# Blogarithm Posts

Last year I had the honor of being asked to write four posts for NCTM’s Math Teaching in the Middle School Blog: Blogarithm (one of the coolest math blog names out there).  They were posted every two weeks from November through the end of December (which just shows that I can post more frequently if someone is reminding me every other week that my next post is due (thanks Clayton).

The four posts are a reflection of a lesson I taught with a 6th grade teacher, in September of last year, who was worried (and rightfully so) that her students didn’t know their multiplication facts.  After a long conference, we decided to teach a lesson together.  I modeled some pedagogical ideas and she supported students by asking questions (certain restrictions may have applied).

Links to the four posts are below.

While you’re at the Blogarithm site check out some other guest bloggers’ posts.  Cathy Yenca has some great posts on Formative Feedback, Vertical Value Part 1 and Part 2, and 3-Act Tasks

# Math: A Fun After Homework Activity

All week long I’ve been asking Connor, my 9th grade son, what he has been working on in coordinate algebra.  Here’s a snippet of a recent conversation:

• Me:  So, Connor, what have you been working on in your coordinate algebra class?
• Connor:  We’ve been graphing.
• Me:  Graphing what?
• Connor:  Graphing different lines.
• Me:  What kinds of lines are you graphing?
• Connor:  Ummmm…
• Me:  Are they linear functions.
• Connor: Yeah, there are linear functions, but we also do curves…
• Me:  Like what kind of curves?
• Connor: Umm… exponents
• Me:  Ok.  Anything else?
• Connor: Umm…
• Me:  Hey, I want to show you something. . .

Versions of this conversation happened several times this week.  Due to soccer practices, games, homework, and Life in general, we never got much past Connor’s last “Umm…”

Until yesterday!  The conversation changed a bit:

• Connor:  We did something cool in class today.
• Me:  Oh, yeah?  What was it?
• Connor:  We had to build a picture using graphs of different lines.  We built a shamrock.
• Me:  That’s what I’ve been meaning to show you all week.  Go grab my laptop.
• Connor:  (playing game of war on an ipad) But I finished my homework.
• Me:  Just take a look at this for a few minutes and see what you think.
• Connor: (heavy sigh)

Enter Des-Man from Desmos.  Once he had gone through the tutorial, he was hooked. . . for a while!  He engaged in this for about 2 1/2 hours.  When he wanted to make something happen, but didn’t know how, he would come to me and ask.  We’d figure it out together.  The best part of this whole experience was when he realized he knew how to create something on his own and went to his math work from class as a reference.

Fast forward to 2 1/2 hours later, when Connor finished his Desman.

To see the picture in detail along with the equations Connor used to create this graph, click Connor Face Graph.

It didn’t stop there.  I had some tabs open and clicked on one with the In-N-Out Burger task from Robert Kaplinsky.  He was curious enough to work through it even after all of the Des-man work.  So, I showed him more by clicking on the Open Middle tab (also from Robert Kaplinsky).  I selfishly pulled up the task that I wrote in collaboration with Graham Fletcher called The Greatest Difference of Two Rounded Numbers.  After making sense of the problem, and a lot of eye opening moments that led to phrases like “Oh, I can make it larger!” He got what he thought was the final answer and we validated his reasoning by clicking on the answer.  A slight smile!

So, we’re looking at close to 3 hours of after homework math investigation that ranged from rounding numbers to graphing equations, and solving problems.  Sounds like a great evening to me.  Great conversations and fun while learning and reinforcing mathematics understanding!  What could be better?  Talking Math With Your Kids – High School Edition.

Feed the hunger of all ages!

More with Connor:  Real Math Homework and Real Learning

# Empowering Students with In-N-Out Burger

The following is a reflection on a 3-Act task I modeled for an 8th grade teacher last week.  The 3-Act is In-N-Out Burger from Robert Kaplinsky and the plan I followed I completely stole from the amazing @approx_normal ‘s blog post on her work with the same 3-Act with administrators last year.

