# Learning

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.

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.

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

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.

Hi, my name is Mike… and I love using Desmos with students.

This is not a bad thing at all.  I’m not giving up time with my family to spend on Desmos. It’s just that whenever I think I’ve exhausted all of the ways to use this fantastic tool with students, the Desmos team adds a new activity or game that I can and want to use right away!  These people know how to keep us wanting more!

Here you can find out what Desmos is all about!

Now, for all of you teachers out there that haven’t engaged your students in this amazing math tool, let me move from a user to a pusher.  4 reasons why you should use this amazing tool with your students:

1. It’s completely free!  (not just this first time – all the time)
2. It’s a graphing calculator that works beautifully online or as an app for students to Model with Mathematics – SMP 4.

This is a screenshot of how my son, Connor, used the Desmos Calculator to make sense of transforming quadratic functions.

3.  When you sign up as a teacher (again, for free) you can assign activities and games (yep, they’re all free to use, too) to your students and you can check their progress from your teacher page.

So, beyond the graphing calculator – which is amazing on its own – as a teacher you can assign an activity to your students based on the content they are investigating. Try Central Park  – it’s my favorite activity.  (If you like, you can go to the student page and type in the code qqbm.  I set this up for anyone reading this post. Feel free to use an alias if you like).

And as far as games go, check out Polygraphs.  It’s like the Guess Who? game for math class. Trust me, your students will love it and there are polygraphs for elementary as well as secondary. The polygraphs are all partner games, so students will need to work in pairs.  I’ve even made a few:

Polygraph: Teen Numbers

Polygraph: Inequalities on a Number Line

Polygraph: Geometric Transformations

4.  As you get sucked in to this tool, you may begin to think to yourself, “Boy, I really wish there was an activity for ______.  If only knew how to create an activity for my students to use on Desmos.” That’s taken care of, too, with Activity Builder and Custom Polygraph (and, yep, you guessed it – they’re free to use, too)

And before you begin to doubt whether you can create an online activity or polygraph, the Desmos team has already taken steps to make this extremely teacher friendly.  Before you know it, you’ll have your own Desmos activity published!

Finally, as a great end of year gift, Dan Meyer blogged about the latest from Desmos – Marbleslides.  If this doesn’t get you to use Desmos with your students. . . well, I’m sure they will think of something else, soon. But seriously, try this out.  I have re-learned and deepened my own understandings of mathematics by trying and reflecting on many of these activities and games, and then having my own kids do them (and then they ask me why their teachers aren’t using them – “Can you talk to them, Dad?”).  The conversations will be happening this semester for sure!

But the best part about all of this is that students get to use the calculator to investigate graphs and compare graphs and equations/functions.  They get to notice and wonder about what matters and what changes a graph’s slope, and y-intercept for linear functions and what changes the vertex and roots of parabolas.  They get to investigate periodics and exponentials and rationals and so much more.  They get to engage in activities and games that have components that ask them to reflect on what they’ve learned in the games and activities themselves.  The students are doing the mathematics.

Then, in class, we get engage students in talking about the math they’ve investigated!  How sweet is that?

You see, as great as Desmos is, it can’t take the place of great teaching.  It’s a tool that can help us become better at our craft and help our students gain a deeper understanding of mathematics!  Sounds like a win-win!

So, I guess I don’t have a Desmos math addiction.  Addictions have adverse consequences and I see none of that here!  I just have – as we all do thanks to Desmos – access to a powerful mathematical learning tool!  Thank you Desmos.  I can’t wait to see what’s coming next!

# 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.

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.

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.

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!

# Unlikely Students in Unknown Places

I recently got back from Santa Fe.  I was attending a conference there for a few days last week and afterward, I drove to El Paso to visit my brother’s family (he’s currently stationed in the middle east so I didn’t get to see him – unless you count face time) for a day before flying home.

Let me preface this story by saying that we all probably have a story similar this, but how we handle it can be a possible game changer.

Somewhere on my long drive, I stopped at a fast food restaurant to grab a quick bite.  So, I went inside and got in line.  The following outlines the beginning of our interaction as I stepped to the counter:

Cashier:  May I take your order?

Me:  Yes, please.  I’d like a number 2.

Cashier:  Large or medium.

