# There is ALWAYS More Than One Way

When I was young, we were told that there was a specific way that we had to show our answers/thinking in math class. We could only show the way that was taught, and we were often times deducted points if we didn’t show the correct algorithm or method. I think that did a great disservice to our generation of mathematicians. I think it limited us in our thinking, and taught us that we were either good or bad at math. There are times that I think math is more about finding creative solutions, than knowing the “right” way.

Take this classic example that happened in my class last week!

We are working on multiplication and division right now, but there is a group of 5 students who pre-tested right out of this topic.  Their knowledge at the beginning stages of multiplication and division was so far beyond the rest of the class, that I have them working independently on my On Stage! Holiday Concert Performance Task.  In this project they are charged with planning a whole entire holiday concert for their classroom.  It is really fun to watch them work through each of the tasks in the problem. I schedule in a time to meet with them daily, and loved this moment I had last week.

Pictured below is one step of the project (there are 6 total). It is my sample invite that I pull out if they are struggling to understand that particular part of the task.  In this part of the project they are asked to find half of an 8.5″ by 11″ piece of paper, draw the rectangle on the given paper and design an invitation for the concert.

I asked the students to describe the way they figured out the “half size” rectangle. I was, and always am, amazed at the many different ways they found it:

• Method 1:  “I just folded a piece of paper, measured each side, and then drew those measurements on the paper.”
• Method 2:  “I figured out that half of 8.5 is 4.25, and that half of 11 is 5.5, so I drew those lines on my paper.”
• Method 3:  “I had no idea how to figure this out, so I folded a piece of paper in half and traced it right on my paper. It seemed like the easiest way.” (LOVE the HONESTY here!!!)

We talked about how each way was just fine for each type of learner.  For someone who understand fractions/decimals, method 2 is great.  Perhaps you don’t have a clue how to use a ruler, then method 3 can work. We discussed that there was always more than one way, even if your way isn’t “perfectly mathematical”, we can then think about how we can do it mathematically the next time. The student who used method 3 became a sponge as she listened to how the other two students described the way they figured it out.

Allowing students to share their ideas allows them to see that there is ALWAYS more than one way!

# Break Those Numbers Apart!

I’m officially trading in my place value blocks for unifix cubes!

Today I was trying to teach a small group of students how to subtract with place value blocks. The problem came when I tried to show how to regroup! As soon as we traded a ten rod for ten ones, eyes glazed over and their minds left me. They couldn’t understand why we’d trade it in and pick up ten ones. I realized that is because it doesn’t really explain the concept, which is that we are breaking the ten into 10 ones, not trading. Instead I tried unifix cubes, which you can actually break apart.

So when I blasted apart a 10 pack of unifix cubes, the light bulbs switched on. The students were so excited to try it themselves, and then, when shown the algorithm next to it, I heard a lot of “OHHHH!!!”

It was the coolest thing. Here is how we did it.

Decide if you can subtract seven from two.  As the numbers stand, you cannot. So…take a ten, and show it on the number below.  (The three tens become two tens.)

Break that ten into ten tiny ones.

Those ten ones are placed next to the two that were already there.

Now we have 12 ones! We write it on the problem below to help us understand why we do that.

Now we’re all situated, and we can subtract.  When you take seven ones away…

You are left with five ones!

Then, you can take one ten from the two tens.

One ten and five ones remain!

The best part is, teaching this way solidifies MY math and number sense.  When I grew up, were were taught how to “borrow and carry” but I had absolutely NO idea why. I learned how to do this as an adult, and it was very difficult to learn. I find that when I understand math more deeply, it is easier to think of different ways to teach these concepts to students who struggle!

# Should We Ignore Them? (Tips for When Problem Solving Gets Tough)

Sometimes I feel like a magnet, with a trail of students behind me as I walk around to conference/help during work time.  We are working on Open Ended Word Problem Challenges right now (I have gone through set one in the first quarter, and we are beginning set two.) These problems include a lot of reading, are many steps, and are open ended.  There can be more than one right answer.

So they hit the panic button right away!

Right now, I am in the middle of training my students to trust themselves, to be okay with feeling a little uncomfortable. I want them to seek the answers to their problem WITHOUT me.  This is very hard for them, especially when we are working on challenging math concepts.

