Math Acceleration or Enrichment

Acceleration v Enrichment

Acceleration vs Enrichment

A while back I was asked a question about student acceleration and differentiation.  The question related to different types of acceleration opportunities for students that master math content before others. This question is often at the heart of differentiation for high achieving students.  I thought awhile about the question and started to brainstorm what opportunities truly exist or if acceleration is needed in those circumstances.   In an education setting acceleration is often associated with a curriculum that is moving faster or happening at a quicker pace than the norm.

In math at the elementary level, concepts are usually built upon one another and acceleration seems to be valued. Similar to a lattice fence, once one concept has been mastered, teachers often move the student to the next row/concept.  The goal is to continually move students in an upwards trajectory towards the next concept on the ladder.


Upward trajectory

When acceleration is the focus, students are asked to master and then move to the next numerical concept. For example, If student A has mastered 2.0A.A.1 they automatically move to the next concept, 2.0A.B.2.  Keep in mind that mastery is often defined by the author of the assessment.  Mastery could be correctly answering a few abstract problems in a row or answering 90% of the answers correctly.  In the author’s mind, the faster this process occurs over time the more the student learns.  This isn’t always the case and the perceived notion of learning might not actually be occurring. This is especially prevalent with online adaptive software programs. This type of philosophy often facilitates minimal understanding and can lead to problems down the road.  Also, students that are accelerated are often asked to answer questions more on an abstract level rather then explore mathematics constructively.  Creating a personal level of mathematical understanding is valuable.  Focusing in on only the abstract doesn’t always lead to a learning experience or a better understanding of math.

I believe acceleration has a place in the elementary classroom, but I don’t think that it should be the default.  Honestly, I feel like accelerating is easier than providing opportunities for enrichment. Instead of acceleration why not emphasize enrichment for students that have already demonstrated mastery? I think the word enrichment gets caught up in buzzword land, so here’s a formal definition:

Miram-Webster defines enrichment as the process that improves the usefulness or quality of (something) by adding something to it.

Enriching math instruction doesn’t necessarily mean that students quickly move from one concept to another, but instead it may focus on practical application and problem solving.  Developing strong problem solving skills enhances the usefulness of mathematics.  I find that students benefit when given opportunities to enrich their understanding of mathematics.  In addition, enrichment provides opportunities for students to practice relevant skills that become immediately useful.  Logical thinking, abstract reasoning, and problem solving can all be part of the enrichment process.  All of the skills that are practiced through enrichment activities can be used cumulatively throughout a math curriculum sequence.  The picture below is just one example.



Students often need to have a foundational understanding of mathematics to be successful at the middle and high school levels. Logical thinking and abstract reasoning skills tend to contribute to the background knowledge for algebra and geometry concepts.  Problem solving is a skill that’s used throughout school and life.  Enrichment opportunities encourage students to use the math learned and apply it to practical situations.  It also enables students to solve problems using trial and error and find multiple solutions.  Perseverance skills are also practiced during math enrichment opportunities.  Instead of completely emphasizing the upward trajectory of concepts, students that experience enrichment opportunities develop skills laterally and may cement a more solid mathematical foundation in the process.  It may also enable students to see mathematics in a new light, not just a lattice of concepts placed in chronological order.  Feel free to review MathwireNRichMaths and Andrew Stadel’s Math Acts,  for a few different examples of how to incorporate math enrichment opportunities.

There isn’t really one right answer to the question found at the beginning of this post.  The solution includes a possible combination of acceleration and enrichment, but immediately leaping to acceleration might not be the best option.

How do you use math enrichment in the classroom?





National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors

Enrichment. 2014. In July 21, 2014, from

photo credit: Filter Forge via photopin cc

Professional Development Conversations


Yesterday I was able to participate in a SAMRiCamp teacher workshop.  Similar to last year, DG58 hosted the event and it was well attended.  It’s great to see so many educators and administrators taking time out of their busy schedules to attend this professional development opportunity.  There were many sessions available and facilitated by educators and administrators in the area.  The sessions provided educators with a variety of options to choose from. For the most part, the facilitators of the sessions had organized presentations displayed on whiteboards that were shared through Google Drive. As usual, the entire conference was paperless.  Discussion generally followed the presentation with the audience sharing feedback with the group.  The majority of the sessions included a hefty dose of teacher conversation.

I find that this type of teacher development model is different than the norm.  This type of model can benefit educators in ways that weren’t possible a few decades ago.  My past teacher trainings generally consisted of specific workshops for teachers within a particular district.  The presenter spoke for the majority of the time with a handout and limited audience participation.  Instead of having one district provide training for their specific teachers, the SAMRiCamp teacher camp model encourages more of the conversation element with participants from multiple districts.  Different perspectives, programs and ideas can be heard when participants offer responses in the sessions.  Gathering teachers and administrators from the local area/state can reap benefits for all participants.

