Conceptual vs. Procedural Understanding Posted on September 9, 2015

Barkelonging down the trejinga, the skeppy flappit floppered the flippit.

You may be asking yourself, “What is that even supposed to mean?” Regardless of what it means – if it indeed means anything – almost 100% of people could answer the following question correctly:

What did the flappit do?

If you said that the flappit floppered the flippit, then you got the right answer. Is it meaningful or has it improved your understanding of the sentence? Not at all. This is exactly what our students experience when we focus solely on the procedure. While many are able to perform the process and arrive at the right answer, the path they have taken has not been meaningful. They are just blindly following the steps outlined for them. Conceptual understanding of the problem is essential for students to develop deeper levels of understanding and be successful with mathematics. Without understanding the WHY behind the HOW, they will traipse along until they stumble over something new. Once that happens, it is very difficult to undo.

Tips for developing conceptual understanding:

  1. Always provide a familiar context when introducing a new concept.
  2. Follow the CPA approach – start with a concrete item and build into the pictorial stage before introducing the algorithm. One such example of this can be seen while using the Place Value Disks to introduce the four operations.
  3. Give students a chance to explore a concept independently or in small groups before providing formal instruction.
Reflecting on the Singapore Math Summit Posted on July 13, 2015

Earlier this month I set out on a six hour drive from Los Angeles, CA to Scottsdale, AZ. I was heading to the Singapore Math Summit. This coming school year I will be making the switch from a multiple subject Kindergarten teacher to a sixth grade math teacher. My hope was to gain some base knowledge of the sixth grade curriculum and some key strategies for instruction. Hands down I can say that I got so much more out of the Singapore Math Summit than I ever expected to. Every session I sat in was incredibly relevant and engaging. I met so many great educators with varying experience teaching a Singapore Math curriculum, and I loved that each session met all of our needs. Not only did I walk away with very realistic instruction strategies from the presenters, but I also received a lot of valuable advice from the veteran math teachers sitting around me. In addition the all of the collaboration that happened at the summit, I was incredibly impressed by how interactive each session was. The presenters had us all engaged in solving common core math problems the way we would ask our students to.  We dove in to the problems, struggled at times, and ultimately worked together to learn multiple ways of solving each problem. I can’t wait to get this school year started, I feel so prepared and excited to teach sixth grade math thanks to the Math Summit. I have already planned out anchor tasks, like the ones we got to experience with Dr. Yeap Ban Har. I have read and heard about using anchor tasks to start a lesson many times before, but watching Dr. Yeap Ban Har lead us in them truly modeled their power. Anyone looking to grow in their math instruction and have fun while learning should check out the Singapore Math Summit.

Math Journals in the Kindergarten Classroom Posted on June 1, 2015

As we pack up this school year and start planning for the next I want to talk about one way to improve your math time in the classroom. Math Journals! If you are not already using math journals for your Kindergarteners I highly recommend putting together a class set as you prepare for next school year.  Math journals are a place for students to take the Common Core standards and your daily objectives and apply them to relatable situations, which is exactly what Common Core wants us to be doing. Not to mention the pride a Kindergarten receives just by having his or her very own journal. I always love seeing my students get excited about their journals.

Math journals should start out with relatable with prompts like:

How many eyes do you have? Draw a picture of yourself and write the number.

How many people live in your house? Draw a picture of your house and write the number of people who live there. Don’t forget yourself!

These sorts of prompts are ones that any student can relate to. There is no stretch for them to apply the math and make sense of it. As the year moves on it is important to include more “real world” prompts that may not be as directly relatable. Maybe a story or situation that your students have to stretch a bit to understand, something that isn’t a shared experience. In my classroom I have included prompts such as “Your mom wants to make an apple pie. She needs 6 apples to make the pie, but she only has 2 at home. How many apples does she need to buy at the store?”  Maybe your Kindergarten student hasn’t baked an apple pie or doesn’t go shopping with their mom or dad. It is important to include these problems because it is exactly what they are going to see on Common Core tests in the upper grades.  Our students need to be able to dissect and understand a word problem for what it is, not whether or not they have experienced the problem.

