Subitizing and Computation


This weekend I started to read a book on how the brain learns mathematics. The first chapter highlights the different ways people develop number sense. One of my takeaways came from a section related to subitizing.  Subitizing involves recognizing a number of items in a collection.  There are two types of subitizing that are communicated in the text: perceptual and conceptual.

Perceptual (glance and find the sum)

Perceptual Subitizing

Perceptual subitizing involves looking at a number of items and recognizing the number without much pause. Generally, the items are separate from each other and a quick glance will often reveal a correct answer.  Perceptual subitizing can remain fairly simple if the digits are close to zero. Larger amount of items often gives way for people to start counting each item. Counting individual items increases the amount of time it takes to find the total.

Group dots together, e.g. 3 groups of 4

Conceptual Subitizing

Conceputal subitizing is a bit different. This type of subitizing relies on the person to find patterns and use those spatial relationships to find a total. Grouping items together (such as 3 groups of 4) would fit into this category. Analyzing the spatial arrangement of the items can lend itself to people using conceptual subitizing.

I’m finding more and more that subitizing plays an important role in the early elementary grades. To a certain extent, I feel like students use subtilizing to quickly identify the number of dots on dominos and dice. Whether students use procedural or conceptual subtilizing depends on the number of dots and the arrangement of the pattern. Students that have a conceptual understanding of subitizing can group items to find sums. Grouping items together with spatial reasoning can lead students to discover additional computation strategies, such as splitting items into equal groups or constructing mental arrays. I see potential in using subtilizing strategies in the classroom.

While researching this topic I came across Steve’s subitizing page. Feel free to browse this amazing resource for more information and classroom resources.

Patterns and Pre-Algebra

Yesterday I was able to get outside and walk around a local park.  While soaking up the sun I started to notice a variety of patterns on the sides of the path.  The patterns changed depending on the vegetation and location.  As I searched for additional patterns I started to find more and and then looked for consistency among the sequences. I took out my phone and started taking pictures of the patterns that I saw thinking that I might use them next school year. After collecting a few I started thinking about how this connects to the math strand of algebra.

What patterns exist?  What about lines of symmetry?

 How does symmetry play a role in the pattern?

Taking the pictures had me thinking of a class I had a few years ago.  I remember reading a district-adopted fourth grade text that introduced pre-algebra to students as patterns and solving for the unknown.  This simple kid-friendly definition was explained to elementary students in a short paragraph. After thoroughly discussing the definition of a pattern (yes, that took time), students took that definition and ran with it.  They started to find patterns (number and otherwise) in and outside of the classroom. If a pattern didn’t seem to exist, students would make a prediction based on the prior sequence.  A completed pattern seemed to make sense and an uncompleted sequence didn’t have meaning.  Students started to put on their “pattern glasses” to identify sequences.  Students would argue whether something was a pattern or not.  I distinctly remember one student saying that to complete the pattern you need to find the missing puzzle piece. These discussions were interesting to observe as students were developing their own rules to the patterns and offering their suggestions to others.

Additional pictures and questions:

What type of pattern exists?

             What type of pattern exists?  Do multiple patterns exist if you zoom in on the picture?

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               Would you consider this a pattern?

After uploading the pictures from the walk I started to think of how students make meaning out of patterns.  This past year my students were able to find patterns in nature, use Which One Doesn’t Belong, and then transition that idea to Visual Patterns.  Understanding the rule or rules behind the pattern can lead to different levels of pre-algebra moving forward. It’s amazing when students start to realize that there can be more than one rule to a pattern or question.  Simple patterns can allow students multiple entry point to access pre-algebra concepts.  Before the school year starts I’ll be pondering the question below.

How do students identify patterns and does that help them become better problem solvers?

I’ll leave you with one more picture:

Does this qualify as a pattern?

Does this qualify as a pattern?

Questioning the Gradual Release of Responsibility Model

gradual responsibility

When I first started teaching I was told from one of my professors to grab Harry and Rosemary Wong’s book and use it as a guide.  The guidance in the book was direct and seemed to be working during my first year of teaching.  I still refer back so some of the pages from time to time.  For the most part my class of fourth graders fell in line with the expectations that I set, which were from the book.  My administrator at that time suggested I use a gradual release of responsibility model with my students.  This “I do, we do, you do” model was heavily emphasized.  Basically, I was instructed to start my lessons with a guided whole class instruction, move to groups or partners, and then have students work on assignments independently.  Student input was limited when I used this model and I didn’t really see a problem with that at the beginning of my career.  As the year passed I found that extrinsic motivation was keeping most students on task.  The pressure of getting high grades and outside rewards moved students in being compliant. As I gained experience my instructional strategies changed .