The conversation and collaborative part of this type of professional development is important. Including time to discuss, ask questions and share ideas can evolve into teacher reflection opportunities.  During these teacher-led conversations, teachers can experience affirmation and may also meet constructive feedback from others that they can bring back to their school.  Pushback, or asking deeper questions that lead to justifying a response can also play a role during these conversations.  Discussions can lead to deeper connections with other teachers outside of their district.  This action also provides opportunities for teachers to expand their personal learning networks.  Being able to candidly discuss matters related to education with other professionals can improve practices. Since many districts are represented, different instructional models and ideas can be brought to the table for discussion. Since educators are both introverts and extroverts, the discussion doesn’t necessarily have to always be verbal.  The conversations and questions could take the form of a shared Google Doc. I believe all teachers have something to share and getting comfortable enough to share can be a positive tipping point in the professional development conversation.  Taking the risk to share/present and receive feedback can benefit all stakeholders in the room.  At the same time, I think it’s important for teachers to be able to say that they don’t have all the answers. The unanswered questions can often help develop an atmosphere of brainstorming, which inturn helps the group.  Reflecting on past practices and sharing/learning from others can lead educators to change their practice for the better.  Feel free to review the #SAMRiCamp tag for a brief overview of what was discussed.



Bridging Procedural and Conceptual Understanding

Yesterday I was putting together a few math projects when a Tweet caught my eye. The Tweet below started a short conversation that I thought was interesting.

David’s Tweet had many responses.  Most responses revealed that educators tend to side with solving one problem ten different ways rather than having students solve ten similar problems.  I started to reflect on how teachers give assignments that ask students to complete repetitive problems that often reinforce procedural mathematical thinking.  I also started to think how in an effort to provide practice, teachers may focus on procedural aspects first and then move towards practical application.  I find this happens frequently with math concepts at the elementary level.  What I don’t find often is the viewpoint that practicing procedural aspects can be embedded in solving specific problems multiple ways.  This type of thinking reminds me of number collection boxes.

Regardless of the assignment I want to be able to give specific feedback.  A larger problem that involves multiple steps can provide opportunities for teachers to pinpoint where misconceptions are and give direct feedback.  This isn’t always possible with ten similar shorter problems.  Below is an example of a few problems that you may find in a fifth grade classroom.  I don’t condone using these types of problems as they are definitely utlized, but I think we need to ask what’s being assessed when students complete this type of problem?  Students are simply asked to find the volume and show a number model.  I appreciate how the problems ask students to show their number model, but these types of problems seem to measure procedural understanding.  Do students know the formula?  Yes, well then they can answer many of these problems, even 10 in a row.



I think the above problems have a place in the classroom, but shouldn’t necessarily be the norm.  Usually these types of problems are found on homework sheets.  The problem below which was adapted from a recent fifth grade test is more challenging, but gives students opportunities to showcase their own mathematical understanding and persevere.  Some would say that these two problems are completely different.  I would agree, but similar concepts are being assessed.  They do look different and the second requires more skills to complete.  Students need to be able to use their procedural understanding and apply it to the situation.  Also, one key element that’s missing from the first problem is the student explanation.  Students are required to show their mathematical thinking in the second problem.  This is big shift and can reveal student misconceptions more clearly than the first problem.  I struggled with the decision, but eventually had students work in groups to complete the problem below.  Students were allowed to use any of the tools in the classroom to find a solution.



At first, all groups struggled with this problem.  Near the end of class all the groups presented their findings.  What’s interesting is that all the groups had different answers and ways in which they came to their conclusions.  I was able to offer opportunities for students to see and ask questions about different math strategies.  During the next class I was able to pull each group and give feedback.  This activity took a good amount of time to complete, but I feel like it was worth the commitment.

Through this experience and others I’m continuing to find that it takes a “bridge” to connect the procedural and application pieces.  At times I feel like there’s an assumption that if students are able to answer 10 similar procedural problems that they will be able to simply apply that knowledge in a multi-step problem.  This isn’t always the case and sometimes the bridge doesn’t fully form immediately.  Performance tasks, similar to the problem above can be one way in which teachers can help the transition from procedural understanding to practical application.  Being able to apply that knowledge to a math performance task can be a challenge for some students.  When teachers focus so much on the procedural, that’s the only context that students see and practice.  A blend between procedural and application needs to be established within the classroom.  I feel like activities like this help bridge this gap.

How do you bridge mechanical and conceptual understanding?