I wish that I had been using Math Journals this whole year. I didn’t start using them until March when we were getting deeper into concepts of addition and subtraction. I have already started creating my journals for next year. I have stocked up on cheap Composition Notebooks and have begun creating a prompt a day. The prompts are small, just a sentence or two typed and printed. I then cut and paste a prompt to the top of each page. This is the perfect project for parent volunteers!  You can always start with just a week or month of prompts in each journal, or you can do the whole thing from the beginning. It all depends on how your pacing goes.

Thinking of prompts can get a little tedious as the days go on so I have found a ton of great resources online. You can find links to these resources at With all the great pre-written prompts out there your math journals will practically make themselves!

What is Lesson Study? Posted on March 19, 2015

As teachers, we are constantly striving to improve our instructional approach. We want every lesson to be direct, efficient and informative while improving a student’s understanding of the concept being presented. We are left to interpret and develop our own approach to a lesson and hope that it connects with students. We then grade our successfulness based on the grades that our students receive and, more often than not, realize we were not as successful as we had hoped. Lesson study helps to address these issues by making lesson planning and evaluation a teacher-directed and teacher-researched event. After completing one iteration of a research lesson, “[o]ne teacher commented that lesson study puts a professional component back in teaching… and treats teaching as a science” that can be analyzed and improved (Lewis, Perry, and Hurd 21).  To fully understand how this conclusion was reached, we must first examine the lesson study process (shown below).

lesson study

“Improving something as complex and culturally embedded as teaching requires the efforts of all the players, including students, parents, and politicians. But teachers must be the primary driving force behind change. They are best positioned to understand the problems that students face and to generate possible solutions” (qtd. in Lewis and Hurd 1). Lesson study provides us with a means of examining the problems and determining the efficacy of our suggested solutions through our own proven research. It begins with a team of teachers examining the concept to be taught and determining what skills students will need and already have in order to complete the lesson. Once planned, one member of the team actually teaches the lesson while the other members collect detailed narratives – how students reacted to the material, what questions were asked, the answers provided, the materials used, the obstacles encountered, and what specifically encouraged student understanding (Lewis, Perry, and Hurd 20). During the final stage, the team members share the collected data and reevaluate how the students learned during instruction. Then the process is repeated in a different classroom.

The important thing to remember is that the intended product of this process is not the lesson itself. The results of this process are a better connection to long-term goals, bringing the standards to life in actual lessons, developing stronger collegial networks, and an overall improved approach to lesson planning. By completing a research lesson, you are providing your own in-house professional development that can be continuously revisited any time a teacher is struggling. “If lesson study is to avoid the graveyard in which so many other once-promising innovations are currently buried, then U.S. educators must understand that lesson study means far more than just walking through a set of specific activities. It means building a set of pathways that enable continual growth of the knowledge, interpersonal resources, and motivation required to improve instruction in the classroom and beyond” (Lewis, Perry, and Hurd 22).


Lewis, Catherine C., and Jacqueline Hurd. “Introduction: What Is Lesson Study?” Introduction. Lesson Study Step by Step: How Teacher Learning Communities Improve Instruction. Portsmouth, NH: Heinemann, 2011. 1-5. Print.

Lewis, Catherine, Rebecca Perry, and Jacqueline Hurd. “A Deeper Look at Lesson Study.” Educational Leadership (2004): 18-22. Web. 12 Mar. 2015. <>.

We Have to Hook Them! Posted on February 2, 2015

Let’s be honest. Americans want to be entertained. American children, especially, want to be entertained. If we want to get their attention during math class, we have to “hook” them at the beginning.

