# Oh My! The Progression of Multiplication

Well, I’ve watched this video three times now and I think I need to watch it at least five more times. I love, love, love how this presented to the audience.

My take aways for when I am teaching multiplication:

1. I need to stop stealing the opportunity to let my students use concrete tools! They should be available every SINGLE DAY.
2. Rushing to the traditional algorithm is a huge mistake. I am thinking we need to have some serious conversations about when to introduce this.
3. I need to let the students explore. Let me say that one again, I need to let the students EXPLORE. So many times when they hit a struggling point I feel this need to jump in and tell…I need a muzzle for my mouth!

What did you take away from this?

# 1 and 2 is NOT 12!

I know you see it all the time, when you ask a student what a number is made of, and they instantly throw place value out the window. That is why so many of us are doing this “Squashing Misconceptions” blog hop about place value.

I was working with a second grader that was struggling to add some two digit numbers (12 and 24).  I was thinking that if we could split the 12 she could add a ten and 2 more to 24.  So I asked her: “What two numbers go together to make 12?”

“1 and 2!”

Now you might be thinking that I tricked her with the question or that it was a matter of not understanding what I was asking, so I posed the question in a few different ways (“What is 12 made of? What two numbers can you add to get 12?”).  Each time, she maintained that 12 is 1 and 2 put together.

So, what do you do when there is a struggle with place value? Get out the MATERIALS.  ALWAYS. Don’t waste time or think that this will make it more difficult or confuse them. Just get out manipulatives, counters or cubes.  I love cubes because you can group and ungroup them at any time.

I gave her some unifix cubes and told her to prove to me that 1 and 2 make twelve. The second she pulled out those cubes her face lit up.  “That’s only 3!”

So I asked her again what makes 12.  Do you know what that little cutie did? She pulled out a ten and 2 cubes without even having to count the ten.  She was so used to seeing those concrete tools that it was a no brainer.  We had to have those materials out to make it concrete, she was not ready for bare numbers yet.

We constantly make those leaps too soon, and then time and time again (myself included!) it doesn’t occur to us to pull out tools, manipulatives or materials.  Our young learners need this, as they are very concrete and it can help them make the connection to number sentences.

So let them have those materials! Have them out for ALL students, so that no student ever feels “babyish” having to use them.  This is essential for students to understand place value conceptually.

Want to read some more about misconceptions in place value? Check out the next stop in the blog hop:

# 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.

# Whose Targets Are They Anyway?

I am going to admit some things that are going to make me sound like a horrible teacher. But I have to admit them in order to explain some of the tweaks I’ve made in the last few years. Here are some truths:

1. When I first started teaching I never, and I mean NEVER wrote objectives, outcomes, learning targets or ANYTHING on the board for my students during a math lesson.
2. When I finally started writing learning targets, they were about 3 sentences long, and I could barely understand my own learning targets.
3. I used to work in total isolation. I had a 6 person grade level team, and so we all wrote our own plans, never consulting one another or even thinking about planning together. It just wasn’t what we did. We all had different learning targets!

There were some things that happened that became game changers (each one corresponding to the number above):

1. I got to see Dr. Robert Marzano speak, and it totally opened my eyes to many of my mistakes, that could be corrected in small ways.
2. I realized I was losing my students with my wordy and often adult-like learning targets.
3. A new teacher was added to my team, and began to ask us to plan together.

So I had to change, BIG TIME. Not just add a new bulletin board in my room with a cool border, I had to change my belief system to the core.  You see, before this epiphany I knew my students loved me. I created fun experiences and I had high energy and we got a lot done.  But deep down they weren’t learning at the highest level, because I wasn’t making their learning clear enough to them.  They were achieving things and learning, but I knew if I could work through this, ALL of my students (including those struggling) could achieve more.

I started to rethink this whole learning target thing, and realized that I needed to really clean up my act. I rethought the teaching in isolation thing.  So I worked with my team to develop common, kid friendly, math and reading learning targets. I knew that just creating them wasn’t going to be enough, I had to actually make them visible for each lesson, have the students interact with them, I had to communicate them to parents, and I had to assess them.

