Teaching

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

 

What Math Teachers Can Learn from Magicians

Yeah, you read that right! I know many of you are now probably thinking about at least one, or likely, a combination of these questions:

  1. What could math teachers possibly have to learn from magicians?
    1. How could there be a connection between these two very different careers?
  2. How would Mike know?

Beginning with the last question probably makes the most sense.  At an early age I developed a fascination with magic, sleight of hand to be specific. Any magician I saw perform – either on TV or live – filled me with wonder. Certainly, some of that wonder was directed toward how the trick or illusion worked, but even beyond that I wondered how I could learn to create this wonder in others. Since I was about 10, I have studied magic and about 7 years later I began performing magic shows at schools, for church groups, and even for a few holiday parties. Once I began my career as a teacher, my role as a magician changed and I focused most of my energy on teaching.  I’ve lived the life of a magician and a teacher and over the last few years, and I’ve begun to notice the similarities between the two.

A magician’s goal is to entertain his or her audience while bringing about a sense of wonder. The means for accomplishing this goal involves the use of any combination of a number of tools including misdirection, psychology, sleight of hand, and story telling. If a magician does his or her job well, the feeling of being tricked doesn’t really enter into a spectator’s mind.  The big idea here is the creation of wonder.

wonder-bwf-quote

That’s the first thing teachers can learn!  It doesn’t take a sleight of hand artist to build a sense of wonder in students.  It takes some creativity and some work and dedication to the idea that all students deserve the chance to wonder and be curious.  All students need that sense of wonder that builds inside them and creates an intellectual need to know and learn.  

This is a great time to be a teacher of mathematics.  Evoking this wonder in students in math classes is extremely accessible because of technology and the online math community know as MTBoS. There are hundreds of math teachers out there at all grade levels and in all areas who have realized the power of making students wonder.  We’ve all been creating 3-Act Tasks and sharing ideas on blogs and webpages, twitter, and youtube or vimeo.  All for free.  They’re there for everyone to use – because we’ve all learned, through using these tasks, that it helps us build student curiosity, engages them in the mathematics and in their own learning, and it helps us build independent, creative mathematical thinkers. Here is more about why you should use 3-Act Tasks.

This brings me to the second thing we can learn from magicians: we can’t do this alone! If we work together, we all benefit!  Most people probably think that magicians are private wizards who lock themselves in a room to practice and never share their secrets.  That’s a bunch of crap! Magicians realized a long time ago that if they work together, they can work more efficiently and become more productive.  Sometimes magicians work on a trick for a while, get stuck and then bring it to some friends they have in the magic community. These other magicians share their ideas, they brainstorm, and try possible solutions.  Then they test the best solution on an audience.  This can be very scary!  Think about it.  This is a trick they’ve never tried – they’ve practiced (A LOT), and maybe even performed in front of small audiences. They must be nervous!  But they go out on stage or wherever their venue is and perform it.  They have to!  It’s how they pay their bills.  Often, some of their friends who helped them are there to provide feedback.  After several performances, and feedback, the script has been adjusted and the magic has been perfected, and it becomes a part of the magician’s repertoire.

Now think about how many math teachers still work. . . alone, in their room, not sharing their ideas.  Magicians realized this was not very productive a long time ago.  Other professions did the same.  It’s time math teachers realize this too!

Take a look at the MTBoS, and see what you think.  Look at some of the sites below and see if you find something you like.  Try some ideas/lessons with your students.  It’ll be a bit scary in the beginning, but soon it’ll become part of your repertoire!  We’re all here to learn from one another because “All of us are smarter than one of us!” ~ Turtle Toms 

What I’ve learned through this whole process is that I get the same feeling of success when I create the sense of wonder in students as I did as a magician creating wonder in an audience. . . but it’s even better with students!

Why use 3-Act Tasks?

The short answer:  It’s what’s best for kids!

If you want more, read on:

The need for students to make sense of problems can be addressed through tasks like these.  The challenge for teachers is, to quote Dan Meyer, “be less helpful.”  (To clarify, being less helpful means to first allow students to generate questions they have about the picture or video they see in the first act, then give them information as they ask for it in act 2.)  Less helpful does not mean give these tasks to students blindly, without support of any kind!

