Teaching

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:

IMG_0940

 

 

 

 

 

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.

Shared Student Work

 

 

 

 

 

 

 

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:

Student Feedback 1 Student Feedback 2 Student Feedback 3 Student Feedback 4 Student Feedback 5 Student Feedback 6 Student Feedback 7 Student Feedback 8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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.

Feed the Hungry

Kim, my beautiful bride of 16 ½ years, does not like for me to go grocery shopping on my own.  Recently she had a procedure and when I brought her home to rest I told her I would do the grocery shopping while she rested and no one would have to go the following day.  I was surprised by the fact that she was totally against this idea.  When I asked her why she wanted to go with me, she told me that I get hungry when I shop and buy a bunch of things that are not on the list, so if I was going shopping, she had to go too.  We both ended up going later that day.

While I was a bit hurt by her reasoning, I couldn’t deny it.  She was absolutely right.  I get hungry when I shop.  Lots of food, free samples, items I like on sale, items I’ve never tried not on sale, eye catching packaging. . . I can’ help myself!  And there’s no pattern to my binge shopping.  It just depends on the aisles, the samples, and my cravings. This got me to thinking if everyone does this.  I think so.

So what does this have to do with math teaching?

Bear with me for a bit.  I started a new position in January as Math Teacher on Special Assignment for our district.  My focus is working with middle and high schools.  I’ve taught elementary school, middle school, high school and even some college courses for pre-service teachers, so I’m comfortable working with students at all of these levels.  But when I started I just wasn’t sure how teachers would react to the support I was offering.  Would they want feedback?  Would they want support in their planning?  Would they want a model lesson or to co-teach a lesson?

As I began my work with these teachers I thought about the different kinds of teachers I would encounter as I move from school to school.  I determined, through my interactions with many that teachers seem to fit into one of three categories:

The bottomless pit.  These teachers are hungry all the time!  They ask for feedback, and resources, plan for co-teaching lessons, conference, and do just about anything asked of them.

The nibbler.  These teachers are willing to take a taste, if it’s not too spicy or too bland.  They want new strategies and will try something if they can immediately see how it will fit within their classroom without dropping something that’s “tried and true”.  If they don’t shop in that aisle, they sure won’t take a taste.

The Pepto-Bismols.  These teachers just ate a three course meal with desert and coffee.  The only thing these teachers might want is an antacid.

At first glance, you may choose one to work with over another, but read on.  There’s more to these categories than meets the eye.

  • The bottomless pits are always eating, but they may be eating things that lead them away from the aisles containing the foods for best practice. So it’s my job to make sure to steer these teachers down the aisle for the food that they need and  They may be devouring number talks, but they may be giving speed tests.  They may be sitting down to the table for a 3-Act Task, but they may not be letting their students come to the table of wonder to eat some for themselves.  Tricky stuff here.  We all have our favorite junk foods, but if it’s all we’re eating we are going to have a lot of problems down the line.

Grocery Aisle 1Grocery Aisle 0

  • The nibblers are kind of tricky. They’re a bit pickier about the food they eat.  Most of the time, these teachers just need an alternative, something that might replace what they’re currently eating.  Like the bottomless pits, these teachers could be in the wrong aisle and nibbling just because it’s easier than walking to the next aisle – even when the food over there is SO much better!
  • The Pepto-Bismols are my favorites. These are the ones who think they cannot possibly eat another bite.  They don’t think they’re still hungry, but deep down they are still craving.  They’ve eaten and their plates are still full.  It’s time to steer these teachers toward the pharmacy aisles.  If we can ease the bloating (often caused by lack of standards-based diet), maybe we can slip in a small piece of gourmet math food here and there (Open Middle anyone?).  When they realize their hunger it isn’t long before these teachers are feeding others!

PeptoUltimately, we all need to help our peers in this global math grocery store realize that they are hungry, feed them a little bit of the math goodness, then teach them how to shop for themselves.  You see, all of us are hungry.  Some of us are just walking down the wrong grocery aisles.

I’m still not allowed to grocery shop alone and I’m ok with that – as long as I can be a Mathmart associate, it’s all good!

Mathmart

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.

keep calm

  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.

Click here and here to learn more about strategy development.  Great stuff from www.nzmaths.co.nz!

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:

Top of Box

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!

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:

Penny Cube Notices

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):

Penny Cube 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 2 inches

How many pennies in an inch

How many pennies in an inch

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

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

How many pennies cover base

How many pennies cover base

How many stacked pennies in 2 inches?

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:

20140919145046-3351215

 

 

 

 

 

 

 

 

 

 

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.

20140916143126-1706895 20140916143218-1880868 20140916144859-1652302 20140916143241-1966871 20140916143157-1917874

 

 

 

 

 

 

 

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!

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:

GA400_Screenshot

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