# Build It = Deep Conceptual Understanding of Multiplication

Being a math interventionist is one of the hardest jobs I’ve ever had. It’s like a constant cycle of diagnose…teach…teach…teach…light bulb on…light bulb off…diagnose…teach…teach…teach…

I’ve been working with a group of fourth graders that were just struggling BIG TIME with multiplication. They weren’t able to see how a fact like 3×7 could help them solve 6×7 by just doubling. The numbers were too abstract, and they had nothing to connect it to.

Of course as I dug deeper I found that they simply had zero understanding of what 3×7 really means.  So as usual, I tried to find a way to connect it to real life, my favorite thing ever. I was trying to think of a way to help them remember the difference between rows and columns.  And then it occurred to me as I was waiting in line at the theater, no one wants to wait in line to sit in the first COLUMN for a popular movie. We all want the best seats in the house, so we wait to sit in the first ROW.

Enter the Movie Theater Multiplication Project.  I went home over the weekend, turned off all distractions and poured days and days into authoring this project. I needed to really think for myself what 3×7 represented.  I was not taught this way and I know that it doesn’t come easily to me.  I also had to think of a way that would be meaningful and that would STICK, since that seems to be the biggest challenge facing students in intervention. The last thing that I really needed to think about was the Concrete-Representational-Abstract instructional approach. Kids LOVE to build. The second they walk into my math lab, their hands are all over my cubes, blocks, counters, etc.  Starting with building means that they can usually connect a pictorial representation to it, and then connect that to numbers.

So I made a movie theater or two or three as I wrote the project. I made theaters that were 3 rows of 4 seats, making 12 seats total (3 x 4=12). I made tickets placing them in the correct row and seat number. And finally I made some mega theater designs so they could learn to use known facts to solve harder facts.  There is a fourth stage to the project also that involves some open ended challenges to calculate profits and revenue. Say hello to my little friends:

Seemed like it might actually work!

So I brought the idea to school. I got out some tools and some little tiny bears and THAT got their attention.  Tiny bears! Seriously, that is all it took?!

The first three days of the project were brutal.  They were making columns instead of rows, they were making rows of 20 instead of 20 seats total.  With probing questions they started to see what was happening.  And THEN, light bulbs turned on…and for several days now the light bulbs have stayed on! Is it sticking?! I hope so, and we’ll find out when they get to the mega theaters and can break down more difficult multiplication. Wish me luck!

If you want to try this out, in the preview you will find three parts of the four stages of the project that are free.  You could totally continue the project by giving them your own specifications!

# An Approach that Works for Struggling Learners EVERY Time

I have been reading about the Concrete-Representational-Abstract Sequence of Instruction for some time now, especially since I began working with our most struggling math students at our school.

I’m hooked and am a firm believer in this approach!

I know you know that moment… where you find students looking at you with the deer in headlights look.  In my intervention groups, I see it several times in 30 minutes! I was desperately searching for more ways to make math meaningful for them when I discovered this approach. And, I will tell you, it works EVERY time. I mean, EVERY SINGLE TIME. There has not been one single concept that I haven’t been able to master with a child when I used this approach.

If you don’t have time to read the article, the approach is summed up quite simply in three steps:

1. When a student is introduced to a new concept or something unfamiliar, you allow the use of tools. (Concrete)
2. When the student can perform the task, they move on to representing the concept with drawings or pictures. (Representational)
3. When the student can master the task with a drawing or a picture they move to using only numbers and symbols. (Abstract)

* Note it is important to keep all three of these ways visible to promote strong connections and deep conceptual understanding.

I realized that this could be even MORE powerful when students could self assess where they are in this approach. I made this poster with them and we refer to it constantly.

They are constantly checking “where their brains are at” when they are struggling through a problem.  When the numbers and symbols don’t make sense, they actually back themselves up to drawings. If that still doesn’t make sense they back up and use concrete tools.

It has been simply amazing, and you must try it!

# Small Steps for Differentiation: Same Task – Different Entry Points

When thinking of differentiation in the classroom, it is easy to fall into the trap of putting pressure on ourselves to perfectly level activities for every student. My mind goes to having a rotated set of groups and centers all perfectly ready to go. In this scenario you never run out of time, every student is exactly where they need to be, AND they are accountable, focused and staying on task the entire time.

