# 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?

# Fly on the Math Teachers Wall – Squashing Fraction Misconceptions

When I was an elementary student, fractions were by FAR my most difficult subject.  I could not ever wrap my brain around them and embarrassingly enough, I still struggle to understand them. The other day I had trouble trying to figure what half of a 3/4 cup of butter would be! My cookies tasted a little more buttery than I would have liked. So today I am talking about misconceptions about fractions, because I’m really a pro!

This question was on a fourth grade test on fractions this week at one of my schools.

Amazingly, many students answered the question by saying they agreed with Molly. Their explanations said things like:

• “4/8 and 1/2 are the same number because they are equivalent.”
• “The diagram below each shows a half, so they are always the same.”

So somehow, somewhere we have a misconception here. Students are missing the idea that fractions can be different amounts if the whole is a different size. After all, one 8 inch pizza is not the same as one 16 inch pizza, right? I’d MUCH rather eat 1/2 of the 16 inch pizza!

My favorite way to clear up misconceptions is to relate it to real life…especially food. Food lends itself beautifully to math in so many ways. Once I brought in the skittles, suddenly light bulbs turned on.  Equivalent fractions may be the same number, but they are not always the same amount.

I think we miss this step very often when we work with students. I think that real life connection is what helps them figure out what the symbols stand for. When we leave that out, students are unable to make sense of a problem.

If you want to see more examples of misconceptions that us math nerds have uncovered, check out The Math Spot:

# 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.

# 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!

# Build A Strong Math Culture With The Standards For Math Practice

With all of the current criticism about the CCSS and “Common Core Math” (I put that in quotes because that phrase has been driving me crazy, now that is another blog post in my mind), I’ve been happy to see that the Standards for Mathematical Practice have been left alone.

I’m glad that they’ve been left alone in the criticism because the Math Practice Standards are all encompassing thinking habits, more than they are standards to be met.  They encourage us to teach mathematics more as a learning subject than a performance based subject. Math absolutely should require lots of messy critical thinking, deduction, discussion and reasoning.

Someone last year said/asked me, “I just don’t understand what these standards are for.  What are they?” I explained them the best way I could, since I had just spent a month researching them to know them better:

1. The math practice standards are a set of math habits, ways in which we should think about math. (That sounds so simple, but the way the standards are worded, it has driven many of us crazy while trying to understand them.  Reason abstractly and quantitatively? Huh?  It sounds like a completely different language!) The standards are all about developing positive habits and attitudes about math.
2. They allow students to explore math as a learning subject.  They begin to understand that math is not about the teaching asking a question, and the student must answer it correctly. Most importantly they begin to see the connection to their lives.  Math connects so beautifully to real life, but because the U.S. has such a worksheet culture, we’ve lost that connection.
3. Math from K-12 has the same underlying theme with these standards. As the years tick by the content standards become more complex, but the practice standards remain the same.  With the Standards for Mathematical Practice, we can develop a very positive culture surrounding mathematics.  A culture of persevering when encountering a problem, making sense of the world with math, using prior knowledge to solve new problems, being precise and reflective, patterning to find faster ways of working, explaining our thinking, understanding others thinking, knowing what tools will best help to solve problems, and connecting the world with abstract numbers and symbols. This all makes us excellent THINKERS.

We’ve had math coaches, administrators and other teachers pass out posters to put up in our classrooms. We’ve seen freebies and posters that we are meant to download and print. We’ve put them up on our walls with very few of us digging in to what they actually mean. I WAS one of those people. I had a poster of the kid friendly standards up for two years, and it wasn’t until last year that I realized one of the standards was completely inaccurate on the poster.  I had never bothered to check, and I assumed that the source knew the standards.  Can you blame me? I didn’t have the TIME to dissect what each one means.  It felt like another thing…another plate to spin…another added responsibility. I truly didn’t understand the importance of the standards to create a culture.

When I decided to figure out what they really mean, introduce them to my students from the start of the school year, work through the problems with them, and embed the language in the classroom, the culture really changed. We became mathematicians who could work through anything. It was remarkable. We put our work on the walls to help remind us that these were to be a part of our classroom daily.  We truly became vicious problem solvers, we worked together and math was about learning.  Math became FUN.

After a month of research I created posters, problems and activities to help myself understand them, but also to help teachers understand them, too.  (Feel free to check out the preview which walks you through the first standard.)

