all of the students out there, Thank you for making your voices heard!

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So, this was the quickest fix.  I just updated the Penny Cube page (all of the coin specifications are still there – in case anyone wants them).

Note:  In this post I share how I changed my approach to teaching the Penny Cube task. 

So, it took a few days, but I’ve revisited some tasks that have been bugging me for a while and I hope it’s for the best.  I know I’ll probably give these another look in the future.  I’ll just need to start in early January to make sure I get the candy I need.

About Strategy Development (and Algorithms)

So there’s this thing going around about algorithms being a bad.

They’re not.  What’s bad is when students learn an algorithm – any algorithm from anyone – without making sense of it on their own.

Enter (what is considered by some) the buzz  word: “Strategy” (Guess what, the strategies being taught now are all algorithms).

I often hear teachers talking about teaching students several different strategies for (insert operation here).  Good, right?

Not so much.  Here’s the thing.  If teachers teach all of these different strategies, without student understanding at the forefront, they may as well teach the standard algorithm.  The worst part here is that students can actually be worse off being taught these multiple strategies without understanding than one algorithm without understanding.

arrrgh

Essentially, students are being force-fed strategies (aka algorithms) that they don’t understand and they feel like they need to memorize all of these steps for all of these strategies.  We’re going down the wrong path here.  Our destination was right, but we took a wrong turn somewhere.

It’s time to stop the madness!

How?  you ask.

Let me tell you a story…

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Back in early fall 2007, when I was still a toddler of a math coach, my beautiful wife’s grandmother passed away and the whole family went to her school on the weekend to help her get some lessons together for the few days she would be out.  Truthfully, I was the only one helping since the kids were 7 and 4 at the time.  Kim gave me jobs to do and I did them with precision and efficiency.  One of the tasks she gave me was to make a 18 copies of a few tasks for her students to complete during her absence.

To help her out, I took my son, Connor (the second grader), with me to the copy room so she’d only have 1 child to keep track of while she was trying to work.  When we got to the teacher work room, Connor watched as I placed the small stack of papers on the copy machine tray, typed in the number of copies (18) needed and then hit the copy button.  Within seconds he asked me (in the most exasperated voice he could muster) “How many copies is that going to make?”

I swear, when things like this happen, mathematicians in heaven play harmonious chords on harps using ratios.  I hear them and respond accordingly.  This time, I brought Connor over to the copy machine screen and showed him the numbers. 

Me: “Do you see that 5 right there?  That’s how many papers, the copy machine counted, and that 18 right there?  That’s how many copies of each piece of paper I asked the copy machine to make.”

Connor:  “Oh…”

Commercial break:  I didn’t really expect much more than an estimate.  This was September and Connor was a second grader.  He may have heard the word multiplication, but likely didn’t know what it meant.  

And we’re back!  His eyes looked up as he thought about this briefly and within seconds of his utterance of “Oh,” he said in a thinking kind of voice, “50…..”

Now, I’m not one to interrupt a student’s thought process – I work with teachers to keep them from doing it.  I actually remember having a mental argument with myself about whether I should ask him a question.  I was so excited in this moment, I couldn’t help myself.  I asked (with as much calm as I could), “Where did you get 50?”

I kid you not, he replied by pulling me over to the screen on the copier and said, “You see that 1 right there (in the 18), that’s a ten. And 5 tens is 50.”

I could hardly contain myself.  Naturally, since I had already interrupted him, I asked what he was going to do next.  I was floored when he said that he didn’t know how to do five eights.  I was floored because he knew how to multiply a 2 digit number, he just lacked the tools to do so.  In the context of this copy machine excursion, Connor made sense of the problem, reasoned quantitatively, showed a good degree of precision, and I’m sure if he had some tools, he would’ve come to a solution within minutes.

As we left the teacher work room, with copies in hand, I asked him to think about it for a bit and see what he could come up with.  When we got back to my wife’s room, I told her all about it.  When I got to the part where he didn’t know how to do five eights, I called across the room to him and asked him if he figured out what five eights was.  As he said, “No.” he paused and thought for a few seconds and said, “Can I do 8 fives?  ‘Cause that’s 40.”  Before I could ask him (thank God), “What about the other 50?”  He said, “40…50…90!”