YEAH RIGHT!

Don’t get me wrong, math workshops are a beautiful thing, but it doesn’t always work as smoothly as we’d like. It is okay to differentiate in small ways, taking small steps to be sure that we are meeting the needs of all children without going crazy ourselves.

This past summer I was lucky enough to read work by Timothy Kanold, and then I was able to work with him in a workshop as well. He proposes that instead of coming up with different activities for every student, we have the same task, but with different entry points. So what does this look like exactly?

Here is an example for a second grade classroom where the learning target would be “I can count money up to a specific amount.”(CCSS 2.MD.C.8):

Task: How many ways can you build 58 cents? Build it and record it.

All of the students in this group are given the same task, but usually all of the students have different levels of knowledge surrounding this task. So instead of coming up with 25 different activities you have only one.  As the activity begins and students begin to work, two things will happen which we all can predict every time. Some students will struggle, and others will fly.  This is when you strategically give certain students more.

For the students who struggle in this case you would lay down another task next to it, where the number is more accessible, and you may also consider telling them the value of all of the coins.

Instead of 58 cents, the students who struggle are working with a more accessible number, side by side with the other students.

In this case for the students that are excelling there are many options: ask them to find the solution if you eliminate one of the types of coins, ask them to show their thinking algebraically using a table, give a different amount and have them predict the number of solutions they may find before solving, or ask them to write a story in which you may need to come up with 58 cents worth of change.

The main thing is that you have to truly be walking your room, listening to your students and conferring with them as they solve. The BEST part of this method, is students are working together and hearing one another’s thinking, elevating the learning for all in the room.

# Release of Materials: More Small Steps for Differentiation

I struggled to understand a very basic thing when I taught third grade, and I see it’s importance all the time now that I am in many classrooms K-4.  Math is incredibly abstract for elementary minds, and the need to provide them with hands on tools and manipulatives is essential for conceptual understanding. So often we skip this step (or at least I did) because we as teachers are developmentally past needing this to understand numbers.  So often we skip this step (or at least I did) because tools are messy, noisy and the students play with them. So often we skip this step (or at least I did) because we run out of time to grab them, or it takes too much time to get them out and put them away. Despite all of this, we really need to allow students to have them so that we can see where they are developmentally.

Which leads me to another small step for differentiating in your classroom.

When introducing a concept, provide tools always to the students. Then, as you notice that some students need more of a challenge, ask them to try it without the materials to see if they can make the connection with just the numbers and symbols. For example, Make 8 Go Fish:

This is a great, easy game for K-1 students. They play Go Fish but ask for pairs that make a given number. In this case I chose 8 for a student that I am working one on one with right now. She struggled to find combinations that sum to 8, so we built them using ten frames before we started the game. Some students may need this support, while others may be ready to do it without ten frames or any tools at all.

Sometimes we force tools on all students, or we just don’t provide them at all to any of them.  Put it in your student’s hands by teaching them how to recognize that feeling of when they do or don’t need them.  Challenge them by releasing those materials when they no longer need them!

# Math is a Learning Subject: More Small Steps for Differentiation

My favorite thing about math is that is a messy thinking subject. It is a learning subject. It should be messy and full of questions. We need to teach kids that it can be glorious when it suddenly is no longer messy and the patterns and the discoveries are right in front of our faces!

We have to model this for students, and more importantly we need to give them opportunities to make math a learning subject. So often we want to give all the answers, and tell them all the patterns, and show them how magical it is, that they lose their passion for discovering math at an early age. They begin thinking that math is a performance subject…teacher asks the question, student gives the answer…25 times in a row…on a worksheet.

Instead we need to give students meaningful explorations that can often run in the background of the school day.  These can often be very simple, and they really allow for differentiation. Some students will take these explorations much further than others.