Classrooms that Encourage Risk-Taking Strategies


Creating a Classroom Environment

Encouraging Risk-Taking in the Classroom

A positive classroom environment often plays a pivotal role in student learning.  Fostering a classroom climate that promotes the learning community can reap benefits for all stakeholders involved.  Feeling a sense of belonging to an organization can increase participation and build confidence.  Primary and elementary grades often spend a good part of the first few days of school focused on creating a classroom community. Building that classroom community can take many forms.  Joy Kirr’s Livebinder provides many classroom community building activities that I found helpful.  A focus on team building, sharing and reflection can all aid in building a productive learning environment that will set a strong foundation for the school year.

This isn’t necessarily easy as there’s always curriculum to cover, but setting aside time to create a classroom climate is worthwhile.  Once established and continually reinforced, it can be a driving force in which students take academic risks in the classroom.  Whether its student council, clubs, art class, or whatever, that sense of belonging often enables students to participate at higher levels as they feel that their voice is truly valued. When I speak of risk, I think of the term in a positive way.  The risks that I’m speaking of often help students move beyond taking a stagnant stance with their education.  Student risk can take many forms in the classroom.

Taking a risk could mean that students:

  • Answer/ask questions more often
  • Are more open to feedback given by peers and teachers
  • Are able to collaborate with others
  • Show perseverance when approaching challenging tasks
  • Take more ownership of their learning
  • Able to explain their mathematical thinking in more detail
  • Take pride in their work more often
  • Reflect on their performance and set goals
  • Rise above their own personal expectations
  • Start to develop leadership skills

For some students a risk is to raise their hand in class. For others, students might engage in mathematical conversations with their peers or use feedback as a learning tool.  Another student might want to take what was introduced in class and start an enrichment project.  Personal risk is truly determined by the student. To make sure that students take academic risks they need to feel as though their community supports them.  Modeling how to approach risk-taking in the classroom is important.  Sharing personal stories and continually reinforcing that making mistakes is part of the learning process can help create opportunities for students to take risks on their own. Teachers can start by creating low-risk opportunities in the classroom (See Reed’s post for examples).  These tasks can be powerful and foster a positive classroom climate in the process.

How do you create a classroom that encourages risk-taking?



Reflections from Digital Leadership

Digital Leadership Takeaways

Digital Leadership Takeaways

About a month ago I started to read Digital Leadership by Eric Sheninger.  His book is full of leadership strategies that are applicable at any school level.  Specifically, he speaks of how to integrate technology in schools and the reasoning to do so.  While reading I took out of my highlighter and it was busy as they’re many gems in the book.   I thought the topics on the role of technology in the classroom and student content creation opportunities were especially intriguing. I’ve outlined my takeaways and reflections below.

1.  Combining pedagogically experienced educators with technology-savvy students can be beneficial

Students often come into the classroom with an average to above average understanding of how to use technology.  Their understanding of technology can benefit a classroom and the learning experiences within.  I like the concept of being able to combine background knowledge of technology-savvy students and pedagogically experienced educators.  Weaving instructionally sound teachers and technology can reap dividends.  Both parties bring an understanding to the table.  Merging both can can turn technology into a tool for learning.

2. Students need to be aware that technology tools are for learning  

I believe that students are aware of the capabilities of the devices that they use, although understanding how they can be used for learning is another story.  I think this is where it’s essential for pedagogically experienced educators to seek avenues to combine  the capabilities of the device with learning opportunities. The transition from a perceived consuming/gaming device to a learning device may take time.  Due to corporate marketing and education success stories, I believe that transition is taking place in the field of education.  Naming them as learning devices also reinforces the concept that technology in schools can contribute to the learning process.  Regardless of the device, the opportunities for learning exist.  Educators and students can benefit from revealing this possibility.

 3.  Students’ learning experiences become more meaningful when they use real-world tools to show conceptual mastery

It’s becoming clear that technology devices can be utilized to showcase conceptual mastery. This year my students created online tutorials and various projects to demonstrate their learning of mathematical concepts.  Based on my end-of-year survey, students found the content creation projects meaningful.  Seeing that they were published online and available for comments provided opportunities to showcase their projects for an authentic audience.  To be honest, not all projects were optimal and I’m going to make changes for next year, but I was encouraged to see students use real-world tools to demonstrate learning.