A recent 8th grade teacher I observed had created a fabulous word problem to introduce a lesson on constant rate. The lesson was during the week before the Superbowl. Her problem involved the running speed of the fastest runner on the Seahawks and the fastest runner on the Patriots. Those students were engaged and excited. From there, all she had to do was reel them in.

What’s Going on in Texas? Posted on January 15, 2015

We started the New Year out in El Paso, Texas. We brought of team of 5 trainers. This was our 8th visit to schools in the Socorro District. These visits have been during summer PD sessions, on Saturdays, on-site visits and workshops. The common thread is the enthusiasm these teachers have for meeting the Texas Essential Knowledge Skills (TEKS). As a state, Texas standards are definitely one of the more rigorous sets of standards in the country. The willingness of these educators to come in on their days off; the support of “Specials” staff coming in on those days to show solidarity says a lot about the dedication they have to their students.

One of the unique standards Texas has set is Financial Literacy which tends to get folded into math. We applaud Texas for adding these standards but in doing so we need to add time to the day to teach them. One of the reoccurring issues we hear from teachers across the country is that the new standards require discussion and thinking time. Many are trying to implement in a 60 minute math class and are finding it difficult. We need to try to expand math across the day and incorporate it into other subjects or have activities handy that do not require prep or set up time to do on the fly. Keep up the great work Socorro teachers!

When Failure is Success Posted on December 4, 2014

Teachers are superheroes! Not only do we have one of the most difficult jobs, but also one of the most rewarding. Seeing a student’s eyes light up when they finally understand a concept they have been struggling with is exhilarating. But what happens when the light seems as if it will never illuminate? SUPER TEACHER to the rescue! Like many who have gone before, we swoop in to save the day and try our best to make our students feel successful. Now, don’t get me wrong, we definitely need to be there for our students, but are they actually benefiting from our efforts? A recent study published in the Journal of the Learning Sciences by Manu Kapur, a researcher at the Learning Sciences Lab at the National Institute of Education of Singapore, and co-author, Katerine Bielaczyc, sheds some light on the subject.

Providing lots of structure early on, until students demonstrate they are able to complete tasks on their own, is a very intuitive teaching model. However, Kapur’s research on “productive failure” shows that allowing students to struggle helps them to learn better than the aforementioned intuitive model. One group of students was taught utilizing intensive scaffolding and instructional support, and they were able to arrive at answers for their assignment. The second group was given the same set of problems and instructed to collaborate without any prompts from the teacher. While they were not able to correctly complete the problems, they did develop their personal understanding of the problems and what solutions might look like. When tested on that concept, the second group “significantly outperformed” the first.

Real world problems are hardly ever neatly packaged and will take some considerable thought to solve. Being able to find the deeper structure of a problem, as the second group was able to do, will allow students to easily apply their knowledge to similar problems. According to Kapur, it is important to “design for productive failure” by intentionally managing the way students fail.

* To see the original article, click here.
SMARTTraining and the Bermuda Union of Teachers Posted on November 7, 2014

The Bermuda Union of Teachers recently held a conference to provide professional development to the teachers of Bermuda. Experts from around the world were brought in to share their knowledge and experiences with the teachers in a wide range of subject bands. SMARTTraining’s presence was also requested. After flying all night on a 9 hour flight with 2 plane changes, Phoenix-based SMARTTrainer, Jennifer McElvania, arrived at the Bermuda International Airport.

“Spending two whole days on the sunny island of Bermuda working on developing teacher insight and math content knowledge was a great experience. On Monday, we discussed Singapore Strategies to be implemented in grades Pre-K through 5. With the largest group of any of the sessions, these teachers showed their dedication by sharing manipulatives, seats and desks as there was nearly standing room only. Tuesday was focused on learning hands-on techniques for Algebra, Geometry and Problem Solving in grades 6-9. With a much more intimate group on this second day, we were really able to dive deep into the content and generate some good discussion. Many teachers said that they walked away with a better appreciation for math and ideas on how to conceptually introduce new topics.”