Here is a third grade example:

The keys to making this work are:

1. Give enough time for the exploration. This one will be 2 weeks.
2. DO NOT, and I really mean this, DO NOT give them the answers. (This is very difficult, I know.)
3. Tell them to work with each other! Isn’t that how we learn best? The second we want to know something we email, text or call someone. Let them teach each other.
4. Make them research it, prove it and let them feel some confusion. This teaches perseverance and also that math is truly a learning subject. Bring in iPads, computers or have them look it up at home. (Hint: Use school tube when searching! Great resource!)
5. Be sure that they understand that the most important part is not the answer they give you, but rather the method they use to solve it and WHY IT WORKS. That is the number one most important thing that they can get out of this inquiry activity.

Will all of the students be able to do this? Possibly…their level of understanding will vary from student to student. But in the end, when you bring them all together let the students do the talking. They will get there, if not now…they will have some prior knowledge for 4th grade.

# Differentiate in Small Steps: Give Them Two Problems

Differentiation is difficult. There is no doubt about it. I’ve been on a mission to find small ways to differentiate without stressing myself out, and without stressing out the teachers I work with.

I often found myself realizing that I was giving one math problem to the whole class when I’d look at my gifted kids faces. You know that look on their face? That boredom in their eyes look…where they’d rather be someplace else than sit and do another problem that they already know how to do. That is what inspired my idea of just putting two problems up, a meets the target (happy) problem and an exceeds the target (stressful) problem. I always explain to the students that if you are able to complete the happy problem correctly, you are meeting the target. It is even more impressive if you can do the stressful problem, but it’s not necessary.

Here is an example. Today in a second grade classroom the learning target was adding 2 digit numbers mentally (without regrouping).  I put up two problems, the happy one was a check for me to see who had it.  (They worked in their notebooks but it’s also great to use dry erase boards.)  The stressful problem is the one that students who need to stretch their thinking just a bit might try after doing the happy problem.

Using two faces makes it both visual and fun for students. Cut them out and reuse them over and over!

I keep those little face headings handy, they go up on dry erase boards, chalkboards, and easels…wherever we are doing math.  If I forget to put two problems, the students definitely remind me. They love to see the stressful problem face, especially the first time when I draw it in front of them.  You can use this method during quick checks, problem solving, mini lessons, practice, mental math…the possibilities are limitless.  This is something that can be done quickly, and doesn’t require hours and hours of work.

# Student Talk Leads to Deeper Thinking

I witnessed this cool thing the other day, the thing that I keep on blogging about because I keep on seeing it over and over. I was in a second grade classroom where students were adding two digit numbers.  The lesson was to add the ones first, then the tens by decomposing numbers. The well meaning adult in the room (me) kept on teaching it according to the lesson.  Students were making mistakes and errors like crazy.  Then, I gave them the freedom to try whatever way they pleased.

I was astounded by their thinking, they came up to share one by one with different strategies that made WAY more sense to each other than what I was preaching. It was pretty amazing to see what they were coming up with. Not only did their ways make sense, but they were also accurate. Another reminder that I need to SHUT UP!

This student made tens, and then added the ones and tens in the order that made sense to him.

So let them talk! The deep thinking and learning that will come from it will be amazing.

# Help! What’s My Error?

Every so often I have one of those moments where I want to stop the entire class to show everyone something a student did. Today was one of those moments when I watched how a student tackled a word problem. He had asked me for help and I was guiding him through it.  He was doing some mental math, realized that something didn’t seem right and checked his thinking. When he realized that he was 10 away from the target number, he very quickly realized that he could adjust his thinking and his answer. (You can see he changed 44 to 34.)

This is perseverance and precision!  This is what we are constantly hoping that kids will do without us having to remind them. The problem is we are running around asking students to do this on an 1-on-1 individual conference. Imagine how powerful it would be if students shared examples like these and learned from them, how much more time would be free up in our classrooms to really dig deep with kids!

Here are some simple ways to share:

1.  Stop the entire class and have the student show their error and how they fixed it.

2.  Build in share time at the end of your lesson for students to tell a story of how they found and fixed an error in their thinking.