4.  The aim is that students move towards creating an actual product.  They need opportunities to show what they’ve learned in a variety of forms

Students in many classes are expected to show mastery of particular concepts through worksheets, usually categorized as unit assessments.  Many times this is mandatory, in the form of district summative testing or state-wide standardized assessments.  Students should be afforded the opportunity to showcase their learning beyond worksheets. Technology devices and apps offer presentation tools that didn’t exist before.  These student content creation tools also give students opportunities to infuse their projects with voice and creativity. This aspect brings student ownership and an opportunity to extend their learning beyond the requirements.  I’ve found that student content creation can showcase learning while providing a lead to engage students in their own curiosity regarding a particular concept. With flexibility and clear expectations, this  type of product can show learning and at the same time be a publishing opportunity for students.

photo credit: Jamais Cascio via photopin cc

Math and M.C. Escher


Math and M.C. Escher

Math and M.C. Escher

During the last week of school my students started to explore topography concepts. Topography usually isn’t the first thing that is thought of when someone mentions the word math. That’s why I find it so interesting.  I truly enjoy teaching this topic because it often brings out the best from my students.  I find that most upper elementary students tend to thrive when given geometric shapes and asked to explore, rotate, translate or even turn them inside out.

I generally introduce the unit with M.C. Escher.  The class learns a bit about the life of Escher and his contributions to the world of art.  Moreover, we discuss how art and math are related. This is often a deeper conversations as students start to expand on the notion that mathematics can be found throughout our world.  Topics like the golden ratio and Pi often get brought up during this time.

After learning about Escher’s life and his influencers, the class looked at his different artistic creations. Usually my students recognize at least a few different creations.  Students seem to gravitate towards his optical illusion pieces or the famous Waterfall work.  As each work of art was discussed the more students found mathematics as an integral part of Escher’s work. After reviewing the different pieces of lithograph art, the class watched a short video on how Escher’s design and math are connected.

After the video the students were asked to have a conversation about how math can be found in most art.  The words symmetry, rotations, slides, translations, reversals, surfaces, and perspective were all brought up during the discussion.  What’s nice is that the vocabulary was brought up naturally as students spoke to one another.   I was able to highlight the words and facilitate the discussion as needed.

Eventually the discussion ended and the class moved to the next activity.  I planned to have the students create their own Escher-like artwork.  The students reviewed how to have “Escher-like eyes” when creating their own pieces.  I was proud of the student responses and the imagination that came forth during this discussion.  The class then reviewed the directions to create their own Escher-like creations.

The students went through the directions and asked questions.  Once the expectations were clear I passed out a 8 inch by 8 inch square to each student.  Students created their own tessellation template.  In the future I’m probably going to cut the square dimensions in half so the patterns become more evident.

This slideshow requires JavaScript.

Eventually the students used the template to create an Escher-like creation.  Students showcased their work to the class using the vocabulary mentioned above.  The students were able to bring their work home on the last day of school.  All in all, this is a lesson I’m intending on using next year and a definite #eduwin in my book.


How do you incorporate art and math?





Using Badges in the Elementary Classroom


During the last #msmathchat the topic of digital badges was brought to the forefront.  The idea of badges in the classroom has always interested me.  I was first exposed to the idea of using badges in the classroom by @mrmatera last summer at a Downer’s Grove PD event. Michael used a form of a digital badge/achievement token while integrating gamification in his own classroom.   His idea spurred on a brainstorming session with another colleague which resulted in the creation of badges for my own classroom.

Looking back, this past year was the first year I decided to use a form of badges within my classroom.  Back in September I decided to research a few different options for using badges.  After much review I decided to use the badge philosophy without a digital component.  Even though there are many digital badge sites, I wanted to start small and simple and using paper badges seemed like the right move.  I decided to create a simple badge template.  The one below is for the app Prezi.

Screen Shot 2014-06-15 at 11.25.48 AM

The  badges were going to be used to show milestones or proficient use of certain skills.   One larger theme in my classroom revolved around the idea of student content creation.  My students were using a variety of apps to showcase their learning through digital means. Since students were using content creation apps, the badges would show proficient use of specific apps.  I found some blank Avery mailing labels around the house and created a simple badge template and then imported it into Word.

Flowboard Badge Label

Flowboard Badge Label

Label Template

Label Template

The title of the class was on the top and the name of the app was on this inside of the logo. Students received a badge when they successfully created a math product with a particular app.  Students decided to put the labels/badges on their personal folders. Individual student folders started to fill up with badges as the year progressed.


Student folder w/badges

Student folder w/badges

Not only were the students proud of their accomplishments in creating mathematical content, but they were able to reflect back on all their badges and growth since the beginning of the school year.  It was encouraging to hear how excited the students were to receive a badge once they finished their project successfully.  Even more powerful was the reflection component that the students recognized as they wrote their final reflections at the end of the school year.  I’m still brainstorming how this idea could transfer to mathematical concepts without turning this into student competition.  Regardless, I’m looking at using a form of a digital badge next year, but using labels is my first step in that journey.


How do you use badges in the classroom?