This sound like a lot of work!

I knew I could do it in small steps.  Here is the first small step I committed to, which made me a better teacher.  I committed to writing my learning targets for both math and reading on the student planner.  The students copied them on the planner (Oh, they were angry with me at first, third graders do NOT like to write) every day.

I can statements, word for word, copied in their planners.

They copied them word for word, and their parents had to sign the planner each night.  I made this a strict routine.  I knew the students were ornery, and I didn’t care. If I put the learning on the planner in the morning, it meant I would commit to my day and remain organized in my teaching. It also meant that the target would be reinforced again when they wrote it at the end of the day, and then once more as they talked over the day with their parents. It turned out to be one of the best things I ever did.  After a month of grumbling, it just became part of US, our classroom community. We were learners! Instead of being some meaningless target on the easel, they began to take ownership of their learning.

I’ll continue to highlight some of the other things I did with my learning targets in future blog posts. Stay tuned as I share the tiniest tweaks, and how they made a big impact.

Today was a totally delightful 20 minutes of math play. It ended up being almost 40 minutes as we created our play situation.

I have to admit that sometimes pretending can be exhausting.  Maybe as I’ve gotten older I’ve lost that spontaneous creativity. So I was happy to find a way that I could “pretend” a real life situation…ordering food at a restaurant. Today, it was almost lunch time and my 5 year old and I decided we were going to make our own restaurant…PB & J. We built the menu, with her telling me the items and prices.  It was fun to think of the different categories and to put together the menu. We grabbed a notepad, marker and some cash and we were ready to go. (Notice some of her choices, “Soda is not very healthy so we can make it very expensive.”)

Right away she wanted to order a Hawaiian Punch juice box, and an appetizer of crackers.  That was when I asked her how much money she had.  She counted her cash, “I have 8 dollars.”  I asked her if she had enough money to buy a lunch.  This was one of those moments where I wish she would think out loud, because she immediately changed her drink order for milk. I bet all kinds of good mathematical thinking was going on there! Now, if I know my daughter, it’s because she wanted enough money for dessert!

Here she is counting her money after ordering her milk and crackers, to be sure she would have enough.

Sure enough, I took her order and it came out something like this:

She quickly realized that she was NOT going to have enough money for dessert, so she sprinted off to go and get her piggy bank, coming back with coins.  She asked me so innocently, “How many of these do I need for a dollar… one?” That was the perfect moment to tell her that a dollar is ten dimes, or four quarters…the perfect intro as to why she needs to know about coins and their values.

Which will lead to many more fun money play sessions!  We can use this same menu, but change up the ways to pay, the amounts and combinations of money.  All of which she will have a strong reason to want to know how to do it.

The best part? I told her that she needed to make sure to leave the server (me) a tip. As I left the room with her dirty plates, she secretly wrote this on a piece of paper and presented my “tip” to me when I came back:

I’ll take that tip over 20% any day!

# Real Life Examples of Geometry

The number of terms that students are expected to learn in geometry is a little crazy.  We counted 30 different new vocabulary words at the end of four days of instruction.  So I checked out an iPad cart and decided to have the students find real life examples of geometry in the world around them. After introducing the symbols, and describing each term’s features…they solidified their understanding of each new word with photos. (We pulled out some of the trickier ones from our minds as well.)

We recorded the findings on a giant chart!

Students captured real life examples of: point, line segment, line, ray, intersecting lines, perpendicular lines, and parallel lines with iPads.

It was both motivating and fun to use technology, as well as promote math talk in the classroom.

# 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.

# Raising Rigor for the Smarter Balanced Assessment

In one year, our students will be taking the Smarter Balanced Assessment in place of our current state wide assessment system. You can click the logo below to find out more.

Our district participated in the pilot test of this assessment, and I was lucky to get a chance to see what we are up against. You can take that first part of the test here (there is not a performance task sample ready yet).

When I peered over the shoulders of my students, my jaw just dropped.  The rigor of this test far exceeds what I’ve been used to for the last 8 years.  There were two parts to the test that I was able to see; a multiple choice/short answer and then a performance task type of test.  The first part was kind of standard, but included a lot of dragging, clicking and typing.