This second grade boy (My Son!) who had never been taught multiplication, what it means, or any algorithm for it, created a strategy for finding a solution to a contextual problem that most of us would solve using multiplication.  He came up with the strategy.  It was based on his understanding of number and place value and he created it.  These are the strategies students need to use — the ones they develop.

I’ve told this story at least 50 times (I’ve even told it to myself while on the road).  Afterward, I often challenge teachers to take their students to the copy machine and watch this play out for themselves.  Some pushback does come out occasionally with comments like these (my responses follow each):

  • That’s because he’s probably gifted.  He is, but that’s not a reason to not do this with any group of kids.  Every student can and will do this when presented with contextual problems and access to familiar tools and where teaching through problem solving is the norm.
  • You probably worked with him on multiplication tables.  Yes, and no.  When Kim was pregnant with Connor and on the sonogram table with a full bladder, I leaned close to her stomach and started reciting multiplication facts to make her laugh (I’m cruel for a laugh sometimes) Other than the 4 or 5 facts I quickly rattled off that afternoon, I’ve never recited them since.  I doubt that did much, if anything, for his math achievement.
  • You must work with him a lot with math.  Not really!  Other than natural math wonders that have piqued my kids’ interests and sparked some discussion, no.  Questions they’ve had, like – “Dad, how many tickets do you think I have in this Dave & Busters cup?” are all we’ve spent any amount of quality time on.  That and puzzles.

So, when it comes to strategy building, it all has to begin at the student level of understanding.  The best way to do that is to let students develop their own strategies, share them with each other, and build more powerful understanding from there.  Then, if they do get “taught” a standard algorithm somewhere down the road, it has a better chance of making sense.

 

 

 

 

Personalized Learning Can’t Trump Content & Pedagogy

The problem I’m seeing with personalized learning (overall and especially as it pertains to math instruction) is the common understandings about what it is, what it can look like, what it shouldn’t look like, and how it works as related to our own learning experiences are fragile at best.

Many school systems, including my own, are looking at personalized learning as a means to improve math instruction, raise math test scores, and increase student engagement. These goals are great and many systems have them in some form or another. However, when personalized learning forces teachers into using sweeping generalized practices that often trump solid content pedagogy, something is drastically wrong.

I don’t think this is necessarily the fault of personalized learning as a concept,  but I do think it is problematic when common understandings become compromised.  These compromised understandings lead to sweeping generalized practices like:

  1. No whole group instruction – ever
  2. Students should be on a self-paced computer program for personalized learning
  3. Teachers have to create new groups of students every day/week to make sure learning is personalized
  4. Teachers should do project based learning several times per unit to engage learners
  5. Teachers need to use choice boards for every standard they teach.

This is not a definitive list – just what I’ve heard from within my own district over the last few years.

I may not have a response to each of these, but I can point out a few sources in addition to my thoughts:

  1.  No whole group instruction – ever – Dan Meyer’s post: http://blog.mrmeyer.com/2014/dont-personalize-learning/  my favorite idea from this is from Mike Caufield: “if there is one thing that almost all disciplines benefit from, it’s structured discussion. It gets us out of our own head, pushes us to understand ideas better. It teaches us to talk like geologists, or mathematicians, or philosophers; over time that leads to us *thinking* like geologists, mathematicians, and philosophers. Structured discussion is how we externalize thought so that we can tinker with it, refactor it, and re-absorb it better than it was before.”

2.  Students should be on a self-paced computer program for personalized learning Personalized learning is not something you get get from the App Store or Google Play  or from any ed tech vendor.

