The Real Number Line

Image by Winnond

Approximately two weeks have passed since the new school year has started and I’m finding that the traditional number line (that many teachers have become accustomed to) needs an upgrade.  My math students are benefiting from the number line, but true understanding of numbers doesn’t come from a number line alone.  For the past seven years I’ve used a “typical” number line from -10 to 100 in my classroom.

Don’t get me wrong … the number line is helpful in teaching many number sense concepts.  In my opinion, the number line offers students a visual/spatial representation of the number system.  I  believe many numeracy concepts are built from understanding the system of numbers.  What is often missed, or not necessarily taught, while utilizing the number line are numbers that don’t fit the category of being whole.  For example, I generally don’t see pi or irrational numbers being part of a number line.

Recently I found a “Real Number Line” poster.  I was fortunate enough to find this poster and have utilized it to teach elementary students about the number system. I think it’s important to communicate that square roots, fractions, percentages, mixed numbers, etc.  should be included on a number line.

I actually created a practical follow up activity in response to this post here.

Instead of purchasing a poster, you could have the students create their own.  A few examples are found below:


I believe that Wolfram Alpha does an excellent job of emphasize the importance of a number line in the answer it provides.  The answer can be represented on a number line.  See the example below.


I’ve been reading How the Brain Learns Mathematics by David Sousa.

David emphasis the importance of the mental number line.  All humans have number sense.  For example:  studies indicate that the brain can decide that 60 is larger than 12, but it takes the brain a longer time to distinguish that 76 is less than 79.  It seems that when the digits are closer in value the response time of the human increased.  Visualizing many different forms of number lines would be beneficial and assist in developing better number sense skills at a young age.

 I thought this quote was beneficial:

“The increasing compression of numbers on our mental number line makes it more difficult to distinguish the larger of a pair of numbers as their value gets greater.  As a result the speed and accuracy with which we carry out calculations decrease as the numbers get larger”

 – David Sousa

Creating Classroom Rules

 Image by Federico Stevanin

It’s only been three days since school has started and so much has already taken place.  In reality, it  feels like three non-stop days of meetings and teaching.  Many teachers are recharging this weekend to begin again on Monday.  After reflecting on the last few days of school, I’m now starting to plan specific learning opportunities for next week.

Many community building activities were emphasized this past week.  My class created classroom rules during the first day of school.  This year I created the rules with my class and they are now posted on a bulletin board.  Most research that I’ve read indicates that when students are part of the creation of the rules, they are more willing to take ownership and model the rules in the classroom throughout the school year.

Here’s the rule creation process that I used this year:

  • Each student is given one Post-it note
  • The students are asked to not write their name on the Post-it
  • Each student writes down one rule that they think would benefit the classroom
  • I collect the Post-it notes and read each one to the class
  • If any student doesn’t agree with a rule they may communicate why it shouldn’t be a rule
  • The class “approves” each rule through consensus
  • After all of the rules have been read, the class starts to categorize the rules (since we can’t have 23 unique rules!)
  • After categorizing each Post-it, my classroom rules look like this:

Ground Rules
Respect Yourself and Others
Follow Procedures
Be Responsible
Do Your Personal Best

  • The rules are posted on a poster in the classroom
  • Students place their signature on the poster and then each individual Post-it note is placed around the poster (like a border)
  • I (and often students) refer to the poster to reinforce classroom expectations throughout the year
I’ve found that this callaborative activity encourages students to participate in creating a positive classroom environment.  It also provides students an opportunity to be responsible for creating the ground rules for the classroom. This activity gives students ownership and slightly shifts responsibility from the teacher to the student.  Detailed directions for this activity are located here.  Practical examples of this strategy can be found below.
The names in the above photo have been blurred.  Having students sign the rules  often encourages accountability.

Additional Resources:

My Job and Your Job – Community Builder

Image by Sheelamohan

No, I’m not talking about classroom jobs, like the all familiar paper passer, pencil sharpener … etc.  I’m talking about using the idea of jobs as a community builder.

