Problem Solving

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_

 

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!

Relevant Decimals Lesson

This is a lesson that I tried with a 5th grade class to give a context to decimal addition and subtraction. Most of the math problems I’ve found involving decimal computation seem “artificial.” They have a “real world” connection, but the connections are irrelevant to most 5th graders. In order to make the connections more relevant (as Dan Meyer posted in a recent blog: students want to solve it) I came up with a context for a problem that had the math content embedded, but also involved the students in the problem itself. Credit for this lesson needs to go to a 3-5 EBD class at my school. The students in this class about 3 yrs ago, loved to make tops out of connecting cubes. They did this because they were told that they couldn’t bring in any toys to class (Bey Blade was the hot toy at the time). Since they couldn’t bring in these spinning, battle tops, they created their own with connecting cubes.

The first time I witnessed these students spinning their tops, the big question they wanted to know, was whose top spun the longest. I filed the idea away until about a week ago when some 5th grade teachers at my school asked for some help with decimals. The following is the lesson I used – thanks to this class of students. It’s written as it was done. I know what I’d change when I do it again. Please take a look. Use it if you like. I’d love to hear about your results and how you change it to make it better!

Standards:

5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

5.NBT.3 Read, write, and compare decimals to thousandths.

a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Materials:

Connecting cubes

Decimats, or Base-ten manipulatives for modeling

Stopwatches (we used an online stopwatch that measured to thousandths of a second)

Opening:

Give students a copy of the decimat and ask what decimals might be represented. Follow up with these possible questions: What might hundredths or thousandths look like? How could you use this to model 0.013? 0.13? Share your thoughts with your partner/team?

Work Session:

The task is to design a spinning top, using connecting cubes, that will spin for as long as possible. Your group may want to design 2 or 3 tops, then choose the best from those designs. Once a design is chosen, students will spin their top and time how long it spins using a stopwatch. Each group will do this 4 times. Students should cross out the lowest time. Students will then use models and equations to show the total time for the top three spins. Students will show, on an empty number line, where the total time for their three spins lies. Students must justify their placement of this number on a number line.

Here is a sample top (thanks for asking for this Ivy!)

Top

Closing:

Students present their tops and their data, then compare their results.

Possible discussion questions:

Whose top spun the longest?

How do you know?

How much longer did the longest spinning top spin than the second longest spinning top?

Show your thinking using a model.

How many of you would change your design to make it spin longer?

How would you change it?

Decimat model 2

Students used models to explain their thinking to each other and construct viable arguments.

Students used models to explain their thinking to each other and construct viable arguments.

Real Math Homework and Real Learning

Had a great night the other night with my (almost) 14 year old son. Connor had some math homework (Pythagorean Theorem worksheet from an outdated math series) and I was just looking over his work, making sure he understood the concept. He was coming along ok, I guess, so he finished up and sat on the couch to veg. for a bit.

Now, at first I thought what happened next was fate, but the more I think about it my subconscious probably took over. I checked my email and saw one from earlier in the week that I wanted to look at. It was from Dan Meyer’s blog and had a couple of links that I wanted to check out. After about 20 minutes of looking at some stuff I hadn’t seen before– including Estimation 180 (great site by the way), I stumbled upon Dan Meyer’s Taco Cart Problem again and began to grin.

Since I was on the couch with Connor by now, I showed him the video. When it ended abruptly, he said, “THAT’s IT!” I asked what was wrong. He said, “I want to know who gets there first.” We started to talk about it and maybe 20 minutes passed.  After realizing this was difficult to do with the limited resources we had on the couch, he asked, “Can we go sit at the table and work this out?  These numbers are too big.” I said, “SURE!” (but in a subdued voice so as not to sound “giddy” in front of my teen-aged son).

We sat there for a while talking about what he needed to know.  He knew he needed to know how far each person needed to travel, but didn’t make a connection about what he knew about the Pythagorean Theorem to solve the problem.  Yes, problem solving should be at the heart of every lesson!  He hadn’t been introduced to any ideas about distance and rate, but he knew he needed to know how fast they walked.  We talked about the relationship of distance, rate, and time and how these relationships can be use to find solutions to problems like this one.

After a little discussion and a lot of questions, Connor got to work.  He stumbled with some of the fraction “mechanics,” but with a little questioning, came through just fine.  Connor did more thinking during this task than I’ve witnessed him doing in a long time.  He was engaged from the start and he would not stop until he figured it out.  This is what students need to do all day in math class!

At the end, it was beautiful! He not only solved the problem, but when I asked where the cart should be for both people to get there at the same time, he was ready to go. He marked a new spot, and figured out the new distance. We had to set it aside, though, because it was getting late. He wants to see how close his placement of the taco cart is for the two guys to get there at the same time. We’ll be looking at it again over the weekend!

Kim, my beautiful bride, stopped at the table and asked what he was doing.  He told her, then she asked him if he finished his math extra credit (he doesn’t really need it – it’s just improving his grade, not really improving his understanding of mathematics). I found it a bit humorous because he was learning more doing this problem, than by doing the extra credit sheet of 19 naked equations.  Context and comprehension mean everything in mathematics!

Just before he went to bed, I asked him what he thought about the taco cart problem. He said he wished he got to do those kinds of problems at school instead of the “stupid problems he gets in class.”

Connor just recreated his thinking through the Taco Cart problem below using the Educreations App.  Enjoy!