Here is what one of those problems looks like!

Here are some ways that I try to raise rigor, and to help students persevere:

1. Ignore them! (What? Are you kidding? How horrible!) Of course the kind of ignoring I am talking about, is the kind where they ask for your help without trying the problem first.  There is nothing worse than when you pass out a tough problem, and the hands go up immediately. This leads to my next tip, a very simple tip.

2.  Make sure the students read the problem three times. Read it once to get familiar, read it a second time to zoom in to what you need to do, then read it even closer a third time to circle key details. The answer to their question is almost always in the problem. Most times I’ll read it out loud!

3.  Encourage students to do what they can in the problem while they wait for help. Sitting there with a hand up, or following the teacher around, trains students that they must rely on the teacher to continue on. When I approach students my first question is always: “What parts did you understand?” They realize that they can do much more than they originally thought.

4.  I teach routines when solving problems. For example, my students cannot actually get up and follow me, rather they wait as I circulate so that everyone gets equal time. Sometimes I’ll have a schedule posted where I meet with small groups.  Knowing that they will all get equal time with me makes everyone relax (including me!).

Teach the students that an “I can do it!” attitude is the most powerful problem solving strategy!

# Tricks are NOT for Kids

Rounding is one of the worst things in the world to teach, right? It is classic every time.   Mention rounding…kids sigh, teachers tense up and everyone’s brains feel muddled. It is HARD to understand rounding when you don’t have strong number sense.

I used to teach every trick under the sun for rounding.  I tried the rounding mountain, I tried the rounding coaster. I even would break it down for kids and have them underline the place, circle the number next to it, check to see if it was 5 or higher.  There were SO MANY problems. Kids couldn’t figure out the place value, where to start, what to change, what to look for or even what the number was in the end.  A few students would understand it, and it would stay that way.

That was when I realized they weren’t understanding the concept behind the tricks. They couldn’t remember the rules of the tricks (not even when it was a rhyme, because they only memorized the rhyme and didn’t get how to use the trick).

So I threw all of it away. I took down all of my rounding roller coaster and rounding mountain anchor charts. I decided to start fresh. I told my students this year to FORGET everything they ever learned about rounding.

Then, we counted by tens.  Not just 10, 20, 30, 40, 50 60, but also 110, 120, 130, 140, 150.  We counted by hundreds in the same way. We talked about what nearest ten and hundred mean. We made number lines that counted by tens and hundreds. THEN, we began rounding numbers.

And this is what we came up with as a class:

It was so simple. If you are asked to round to the nearest ten, begin counting by tens by making a number line. Find the closer ten by checking how many spaces away it is from your number.  As we worked with more numbers, they uncovered the mystery of the 5.  “What do we do if it is in the middle?!” We round to the higher ten, because that is what the world decided to do so that everyone does it the same.

We could also round to the nearest ten in numbers in the hundreds, because we practiced counting by tens in the hundreds.

We can also round to the nearest hundred in this same simple way!

This whole entire thing was AMAZING.  After several days of practice students weren’t drawing out number lines anymore, they were able to picture them in their heads.

On my pre-test I had 5 students who could round to the nearest ten and hundred.  By the post-test every student except for ONE student could do it.  Only one! I have been able to sit down with that student each day as he comes into school, and he’s getting it now, too.

It made me realize that tricks really aren’t for kids. When students don’t understand the WHY behind the trick, the tricks don’t work. If they don’t understand the number sense behind rounding, no matter what number they underline or look at, it won’t be solid conceptual understanding, and the trick will get mixed up.

# Struggling to Learn Their Math Facts? One Way to Help

I know you have the student (or maybe you have many of them) that I am about to describe.

1. They count on their fingers as their only strategy to recall their math facts.
2. No matter how many ways they try to learn math facts, they come very slowly.
3. They are at least a grade level behind in math fact fluency.
4. They feel helpless, like they will never learn their facts.

There is this strange attitude with these types of students, as if they think they will never learn their facts. Their parents or siblings will say “I never really was very good at memorization, or learning my facts.” Those comments validate the student’s struggles and they feel like they, too, will never learn their facts.

Nonsense!