Cashier:  (after pushing more buttons than is conceivably necessary to enter my choice of “medium”): Your total will be \$6.05.

I dug through my wallet (receipts from the trip and everything) and found that all I had was a \$10 bill, so I handed it to her.  She entered \$10.00 correctly and the correct change of \$3.95 showed up on the little screen.  At just about that point, I remembered that I had a bunch of change in my pocket and said quite enthusiastically, “Oh, wait, I think I have a nickel.”  Who wants to carry around \$0.95 in change in their pocket.

The cashier didn’t miss a beat, and said, “So, your change will be \$4.00 even.”  I kind of smiled as I continued to look through my change, proud that she had a mental strategy to adjust to the situation and that she seemed quite confident and comfortable using it in this situation.

Unfortunately, I didn’t have a nickel, but I still didn’t want change falling out of my pocket into the depths of the rental car, never to be seen by me again.  So, I told her, “Oh, I’m sorry, I don’t have a nickel, but I do have a dime.”

As I handed her the dime, I saw her face morph from a confident smirk to a confused, almost terrified look of despair .  I had just taken her from a mathematical point of “Yeah, I can do this math stuff.  I may not use the computer for the rest of my shift” to “Holy \$#!+, what the #=|| just happened!”

I went into math teacher mode and waited patiently for her to begin breathing again.  And then I waited for her begin thinking.  She adjusted my change with my introduction of the idea of a nickel, why not a dime? After what seemed like 5 minutes (probably more to her), it was painfully obvious to all around that her anxiety in this situation was taking over her ability to tackle this problem. So, I tried to think of a “least helpful question” to ask.  Now I put myself on the spot.  If she only knew that we were both now feeling some of this pressure.

So, I finally asked her my question and she gave me the correct change within a few seconds.  She smiled as she gave me my change and my new “unknown” student and I parted ways.  I know I felt good about helping someone develop a strategy outside of the classroom.  I hope she had a similar feeling about learning to make sense (no pun here) of making change.

Being a math teacher is a 24-7 job sometimes and we can find our students anywhere – even in a fast food restaurant in New Mexico!

What you would have asked the cashier in this situation.  I’d love to hear what your “least helpful” question would have been. No pressure, take as long as you like.  No one is waiting in line behind you!

Feed the hungry!

Oh, here’s my question:  “If you could change the dime into some other coins, what would you change it for?”

# 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

# Connecting Percents and Fractions

Not understanding mathematics can be extremely frustrating for students.  As a teacher, figuring out how to help students understand mathematics can be just as frustrating.  My primary go-to resource for these situations is Teaching Student-Centered Mathematics, by John Van de Walle et. al.  because it’s all about focusing on big ideas and helping students make sense of the math they’re learning in a conceptual way.

Recently, I was asked to model a lesson for a 6th grade class who was having difficulty working with percents.  So, I turned to my go-to resource, and during planning, I realized that I didn’t know anything about these students other than that they were struggling with percents.  So, I couldn’t assume anything.  I ended up creating three separate lessons and combined them into 1.

First, I handed groups of students a set of Percent Cards and Circle Graph Cards.  Their task was to match the percent with the corresponding circle graph.  As students were working on this, I heard groups reasoning about how they were matching the cards.  Many started with benchmarks of 25%, 50%, and 75%, while others started with the smallest (10%) and matching it to the graph with the smallest wedge.  As groups finished, they were asked to find pairs, using the matches they made, that totaled 100%.  Once finished, a discussion about their process for completing these tasks revealed a solid understanding of percent as representing a part of a whole.

Now to shake their world up a bit.  I asked them to leave their cards because they would be using them again shortly.  I introduced these Percent circles and asked them what they were.  A brief discussion revealed some misconceptions.  Some students said they were fractions, others said they were wholes because nothing was shaded.  I altered my planned line of questioning to questions that eventually led to a common understanding of what fractions were and how the pictures of the fraction (percent) circles really showed wholes and parts (fractions).

Their next task was to match their cards with the equivalent fraction circle.  This was incredibly eye-opening. Groups began to notice that some percent card matches could fit with multiple fraction circles (50% could be matched with the halves, quarters, eighths, and tenths).   Thirds and eighths were the last to be matched.  But their reasoning didn’t disappoint.  One group noticed that the percents ending in .5 all belonged with the eighths because they were too small to be thirds (the other percents with decimals).