This entire process will likely cause some anxiety (for all).  When jumping into 3-Act tasks for the first (second, third, . . .) time, students may not generate the suggested question.  As a matter of fact, in this task about proportions and scale, students may ask many questions that are curious questions, but have nothing to do with the mathematics you want them to investigate.  One question might be “How is that ball moving by itself?”  It’s important to record these and all other questions generated by students.  This validates students’ ideas.  Over time, students will become accustomed to the routine of 3-act tasks and come to appreciate that there are certain kinds of mathematically answerable questions – most often related to quantity or measurement.

These kinds of tasks take time, practice and patience.  When presented with options to use problems like this with students, the easy thing for teachers to do is to set them aside for any number of “reasons.”  I’ve highlighted a few common “reasons” below with my commentary (in blue):

  • This will take too long.  I have a lot of content to cover.  (Teaching students to think and reason is embedded in mathematical content at all levels – how can you not take this time)
  • They need to be taught the skills first, then maybe I’ll try it.  (An important part of learning mathematics lies in productive struggle and learning to persevere [SMP 1].  What better way to discern what students know and are able to do than with a mathematical context [problem] that lets them show you, based on the knowledge they already have – prior to any new information. To quote John Van de Walle, “Believe in kids and they will, flat out, amaze you!”)
  • My students can’t do this.  (Remember, whether you think they can or they can’t, you’re right!)  (Also, this expectation of students persevering and solving problems is in every state’s standards – and was there even before common core!)
  • I’m giving up some control.  (Yes, and this is a bit scary.  You’re empowering students to think and take charge of their learning.  So, what can you do to make this less scary?  Do what we expect students to do:  
    • Persevere.  Keep trying these  and other open problems.  Take note of what’s working and focus on it!
    • Talk with a colleague (work with a partner).  Find that critical friend at school, another school, online. . .
    • Question (use #MTBoS on Twitter, or blogs, or Google 3-act tasks).  
    • Write a comment below. 🙂

The benefits of students learning to question, persevere, problem solve, and reason mathematically far outweigh any of the reasons (read excuses) above.  The time spent up front, teaching through tasks such as these and other open problems creates a huge pay-off later on.  However, it is important to note, that the problems themselves are worth nothing without teachers setting the expectation that students:  question, persevere, problem solve, and reason mathematically on a daily basis.  Expecting these from students, and facilitating the training of how to do this consistently and with fidelity is principal to success for both students and teachers.

Yes, all of this takes time.  For most of my classes, mid to late September (we start school at the beginning of August) is when students start to become comfortable with what problem solving really is.  It’s not word problems – mostly. It’s not the problem set you do after the skill practice in the textbook.  Problem solving is what you do when you don’t know what to do!  This is difficult to teach kids and it does take time.  But it is worth it!  More on this in a future blog!

Tips:

One strategy I’ve found that really helps students generate questions is to allow them to talk to their peers about what they notice and wonder first (Act 1).  Students of all ages will be more likely to share once they have shared and tested their ideas with their peers.  This does take time.  As you do more of these types of problems, students will become familiar with the format and their comfort level may allow you to cut the amount of peer sharing time down before group sharing.

What do you do if they don’t generate the question suggested?  Well, there are several ways that this can be handled.  If students generate a similar question, use it.  Allowing students to struggle through their question and ask for information is one of the big ideas here.  Sometimes, students realize that they may need to solve a different problem before they can actually find what they want.  If students are way off, in their questions, teachers can direct students, carefully, by saying something like:  “You all have generated some interesting questions.  I’m not sure how many we can answer in this class.  Do you think there’s a question we could find that would allow us to use our knowledge of mathematics to find the answer to (insert quantity or measurement)?”  Or, if they are really struggling, you can, again carefully, say “You know, I gave this problem to a class last year (or class, period, etc) and they asked (insert something similar to the suggested question here).  What do you think about that?”  Be sure to allow students to share their thoughts.

After solving the main question, if there are other questions that have been generated by students, it’s important to allow students to investigate these as well.  Investigating these additional questions validates students’ ideas and questions and builds a trusting, collaborative learning relationship between students and the teacher.