As the years passed I started to let students make a few decisions in the classroom.  I offered students a chance to sit where they wanted at the beginning of the year.  Students also had options in what projects to complete.  This happened rarely, but I found that the choice opened up a new realm of student responsibility.  When students had a choice they often performed better and with more enthusiasm.  The reward for accomplishing a task started to become more intrinsic.  From there I surveyed students and included plus/delta charts throughout the units that I taught.  The more students offered input and felt like their voice was being heard, the more active they became in their own learning experiences.  Now that students were offering input I gave them opportunities to reflect on their learning and had them set goals.  Last school year students participated in genius hour.  I was truly amazed at the projects that were created by the students and the passion that I could visibly see as students presented their projects.  Students happily took advantage of these opportunities.  Students were asked to think about their own thinking, which was a new experience for students.

This opened up a new realm of possibilities for students as I felt they were realizing teaching wasn’t being done to them.  Instead, students started to realize that they were an intricate part of their own learning.

All this is good, but this type of thinking didn’t happen until the last third of the school years.  I scaffolded the gradual release of responsibility model until I felt confident to let the students take on more responsibility.  My confidence in students was conservative and I didn’t take the risk in allowing them to take control until later in the school year.  I’d like to change this next school year.  Allowing students to be responsible early in the school year can lead to dividends throughout the school year.  One book that has influenced me in this thinking has been Paul Solarz’s book.  Students should be given the opportunity to take the lead and be empowered in the classroom. One strategy that Paul highlights is his “give me 5″ technique.  I’d like to start this early in the school year.  I’ve also questioned my own thinking regarding how students should be expected to proceed with a gradual release of responsibility philosophy.

I still adhere to the philosophy although I’d like to tweak my perception of it.  Instead of providing constant scaffolding to release responsibility, I’d like to start off the school year with student empowerment opportunities.  Waiting too long to give students responsibility can be costly.  Giving students opportunities to lead with support and guidance from the teacher can lead to positive results. I’m assuming there will be times where students will speak out of turn or take advantage of the empowerment opportunities, but I’ll take that risk.  With direct teacher support and feedback, I feel like students will become better at taking responsibility for their own actions.  There is a risk, but I feel there’s so much potential in empowering students to become part of their own learning experiences.

Student Surveys and the Reflection Process

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Yesterday was the last day of the school year for my students.  The end of the school year tends to be filled with excitement and pride as students transition from one grade to another. During this time of the year I usually give my students a feedback survey. I tell the classes that I’ll be using the information to change next year’s classes for the better.  I’ve been using this method for the past few years and find it valuable in preparing for the fall.  Most of the questions that I ask tend to stay the same while I add a few others depending on what I’m focusing in on for the year.  This year I asked a few questions related to feedback and student refections.  These particular questions stem from some of the district’s initiatives, as we’re emphasizing Hattie and Dweck’s research.  Next year we will be focusing on them even more and I believe they’ll be part of a formal walk through process.  So I gave the survey to 50 3 – 5th graders and collected the data.  The survey that I used can be accessed here.

I took the 50 students responses and had Excel calculate the averages for all of the questions. Below are few highlights from the feedback and reflection questions.  I used a 1 – 10 rating, with 1 being all the time and 10 being never.

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My takeaways:

I have to keep in mind that elementary students are taking this survey.  It’s valuable, but I feel like a third grader will perceive a question possibly different than a fifth grader.  Regardless, the data is valuable in my mind.  I looked over the numbers and shared this information with another class.  After showing the data we had a great conversation about reflecting on our learning.  Our conversation looked at the connection between allowing reflection opportunities and how they impact our learning.  We started connecting parts of the survey as a cause/effect scenario.  The conversation wasn’t too deep, but worthwhile as students made connections.  We decided that reflecting on our learning can be impactful, but not necessarily help a person understand a particular concept.  Feedback, reflection and opportunities to take action need to all be place. What seemed to be lacking this year were opportunities for students to reflect AND take action based on that reflection.  It’s important to reflect, but without any action or change in perception the act might not be reaching its full potential.  I decided to write an informal flow chart indicating the process that the classes tended to use.


I told the students that one of my homework assignments over the summer is to provide ways to make student reflection opportunities more efficient.  This is something I’ll be revisiting in the fall with my new classes.

Content Creation Tools and Learning Curves

How do I use this?

During the last few weeks of school my students have been working on their last projects. These projects are student-driven, as students ask a question, conduct research and present their findings to the class.  Through this project I’ve attempted to add a genius hour philosophy to math class.  For the most part students are excited to participate in this learning experience. At the beginning of the process I gave them options in how to present their findings.


Most of my fourth and fifth grade students are presenting their projects using Explain Everything, a poster, Power Point, or iMovie. Even though all of these options were given to grades 3-5, all of my third graders chose one way to showcase their learning. Last year, third and fourth grade students used many different apps and content creation tools to showcase their learning. Each unit had a content creation component and students took full advantage as they explored the different features of the apps/programs. So now these particular students are in fourth and fifth grade and have had the experience of using a variety of presentation tools.

Regardless of the grade level, I allowed students to pick a presentation tool for their projects.  Every single third grade students decided to use Power Point. I don’t feel like this is negative, but it also has me wondering if their reluctancy to try a different tool resides in not understanding its functions. It also has me wondering what happens if students aren’t introduced to the tool? My current third graders haven’t had the opportunity to use other technology presentation tools. I think that’s part of the reason, but not necessarily the entire picture.  What happens if that exposure is limited or doesn’t happen at all?