Students were eager to share their thoughts about what they learned about fraction circles and percents:

• Fractions and percents are the same because the pieces look the same.
• 1/4 is the same as 25% and 2/8
• I don’t get why the eighths end in .5.
• The percents all can be fractions.
• 1/8 is 12.5% because it’s half of 25%

Finally, I asked students to solve a percent problem (now that they’ve all realized that fractions and percents can be used interchangeably).   I gave them the m & m problem from this set of percent problem cards.  The only direction I gave was that they had to solve the problem using some representation of the percent in the problem before they wrote any numbers.

My bag of M&M’s had 30 candies inside.  40% of the candies were brown.  How many brown candies is that?

While this was problematic at first, students looked at their fraction circles and percent cards and realized they could use four of the tenths since each tenth was the same as 10%.  Most students needed just one “least helpful” question to get on the right track:  Where do the 30 m & m’s belong in your representation?

Most groups were able to make sense and persevere to solve the problem correctly, and explain why they “shared the 30 m & m’s equally among the ten tenths in the fraction circle” and why they “only looked at four of the tenths because that’s the same as 40%.”

My beliefs that were reinforced with this lesson:

• We can’t assume understanding from correct answers alone.  We need to listen to students reason through problematic situations.
• Students really want to share their thinking when they realize that someone is really interested in hearing it.
• Students crave understanding.  They really want to make sense.
• Procedures are important, but not at the expense of understanding.
• Empowering students by allowing them to build their own understanding and allowing them to make connections allows students to feel comfortable taking risks in problem solving.

Please take a look at Jenise Sexton’s recent blog about percents with 7th grade students for some fantastic ideas about students using number lines and double number lines to solve percent problems.  It’s SWEET!

# 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.

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.
• 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.

# Math Students are Bleeding Out!

Let me explain.  There’s a math epidemic (remember Ebola 2014+).  Students are bleeding out from the gashes of their misconceptions of mathematics.  The lack of teaching conceptual understanding along with sacrificed opportunities to make mathematical connections is the double edged sword.  This is an epidemic, and some teachers, school systems and educational leaders are treating it like it’s a tiny scratch, instead of the pervasive threat to mathematical achievement that it is.

Here’s a familiar scenario:  A school’s test scores come back after the spring testing season (or mid-terms).  The scores show little growth from the previous year in the area of mathematics, and any change is not in a positive direction.  The knee-jerk reaction to the valid question, “What can we do to fix this?”  is to look for programs and technology that will fix the problem.  These are the same individuals who, way back in August, looked us all in the eye and, with the greatest of sincerity, reminded us that the single most important factor determining student success is the quality of the teacher.  Not new programs.  Instead of growing the quality of teachers, we get programs that:

• push speed over comprehension (imagine if we taught reading this way).
• define fluency based on digits rather than efficiency, flexibility, and accuracy.
• use technology to separate us from our students when we know that what we really need is to spend more time listening to them and creating an interactive classroom with technology as a support for this human interaction
• are essentially Band-Aids

I often hear the phrase “back to the basics” in times like these.   I’ve heard parents, administrators, and even a few teachers say this.  I think everyone would agree that “back to the basics” should mean that students become computationally fluent.   The idea of going back to this implies that we were doing something right before.  And we all know that’s not true.  After all we have generations of adults who are not computationally fluent and/or have extreme math anxiety.  And how did that happen?

Answer 1:  Timed tests.  My sixth grade teacher called them speed tests.  We did them every day, right after lunch.  (I was never in the top 10).

Answer 2:  Algorithms memorized by students with no understanding, presented by teachers with little understanding other than from a teachers’ edition.

Answer 3:  Little or no real problem solving.  Naked computation all around.  No wonder students were turned off by mathematics!

Answer 4:  No interaction.  Math is a social activity.  If you talk to any engineer, designer, architect, mathematician, statistician, etc.  They aren’t doing their work in an office silently sitting in rows.  They’re constantly talking to one another about the mathematics they’re using.  The idea that all of us are smarter than one of us makes so much sense in the real world and it should make sense in the classroom as well.