This past Thursday was the day we agreed on to model the lesson.  So, this group of 8th grade students, who have never even seen me before, are wondering who this guy is that’s about to teach their class.  And, just as planned, they were giving me weird looks when I showed them the first cheeseburger picture and asked them what they noticed.  I believe one of them even asked, “Are you a teacher?”

Fast forward through to the “What do you wonder?” piece and the questions were amazingly well thought:

• “How much weight would you gain if you at that whole thing (100×100 burger?)”
• “How much do the ingredients cost for it (100×100 burger)?”
• How much does it (100×100 burger) cost?
• “Why would someone order that (100×100 burger)?”
• “Did someone really order that (100×100 burger)?”
• “How long did it take to make the (100×100 burger?)”

There were just a couple more, and they all came up very quickly.  The students were curious from the moment we started the lesson.  They are still working on precision of language.  The parentheses in their questions above denote that this phrase was not used in the question, but was implied by the students.  We had to ask what “it” or “that” was periodically throughout the lesson as they worked and as time went on, they did become more consistent.

The focus question chosen was:

• How much does it (the 100×100 burger) cost?

Students made estimates that ranged from \$20 to \$150.  We discussed this briefly and decided that the cost of the 100 x 100 burger would be somewhere between \$20 and \$150, and many said it would be closer to \$150 because “Cheeseburgers cost like \$1.00, and double cheeseburgers cost like \$1.50, so it’s got to be close to \$150.”  That’s some pretty sound reasoning for an estimate by a “low” student.

As students began Act 2, they struggled a bit.  They weren’t used to seeking out information needed, but they persevered and decided that they needed to know how much a regular In-N-Out cheeseburger would cost, so I showed them the menu and they got to  work.

I sat down with one group consisting of 2 boys (who were tossing ideas back and forth) and 1 girl (Angel) who was staring at the menu projected at the front of the room.  She wasn’t lost.  She had that look that says “I think I’ve got something.”  So, I opened the door for her and asked her to share whatever idea she had that was in her head.  She said, “Well, I think we need to find out how much just one beef patty and one slice of cheese costs, because when we buy a double double we aren’t paying for all of that other stuff, like lettuce and tomato and everything.”  The boys chimed in: “Yeah.”  I asked them how they would figure it out.  Angel:  “I think we could subtract the double-double and the regular cheeseburger.  The boys, chimed in again:  “Yeah, because all you get extra for the double double is 1 cheese and 1 beef.”  “Well done, Angel!”  You helped yourself and your group make sense of the problem and you helped create a strategy to solve this problem!  Angel: (Proud Smile)!

We had to stop, since class time was over.  Other groups were also just making sense of the idea that they couldn’t just multiply the cost of a cheeseburger by 100, since they didn’t think they should have to pay for all of the lettuce, tomato, onion, etc.

They came back on Friday ready to go.  They picked up their white boards and markers and after a quick review of the previous day’s events and ah-ha moments, they got to work.  Here is a sample after about 15 minutes:

Many groups had a similar answer, but followed different solution pathways.  I wanted them to share, but I also wanted them to see the value in looking at other students’ work to learn from it.  So I showed this group’s work (below-it didn’t have the post-its on it then.  That’s next.).  I asked them to discuss what they like about the group’s work and what might make it clearer to understand for anyone who just walked in the classroom.

Here’s what they said:

• I like how they have everything one way (top to bottom).
• I like how they have some labels.
• I’m not sure where the 99 came from.  Maybe they could label that.
• Where’s the answer…

During this discussion, many groups did just what @approx_normal saw her administrators do when she did this lesson with them.  They began to make the improvements they were suggesting for the work at the front of the room. It was beautiful.  Students began to recognize that they could make their work better.  After about 5 minutes, I asked the class to please take some post-its on the table and do a gallery walk to take a close look at other groups’ work.  They were to look at the work and give the groups feedback on their final drafts of the work using these sentence starters (again, from @approx_normal – I’m a relentless thief!):

• I like how you. . .
• It would help me if you. . .
• Can you explain how you. . .