The second part was what really got me.  Nothing in our textbook would ever prepare students for this!  The first section of the problem was a full page of reading that they could go back and refer to over and over.  The second section contained about 20 questions related to that reading passage that built on each other in difficulty.

Notice, this was a MATH test, not a reading test.  If you can’t read you really can’t do well on this test.  If you can’t persevere, you really can’t do well on this test. If you can’t sort through massive amounts of information, or use appropriate strategies and tools, or PERSEVERE when the going gets tough…you will not do well on this test.

In come the Standards for Mathematical Practice. If you aren’t familiar with them, you should get familiar. They are amazing, they are exactly what we’ve needed and wanted students to do since the dawn of time.

I have been thinking about some ways to prepare students. Here are some things I am thinking for the coming school year:

1.  My plan this year is to incorporate deep problem solving DAILY.  I will use mini lessons and structure to help them get started, and then get them working independently. We will do this every day for 15-20 minutes per day, I’ll even set the timer!

2.  I plan to incorporate and embed those Standards for Mathematical Practice. I will explain that they are behaviors that great mathematicians do! I created a systematic way to introduce them to students.

3.  I am going to let them struggle. Sometimes we learn best with our mistakes.  Struggling can help us do our best critical thinking.

4.  I will help them read for understanding, I may even incorporate reading math problems during Readers Workshop. We’ll talk about how math reading can be different than reading for enjoyment. I am going to use these open ended problems that are full of reading, encourage persistence, and are real world.

5.  I’ll blend in reading, writing and math content into my problems so it becomes seamless.

To prepare them for the task coming (which they very well SHOULD be able to do), we’ve all got a lot of work ahead of us! I hope we can join together and re-ignite student’s passion for projects, hands on activities and rigor.

# The Float Challenge

The amount of vocabulary that is embedded in the measurement and geometry Common Core State Standards is staggering. Along with most things in math, I feel that more than ever students need to see it, use it, or touch it in the real world to connect to these words.

This occurred to me halfway through our measurement unit, as I was teaching metric units to measure mass. The day before, we had covered customary units for mass. Adding more vocabulary the next day felt like I was pushing too much information into their brain.

So I tried a little something. A Float Challenge.

I gave the students a 1′x 1′ piece of aluminum foil, and asked them to create a flotation device. This device had to float/carry the amount of mass weights (in grams) as high as they were willing to bet. The person who could successfully float the greatest mass (had to stay afloat for 5 seconds) would be the winner, but if they bet TOO high and their boat sank, they were disqualified. The mass weights that I had came in increments of 1, 2, 5, 10, 20 and 50 grams. They could decide how they wanted to distribute their weight.

The bets came in slow and steady. Some students wanted to see how others would bet. Others worked on their flotation device right away. A few students went for it all, hoping their flotation device would carry 80-100 grams. Every student came to the table at some point in the design process to feel what the weights felt like in their hands. After about 15 minutes of engineering we began the floating. It seemed like every student was hanging on the edge of their seat as the flotation devices came forward. Near the end they were trying to stand on chairs to see the final two. The results as we proceeded were recorded on our easel chart.

The reflection that followed was powerful and meaningful. Many wished they had bet more weight, others expressed an interest in changing their float design. Some explained that they would have liked the opportunity to test their float ahead of time. Other students felt that they had over-thought their design, and wanted to do it all again.

By the end of the lesson, every student in the room could tell me what 1 gram felt like. A week and a half later, they could STILL tell me what 1 gram feels like, AND they are connecting it to other things in the world around them. Seeing the success of something that was so simple, and that truly took only 25-30 minutes makes me see that I need to do this more. I need to connect it more to their lives, allow them to see it, touch it, (maybe even taste it!) to make it meaningful.

You can check out my Float Challenge in the Resources for Educators section, or just click here! In it is the lesson in GANAG format (Jane Pollock’s amazing lesson planning format) and includes goal setting, the guidelines for the contest, and formative assessment at the end. Happy Floating!