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!


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!


A Math Book to Change Your Teaching

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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.

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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.



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.

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!



Build Your Own Restaurant

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.”)

Our menu, cash and the ordering pad for the server.

Our menu, cash and the ordering pad for the server.

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.

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:

real-life-math-playShe 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!



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:

Hans On Real World Math

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.


Third Graders are Googling the Answers to Math Assignments?

My jaw literally dropped when I received this email the other day:

Assign Rigorous Problems

Let me explain two things to give you a little background:

First, I have a “contact me” section on this blog, for anyone who may need to get in touch. I received this email through that form.

Second, I recently put out a free resource called Doggy Dilemma for teachers.  It is an open ended problem that requires a lot of reading, writing and thinking.  There is no immediate answer, and all students would have a different answer in the end.

So…by the powers of observation and inferencing I can only conclude:

  1. This person who contacted me is a child that has received the assignment in class. (I am thinking this due to the lack of punctuation, capitals and misspellings. The “voice” of the writer seems very young, also.)
  2. This person is likely a 3-5th grader (since that is the target age group of the problem).
  3. This person is incredibly resourceful and bold. Not only does she google the problem, but she thinks to contact the author of the problem for an answer!

After I got over the shock of receiving this message I thought to myself, THIS is why I create what I create. No child should be able to google the answers to a great math problem.

What do you think about this?! I would love to know your thoughts.