Some of the feedback (because the picture clarity doesn’t show the student feedback well):

• I like how you showed your work and labeled everything.
• I like how you broke it down into broke it down into separate parts.
• It would help me if you spaced it out better.
• I like how you explained your answer.
• It would help me if it was neater.
• I like how you explain your prices.
• I like how you wrote your plan.
• I like how you explain your plan.
• I like how you told what you were going to do.
• Can you explain how you got your numbers.
• I like how you wrote it in different colors.
• It would help me if you wrote a little larger.

Some samples with student feedback:

Not only was the feedback helpful to groups as they returned to their seats, it was positive.  Students were excited to see what their peers wrote about their work.

Now for the best part!  Remember Angel?  As she was packing up to leave, I asked her if her brain hurt.  She said, “No.”  After a short pause she added, “I actually feel smart!”  As she turned the corner to head to class, there was a faint, proud smile on her face.  Score one for meaningful math lessons that empower students.

Please check out the websites I mentioned in this post.  These are smart people sharing smart teaching practices that are best for students.  We can all learn from them.

# Re-viewed: Children’s Mathematics. . . It’s a Beautiful Thing

About a month ago, I was asked to preview the new edition of Children’s Mathematics and write about it on this blog.  I was more than happy to oblige!  Children’s Mathematics is one of a select few books that I’ve read in the past decade that have really had an impact on how I teach mathematics.

Let me begin by saying that no matter which edition you read, it’s worth it.  If you’re a teacher (or parent) and have an unread edition of Children’s Mathematics sitting on your shelf (for whatever reason), do yourself and your students (or children) a huge favor and read it.  Then do exactly what it says to do!

The research-based approaches to teaching mathematics you’ll learn from the contents in this book are invaluable.  Cognitively Guided Instruction.  That’s what it’s called.  And it’s a beautiful thing to see in action.  And it’s even easier to see in action as you read the book (more on that later).  Empowering students to think, make sense of, and solve problems based on their own understanding.  Why don’t we all teach this way?  It makes so much sense.

To be clear, Cognitively Guided Instruction (CGI) is not a program or a curriculum.  It’s an approach to teaching and it’s based on research on children’s mathematical thinking and how it develops.  The idea that’s most intimidating to teachers (and parents) is that no direct instruction is used before giving students a problem.  Many would argue that students won’t know how to solve the problem unless they are shown how first.  This is so NOT TRUE.  The contexts of the problems give the students all they need to jump in to the problem.  Their pathways to solutions are defined by their own understandings.  For example, students may be given a problem such as:

Luke had 7 toy cars.  His friend gave him some more cars for his birthday.  Now Luke has 12 cars.  How many cars did his friend give him?

Students given this problem may solve it by counting down from 12 or up from seven,  They may begin by choosing 7 objects to represent the cars, then counting some more to get to 12.  They may even make two sets (one set of 12 and one set of 7). There are multiple ways students can represent the problem.  All of them valid.  Some are more efficient than others, but regardless of the strategies used, it’s a beautiful thing.  It’s especially beautiful when students share their strategies and learn from each other.  When we listen to students’ thinking we best know how to work with them in order to move them along their own mathematical journey.

Now this is all great, but you can get this and more from any of the earlier editions of this book.

Here’s some of the new goodness you get from the latest edition (out later this month):

• A chapter dedicated to Base-Ten number concepts – this was nice to see, since base-10 understanding is a huge part of elementary mathematics.
• Quotes from real teachers using CGI in the classroom.  These can be found at the beginning of each chapter.  Its a small part of the new edition, but really it’s one of the things I really enjoyed!
• Video clips that you can watch as you read! No more CDs to have to load, or lose or break.  When you’re reading and want to see the accompanying video, just scan the QR code in the book with your phone.  It just pops right up!