3.  Here is my favorite idea…make a “What’s my Error?” chart!  This is a simple chart where students (while they are working on an assignment) could put up problems that they are stuck on.  We’ve all been there before where we keep on getting the same answer, but we know that something isn’t right.  Other students during a share time could help figure out the error and write their thinking on the chart.  So often adults turn to others for help when we need it (for technology, for many things), but often in math class we leave students to figure out these things alone.  A “What’s My Error?” chart could help students explain their thinking AND help them to be more interested in finding the error in their ways in the future.  Like all things, you have to manage it by making it a routine and having general expectations (imagine the students fighting over the markers, crowding around the chart), but isn’t that a good problem to have?

Let me know if you try it I’d love to hear how it goes!

# Math is Messy!

Today I filled in for a teacher who had to be out suddenly. It was so exciting! I’ve been coaching now for several months and I was anxious to slip back into teacher mode.

These were fourth graders who had been introduced to the distributive property the day before. The teacher was looking for something to both review and solidify their knowledge. I decided to try to connect their past experiences with multiplication to their new thinking from the day before. I wanted it to be a little messy!

The learning target:

I can solve multiplication sentences in more than one way.

I had the students rate themselves based on this success criteria:

0: “What is multiplication?”
1: “I’ve heard the word multiplication before but I do not have any strategies to solve it without help.”
2: “I can memorize facts, but I still don’t know what multiplication really looks like.”
3:  “I have more than one strategy to solve a multiplication sentence like 6 x 8: number lines, equal groups, repeated addition, etc.”
4:  “I can apply what I know for strategies to harder problems, like 16 x 8.”

Then, I put them in groups to solve the following two facts as many ways as they could on giant paper…6×8 and 17×5. The results were remarkable, AND messy!

Math is messy!

One student wanted a new paper, but I explained that mistakes feel kind of awesome. We should expect to make mistakes and try again, it is all about perseverance.

This messiness reflected the thinking of the students. I had to suppress the urge to organize their thinking into my own pretty little anchor chart. I had to remind myself that this wasn’t about an anchor chart, it was about letting the students know that math is messy and that they could learn from each other’s thinking.

After giving them about 10 minutes, it was time to share. We rated ourselves against the success criteria to see where each group was. Then, I asked them to think about which strategy was most efficient on their poster.  They laughed pretty hard about the inefficiency of drawing 85 circles/tallies.

Here were some of the favorite efficient strategies that students shared:

A nice notation of the distributive property, where the student broke the 17 into 10 and 7.

Classic repeated addition (bottom), and another way to note the distributive property on the top.

This person used the distributive property, but broke the 17 into 8 and 9! Very cool. I also appreciated how they showed their thinking below the number sentence.

He broke the 10 into two fives! That is someone who really understands what he is doing.

It was wonderful to get them talking and sharing.  The person doing the talking is the person doing the learning!

# A Math Book to Change Your Teaching

This is a funny title for a post because I have to say that this math book hasn’t “changed” my teaching exactly, but it has opened my eyes to how we can simplify our teaching.

I’m reading Building Mathematical Comprehension: Using Literacy Strategies to Make Meaning by Laney Sammons. I chose to read this book because with all the demands in our teaching profession, I was seeking a way to simplify things.  We cannot get everything done in a day, a week or even in a school year, so I keep thinking that there has to be a way to integrate things so that we aren’t going crazy every day.

Now literacy and math are not the same. But they have similarities, check this out:

The entire book focuses on ways to use literacy strategies in math.  And when you read it, you’ll be thinking “Oh my gosh! This just makes sense!”

One thing I’m going to try this year as a math coach, is to model some problem solving lessons in classrooms. In this book she talks about using comprehension strategies before, during and after solving a problem. Since math truly is all about problem solving, I’m envisioning an anchor chart adapted from my reading (pg. 35-38). I think as we work through problems, it may help students get “unstuck”.  It’ll look something like this:

I say that it’ll look “something” like this, because I want to come up with the anchor chart WITH the students. I think doing a problem as a whole class (and using a think-aloud strategy) would help them see the kind of thinking that should be going on in your mind while problem solving.

This book has even more literacy strategies that you can use in your math classroom.  Using some of the same vocabulary in your math and literacy blocks can help students make great connections!

To win a copy of this book, you will have to enter by clicking the link below.

There are a bunch of us talking about great books that have changed our teaching, so don’t forget to check out Mr. Elementary Math’s book, he is next in this blog hop.