They CAN and they WILL learn their facts. They just need some good tools.

First and foremost we must teach strategies. There are many ways that students can interact with their facts. They can use ten frames, number lines, fact families, pattern work etc.  Time tests alone do not teach students their math facts, and we cannot rely solely on flashcards/memorization.  We must first saturate their minds with a variety of ways to work with and look at their math facts. That is when memorization works, after they’ve seen the fact so many different ways they are sick of looking at it. This works for the majority of students as they learn their math facts.

But for some students with learning difficulties, it STILL won’t work. THEN, we must boost children up by giving them intentional memorization strategies.  The strategy I am referring to is called incremental rehearsal.

It is adapted from a strategy that I learned over at Intervention Central.  Their strategy is amazing as well, but I found it was overwhelming for my struggling students to have so many facts involved (Instead of 10 but I’ve narrowed it down to 5 at a time).

There are a variety of ways I’ve used this strategy:

1. Parent volunteers come in to work 1 to 1 with my most needy students.
2. I’ve had peer tutors from upper grade levels come in to work with students in a math fact recess club. (Bring candy or treats, they’ll feel amazing to be a part of the “club”!)
3. I’ve worked with small groups in intervention block with specific facts.
4. If all else fails, I’ve sent home the facts to work on in baggies with the instruction page (see below).

Some students truly are going to take longer to learn the facts…but they WILL learn them.  Especially if we never give up on them.

Thanks to Charity Preston for the link up!

# Data Doesn’t Have To Be Overwhelming!

The idea of collecting data has really been getting a bad rap lately. Have you heard these comments (or comments like these) in your building?

• “We are drowning in data!”
• “All we do is test our kids.”
• “I feel like our kids are just numbers, like we are ignoring that they are PEOPLE.”
• “We collect all this data, then we have no time to analyze it and use it.”
• “I don’t have time to enter in all this data.”

I think we’ve probably all heard some version of this at some time or another. Some of these comments might actually be true in some districts. I’ve heard horror stories about schools that are doing so much test prep, that they really aren’t finding time to intervene and help their students when they struggle. I feel lucky to work in a district that believes firmly that our data should drive instruction, and if it doesn’t, we shouldn’t collect it.

I had an amazing moment about 5 minutes ago. (Yes, I was working on a Saturday night, not totally uncommon around here!) I was looking at some problem solving we did for report card purposes/parent teacher conferences (THAT explains why I’m working on a Saturday night), and I noticed something amazing.

I use multiplication and division word problems in my classroom about 3 days out of the week. I am a firm believer that students need simple problems to try out before diving into difficult and complex ones. I use Practice Problems for Multiplication and Division because according to the Common Core these students must have multiplication and division mastered by the end of grade 3.  There are also 9 word problem types that are broken down in the common core for students to master in grade 3 so why not teach these things together?

Instead of just diving in and giving everyone all the problems in the entire booklet, I first assess each student on each problem type by giving them the first one. Then, I set up a spreadsheet to look at my results.  I noticed that my class was really struggling with the Type 2 problem, only 8 students got this problem correct. (I score this according to a standards based grading scale, it could also be scored pass/fail.)

My chart:

So we practiced this problem type. We tried this type of problem five times, each time giving students a chance to come up to the chalkboard to explain their thinking.  Awesome strategies were shared, and students asked many questions to see how they solved it.

The student on the left has pretty good knowledge of multiplication, while the student on the right is just beginning. Both strategies are successful.

Now we fast forward to tonight.  For parent teacher conferences, I decided I wanted a sample problem solving exemplar to show to parents (it also made sense to have the latest info for report cards!). Naturally I choose this same problem type, since we’ve worked so hard on it. I just finished scoring them, and noticed that 21 out of 25 students got it right this time! I started yelling to my husband (who probably thought I was crazy) that I was so proud and excited for my students.

It really DOES work.  Using data to target instruction is a much more focused way of going about planning instruction.  I haven’t wasted any time on problems that students already knew, and now I know exactly who needs a little intervention work with me! The best part is, this entire process was so simple.  (Instructions are included in the Practice Problems for Multiplication and Division resource. You could also set this up with your own problem types!)

I can’t wait to share this awesome news with my class. I am hoping celebrating our success will motivate them to continue to work hard.