Overall, we’re trying to help our students mathematize their world.  We’re best able to do that when we use situations that are relevant (no dog bandanas, please), engaging (create an intellectual need to know), and perplexing .  If we continue to use textbook type problems that are too helpful, uninteresting, and let’s face it, perplexing in all the wrong ways, we’re not doing what’s best for kids; we’re training them to not be curious, not think, and worst of all . . . dislike math.

More Resources like 3-Act Tasks:

 

 

 

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!

 

Moving Decimals!?!?!?

Why are there teachers out there still teaching multiplication (and division) with decimals where the decimal is moving?

The answer to this question is simple.  It’s easy.  It doesn’t take as long to teach (though when you look at all of the time spent on remediation, I tend to disagree here).  And it takes little preparation.

Let’s take a look at this.  First:  It’s easy for teachers.  It is!  I agree.  If the teacher has this procedural understanding down, all they need is to find a set of computations (usually in the form of a worksheet with no context) Unfortunately, this procedural understanding breeds more procedural understanding and neglects the sense making necessary in learning mathematics.  Teaching  any mathematical procedures at the expense of making sense is like teaching only phonics with no connection to literature and comprehension.

By teaching moving the decimal, teachers are undoing any understanding of place value (and this is often surface understanding).  Think about it.  In first grade, students learn that when you get too many (10) popsicle sticks they need to get grouped together to make one ten.  This requires sense making on the students’ part.  The students are beginning to think of the group of ten as a unit.  The “ten” is a unit and they can work with that unit in much the same way as they work with a “one.”  In terms of place value understanding, the physical grouping and the representation on a place value chart help students make the connection between the digits and the values of those digits due the quantities of popsicle sticks (or any other material).  The digits are moved to a different place value based on the quantity.  Quantities connected to groupings connected to place value.  It makes sense to students when they experience it consistently.

  • Two popsicle sticks are represented by a digit 2 on a place value chart
  • When we get to 20 popsicle sticks (10 times as much as 2), that digit 2 that was in the ones place is now moved to the tens place.  The digits are placed based on the quantity they represent.

Flash forward to 5th grade (for example).  A student is learning to multiply decimals and the teacher is teaching procedural methods where students are told to move the decimal.  What if the student gets the incorrect product?  Do they know?  Are they aware that their computation is off?  Most likely not.  They have been taught to follow procedures (often blindly) and if they do, they’ll get the correct product.  So, when they do make an error, they are not concerned, because they’re being taught to be robots.  Follow these steps and you’ll get the right answer.  Here’s how it might sound in a classroom:

Problem on the board:  10.030 x 0.03

Teacher to student:  You made a mistake.

Student:  (answer 0.03009) But I followed the steps.

Teacher:  You made a mistake.  Please check your work.

Student:  Ok. (after a few minutes) I got the same thing.  I checked my steps.

Teacher:  Did you check your multiplication?  Maybe your error is in the facts.

Student:  Yes.  I checked the multiplication – all of my facts were correct.  I don’t know what I did wrong.

Teacher:  Let me see. (a few minutes pass) Right here.  Your decimal is in the wrong place.

Student:  But I counted the places and counted back.  Why did I get the wrong answer?

Teacher:  The answer is 0.3009.

Student:  But if you count the decimal places, the decimal should go 5 places back, not 4.

Teacher:  Hmm?  Thank you for bringing this to my attention.  I’ll take a look at it. . .

This scenario is very informative.   First, it’s obvious that no one in this situation “owns the math.”  The teacher is trying to be the owner, and in the student’s mind, it may be the case – as soon as the teacher says, “let me see.”  The student is trying to make sense (once the teacher corrects him), but can’t and doesn’t even know where to begin, due to the limiting procedural understanding in place.  Based on the “rules” the student learned, he is correct. So why is the answer incorrect?

One thing that the teacher did well is admit that he doesn’t know and that he wants to try to make sense of the situation, but that’s really just the beginning.  Students should also make sense of why the rule fails here.  Instead of blindly following rules, students should be estimating and using what they know to make sense (about 10 x 0.03 = 0.3 so my answer should be really close to 3 tenths).  This should be a part of every student’s math day.  It can’t just be told to students.  They need to experience the value of estimation through problem solving situations on a daily basis.  Over time, students adopt this valuable strategy and use it readily in multiple situations.

Learning (and teaching) mathematics is about making sense, not just procedures. There’s no better time to start than the present!