If we went back to teaching math like we did 20-30 years ago (I think that’s what some of these folks were implying when they said “back to the basics.”  We’d still be in the same boat.  Anyone ever watch How old is the shepherd?.  That was popularized over 20 years ago and the results haven’t changed.  Going back is not an option.  Building fluency is.

So what do we need to do in math classrooms?  I have a few ideas to stop the bleeding and these are certainly not original to me.

1. Apply pressure to the wound. Give up on the ineffective treatment, not the patient.  Apply pressure to stop the bleeding.  Focus on tasks and activities that build number sense.  Number Talks, Math Talks, Estimation 180, Visual Patterns any or all of these can be put in place at any level.  And the best part is, students can easily be trained to begin to apply the pressure themselves.  They have the power to stop the bleeding!
2. Close the wound. This can only happen with stitches.  And it takes time to get the hang of it.  The wound has to be closed with the thread of understanding.  We can’t understand for them, so the wound has to be closed with the help of the students.  The students create this thread as we stitch and we can’t do it without them.  How do they create this thread of understanding?  We have to stop telling so much and instead “be less helpful.”  If we tell students too much, the thread breaks.
3. Treat any symptoms that may show up after the initial treatments above:
 Symptoms Name Treatment Students may begin to rely on rote procedures with no foundational understanding Sometimes unintentionally caused by parents & other adults trying to help. Misconceptionitis Identify the misconception(s) and re-build understandings using the CRA (Concrete Representational Abstract) model Students are finding unreasonable solutions to tasks & problems and they often seem unaware; clueless Unreasonableness This is often attributed to students just not thinking enough.  Treatment should include a DAILY diet rich in estimation – prescribe www.estimation180.com Students count (often on fingers when computing or rely on a calculator for the simplest of calculations and even then, they can get incorrect answers. Influencia This is often diagnosed along with unreasonableness (see above).  Its roots lie in naked computation and memorization of facts rather than allowing students to build strategies and practice those strategies until they become fluent.  First, counting strategies are the lowest level strategies.  Students need to build more efficient strategies by exercising with  investigations of number relationships through number talks, math talks, and strategy building.  Stop giving speed tests. Students have strategies for computation, but are not applying them in problem solving situations No Solvia Students need a heavy dose of problem solving every day.  This must involve students engaging in the Big 8 Standards for Mathematical Practice.  Problem solving tasks every day.  Hydrate often with student reasoning.  Adopt the classroom mantra: “The answer isn’t good enough.” Begin new concepts with a problem before any formal instruction on the topic.  See what students can do before assuming what they can’t do.

I’m a teacher and I know many of you reading this are the choir that need no preaching to.  If you’re interested in saving the patient, stopping the bleeding, and raising math achievement, click on some of the links in this post.  There’s so much to learn from those smarter than me.  Also check out #MTBoS on Twitter.  Lots of math goodness from the best out there.

# Perplexing Donuts

A good friend and colleague, Krystal Shaw, tweeted this article about Krispy Kreme Donuts in the UK a while back and it immediately got me thinking. . . so I really liked it and wanted to use it with kids.  To plan for the lesson, I started to take myself through this problem as if it were a 3-act task (I wasn’t sure it would become one, but I wanted to see where this would lead).  I looked at the picture:

and jotted down what I noticed. Then I began wondering:

• How many donuts are in that big box?
• What are the dimensions of the box?
• Is there more than one layer of donuts in the box?
• How many rows of donuts are there?
• How big is (What is the diameter of) a Krispy Kreme donut?
• When I was finished (or thought I was finished) wondering, I began to seek the information needed to answer my questions.

I found some nice strategies for determining the number of donuts in the box.  Strategies accessible for 4th grade students.  I was happy, so I moved on to the next question: What are the dimensions of the box?

This is when it happened.

I was stuck.

Perfect.

Challenge accepted.

I looked at the pictures, found the information in the article, then began to question that information (and myself) as well as some critical friends.  This problem was getting better and better as I walked myself through it.  Fantastic!  SMP 3: Construct viable arguments and critique the reasoning of others, such as a Krispy Kreme representative from the UK or a USA Today reporter.  Maybe this question won’t have a third act, but the estimation and reasoning used to solve this could be extremely empowering for kids.