# When Differentiation Feels Impossible

I know that differentiation is SO important.  I know it is the right thing to do. But sometimes it is SO difficult to make sure I am meeting the needs of all learners.

Right now we’re plugging away with the concept of regrouping when adding 3 digit numbers.

There is a group of 6 students in my class who’ve gone BEYOND mastering that concept. I know that I can’t spend an extra day with their bored faces staring up at me.  I just can’t take it, and I know in my heart it isn’t right to continue whole group teaching when they need the support of a higher level thinking challenge.

So…meet Open Ended Problems! I’ve used them before when students have mastered content, these problems get them thinking and are REALLY challenging and complex. Great, right?

Well today something went wrong. In a perfect world, I’m instructing the students who are struggling with adding/regrouping as a whole group, while the other students are working on the open ended challenge at their desk. That didn’t happen today. The students who were struggling with adding/regrouping were frustrated…and the students working with the open ended challenges were frustrated. Everyone in the room at some point had red cheeks, tears in their eyes at times, scrunched up faces, and a general lack of unease.

Why the frustration?  What was happening?! I realized that while I was working with my group, we were being interrupted by the students who were working on the open ended challenges.  They weren’t coming up all at once, rather, they were coming up one at a time, every minute or so, interrupting the thought process of the group. They weren’t interrupting to be rude, they simply ran into a problem while they were working, and came to ask me what to do.

That was when it hit me that we are sorely lacking in the Math Practice Standard 1: I can make sense of problems and persevere in solving them.

Instead of making sense of the problem, those 6 student’s first course of action were to come straight to the teacher. Not used to feeling challenged, they were “stuck” because they are used to answers coming quickly.  They weren’t used to having to read the problem more than once, and didn’t even realize that their questions that they asked were answered RIGHT in the problem they hand in their hands.

After about 5 minutes of these interruptions, I explained to the group of students working on the challenges that they would be meeting with me in 15 minutes.  Once they knew this was coming, the tension in the room released. ALL of the students began to breathe and feel more comfortable.  That simple gesture of letting them know that they’d get their time with me was the solution.  The students in front of me relaxed, the students at their desks relaxed and we were able to go on with our work. By the time I got to that group of 6 students, most of them had figured out the answers to the questions they had!

Sometimes, differentiation feels impossible. It is difficult, but it is RIGHT.  Everyone in the room had a challenging task, and that is how it should be.  Now, I’ve just got to reteach and work on my expectations and routines for small group work.

Differentiation looks different in all subject areas, but I find it to be the MOST challenging during math.  How do you differentiate and hold students accountable? Do you do centers, or a math workshop style? I would love to hear suggestions, comments, questions and more!  The more we share the more we learn.

# Problem Solvers Aren’t Born, They Are MADE

My first year of teaching, I was the queen of teaching problem solving. I would stand up in front of my students each day, and show them my beautiful strategies for solving the problems. Over and over, they would see my drawings, my number sentences and my solutions. I would ask them to copy them down if they couldn’t figure it out, so that they would have an idea for the next time. As I’d look at my data, I would notice that I had a top group of problem solvers who could always solve it, a big group of solvers that would typically get the problem correct, and a group at the bottom that would NEVER get the problem right.

I was foolish enough to believe that this was okay. I thought that some kids just weren’t very good at problem solving. I was SO wrong, and I am SO embarrassed to admit this now.

My second year of teaching, I heard this quote: “The person doing the talking is the person doing the learning.” I honestly felt sick to my stomach, because I realized that I was doing WAY too much of the solving, working and showing. I needed my students to take ownership, stand up, share their thinking in kid speak and start to GROW. I learned a lot that year, that students aren’t born to problem solve. It is something that requires an immense amount of practice.

I’ve come a long way since then, and would like to describe what I have done to be SURE that all of my students are getting this problem solving thing down before they leave my classroom. First of all, we take TWENTY minutes per day, every day to practice problem solving. This is something that is a priority during my math block. Then, I follow the steps below (UGH, I realize this looks like a TpT commercial, and I don’t mean it to be! You can do all of these things with your own resources.):

1.  I assess what problem types the students can solve. Did you know that there are 9 problem types for multiplication and division in the common core?  I use a series of multiplication and division problems, administering the first in the set to see how they do on each of the 9 problem types.  I compile the results in a data table to see which ones the class struggles with as a whole.  Then, we attack those problems one by one throughout the school year. I assess them again at the end with the last problem in the set to measure growth.