The new school year is here.  I’m once again finding myself digging up lessons from the past.  For the past six years I’ve used an activity that always seems to generate student interest and builds a positive learning climate.  This activity can also be utilized and referenced at back to school night.  I’m referring to the activity My Job, Your Job, Our Job.  Here are my subjective steps to implement:

1.)  Pass out a Post-it note to each student in the classroom.  Ask the students to write down 2 – 3  sentences (or it can be just a few words) that describe their teacher’s job.  In other words, what is the teacher’s job?  Often, you’ll get a few surprised looks and then the students get busy writing down their ideas.  I try not to model too much during this, as I want the students’ original thoughts and ideas.  I then ask the students to place all of the Post-it notes under the “Teacher Job” category on the whiteboard. You can write out the answers or just use the Post-it notes, as it adds to the authenticity of the activity.

2.)  Follow step one, but instead of writing about the teacher’s jobs, the students will describe their job.

3.)  Students will describe their parents’ jobs at home (not their employment).  This may require a bit of modeling, as some of the answers may be way off base (although that may add value and humor to the activity).  I remember one student of mine a few years ago wrote down that her parents’ job was to provide health insurance for her.  The parents thought that was hilarious, especially coming from a second grade student.

4.)  In my opinion, the “Our Job” portion is extremely important and what I generally emphasize in class. I ask the students to think of a common goals that all participants (teacher, student, and parent) share.  I ask the students to write down their answers and the class starts to conclude that all stakeholders seem to be working together (for the success of the student).  This is a unique learning experience and also provides a critical foundation during the beginning of the year.

5.)  Optional – Take a digital picture of the poster that was created and show it to the parents during back to school night.

A few examples are below.



Or students can fill out their own individual sheet …

Additional Resources:  MCPS Baldrige and Word template for activity.

Math and Art

Image by Graur Codrin

I have found that students enjoy and often thrive when presented with a challenging real world problem (often outside of the textbook). This can be observed during a problem based learning activity.  When students come to the conclusion that their isn’t one specific right answer, they are more willing to communicate their ideas and opinions to one another. Many practical problems outside of the K-12 education realm have more than one “right” answer.  When students are faced with problems that have multiple solutions the class community asks questions that often spark additional questions.  Active learning often comes to fruition through these activities.  In fact, the #realmath hashtag provides practical resources / images related to math found in the real world.  I have provided one example of using math through art below.  This idea comes for a PD opportunity that I participated in last year.

1.)   Begin by showing a small section of a picture.  Ask students the questions below or add your own (my questions are based on an elementary classroom).  Attempt to stay away from yes or no questions as students offer their own perspectives.  If you ask a yes or no question, follow it up with a why.  You might want to tell the students that some of the images don’t fit like a puzzle, as some pieces were cropped at different zoom levels – this adds to the complexity of the activity. I keep a separate chart on the board to write down math vocabulary that is used during this class activity.  Keep in mind that each picture is displayed one at a time, generally in a presentation format.  You may want to randomly have students answer the questions below.

Picture one questions:

  • What do you see?
  • Does this picture remind you of anything?
  • Describe the polygons in the picture.  Where are they?
  • What type of math vocabulary can you use to describe this picture?
  • Why is one rectangle in the picture lighter?
  • Where/when do you think this picture was taken?

2.)  Now show a small section of another portion of the picture.  Follow the same guidelines as step one.

Picture two questions:

  • Using math vocabulary, what do you see?
  • Using fractions tell me more about this picture.  Where can fractions be found in this picture?
  • Where/when do you think this picture was taken?
  • What similarities can be found between this picture and the first picture

3.)  Now show another section of the picture.  Follow the same guidelines as step one.

Picture three questions:

  • How does this picture similar to the first picture?
  • Why are some of the flags horizontal?
  • What type of information do you think the builders needed to construct this building?
  • How tall do you think this building is?  Why?
  • Do you think the building in the very front of this picture is the tallest?  Why?
  • What direction do you think the sun is shining?  Why?
  • What part of the picture do you think is missing?
  • Where do you think this picture was taken?

4.)  Reveal the full picture

Picture four questions:

  • How accurate were your initial predictions?
  • What are the differences/similarities between pictures one, two, three and four?
  • What additional math terms can you use to describe items in this picture?
  • What materials would you need to construct a building like the one in the picture?
  • Why is there a reflection on the building on the right?
  • How could you estimate the height of the building in the center of this picture?
  • Optional -Reflect on today’s activity in your journal.  Describe your reaction and what you learned during this activity.
  • Optional –  Similar to this example, students could take pictures around the school and create their own presentations on finding math in art.  Students could be given a rubric and work in collaborative groups and present their findings to the class.

Of course feel free to modify or change any of the steps above to meet the needs of your specific students.  My example is only a general template.  I’ve used this in elementary classrooms to introduce specific topics.  You could use a variety of images for this project, or have students create / take pictures on their own.  What about using Escher’s artwork below? As you can see, there are a lot of possibilities.



update 12/29 – An additional resource – Mathematics Meets Photographs

Feedback … from students?

When are students asked for feedback? This often happens at the college level, but not so much during the K-12 experience.  I believe students, at all grade levels, need an opportunity to express their opinions and ideas in the classroom.

I would assume that most professionals in the education field would agree that when students feel safe in school they are more likely to learn and achieve at high levels.  Many teachers perceive the beginning of the school year as the starting point in building a collaborative environment for learning. Teachers will use a variety of methods to get to know their class.  Some teachers will utilize a puzzle strategy while others use collaborative games. Sometimes it’s a challenge to continually remind students that the classroom is a community (especially around breaks and near the end of the year!).  In my experience, I’ve found that plus/delta charts are a great tool to remind students that their input is valued. These types of charts are also utilized outside of the education realm, as seen here. I’ve found that at the very minimum, plus/delta charts are a valuable community building tool.  My practical steps to incorporate plus/delta charts in the classroom are outlined below.

1.)  Start out by drawing a chart with a + and a triangle near the top of  the writing surface

2.)  Ask the students for positive happenings in the classroom

The + represents the positive aspects of the class that the students enjoy.  The + could pertain to certain activities or projects that were assigned.  It could also represent class goals that have been achieved.  All of my examples below include “we” meaning the class.

Examples (more geared towards elementary):

+ The class was respectful during the field trip

+ We worked well in groups today

+ We enjoyed the music being played during independent work

+ We brought all of our supplies to class for the past month

3.)  Move on to the delta or negative aspects

The triangle represents items that the class needs help with.  For example, students might feel that talking when the teacher is talking is disruptive. Or students might comment that the class needs to become better at turning in assignments on time. The delta starts to become more of a problem solving piece if this process is used on a regular basis.

Examples (again, geared towards elementary):

– We were a little loud during line-up today

– We forgot to complete the homework

– We need to put more effort into our work

– We didn’t listen to the teacher’s directions before starting the assignment

4.)  Students can set goals based on the plus/delta chart.  

This can be accomplished by utilizing goal setting strategies.

If you read this far into this blog post then you probably want to see practical examples of plus/delta charts.  Here you go:

If you’re looking for a possible template to use, click here.

5.) (Optional)  Students reflect on what was discussed in class through a self-reflection journal activity.  The journal activity could actually be integrated into a language arts connection.

6.)  Hang up the plus/delta chart in the classroom as a reminder tool and refer to it as needed

After a debriefing session, I generally cover the old chart with a new chart.  Students are able to view what progress was made over time by comparing the two charts.  A chart is completed every month in some cases, but the time elapsed between charts really depends on the teacher’s preference.

One Way to Personalize Learning in the Classroom

                                                                                                                                                                                                                                                                                             Image by Sujin Jetkasettakorn


I recently participated in a chat about personal learning.  After the chat I spoke with colleagues about what is the best method (or one effective method) to integrate personal learning in the classroom.  I also read an insightful blog post by Darcy Mullin regarding that same topic. After clarifying the term personal learning, which took some time, my colleagues suggested that any integration needs to start at the beginning of the school year. So that’s where I would like to start …

While looking at my resources, I found that @tombarret has some great ideas in getting to know your students early in the school year.  Once the concept of community is cemented, how does an educator personalize students’ learning experiences in their own classroom?  The five actions below enable teachers to take risks and become more student-centered.  Students start to become more interested and responsible for their own learning when learning is personalized.

Five Actions (not steps) * Keep in mind that each action requires modeling

1.)  Survey stakeholders near the beginning of the school year

By surveying students and receiving honest feedback, educators will be better able to understand students’ needs (not just academic).  The background knowledge gained will enable educators to differentiate the curriculum based on specific needs.  If you’re looking for a good survey tool, check out the Google Docs survey creation tutorial.  Also, surveys give educators a heads-up to possible concerns early in the school year.

2.)  Analyze Formative Assessment or Pre Assessment Data

Students are generally given some type assessment related to the topic of study near the beginning of the school year. Give students an opportunity to analyze their own data and write a reflection on how they feel about their score.  It’s important to communicate that the reflections will not be graded in order to gain reliable information.  Instead of having students say I did well, you could direct the student to describe what in particular was exceptional.  Students should also reflect on the teacher’s feedback.

3.)  Students set goals

After reviewing the reflections and teacher feedback, students set attainable goals for themselves. The key is to have the students create goals that are attainable. Don’t underestimate the power of student goal setting.  I’ve found that student goal setting often leads to more responsibility.  Modeling is vital for this action.

4.)  Offer Choices

After analyzing the assessment data and reviewing personal reflections, students are given the opportunity to choose how they will be assessed.  I understand that some assessments are not optional, but the teacher can utilize a variety of formative assessments designed to meet the students needs. The students will choose one assignment (could even be a collaborative group assignment) to complete from a variety of potential assignments.  The assignments should vary and could even target specific learning styles. After the assignments have been completed, students will reflect on their learning and the teacher will use a rubric to score each assignment.  Students feel (and are) in more control of their learning when they can choose their assignments. This action helps put students in the center of the learning process.

5.)  Offer Student Conferences

Students may view their assessment results and reflect on how much they have learned.  Students participate in individual conferences with the teacher to monitor the learning that has (and is) occurring.  This is also a good time to check-in to see how the student’s goal is progressing.  Students can also suggest alternative curriculum topics to explore.



Remember Constructivism?

What do you remember learning during your K – 12 experience?  You might cringe a bit as nostalgic memories come to mind…..  Now think about what you learned academically during that same time period. If you’re like me, most of what is remembered is attached to some type of positive (hopefully) engaging learning experience. Those memories have stood the test of time for some reason.  Educators understand that teaching is a process of making personal meaning. Those “personal meaning” experiences were most likely created by teachers who planned interactive lessons that engaged students.

I may be in familiar company, but I assume some learning experiences have left my memory banks altogether.  Here’s a brief list of what I remember:

  • Working in groups with other students – collaborate group assignments
  • Interactive projects that were presented in class
  • Concepts learned in school were tied to school / community based projects
  • Using technology in some form to create projects
  • Using games to learn

Even though it’s been many years since my K-12 experience, surprisingly, I still remember the concepts associated with the instruction.  From what I remember, most of what I enjoyed (or decided to actively learn) during my K – 12 education was generated by teachers who utilized some version of Constructivist teaching theory. Constructivist teaching has been in the news recently, specifically in education circles.  In fact, billionaire philanthropist Bill Gates has been quoted to have an opinion regarding Constructivism in the classroom. You can find his opinion in the second paragraph of this article.  Despite recent media coverage, Constructivism is still valued by educators and is utilized in many classrooms around the world.  Not one tool, even Kahn Academy, will improve the education system overnight. I think most educators would agree that how the curriculum is communicated is one factor, among many, that impacts student learning.  If utilized correctly, Constructivist teaching strategies can be a terrific tool, enabling teachers to developing engaging lessons that improve student learning.

Additional Resources on Constructivism can be found below.

What does a Constructivist classroom look like?

More examples

Goal Setting for Students

We all set informal / formal goals, whether it’s to get through today’s workout on the elliptical or to have a smooth school year.  Our goals are usually something we strive for, an end to some type of means.  I’ve found that goals change as people change.  Goals can be placed into different categories, such as academic, fitness, health, financial …. the list goes on and on.  Have you ever made an academic goal?  For some, the answer is a hesitant … yes, I think so … to get through graduate school or something like that.  Effective educators need to be able to communicate the need for goal setting.  Why is goal setting for students important?

1.)  Gives students responsibility for their own learning

When students analyze their own data (assessments, homework, class participation, etc.) they often become more interested in the analysis because it’s relevant to them.  While reflecting on the data, students have an opportunity to set goals for themselves.  Teacher modeling is a vital component of this procedure, although when left to look  at their own data, students often make essential connections and can relatively pinpoint where they personally struggle.  While introducing the concept of student goal setting, teachers can model from their own lives when they’ve had to overcome a goal.  Britt Pumphrey and Jonathan Ferrell’s blog has a few practical visuals that can assist in communicating student goal setting.  Students seem to express interest when they see that their own teacher has had to overcome some type of obstacle and it relates to the topic being discussed.  After students set their goals, they develop a plan to achieve their goal. The teacher and parents are all aware of the goal and help support the student through this process.  By creating goals, students  start to take on more responsibility for their own learning.  In the example below, new goals are created every 2-3 months.

Math Example 

After a general math pre-assessment or assessment, students are given the opportunity to analyze their own data to see which concepts they understand and which concepts need strengthening.  A student might observe that most of the problems missed are related to multiplication and division concepts.  The student decides that the goal is to improve the efficiency and accuracy of solving multiplication / division problems.  The student sets a goal to improve in that specific area.  To achieve the goal, the student decides to practice multiplication / division problems twice a week for 20 minutes on the computer and to create and solve two practical word problems a week relating to the goal.

 2.)  Shows students that effective effort leads to achievement

When students analyze their own data, they can observe over time that appropriate effort (i.e. practicing computation math problems / creating world problems / other factors) leads to achievement.  The students will will also observe that practicing good habits (following through with their action plan)  positively affects the outcome of their goal.

3.)  Gives the student a skill that they will need as adults

Educators and administrators set goals and this should be modeled for students.  The students that we educate today need to understand the importance of setting goals, and more importantly, how to achieve them.  Not only is this academic related, but this is also as skill that will help prepare our students for life outside of the classroom.

Twitter and Professional Development

I’ve observed and participated in a number of Twitter chats this year.  To be honest, I’ve expected a conversation with individuals who may be part of a PLN that are willing to express their perspectives on education. Scheduled chats generally have moderators and participants are free to express their opinions and may even ask questions aligned with the topic. Constructive debate is sometimes encouraged as educators often question the norm (or are at least not satisfied with the status quo).  At times, resources might be shared and links bookmarked.  While contributing, I share background knowledge and resources that have improved my teaching practice. Generally I come away from the chat with additional resources and ideas that I can practically utilize in the classroom.

General Chat process (informal list)

  • Moderators ask questions / introduces topic(s) to start the chat
  • Participants offer their opinions / experiences on particular topic
  • Affirmations / connections become evident
  • New ideas / resources become available via comments or links
  • Ideas on how to apply newly gained resources / perspectives become evident
  • Participants express interest in next chat topic

Here’s an example:

My latest Twitter experience at #elemchat  followed the above process, yet challenged participants to take purposeful action after the chat.

The topic during this specific night was about social bookmarking.  The moderators did a fine job asking questions and guiding the discussion accordingly.  Background knowledge was expressed by individuals who have had experience using social bookmarks.  Social bookmarking links and student examples were shared during the chat.  Diigo, Delicious, Symballo, Google and Scoop bookmarks (there may have been more, but I’d have to look at the transcript) were all discussed and analyzed during the chat.

One participant even stated:

“Can’t believe all the SB (social bookmarking) sites I was totally unaware of!”

In my opinion, what was said above is intriguing.  Gaining a better understanding of how to practically apply social bookmarking was one of the purposes of the chat.  Not only were participants gaining knowledge, but they were able to find ways to apply learning to improve their own practice.

Approximately half way through the chat, one participant thought that an#elemchat bookmarking site should be created.  Another participant created the bookmark on Diigo and since it’s been created a number of people have added valuable resources that educators can utilize to improve student learning.  My latest chat gave me a sense of how a scheduled chat can transform into a professional development opportunity.  Administrators and educators alike can see the value of Twitter chats.   If you’re on the fence and wondering if an educational chat is actually worth the time and effort, I would recommend getting your feet wet and become an active participant.

The New Community of Learners

As many (or few?) of you know, a storm passed through the Chicago area recently.  The storm produced massive winds that caused devastating damage.  Electric power and internet has been inaccessible in some pockets of the Chicago region for the past few days.  The above image jumped on my computer screen shortly after the storm passed.

I’m always amazed at the amount of community building that occurs when power is cut from a subdivision.  As I look outside, neighbors from across the street come and join other neighbors to chat.  Generally, the conversations revolve around the recent lack of electricity and then proceed to how’s the family … kids.. etc.   The community seems more connected during these times, almost out of necessity to find familiarity and the need to acquire information on the whereabouts of the electricity.  It was quite refreshing to see the neighborhood come together during this time of need.

This made me think of how people view community.  Part of my personal community was inaccessible during this time because of the lack of internet.  Since I didn’t have access to the internet (or power) at home, I left in search of a WiFi location.

 I found a local coffee shop.  Approximately 80% of the patrons had laptops and were standing and sitting on spaces on the floor attempting to communicate with clients and bosses over email.  Eventually the coffee shop’s internet buckled and shut down completely because bandwidth issues.  It seemed like I wasn’t the only one who needed the internet. People / businesses /schools have a challenging time functioning without the use of the internet / technology.

About five years ago I read a book titled, Campfires in Cyberspace.  Not necessarily on the NY Bestseller List , but it was a good read. This book gave me a few ideas on how to integrate technology in my own classroom.

This book spoke of the learning that can occur via technology.  This book was published before Twitter, Google+ and web 2.0 tools grabbed any type of traction in the education sector.  Actually the “new” learning tool for that time was Webquests.  I still enjoy utilizing Webquests in my classroom.

During the time I read this book, the words “Digital Native” started to become more prevalent in schools across the country.  If you’re still wondering what “Digital Native” is, take a look at Jodi Harrision’s blog and graphic. Just like adults have communities, I believe students have an opportunity to create and be a part of their own school community by utilizing technology.  ASCD has a brief article that outlines how to establish and suggests guidelines in creating an online student community.

 If adults seem uncomfortable without a day of internet access, how do you think students (that use technology daily) feel when they spend their entire school day without the web or using technology?  

By  utilizing technology, Educators will be able to open the box of  untapped potential to improve student learning. Now, more than ever educators / administrators need to integrate technology in the classrooms.

If you’re hesitant or want more information, follow #edtech on Twitter. Also, check out the following tech blogs if you’re still curious.

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