Overall, this new edition has some updated content and makes it easier (thanks to technology) to see in action.  As one teacher from the book put it:

The better I get at listening to children, the clearer I hear them tell me how to teach them.

Have I said this already?  Beautiful.  Absolutely beautiful!

What are you waiting for?  Go out and get Children’s Mathematics and read it.

And then, go out and get the “sequels”:

Extending Children’s Mathematics: Fractions and Decimals

and

Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School

Check out the new edition here:  http://heinemann.com/ChildrensMath

# The Penny Cube

I just finished a 5th grade 3-Act task called Penny Cube that I created last spring.  I tried it then, but just to get some feedback from students and see what I might need to change about how the task should be presented.  Now, after completing this task with two groups of students (at two different points in the year), I’ve learned three things:

1. Students see a video and notice a bunch of things that teachers don’t even realize are there.
2. The curious questions students ask first are often “why” questions.
3. There’s no way to predict everything a group of students might wonder.

I’ll take this reflection from the beginning.  First, I let students know that I was going to show them a video clip.  I also told them that I was going to ask them what they noticed when it was finished playing.  I gave students a chance to brainstorm ideas about what they could do to make sure they would be able to share what they noticed once the clip had finished playing.  Their ideas were amazing:

• We could look for expressions (on faces – I found out later that this student was thinking about context.  A person’s facial expression can tell a lot)  Unfortunately, there were no facial expressions in this video.
• Listen carefully (they might be able to hear something that might give them a clue about what was going on – these students were already expecting a problem situation!)
• Stay focused on the clip.
• Take notes.
• Try to remember as much as you can.

I had never done this before, but after hearing their ideas, I will be using this again.

After showing the video clip for Act 1, I immediately had them talk about what they noticed with their groups.  Then, they were asked to share with the whole group.  Here is what they noticed:

What’s missing from this picture is the wonderful reasoning given for some of these.  For the last bullet, “container is open in the front,” the student told the class that it was open in front so the pennies could be placed in the container more easily (I never thought they’d see or think about that).  They even began to wonder a bit here – “it might be an expression or it might be counting.”  My favorite, though, is the estimation by the girl who said “it looks like 100 pennies in the stack (\$1.00).”  This was particularly interesting to me because of what happened when they were asked to estimate for the focus question.

The wonders were typical from what I usually get from students new to 3-Act tasks, but I handled it a bit differently this time.  Here are their wonders (click here for a typed version of Penny Cube Notices&Wonders):

In my limited (yet growing) experience with teaching using 3-Act tasks, I’ve noticed that the wonders are initially “why” questions (as stated in number 2 above).  I told the class that I noticed that the questions they were asking were mostly “why” questions.  I asked them what other words could be used to begin questions.  Rather than trying to steer students to a particular question, I decided to focus the students’ attention on the kinds of questions they were already asking, and guide them to other types of questions.  It didn’t take long!  Within about 5 minutes, students had gone from “why” questions to “how many . . .” and “how much . . .” questions which are much easier to answer mathematically.

The students were then asked to figure out what they needed to solve the problem.  From experience with this task, I knew that most students would want pennies, so I had some ready.  I didn’t give them the Coin Specifications sheet, because no one asked for it.  I did have it ready, just in case.  Every group asked for pennies and rulers.  I wasn’t sure how they would use them, but I was pleasantly surprised.

Here’s what they did:

How many pennies in 2 inches

How many pennies in an inch

How many pennies fit on a 6 inch edge of the base

How many pennies cover base

How many stacked pennies in 2 inches?

The students all started in a place that made sense to them.  Some wanted to figure out how many in the stack, so they stacked pennies and quickly realized (as I did when filling the cube) that you can’t stack pennies very high before they start to wobble and fall.  So, they measured smaller stacks and used that info to solve the problem.  Others wanted to find number of pennies along an edge to find how many cover the base, then work on the stacks.  Students were thoroughly engaged.