2.  I introduce the Standards for Mathematical Practice. These standards are SO important for students to develop as math habits.  They cannot be stressed enough. It takes us about a week and a half to get through them all, but it is worth the time. I post our work daily on the wall. The vocabulary from these standards becomes a part of our every day language.

3.  I start small, with practice problems that are simple that they can relate to. Then, we go BIG.  I have three types of problems that I use juggle through and use.

Simple Problems:  There are so many problems out there about trains arriving and leaving on time, or other topics that students cannot connect to.  I finally broke down and created problems over the years that would allow for practice of multiplication and division concepts. The problems are about things that students can understand. These are done on most days, with other problem types sprinkled in from my current math series.  These simple problems are NOT done every day.  That is not enough for students to become strong problem solvers.

Open Ended Problems: These problems require more reading, more steps and are much more complex. There are times that these problems require two 20 minute class periods to complete. These are the types of problems we will find on the Smarter Balanced Assessment next year.

Here is an example of an open ended problem:

Project Problems: These are always my student’s favorite type of problem.  They spend several days on these problems and are a bit more out of the box. I always begin with the Book Order Proposal and go from there.

4.  I make manipulatives available to them from the start, and I encourage their use.  The idea that hands on problem solving is for young students only, or for struggling problem solvers is incorrect.  Manipulatives are wonderful for any level of problem solver, it promotes deep thinking of the math concept you are working on.

5.  I allow students to model their thinking in front of the class.  More about how I do this you can find in this post. They solve it, explain it to the class and accept questions and compliments from the rest of the students.  This is where the students do the talking, the questioning, the complimenting. They are seeing multiple strategies each day, they can “steal” ideas from each other and are held accountable for their work.  I keep a tally chart right on the chalkboard so that students can see how many times everyone has been up. We try to make it equal, even though problem solving comes more naturally to some than others. This helps everyone know that they are ALL welcome up to the board, even if their solution is wrong.

6.  I intentionally plan out which problems to do and when.  I carefully monitor my students to be sure we are using our time effectively. I watch to see how we do as a class as we solve the problems.  When the majority of the class is getting the problem type, I’ll switch to a similar problem type that requires a tiny adjustment in their thinking. I always incorporate a problem from the book that has to do with the concept we are studying from time to time as well. A two week plan might look like this (and it is always flexible):

• Day 1: Equal Groups (Unknown Product)
• Day 2: Problem from math series covering current concept.
• Day 3: Equal Groups (Unknown Product)
• Day 4: Equal Groups (Number of Groups Unknown)
• Day 5: Problem from math series covering current concept.
• Day 6: Equal Groups (Number of Groups Unknown)
• Day 7: Equal Groups (Number of Groups Unknown)
• Day 8: Equal Groups (Unknown Product)
• Day 9: Open Ended Problem – Day 1 of 2 (complex, many steps)
• Day 10: Open Ended Problem – Day 2 of 2 (complex, many steps)

7.  I keep accurate records for myself. I have a class list so that I can see when students are getting the problem correct.  I keep the problem as our daily focus until 90-95% of the class has mastered it.  I have an answer key that allows me to check off when we’ve done the practice problems so that I don’t accidentally repeat the same problem.

8.  I intervene with students when the problem type is a struggle.  I pull small groups during our math work time, in the morning when students come in, during recess, during our intervention block time, whenever I can to get those students up to speed.  Many times they just need more one on one support to be successful.  I don’t wait any more for them to figure it out on their own. I intervene as soon as I notice the struggles.

It sounds like a lot, but once we get in the groove, and routines are in place things get ROCKING!  I didn’t realize how much students love this process until we had a substitute teacher in for a day.  The teacher worked the problem out on the board much to the anger of my students! The next day, they were SO fired up and upset that she didn’t give them time to work it out on their own.  That is when I knew that the students in my classroom were finally owning their own learning.