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Some other comments from Dan Meyer:  Personalized Learning Software: Fun Like Choosing Your Own Ad Experience  and from Benjamin Riley:  “Effective instruction requires understanding the varying cognitive abilities of students and finding ways to impart knowledge in light of that variation. If you want to call that “personalization,” fine, but we might just also call it “good teaching.” And good teaching can be done in classroom with students sitting in desks in rows, holding pencil and paper, or it can also be done in a classroom with students sitting in beanbags holding iPads and Chromebooks. Whatever the learning environment, the teacher should be responsible for the core delivery of instruction.”

3.  Teachers have to create new groups of students every day/week to make sure learning is personalized – I’m not sure this is the case.  If teachers really know where their students are in their mathematical progressions (lots of ways to do this – portfolios, math journals, student interviews (GloSS and IKAN from New Zealand, etc.)  These types of data are much more effective that computer testing programs because teachers are able to see and hear students’ thinking as well as their answers.  In my opinion, you can’t get more personalized than that!

4.  Teachers should do project based learning several times per unit to engage learners – anyone who has had PBL training knows that 1 per year is a good start!  PBL takes time – to plan, and plan some more (most often with other content areas).  If anyone expects more than one per year or semester initially, it’s time to have some Crucial Conversations!

5.  Teachers need to use choice boards for every standard they teach – student voice and choice does not have to be a choice board.  And really, how much of a choice do students have if we’re giving them all possible choices with no input from them?

To sum up: In order to really improve those goals of improving math instruction, increasing student engagement, and raising math test scores one thing is certain – an investment to increase teacher content and pedagogy knowledge must be at the forefront.  There is no other initiative or math program that will help districts reach these goals more effectively than this!

 

 

 

 

 

 

 

Filling Gaps: Buy a Program or Help Teachers Grow?

This post actually started as a rant as I was sitting through meeting after meeting with really nice people trying to sell products to “Fill the Gaps.”  So, if it has a rant-y feeling, just know where I’m coming from.  If no one really likes this, that’s ok.  At least it’s out of my system for now.  You see, when you’re “invited” to attend meetings to raise student achievement, you really need to show up, or who knows what will  happen.  So, in the effort to stand up for teachers and students, I attended all of them.

Man shouting, pulling hair

These were really nice people presenting to us, and they were very passionate about their products.  I even largely agree with several of them on their basic philosophy.

At least one of the people listening with us in the room was sold on many of the ideas before we even started these meetings.  Every slide or picture shown was met with a “That’s good!” or a “That’s really good!”  I think if they showed us a shiny, new penny, this person would have said, “This is what our students need!” with the same reaction!  The pictures of bulletin boards showing concept maps and vocabulary word walls and even students working may be good – or may not.   Really, there’s no way to tell – especially with the picture of the students working.  What were the students saying?  Were they discussing mathematics?  Were they using the vocabulary on the bulletin board?  Were they making connections to the concept maps?  Did they give and receive feedback about their work?  Let’s see some video, so I can see how this is really working.

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Again, philosophically I agree with their framework of instruction.   However, the product is not really necessary if the PL these companies are willing to provide is effective.

Now, on to the PL.  Lots of good strategies offered here.  And more pictures of students “engaged.”   My question:  what are the students engaged in?”  Are they engaged in the mathematics or the product?  My initial response to this self-posed question was:  Does it really matter?  The students are working.  After  thinking about this for just a few seconds, though, I can say without a doubt that it does matter!

Engaging students can be tricky.   A passerby, seeing students working silently in their seats, might conclude student engagement in a task.  A passerby, seeing and hearing students discussing a task, may conclude non-engagement in a task as well as lack of classroom management.  Really it’s hard to tell, in either case, whether there was any engagement or what kind of engagement there was.

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Students in the sixth-grade Harlequin Team from Paris Elementary School work on a math problem. Clockwise, from front left, are Abby Steeves, William Dieterich, Annie Choi, Katerina Crowell, Halie Page and Sebastian Brochu.