After three 1 hour classes, students were wrapping up their solutions.  Some groups were still grappling with the number of pennies in a stack.  Others were finished.  A few were unsure about what to do with some of the numbers they generated.  All of this told the classroom teacher and me that there were some misconceptions out there that needed to be addressed.  Many of the misconceptions had to do with students disengaging from the context, rather than integrating their numbers into the context:

• One group was unsure of whether to multiply  the number of pennies in a stack by 12 (6 inches + 6 inches) or to use 64.
• Another group found the number of pennies to cover the base and multiplied it by itself to get their solution.
• A third group found 37 pennies in 2 1/2 inches and was having a difficult time handling that information.
• A fourth group had come up with two different solutions and both thought they were correct.  Only one could defend her solution.

Eventually, several groups arrived a solution that made sense to them.

Time to share!

I chose one group to share.  This group had a reasonable solution, but their method and numbers were different from many of the other groups, so this is where we were hoping for some light bulbs to begin to glow a bit.

This group shared their work:

I asked the class what they liked about the work.  The responses:

• The math (computations) are written neatly and they’re easy to follow.
• I know what their answer is because it has a bubble around it.
• The question is on it.
• It’s colorful.

All good.  Now, for the best part:

What questions do you have for this group?  The responses:

• Where did you get 34?
• What does the 102 mean?
• How about the 64?

Any suggestions for this group to help them clarify their work to answer some of your questions?

• Maybe they could label their numbers so we know what the numbers mean.
• Maybe they could tell what the answer means too.  Like put it in a sentence so it says something like “6,528 pennies will fit in the container.”
• Maybe they could have a diagram to show how they got a number like 64 or 34.  I know that would help me (this student had a diagram on his work and thought it was useful).

The light bulbs really started to glow as students began making suggestions.  As soon as a suggestion was made, students began to check their own work to see if it was on their work.  If it wasn’t, they added it.  All of the suggestions were written on the board so they could modify their work one final time.  The best part about this whole exchange was that students were suggesting to their peers to be more precise in their mathematics (SMP 6 – Attend to precision).  And, they really wanted to know what 34 was because they didn’t have that number on their boards (which is why I chose this group).

Now for the reveal!  When I asked the class if they wanted to know how many pennies were in the cube, they were surprised when I pulled up the reveal the video.  I guess they thought I’d just tell them (that’s so 1980’s).  They watched to see how close they were and when the total came up on the screen, many cheered because they were so close!

The students in this class were engaged in multiple content standards over the course of 3 days.  They reasoned, critiqued, made sense, and persevered.  It’s almost difficult to believe that this class was a “remedial” class!

Below, I’ve included a picture of each group’s final work.

Finally, one of the conversations witnessed in a group was between a girl and a boy and should have been caught on video, but wasn’t.  This group had an incorrect solution, but they were convinced they were correct, so to keep them thinking about the problem, I asked them how many dollars would be equal to the number of pennies in their answer (3,616).

• Girl:  There are 100 pennies in a dollar.  So 600 pennies is . . .
• Boy in group: \$6.00
• Girl (after a long pause):  1,000 pennies equals \$10.00
• Boy:  So that’s . . . um. . .
• Me:  How does knowing 1,000 pennies = \$10.00 help you.
• Girl:  We have 3,000 pennies, so that’s \$30.00.
• Boy:  \$36.00
• Me:  Share with your group how you know it’s \$36.00
• Boy:  Because \$30.00 and \$6.00 is \$36.00
• Girl:  And the rest (16) are cents.  \$36.16!

And they didn’t even need a calculator!

Math really does make sense!

# How Old is The Shepherd? Revisited in 3rd Grade

Over the past few weeks, I’ve shown the How old is the shepherd? problem to both of my kids and then shown them the video  from Robert Kaplinsky’s blog. Both were shocked at how many students don’t pay attention to what is happening in the problem.  Connor even said, “I guess I’m not one of the 24.”