# Just Make it Real World

I am not sure if you’ve ever had that “moment”.  The moment where you are at a frustration level with why things aren’t working. I used to look for extra worksheets to give more time for tricky math concepts to “stick” with students. I looked online for further practice activities, I asked colleagues for their extra resources for more practice, I looked for games. I felt like I’d tried everything.  That’s when I read some research that making math real world, connecting it to student’s lives was REALLY good practice.  So a few years ago I started to create real world problem solving projects to help this problem.

That was how the Book Order Proposal project started (for my gifted and talented students I’ve used The Housing Market Analysis). I knew that I needed to continually reinforce the concept of rounding/estimation, comparing numbers, and mental math addition strategies. I gave my students the chance to do just that by offering to buy books for our classroom library. I decided to coincide this project with my parent Scholastic Book Club order.  Here is how it worked:

• I made it my problem of the day for 4 consecutive days, giving 20 minutes each day for the project.  The first three were days for them to work (with a mini lesson or two if needed), and the fourth day was the peer review day.
• All students were given a budget of \$50 (bonus points offset this cost-I was able to get all of our books free this last round) to look through three Scholastic flyers.
• The students had to put together a proposal, thinking about their classmate’s reading interests, as well as thinking of what we currently have in our classroom library.

What happened was kind of interesting. The majority of the students got within \$2 of the \$50 budget.  A few of the students tried to hand in proposals that were \$1, \$30 or \$25.  When we talked as a class on the second day, I asked my students if it was okay if someone didn’t get close to \$50.  The resounding answer was “NO!”.  When asked why, they explained that it would be a waste of money if they didn’t spend it all, especially since they would become THEIR books for their classroom.  I handed back those few papers and asked them to start again. (Now that is what we call peer accountability!)

At the end of the project we laid the papers out and did a gallery walk. Students voted on their top 3 favorite proposals. The proposal with the most votes actually got ordered!  It was such a fantastic way to end the project.

My favorite part though, the very best part of the entire project, was the gallery walk and natural reflection. Students could see how others choose to put the proposals together. Some were neat and organized, others were missing information, some of them had a hard time with their handwriting, and other student’s numbers didn’t quite add up.  It led to great discussion, and the students wrote goals on their proposals for the next time we have to present information to our peers.

It has been clear to me that making math real world, and connecting it to their own lives is a powerful thing!

# Purposeful Play: Explore Concepts Before You Name Them

In the beginning of my teaching days, I was terrified of manipulatives. I thought that students would mess around with them, blocks and chips would be flying through the air and no one would be learning. I’ve learned the last few years that the opposite is true. Manipulatives have become a way for students to both play and discover math concepts in my classroom.

Today the learning target was: I can discover addition rules.

In the middle of the floor I put out cups and two color counting chips. I showed them two cups with a few chips in each one and asked them how I could find out how many chips I had. I had 4 chips in one cup and 5 in the other. Many students scoffed (exclaimed “this is easy!”) and told me to count them. So to get at the properties of addition naturally, I asked how many different ways I could count them.  I asked them to write some number sentences in their math journals. The students came up with:

4 + 5

5 + 4

(3 + 2) +4

9 + 0 (They made me pour all the chips in one cup!)

It was awesome to see how they were thinking.  In just a minute or two, they figured out the commutative, associative and identity properties of addition. I walked around to each student, dropped counters into their cups, and asked them to record as many number sentences as they could with their chips.   I was able to differentiate easily by dropping a small number in the student’s cups who were struggling, and larger numbers in those gifted student’s cups. Some of them asked for 3 cups, others stuck with two. They were sorting and dumping the chips, writing number sentences as they went along.

After about 5 minutes I asked them to share some addition rules that they discovered.  They reported to me (in kid language):

• In every number sentence I always had the same sum.
• No matter how many ways you break the numbers you still get the same sum. (This is the associative property!)
• It didn’t matter which cup I added first, I always got the same sum. (This is the commutative property!)
• If I had a cup with zero in it, the other cup was the sum! (This is the identity property!)

Tomorrow I will name these things for them. We will learn that those exact “rules” they came up with are actually properties of addition, which can then translate to multiplication later in the year.

I now know that purposeful play is an important part of the learning process.  My goal is to integrate this more and more each day!