So, what does engagement mean?  It depends on what you want.  One of my goals year after year is to engage students in the mathematics they’re studying.  When I first started teaching, I wanted students to just be engaged, no matter what.  As I think back, they were engaged – probably in my educational “performance.”  I was the “fun” teacher that did crazy math lessons.  As I grew professionally, my lesson focus evolved to take the students’ engagement away from me and toward the mathematical content.  So, why is it so important?  If students are engaged in creating the product (creating a poster, making a presentation, etc.)  they may be learning mathematics, but how do we know.  I’ve seen students engaged in creating beautiful products and walk away with little mathematical understanding.  I’ve also seen students engaged in mathematics and creating not so beautiful products, but beautiful understandings and mathematical connections.

So, for all of the professionals in the room thinking this (or any of the other presentations we’ve seen) is the silver bullet. . . It’s not.  The only silver bullet out there that’s going to raise student achievement is teacher PL grounded in  understanding mathematics conceptually and building teachers’ pedagogical understandings and strategies.  If we want high achieving students, we have to help teachers achieve their greatest potential.  No program out there will do that, but if you really want to become a better math teacher, Twitter and the #MTBoS are a great place to start!

Change vs. What Worked in the Past

So, I’m at this Standards Setting meeting in Atlanta this week.  I’m working with people I’ve never met before.  As we settle into our assigned seats, we begin the small talk:

I introduce myself (since I seem to be one of the last ones to arrive).

Others at the table introduce themselves as well and before long we’ve found some common ground (many of us are in a coaching role) and start building a professional relationship.  I love this part of attending professional learning sessions at a state (and national) level.  “All of us are smarter than one of us.”  By day two of our work, you would think we worked at the same school.  Our conversations, while still mostly professional, are much more relaxed.

During one of our breaks, we begin talking about some of the teachers we work with who are “stuck in their ways.”  The question bouncing around (at least in my head) is “Why?”  Why are they so stuck?  As we talked at our table, the “reason” that seemed to dominate the conversation was one that many of us have heard before:

The teachers “reason” is that teaching this way has worked for the past “y” years so why should I change now?

Our conversation then takes an interesting turn.  A “what if” turn.

What if Apple thought the same way.  How would our world be different?  Would we still have iPads, iPhones, Apple TV, etc.?  It’s unlikely.  We’d probably have something that looks like this:

First Apple Computer

because what worked in the past should be good enough now, right?

What if Ford Motor Company thought the same way.  How would our driving experience be the same?  I doubt we’d have radios, or even seat belts.   Our new ride may look like this:

1910Ford-T

because what worked in the past should be good enough.

What if we wanted cataract surgery?  How would that look, if the surgeons of today had the same attitude about what works best?  Did you know that cataract surgery goes back to ancient Egypt?  Would you rather have Lasik or have a surgeon come at you with one of these?

Ancient Eye Surgery

If the only thing keeping you from changing is because it’s the way you’ve always done it, then it can’t be the best.  We’ve been growing and changing the way we do things because we are always searching for the most efficient way, or the more cost-effective way, or the safer way, or the way that will improve our lives.  Have we done that for students.  Is the way you’re teaching mathematics what’s best for your students?  Is your pedagogy guided by what’s proven through research or just what you’ve done for years?

Our conversation ended abruptly because we had to get back to work on standards, but as I met and reconnected with others at the workshop, this same conversation came up multiple times.  My thoughts on this are below, but I hope others chime in here with their own thoughts on this.

Steven Leinwand wrote something several years ago that I think relates well to this.  What he wrote was (and I’m totally butchering this, I’m sure) that we shouldn’t expect more than 10% growth/change per teacher per year.  On the flip side of that, he also said that teachers should strive for more than 10% growth/change per year.

This is something I’ve really tried to work on in my coaching role with teachers.  When learning something that seems daunting to a new or veteran teacher (moving toward a standards-based, student-centered approach to teaching mathematics for example), I suggest teachers choose one thing, one piece of what we’ve discussed that they think they can become really good at over several months, rather than trying to make everything fit at once.

Letting teachers know they are not expected to become experts all at once is great, but following through is even more important.  Without constructive feedback, teachers will likely fall back to their comfortable habits.  Just like teachers need to really listen to students, coaches need to listen to teachers.  We need to model what we expect.

If we don’t, we may end up with this 20 years from now: