First Few Days of School


In less than a week my school year starts. The first week is so important in helping set the tone and stage for the school year. Usually I take out my lesson plan from last year to start planning out the present school year. Some of the activities are the same from year to year and others I tend to ditch. This post/plan is by no means set in stone, but it’ll be helpful in planning as school is just around the corner. Ideally, I’d like to get to everything noted in this post, but honestly I doubt that will happen.  Flexibility is key here and this is a rough outline.

Keep in mind that I usually see four different groups of students during the first day of school. Each group stays for their math block, which is about an hour.

Day One

For the past few years I’ve always had music on as students enter the classroom for the first time. This year will be no different. Students will enter the classroom and find their own seat. The seats aren’t marked. Once everyone arrives I’ll quickly introduce myself and ask the students about their summer. I ask the students to write down one activity that they participated in this summer that they’d like to share.  Students write this down on a Post-it note. I then take all the notes and read off the activities. Each student then claims their activity and tells the class a bit more about their experience.

The class then reviews the arrival / dismissal flow chart.   This is a time where I open up the floor for any questions. We then have a conversation about procedures within the classroom. This takes about 10 minutes. The class then participates in a hands-on geometry game. It’s similar to a Simon Says, but with geometry terms and movements. The students tend to enjoy this and it’s a time for them to get out of their seat and engage in a different activity related to math.

After a few rounds of the game we all find our seats again and I pass out the student consumable math journals. Students then take out their math supplies and start organizing their accordion file.  I model how the accordion file should look and place the tabs in the correct places. Students label their accordion file tabs and organize their materials. I give each student a class information sheet, curriculum guide and contact sheet. Students get all business-like and start organizing their files.

Then it’s picture time! We all line up in the front of the class and take a class picture. The picture is then usually used during Back to School Night.

Following the class picture students start filling out their hand. Students use a Sharpie and write their name on the hand and place it on the door. It remains there for the entire school year. In some sort of small way I feel like it also encourages ownership.


Completed hand project

After all the hands have been tapped up on the door we move to the next activity, the puzzle piece community builder. This has been a staple activity for years. The puzzle starts like the picture below.


I then cut out the pieces and each student creates their own according to the directions.  Students place their name, favorite place to visit, favorite math topic, an interesting drawing or whatever you’d like them to place on the piece.  All students in the class create a puzzle piece and then the puzzle is put together once everyone finishes. Once it’s finished it hangs in the room for the year.


Students usually have around 10 minutes or so to work on the puzzle piece before they leave to their next class. Near the end of class I remind students of the dismissal flow chart as they leave.

Day Two

While students enter the classroom I’m planning on having the arrival / dismissal flow chart clearly visible. Today students will help create expectations for the classroom. This takes up a good part of the class, but I feel like it’s worth the time commitment. Once the expectations are established, students sign their name and this document is posted on a bulletin board for the year.   I’m planning on having students practice logging into their online math accounts today. This is important because the math student reference book is only online.

Students will also continue to work on their puzzle piece. Today I’m planning on introducing Estimation 180 and the student recording sheet to the class.  I haven’t yet decided on what picture to use, but I’d like to incorporate this periodically throughout the school year.  By end of the class students should (emphasis on should) have finished their puzzle piece. Today I’m also taking pictures of students as they work. I’m looking forward to using our class Twitter handle and Instagram to document our learning journey.  Students will be asked to compile Tweets in their own words that I will send out throughout the year.  This is another way to document our shared math experiences.

Day Three 

Again, students will follow the flow chart that’s posted. I’ll remind students of the expectations that were created yesterday. Students will start to compile the community puzzle of the classes. Today I’ll introduce the math journal to the students. Students will write about their past experiences with math and maybe even write a short version of their math autobiography.   This is a good opportunity to talk about the learning process and how mistakes are valued in this class. I want students to be able to use the math journal as a reflection tool and a place to record their mathematical learning. While students are writing in their journal I generally play sometype type  of music in the background. Students find a comfy place in the classroom to setup their journal time. Once finished, the class will move to a math game/station discussion. Each grade level will play a math game related to their current goal. Some of the more regular games that we play are Angle Tangle, Factor Captor and Name that Number.

Day Four

Today is dedicated to the Marshmallow Challenge. Before completing the activity the class will have a discussion about the importance of being part of a community that’s supportive. We also discuss the math implications of building a tower out of food items. At the end of the time the class will measure all the towers. We then fill out a plus/delta chart indicating what worked and didn’t work. Students usually end this class by having a conversation about team work and building a classroom community of support/trust.

Day Five

Students will delve deeper into their mathematical understanding by completing different types of open-ended/response problems (similar to 1 or 2) in small groups. Students will be asked to explain their thinking and find a solution. Student groups will present their solutions to the class. Many of the open response problems have already been compiled and are found in the district-adopted curriculum. Afterwards, students will be asked to document their experience in their math journals. Students will also login into their Showbie account on their Ipads. Students will be using the iPads to turn in certain math projects throughout the year. Students will be asked to take a picture of their work, annotate their picture and turn it into their Showbie account.  This will also provide students with an avenue to share math work with others.




Thoughts on Questioning Techniques in the Classroom

photo credit: mag3737 via photopin cc

photo credit: mag3737  cc

Every year I find that pairing the right math activity while asking specific questions can yield some amazing student learning experiences. I would assume that most math teachers would agree that only giving a specific solution to a student doesn’t necessarily help them understanding concepts. Offering solutions without feedback or questions can encourage students to care only about finding the answer. The act of “answer finding” limits understanding and diminishes curiosity.

When I started teaching I spoke constantly. I would give examples and statements that I thought would help all my students. Looking back, I spoke more than I should. As I progressed in my career I found that constructing a mathematical understanding doesn’t always ignite from just listening to the speaker.  There’s a time and place for listening, but being engaged in the learning process is vital.  I soon found that a balanced instructional approach was needed so I decreased the amount of talking and started to ask math related questions instead.

Although statements are beneficial, effective questioning techniques can provoke a response from the student. Offering guiding questions, or questions that encourage students to delve deeper in their explanation benefits the student. I feel like part of my job is to create an environment where students are able construct mathematical understanding. When students struggle with that understanding, questioning techniques can be another tool that teachers utilize. Questioning also helps students think more independently and explain their mathematical reasoning in a verbal or written form. Students need to be able to explain why and how they find solutions.  This type of communication is an important skill to develop.  Before planning on using questioning techniques in the classroom there are some important points to consider.

The environment

Students have to be open to answering the questions that are posed. In order for questioning techniques to work, students need to feel comfortable enough in the classroom to offer their ideas and explain their mathematical thinking. This environment is often intentionally built by creating a positive classroom learning community early in the school year.  Students will often participate less if they feel as though their input isn’t valued.

The timing 

Teachers can spend extensive time planning, but I find the best times to use effective questioning techniques are in the moment. Learning can be messy and teachers need to be able to have questions available depending on where students are in their mathematical understanding.   I’ve seen great question techniques used in whole class and small group settings.

 The questions 

The questions that are posed truly matter. When I started teaching my questioning techniques were less than stellar. Through time I’ve learned to expect more from my students. When given a chance, students are fully capable of expressing their thinking. Teachers need to allow students opportunities to do just that. The questions should prompt a response from the student beyond yes or no.  I want to get the students talking about their math process and learning.




Other classroom questioning resources are below.

Effective Questioning Techiques
Asking Questions
Using Questioning to Stimulate Mathematical Thinking
Leveled Math Questions












Teaching and Scuba Diving


Last week I took some personal time to disconnect and spend time with my family before school starts up again. While away I decided to swim and snorkel. I’ve always enjoyed snorkeling and watching the life under the ocean. We stayed at a particular facility that had a scuba diving class. My wife and I decided to take the class and learn about the scuba diving experience. The class was great and the instructor gave a basic overview of the equipment and we practiced different skills in a pool before heading out into the ocean. The ocean experience went so well that we decided to pursue a scuba certificate.

The instructor communicated that a certificate would require us to read an instructional book, take a test, practice skills in a pool and then show mastery of the skills in the ocean. That evening I read through the book. The book began with a section on performance-based mastery.  The section explained that students will be instructed using student-centered learning strategies.  Meeting the objectives are what’s important – not how long it takes.  As I read this I felt a bit of weight come off of my shoulders.

Each section had questions that required an answer and then you checked your own answers on the following page. At the end of each section there was a review of the content. After a few hours the book was complete.  The next morning I reviewed the book and my responses with the instructor. Time was given to ask questions. I was then given a 30 question open-book test. In all honesty, I have to admit that I used the book to find some of the answers. The test wasn’t timed and I didn’t feel anxiety during this process. After the test was complete the instructor quickly reviewed the answers and then we jumped in the pool.

The instructor went through specific skills that he showed me in advance. We learned how to clear a mask, check air gauges, and establish buoyancy.  There were many other skills, I just can’t remember them all as I write this post. I’d like to say that I perfectly practiced each skill on the first try, but I didn’t. In fact I commend the instructor for his patience with me. After each failed attempt the instructor gave feedback and we tried again. After about 3 hours we were finished and ready to show the skills in the ocean. My wife and I then went down to the bottom of the ocean to demonstrate the skills needed for a certificate.


After the ocean dive, I reflected on this learning experience and came up with a few takeaways:

  • I never received a grade or even an official percentage during this process
  • The feedback I received during this journey was clear and could immediately be used
  • I practiced the skill until I was able to independently complete it on my own
  • I was given time to reflect on my performance and ask clarifying questions
  • My documented proficiency was based on my last dive

I also thought of how much more pressure I would have felt if I was graded on each section of the certificate process. I would most likely fixate on each individual grade and not necessarily the skill. Instead, this process had me focus on the skill and the proficiency of that particular skill. I was able to fail, reflect, ask question and retry until I became proficient.  I found the low-risk opportunities to practice were beneficial during this learning experience.  This led me to ask …

How often do educators utilize these strategies in the classroom?  

This is just something I’m considering before school starts in a few weeks.

By the way, we passed the final dive and are looking forward to diving at some point in the future.

photo credit: CaptPiper via photopin cc




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?