Build a culture by introducing, working with and revisiting the Standards for Math Practice.

Even if the Common Core goes away (which it most certainly will, and already is in many states), I will always keep the Standards for Mathematical Practice.  It is a foundation in which we can all build upon, year after year!

# Crank It Up a Notch: Add Something They Can Touch When Problem Solving

One of my favorite ways to amp up problem solving is to throw something into the mix that they can touch.  This makes the project or problem so much more interesting to students in one instant.  We are working on the Design a Dream Bedroom project, so I picked up some free samples from the hardware store:

Give them stuff to touch when they are working on real world math activities.

Of course you can be sneaky about introducing the materials.  Before I even went over the problem during math, I spent the morning organizing the materials on a common table when they were arriving for the day. I got about a million questions, and hands were reaching out to touch the carpet and flooring samples before I could even get them in the bucket.

That is all I needed to do to get them interested in the problem.  After I read through the introduction with them during math problem solving time, the students literally leaped out of their carpet spots to run up and grab the problem from me.

That is what we want problem solving to be like…exciting, engaging, rigorous and motivating! Putting things in their hands to make it real world has worked every time.

# Concrete Learners: Hands On and Real Life…Every Day

In teaching division this year, I’ve never before used so many counters for so many consecutive days in a row.  I’ve got a core group of students who feel really great about division, some of them have even been memorizing multiplication at a very fast rate, which allows them to make better sense of division.  But I also have the exact opposite end of the spectrum as well.  As soon as they see that division symbol, their eyes glaze over and they become fearful of the problem. They worry about what to do and they think they cannot divide (even through they’ve been dividing all of their lives, they just haven’t seen the number sentence for it).

To help struggling learners, I’ve been trying to make it more concrete. Our youngest learners often need to see visual representations of numbers so that the concept is not so abstract.

In this set of problems, fruit was being divided equally into bags. I decided to lay out paper bags for this student.  The problem was 14 divided by 2, and he could easily solve it. He was both proud and excited to write his answer.  Win!

Another student though, had a little more trouble. The problem was 8 divided by 4.  Because she had done 10 divided by 2 right before this one, she forgot to put away the 2 counters to start. She hit a wall very quickly and gave up. Instantly, she had her hand up for more help.

To help her, we re-calibrated a bit by checking the problem again.  She very quickly realized that her counters started out wrong, and she was able to fix it. That check back to the problem is what I want her to do in the first place, great mathematicians do that without any prompting.  It was clear to see that she wasn’t connecting the number sentence with the manipulatives at all. It was a quick 1 minute conference on the importance of paying attention to detail/being precise, a math practice standard that many students struggle with. I told her that the next time she should try that strategy before asking for help.

Giving these quick teaching tips while conferencing with students makes WAY more sense when those tools are right there in front of them.  I used to try to draw on student’s papers, help them extend patterns etc… but the most struggling students need those tools in their hands to actually act out and see the problem. That is when I’ve noticed teaching tips given to them have become ultra powerful!

# How Much is Enough? How Much is TOO Much?

I must ask the question to myself over and over again, almost daily. How much practice is enough practice for my students? Today I struggled as a math teacher. Writing this post hurts my heart a little.  I am wrestling with a new math series, trying to give it a try to have it be at fidelity while still balancing the needs of my classroom.

We all have learners that fall on the spectrum of different levels of understanding regarding the learning target. I am prepared for that daily.  What sometimes gets me though, is the way that our new math series assigns a number of problems to a student.  In one math lesson, the students are to solve 6 review problems, a large hands on problem, 30-35 independent practice problems from the book and a 20 problem homework page.

This is absurd in my mind for two reasons:

1.  The students who understood the learning target, certainly don’t need to do it 50+ times.

2.  The students who didn’t understand the learning target, absolutely CAN’T do it 50+ times.

This is what the book page looked like (only the bottom half):

We are only beginning division. Asking students to learn the concept of division, and then 3 days later have them divide by 8 and 9 is just crazy.  I know that as their math teacher, I can step in and give modified assignments, lessen the number of problems and the level of difficulty. I know that is why I am the professional in front of them, giving them what their brain needs.  But this isn’t just this particular lesson this particular day, it is just about every lesson, just about every day. Having to modify everything can be exhausting, and it makes those struggling students feel sad that they can’t do it all.