We’re excited to say that they have already asked us back for additional training days this spring!

KinderCorner – Back to School Posted on September 2, 2014

The sun is shining, apple décor is flooding the shelves of every craft store, and snotty noses are walking through my classroom door. Yup! It must be the beginning of a new school year. As I sit at my desk after a long first week I find myself looking more often than usual at a quote posted on my desk. The quote from Ruth Beechick reads, “A teacher who loves learning earns the right and the ability to help others learn.”

This quote is going to be my motto for the 2014-2015 school year. It just has to be. In everything I do this year I want to convey to my students an absolute love and excitement for learning. As the years go by our students are getting more and more interested in their iPads and Netflix and less and less interested in practicing basic skills. And as the saying goes “If you can’t beat ‘em, join ‘em.”

To keep the learning fresh and new I have decided to create superhero type characters this year that focus on different basic math skills. For example, during the first couple of weeks of Kindergarten we focus on color and shape recognition. My student’s current favorite superhero is “Mr. Vertice”. Mr Vertice is a boy in a green cape who is an expert at counting the number of vertices on each shape.  The students are so engaged in the lesson and being able to use “Mr. Vertice” that during independent practice they don’t even realize they are completing a basic skill practice sheet. Not to mention to rich vocabulary they are learning as they chant “Go Mr. Vertice, go!” I also have “Rainbow Girl” who identifies colors around the room. I just put a little picture of “Rainbow Girl” on each color worksheet and it’s like they can’t get to work fast enough.

With Common Core in full swing it is more important than ever that we are teaching these basic foundational skills in Kindergarten. If a child is expected to complete a conceptual math problem that incorporates multiple skill sets then they need to have a firm foundation. While we can choose to be discouraged by how little our Kinders know at the beginning of the year, we can also choose into projecting a love for learning. Once the excitement is there, the kiddos are motivated. And once the kiddos are motivated, they cannot be stopped!

Superheroes may not be your “thing”, but I encourage you to find a theme that does work in your classroom. Even the consistency of using one theme throughout all your math lessons help students build a stronger foundation. Animals, popular TV shows, sports, and music are all ways to engage your students. I would love to hear what you use at your school to make learning fun from Day 1!

Teach Less, Learn More Posted on August 28, 2014

From the title of this article, you are probably thinking, “That’s not possible.” According to David Ginsburg*, it’s the only way for students to truly succeed. What is meant by success in this context is that students should be prepared for the independent and resourceful thinking required in high school and beyond.

We as teachers are very caring and giving when it comes to our students’ success. This means that, more often than not, when our students come to us for help, we jump right in and try to save them. Yes, WE. I am also guilty of this at times. But consider this: in college or the workplace, will the professor or employer jump in every time they struggle? Did yours? The answer is most definitely, no. We need our students to be able to help themselves.

How can we accomplish this? Teach less! Now this doesn’t mean step back and never help. What this means is, make students work for your help. Give them resources in the classroom that they can turn to before they come to you. This could include notes, textbooks, technology, or even each other. To summarize Lev Vygotsky, a pioneer in the field of social learning, what children can do in collaboration today, they can do independently tomorrow. Isn’t this one of our ultimate goals?

This is going to be a change for your students as well as yourself, and isn’t something that will happen overnight. You will be met with resistance, by both students and their parents. However, according to Ginsburg, the risks are well worth the reward. Ginsburg’s “conviction around cultivating resourcefulness in students is stronger than ever because of the effect [he has] seen… Higher test scores? Absolutely. But more important, better preparation for future academic and employment challenges.”

Challenge yourself this school year to teach less. Use your lesson planning time to develop questions and activities that will foster student understanding through exploration. Try to limit how often you “answer” questions posed to the class and script out questions that you can use in response to student inquiries. “To do otherwise is to hurt students rather than help them” (Ginsberg).

* To read David Ginsburg’s original article, click here.