I challenge you to solve this problem with your class as well and share your results.     Challenge yourself and your students to construct a viable argument and critique the reasoning of others.  Does your math challenge the information in the article or support it.  Either way, integrate writing into math class in a meaningful way:

write to the reporter, Bruce Horovitz or Krispy Kreme UK: helpdesk@krispykreme.co.uk and tell them what you  discovered

Time for me to give this a try!  More in about a week.

By the way:  Krystal Shaw gave her amazing Mathletes after school club the task of writing a 3-act math lesson for their teachers to teach.  I think she should post it on her blog to share with the MTBoS!

# So…Have You Always Taught Math This Way?

I’ve been asked this question several times over the past 15 years or so.  Most recently at a workshop I facilitated for middle school teachers.  The short answer is no.  My teaching has evolved.  I strive to improve my practice every day.  Below, is my response to the group of middle school teachers.

When I first started teaching, I used what I learned in college about teaching mathematics – you know . . . using manipulatives, group work, classroom discussions.  All of those things that I still use today.  But, when things didn’t go the way I anticipated, I seemed to always fall back on the way I learned which was primarily stand and deliver.

At the end of my first year, I spent some time in my room, at my desk and wrote down all of the changes I wanted to make and how I planned to make them.  This was probably the best idea I ever had!  Throughout the summer I reread that list and, when necessary, created things that would help me reach my goals.  I didn’t reach them all, but the next year was much more successful.  Couple that with the summer PL that I took and the way I was teaching math was really beginning to change.

One of the first changes I made was to incorporate children’s literature into my lessons.  One of the PL’s I took that summer was a Marilyn Burns workshop where we  learned that there are a tremendous number of books with mathematical connections.  We learned how use the books to introduce mathematical concepts and problem solving, how to ask better questions, and one of my big “take-aways” was to listen more!

Over the years, I’ve continued to look at literature as a place to begin lessons.  And all was going well, but I still wasn’t getting the the amount of  buy-in from my students that I wanted.  I was excited about the math, but they weren’t.  Then, one morning, I was riding to work with my wife, Kim.  We were listening to a morning radio show in Atlanta on 99x called the Morning X with Barnes, Leslie, and Jimmy.  On that morning, November 10, 1999, Jimmy was laughing about a news story that he couldn’t wait to share.  As he was reading, I was scrambling to write it all down!  The story went like this:

Earlier this morning a man held up a GA-400 toll booth.  His stolen getaway car broke down and he is now on the run with a 58 lb. bag of quarters.

When I got to school, I turned on my computer, printed the story out on a transparency with a picture of some quarters and put it on the overhead.  Here’s a sample of what happened:

Several students as they entered the classroom:  Mr. W., what’s that on the overhead?

Me:  I heard that on the radio this morning and wanted to know what you all thought and if you had any questions.

Multiple student responses:  “Oh, ok.”   “That guy is stupid.”   “What kinds of questions do you want?

Me:  Whatever questions come to mind.  You can write your thoughts and questions in your journal.

What I got from these 5th grade students at the beginning of class amazed me.  They were totally engaged in the problem.  The problem context had them so curious, they wouldn’t let go.

Some of their questions:

• How many quarters is that?
• How much money is that?
• How tall would a stack of 58 lbs of quarters be?
• How far could you run with a 58 lb bag of quarters?
• How big is the bag of 58 lbs of quarters?
• How long would a trail of 58 lbs of quarters be if they were laid end to end?

This one context from a morning radio show kept my students focused on the mathematical concepts of weight, length, decimal computation, and time for over a week.  More questions came up as new ones were answered.  They had developed not only a curiosity, but an intellectual need to know.

This is what I had been searching for.  A context that engaged my students in mathematics so deeply, that they wanted to figure out the answers to their own questions.

It wasn’t easy to find stories like this back then.  But now, they’re everywhere.  Just Google bizarre news stories. Since then I’ve learned, along with a whole host of others (check out some of the people I follow), that I can create these contexts using all sorts of media to get the same results (3-Act Tasks).

Below is a copy of the original context I used with my students.  The image has changed over the years, but it is essentially the same document.  And it works just as well today as it did 15 years ago!  I just wish I had a recording of the news story!  If you decide to use this, please share your experience.  I’d love to hear about it!

GA400 toll problem