Classroom Discussions

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

 

Understanding Decimals

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 meaningless to most 5th graders. In order to make the connections more meaningful, 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:

MCC.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.

MCC.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

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.

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?

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.

A Number Talks Reflection – A Look Back Over the Past 3 Years. . .

 Before reading this post, you may want to check out the GloSS and IKAN diagnostic math assessments from nzmaths at this website:   http://www.nzmaths.co.nz/mathematics-assessment.  These assessments are diagnostic interviews that teachers use with students.  When students respond with answers to the problems posed by the teacher, the teachers have to listen to students’ reasoning, not just the answer.  The information gathered is incredibly powerful and has driven our teachers to ask for resources and strategies that will help their students grow and progress through these developmental stages.

I introduced number talks to my school during pre-planning three years ago.  I read the book the previous summer and knew it would be a success if I could just get my teachers to try it.  The challenge for me was to find a reason for them to want to try this new thing called number talks in the midst of all of the other new initiatives.  I looked at what we had been doing over the past year and a half and tried to find where these number talks would fill a need.  When I discovered that need, several (more than I expected) teachers wanted me to introduce number talks with their classes immediately – during the first week of school!

 

The need I found was to improve strategies for computation to help students achieve higher strategy stages on the GloSS assessment.  Teachers had noticed that students were getting stuck on stage 4 (basically, the majority of students – even those in 5th grade – had one strategy for everything, counting on).  They were stuck because we continued to assess, but hadn’t looked at the data gathered from those assessments to come up with a course of action to help students.  The ideas were out there and we had discussed strategies before, but few teachers were implementing these ideas daily.  We wanted the pig to grow, but we were weighing it instead of feeding it!

 

When I started introducing the number talks, teachers were very interested and many were excited about out how this would work.  I worked with each teacher/class for an entire week.  For four days, I would model the number talks.  On the fifth day, the classroom teacher would take over and I would observe.  We would meet after to talk about the experience and we would discuss how the teacher would move forward from this point.  Sometimes these were difficult conversations.  What I learned from these discussions was that many teachers thought of this as a magic bullet, where teachers would talk about strategies first and then have students practice a few verbally.  This myth was dispelled as soon as I walked into the first classroom.

 

I introduced number talks to every class that year.  Some teachers wanted to see them, and then decide whether to use them.  Some knew they wanted to use them, and some just wanted 4 days with someone else teaching for 20 minutes.  And there was one skeptic, who did number talks with the expectation that they would not work.  And that was ok.  It wasn’t mandatory, just a strategy.  A tool to use to help kids help themselves.

Number Talks Assessment from 3rd grade with teacher commentary (September)

Number Talks Assessment from 3rd grade with teacher commentary (September)

I would check up on teachers every so often to see how teachers and students were doing with their number talks.  Some had stopped doing them after a while, some only did them 3-4 days a week, but there were some… Some who saw the value right away and did them religiously (I apologize for this blatant disregard of separation of church and state) every day.  These teachers took number talks and ran with them!  They not only used them to help students develop strategies, they used them to assess those strategies. They were asked to share. And they did.  During professional learning, faculty meetings, and through emails, other teachers began to notice that the number talks were beginning to show results.

 

Teachers were amazed, and so was I, when one month after introducing number talks to a third grade class, I walked in just to see what was happening and saw student after student mentally adding two three digit numbers using strategies based on place value, friendly numbers, and compensation.  These were a mixture of Special Ed., EIP, Title, and Gifted students.  They were all at different places in their understandings of the strategies they were hearing and using, but because they were developing the strategies, they were empowered to keep trying to use them and develop new strategies that were efficient (quick, easy to think about, and work every time).

 

The teachers who did the number talks consistently and with fidelity were the ones whose students reaped the rewards.  When the teachers assessed with the GloSS at the end of the year, those teachers were the ones tracking me down to tell me their stories.  I heard things like:

 

“All but two of my students went up 2 strategy stages.  The others went up 1.  It has to be the number talks.  That’s the only thing that really changed this year.”

 

“Number talks was a great way to really listen to my students and hear what they know.  The GloSS makes more sense now.”

 

“I can’t believe what my lower students said during the last GloSS assessment.  They really used what we did in those number talks.”

 

“Number talks really helped my kids with their strategies, and it shows in their other math work.  I love number talks!”

 

Number talks have been a huge success for all teachers at my school who have used them with fidelity.  We’ve hired some new teachers this year and they seem just as eager to learn about number talks as the teachers I worked with a few years ago.  Now, with all of this experience and several number talks experts, our school can offer more support than ever to these new teachers.  We’re all expecting the best.

 

Oh, and remember the skeptic. . . well, she’s one of the experts now!

 

Lesson Opening Takes Over

10/21/13

5th grade-Decimal understanding and comparing.

I went into the class to model a lesson where students use

models to understand and compare decimals. My opening

was an empty number line with 11 hash marks – zero on the

far left and 1 on the far right.

I asked students if they knew what any of the hash marks on

the number line should be labeled. Only a few students raised

their hands, so I asked the class to talk about  this at their

tables for a minute.

After a quick discussion, a boy was chosen to come to  the front.

I asked him to point to the hash mark on the number line that

he thought he knew the label for. He pointed to the  middle line.

What I would’ve done 15 years ago, is ask him what it should be

labeled and move on with the lesson.  Instead, I asked him to

whisper what the label for the hash mark should be.

I thanked him and asked all of the groups to focus on the middle

hash mark on the empty number line and see if they could agree

on what it should be labeled.

This teaching strategy never ceases to amaze me – and neither do the

students.  The conversations were incredible.  Just allowing students

to share their ideas with each other and try to make sense of numbers

(fractions and decimals) on a number line.

In the beginning of their discussions, most students thought 1/5 (the

same thing the boy whispered in my ear). Their reasoning was that

there were 11 hash marks and the middle one was the fifth one over.

It made “perfect incorrect sense.”  But I learned what misconceptions

were prevalent in the class.

As I talked with each group, students began to question their own

reasoning.  One group, while defending the idea of 1/5 said, “Yeah,

the fifth one over is in the middle and . . . well, it is in the middle, so

it could be 1/2.” This was my time to ask, “Can it be both 1/5 and

1/2? You have 90 seconds to discuss this and I’ll be right back.”

By the time I got back, they had decided it had to be 1/2, because

they “knew” that 1/2 and 1/5 weren’t the same.

When we came back as a whole group, many of the students had

shared that they had thought it was 1/5 at first, but many changed

their minds because of the idea of the hash mark being in the middle.

Many changed their minds to 1/2, but not all.  Some had decided that

since our standard was about decimals, the hash mark should be

labeled 5/10.  The next discussion lead to proving that 1/2 = 5/10.

Once students were comfortable with the decision that 1/2 = 5/10, I

asked them to label the hash mark to the left of 5/10.  The discussion

was quick and efficient.  They knew it was 4/10 because there were

ten “sections” on the number line (no longer 11 lines), and that hash

mark was the end of the 4th of the 10 sections.  They were thinking of

the number line as an equally divided line (fractions).

Finally, the students were asked to draw a number line in their journals,

like the one at the front of the room, and label all of the hash marks

with fraction and decimal notation.

It’s important to note that this opening to the lesson (that ended up

becoming the whole lesson) would not have been possible if the teacher

hadn’t developed group norms with the students at the beginning of

the year.  This class knows, after 9 weeks, how to talk to each other,

discuss their thinking, and work together toward a common goal.