Here is my son, Connor, with his response to the problem:

Unfortunately, his first statement, “That’s stupid!” was not caught on video!

My daughter, Lura, with her response:

Last Saturday, after ambushing one of my daughter’s friends with the problem while she was visiting, Kim (my wife) became more curious about the problem, so I showed the video to her and shared some of the data on Kaplinsky’s blog. She was also shocked at the results.  We had a brief conversation that went something like this:

Me: This is why we need to teach math content through patient problem solving and sense making!

Kim: Ok. (with a look that says, I know you’re passionate about this, and that it’s important. We’ll talk later. Go make a 3-act video and post it to your blog.)

Me: Ok.

It was left alone until this morning. It’s just me, but I like to think we would’ve talked sooner if I hadn’t been fighting a cold. She texted me and asked me to send her the Shepherd problem. I did, but only with the requirement that she share what she does with it.

Kim (and her co-teacher) gave the problem to each of their students and I just received the results:

• 3 out of 19 students made sense of the problem (15.8%)
• One student added 125 five times.
• One student reasoned that by the time you had 5 dogs and 125 sheep, you have to be in your fifties.
• One student divided 125 by 5.
• 6 students added 125 and 5 to get 130.
• 3 computed an operation with the two numbers incorrectly
• The other students guessed or showed no reasoning.

Now the good stuff:

• One student (an autistic child) shared his reasoning about the problem with his classmates:

“The shepherd has no-o-o-othing (said as a sheep might say it) to do with the sheep and the dogs.”

• Both teachers lost it!

Take aways from this:

• It’s best that we start teaching math content through problem solving early and consistently K-12 and beyond.
• Making sense of mathematics needs to be a priority for all students. (SMP 1)
• All students bring something of value to a classroom.
• Stories like the student who shared his reasoning sometimes get us through days that are not so much like this.

Below, you will find some of the students’ reasoning.

How old is the shepherd_

# Are Your Students Doing Mathematics?

It seems like a silly question, really.  The answer, we would expect, is “Yes, every day!” Unfortunately, I’m not sure this is the case.

For those of you about to first step foot on the exhilarating math train that is teaching mathematics, it’s probably a good idea to share a few facts and myths about learning and doing mathematics.

Myths:

• math is equated to certainty (sadly, this belief is held by many!)
• knowing mathematics means being able to get the correct answer – quickly (again, this belief is held by many)
• mathematical correctness is determined through the use of a teacher or an answer key.

Facts:

• mathematics  is a science of pattern and order (this was taken from Everybody Counts)
• math makes sense (teachers cannot make sense of mathematics for students)
• doing mathematics requires students to solve problems, reason, share ideas and strategies, question, model, look for patterns and structure, and yes even fail from time to time.

If you walk into a math classroom – at any level – students are doing mathematics if you see/ hear students doing the following:

 Explore Construct Justify Develop Investigate Verify Represent Describe Conjecture Explain Formulate Use Solve Predict Discover Discuss

If teachers are doing most of these, a shift needs to happen.  All students can do these things.  All students can learn and do mathematics.  All students can make sense of mathematics because math makes sense.

As I reread what I’ve written so far, it tends to read a bit negative.  That was not my intent.  I just wanted to point out that wherever you are in the vast range of stakeholders of math education, please be aware that just because there are students in a math class, does not mean they are necessarily doing mathematics.  That wasn’t much better!

This might be a better way to end this post:

There are many of us (more than I thought when I first started this blog) who are making the case for teaching mathematics for understanding through engaging tasks.  Dan Meyer, Andrew Stadel, Fawn Nguyen, Graham Fletcher, Jenise Sexton, and Robert Kaplinsky, just to name a few, use their blogs to share their thoughts, lessons & tasks they create, and their thoughts on what it means for students to learn and do mathematics.  These, and many others, continue to push all of us to become a better math teachers.  Personally, they strengthen my resolve, knowing that our numbers are growing along with our minds and the minds of our students!