Giving students options is important, but if they don’t know how to use all of the options then that often reduces choice. In this case, familiarity trumped the intimidation of using something for the very first time.  Now I could have had a brief introduction to each tool and then maybe some would try to use it, but I didn’t try that.

Why this post? I feel like students need to be given the opportunity to use a variety of content creation tools in the classroom.  Not for the sake of using THE tool, but the experiences using the tool often empower students.  Students also become more confident and find different ways to showcase their learning in doing so.  I’ve found certain tools to be valuable in showcasing student learning far beyond a typical bubble test.

Although the technology opens up more doors, I feel like Bill makes a great point by saying, “technology is a tool, not a learning outcome.” At some schools these types of tools are introduced in the media center, at others the introduction depends on the teacher. The tools don’t have to be digital or complex. Even a brief amount of exposure on the functionalities of the tool can go a long way in encouraging students to move outside of their comfort zone and create. Despite being digital natives (I’m not a huge fan of this term), at times, students struggle with being able to adapt a tool to fit their project. Students may know how to use an Ipad for enjoyment, but not so much for editing, creating, re-editing, exporting, sharing, or saving. Being able to use technology tools for a specific purpose is important. Direct exposure to these tools may provide long-term benefit. With everything expected, it’s a bit silly to put the responsibility of this on just one teacher. I feel like a collective effort is needed to provide students with opportunities to explore different content creation tools.

Sidenote: I did have a few third graders ask what these “new” tools were, but they decided to go to Power Point when finding that they’d have to research the tools on their own.

Surface Area and Conceptual Understanding

Surface Area

My fourth grade class has been exploring a measurement unit for the past few weeks. We’ve been discussing the difference between area and volume. This has been a bit challenging as many students can apply area and volume formulas but struggle when finding surface area. Students were confusing area and volume and weren’t sure when to use a specific formula.  The idea of area being squared and volume cubed has been emphasized but still not cemented.  It seemed that students knew much more about the formulas and not as much about the conceptual understanding. To strengthen students’ understanding my class started a surface area activity late last week.  Click the image below for the template.


Students were asked to pick one box in front of the classroom. I had many different boxes to choose from. Many of them were board games or boxes I borrowed from other teachers. It’s near the end of the school year and some teachers are moving classrooms so there were plenty of boxes. All of the boxes were rectangular prisms. Once students picked a box they took a picture and then found the dimensions. Students then took one piece of butcher paper and created a net based on the dimensions found earlier.

Photo May 20, 11 22 59 AMPhoto May 20, 11 23 08 AM

Students created the net and then wrapped up the box. Students were able to immediately identify whether their measurements were off or on target. It took some groups multiple attempts to find a correct solution. After students wrapped up their box they took a picture. Before and after pictures were sent to me via Showbie. I printed them out and the students placed them on their sheets.


It would have been great to print these out in color, but at this time of the year our school’s color printer is out of ink.  After the activity students reflected on how their perception of area has changed over the past week.  After listening to a few student reflections I’m deciding to keep this activity for next school year.

Creating Common Assessments

Focusing in on Common Assessments

Focusing in on Common Assessments

Yesterday was a teacher institute day. Along with middle and high school teachers I took part in a session dedicated to discussing common assessments. The session covered topics of what role common assessments play and why they should be given. We discussed what qualifies as a common assessment and the need for teachers to be involved in the creation process. As we delved deeper into conversations I found that many of middle and high school colleagues create their assessments since there isn’t really a textbook that covers all the standards that they teach.

This often isn’t the case at the elementary level. In math, I find that the content publisher creates assessments and teachers rely on giving that piece to students in the form of quizzes/tests. Although the publisher-created content is decent, it can sometimes provide little value to the teacher beyond writing a score in the grade book. In addition, the teacher may have been required to give the assessment per district protocol.  In many cases teachers might not have any type of ownership to the pre-created content.

Later in the session the participants were given the opportunity to create their own common assessment. During the process we filled out a common assessment mapping tool. The mapping tool included fields for the learning objective, item number and type (i.e. multiple choice, short answer …) , item domain (i.e. informational or skill item), item depth (i.e. recall, understanding, strategic thinking, evaluation/creating), and point value.

While filling out the map we had to keep in mind what type of question was being asked. We eliminated some of the multiple-choice questions and decided to add questions that give students opportunities show their mathematical thinking. After picking the questions we looked at the item depth. The item depth determines the depth of understanding that the teacher is seeking. At the end of each question the team decided on a point value for that assessment item. Near the end of the session our mapping tool looked something like this:

mapping tool

I could see my team using this mapping tool for additional common assessments. Not only does it give our team more information that can be used to adjust/inform our instruction, it’s also valuable to the student. After completing the common assessment, students can reflect, set and make goals.  Creating common assessments may also provide opportunities for teachers to take more of an ownership role because they helped in the creation process.