Why do math textbooks have this almost mindless repetition for students who get it, and then induce pure panic in those that don’t?  Watching students who struggle try to answer even one problem is heartbreaking, leaving them feeling broken when they see that there are 29 more to go.

This is why I’ve gone away from the traditional model of math, in favor of deeper problem solving and a more project based feel.  Watching student’s faces drain of the love of math is totally heartbreaking and I really can’t do it anymore. Math is an amazing, beautiful subject and I know I can help my students grow.  I can prove it with data, and I can prove that I don’t need to do it with endless worksheets.

Even with a new math series that led us to believe that things would feel different, I am realizing that being an effective teacher and lifelong math learner is truly the only way to help students grow. There is no magic textbook, no perfect program.  Instead, we must search and find the best tools, and tailor our instruction to every student at every level.

I hope you can share the best ways that you know how to do this with me (all of us!) as well.

# Sneak It In All Day: Make Math Real World To Increase Minutes

Children are surrounded by words and language all day long. It is everywhere around them, on signs, all over their technology, in their books, and it takes up a large portion of the minutes in an elementary classroom. In an effort to get more math minutes in, I’ve been trying to sneak math in during other times of the day. Here is an example:

When my daughter was ill last week, I needed to take an unexpected day off.  My class this year can be a little shaky with a substitute teacher.  Like most classes, upsetting their routine can cause a lot of anxiety and disruption. Whenever I return the next day, I like them to reflect on their behavior the next day.  I put up the chart on the easel below, and had them vote as they arrived in the morning.

We turned it into a 5 minute lesson during morning meeting on how to calculate percentages. It was perfect for my gifted students who are working on some presidential birthday data, they can now extend this thinking to their project.

This directly connects math to the real world, and it gives us data to reflect on.  It is also a really visual way to look at the day! We certainly want the orange part of our graph to be at 100% next time! We talked about where we went wrong, and set goals as a class for the next time a guest teacher is in:

1. Listen to the speaker when they are talking.
2. Focus on your own work, not what others are doing.
3. A level zero voice means level zero!
4. Respect other people’s space.

Putting this type of work into all parts of the day can really help students understand the importance of math in our every day life!

# The Fishbowl: A Peer Modeling Strategy

Whenever I notice problems in the classroom, I pull out one of my favorite instructional strategies, the fishbowl.

The fishbowl strategy is an awesome way to showcase model behaviors in ANY subject. I particularly like to use it when I notice students are skipping steps or struggling with a certain activity. I have used it to model word work strategies, reading behaviors, problem solving, computation and even science class behaviors.

Here is an example of how I used a fishbowl last week to help model effective problem solving and group work behavior.

We are working on some collaborative problem solving, which requires an intense amount of group work.  The problems are differentiated so they are just right for the groups, but that means they require some really tough thinking.  Often, I’ve noticed that the behaviors of group work can get in the way when they are pushed like this. In an effort to prevent this, I set up the fishbowl before even beginning to problem solve collaboratively.

1.  First I put up a t-chart that had 2 columns. We filled in the left column as a whole group activity. I recorded their answers on the board as they shared them with the group, and they recorded them on their paper.  I purposefully left the right column covered until the next part of the activity.

2.  I asked my strongest problem solvers (I know they are strong problem solvers, the other students do not) to go into the middle of the circle, while the rest of us surrounded them. I told them that they were going to be solving a very difficult level four problem, and that the rest of us were going to take notes on their behaviors. The students taking notes had their t-charts in front of them, ready to go.

3.  As the students in the middle began to problem solve, the observers on the outside of the fishbowl started writing down everything they did. They wrote what they did with their bodies, how they sat, what they said, the voice levels they used, etc.

4.  Then, after a few minutes, we debriefed and shared what we noticed up on the t-chart.

5.  The rest of the students were now dismissed to work on their problem, with the expectations all laid out for them.  I explained that if a group encountered problems working together, they could call a “timeout” and recheck the board to see how they were doing.

As they began working, it started out great. Students were problem solving, everyone was engaged.  Then, like usual, I noticed in a few groups a couple of students starting to drift off and let the others do the work. In other groups the voice level was rising. That is when I called a timeout and asked them to look at the board. They quickly adjusted their voice levels, how they were sitting and re-engaged themselves in the activity.  They very badly wanted to be just like those students they watched in the fishbowl. Peer accountability is huge!

I love this technique, and the best part is that it becomes an anchor chart for the wall whenever we do any collaborative problem solving!