Getting Students to Think and Understand Mathematically

Climbing the ladder to higher understanding … this is something both me and my students are constantly working on to improve!
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The longer I teach, the more I philosophize about what good teaching truly means to me. I am passionate about teaching students to think and actually show their understanding. I don’t just want the students to regurgitate some math fact or memorize the steps to solving a problem; I want them to be critical thinkers and be able to explain their thoughts mathematically.

For the past four years, I have worked in 1 to 1 international schools, therefore have taken advantage of both Microsoft Excel and Pages (iWorks) during the statistics/graphing unit. The typical approach I’ve adopted when using these programs is to teach the step-by-step process for how to graph data that’s been provided, or possibly collected by the students. I have been known to do this every day, for several days, until we have learned how to make a line graph, bar graph, circle graph, pictograph, etc. Then, we might answer a few questions about their graphs to check for understanding. But does this process truly check for understanding?

This year, I wanted to approach the teaching of this unit differently. I wanted the students to be critical thinkers and be a part of the learning process.

We started the unit with a pile of magazines and newspapers, and my only direction was to cut out anything that looked like a graph (I didn’t clarify beyond those directions). They worked at tables consisting of 4 people, and each group found at least 30 graphs. From there, I asked them to try and classify the graphs that their small group cut out. Once again, I didn’t give them any further instructions. When every group was finished with that exercise, they presented the classification system they used to the class. We identified similarities and differences amongst the groups. Now for the hard part … we needed to come up with a classification system for the class and reorganize our graphs. Well, it wasn’t really that hard – the students sorted it out in less than 10 minutes (bar graphs, circle graphs, line graphs, pictographs, frequency tables). Each similar group of graphs were then glued on a large poster board in preparation for the next activity.

During the next portion of this activity, the students rotated from one group to the next, making observations about the graphs they saw glued on the poster boards. They were asked to find the similarities, differences, and anything else that they observed within a common group of graphs. They each had their own editable PDF document (that I created but made accessible to the students) with a chart to record their observations. They rotated around all five groups and then had an opportunity to share their observations with a small group of 4. It was fascinating listening to the mathematical language being used amongst the students. After all members shared, they had one last step to complete as a group: to go back and review their lists, and highlight the key points (the ones they felt were the most important) that they would want to share with the class. This step required them to reflect on what they had written, really question the validity of each point, and discuss which points were the best mathematically. As a class, each group presented their findings and this is what they came up with:

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The mathematical connections and language that the sixth graders used to describe each type of graph truly amazed me! This piece was the connection to everything the students learned in the unit. And this year, compared to all others, the students had a much better understanding of graphs because they were part of the learning process.



Who Should Teach Tech Skills?

I am in my first year of teaching at JIS (Jakarta International School) and we are launching our 1-to-1 program this year in grade 6. By next year, all students in grades 6 through 12 will have their own laptop. It has been a bit of a “going back in time” experience for me since I’m coming from a 1-to-1 school where my students have had computers in my classroom for the past 3 years. Our school is going through the natural growing pains that come with such a shift in thought and approach. With this initiative becoming a reality, the school has decided to limit the number of technology classes it is offering, eventually leading to eliminating all technology classes.

What does this mean to the classroom teachers? Should we be worried? Absolutely not!

As David Warlick stated on his blog, ‘we should not be teaching computer applications, but rather computer application.’ He goes onto say that we, as teachers, don’t even need to teach our students specific tech tools but rather teach our students the skills to “simply learn to apply computers to solve problems or accomplish skills.” If a student doesn’t know how to use a specific tool, they will probably have gained the skills and confidence necessary to teach themselves. Here’s a great quote from another blog I stumbled across: “Simply put, we can’t keep preparing students for a world that doesn’t exist. We can’t keep ignoring the formidable cognitive skills they’re developing on their own. And above all, we must stop disparaging digital prowess just because some of us over 40 don’t happen to possess it. An institutional grudge match with the young can sabotage an entire culture.”

Students today have fantastic skills on the computer, and what they don’t know how to do, they generally pick it up very quickly. If my lesson requires a specific piece of software, I take the time to teach the ins and outs of the program to my students as best as I can, while also asking ‘student experts’ to help me in the process. Working as a team with the students in the classroom, and providing them with some ownership, creates a united team. It’s the letting go that we need to be able to do as teachers … that’s the difficult part.

Beg, Borrow & Steal

As a teacher in today’s age, and working at a 1-to-1 school, I am constantly using my creative juices to develop lessons and activities that will best reach the learners in my class. I tend to be the type of educator who sways away from the textbook and instead creates “fun” activities that build strong connections between math and real life. Teaching in this manner takes a lot of extra time, effort, and energy but is well worth every second of it.  

Over the course of my 17 years of teaching, I have become familiar with the saying “beg, borrow and steal’, just as most teachers have. As a profession, we are great at sharing our ideas with others, but also love to borrow or steal when we find something good that has been created by someone else.  

In recently trying to get my head wrapped around the legalities of digital resources, this has become a bit of a concern to me. I still don’t truly understand all of the rights of academic institutions versus individuals, even after reading this article titled Intellectual Property Rights, but I do know that someday I will be exploring other regions of the world as an educator and would really love to take everything that I have created with me. Who owns it? Do I or does the school I work at when I created it? Who would have ever thought that our world, our education system, would get to this point one day?  

I feel very strongly that I am the owner of my own creations. All of them have come from MY brainpower, MY time, MY creativity, and MY energy, therefore they are MINE. And as with most educators, I am willing to share everything I have ever created, at absolutely no cost, thus the reason I post most of my creative lessons/activities on my blog … so please help yourself.  

Since most educators are already familiar with the phrase “beg, borrow, and steal”, and most of us actually do this, it is shocking to hear that some teachers actually sell their lessons for cash. (NOTE: There have even been songs written about begging, borrowing and stealing – check out this video of the New Seekers in the 70’s). YouTube Preview Image When searching on the web for ideas to use in my classroom, I am constantly diverted to Teachers Pay Teachers, a website which allows teachers to post lessons and/or activities for a cost. And recently, I came across this article in the New York Times titled Selling Lessons Online Raises Cash and Questions. It makes an absolutely brilliant point – that when teachers sell their lessons, it “reduces the power of the learning community and is ultimately destructive to the profession.” We, ultimately, need to pull together as a profession and help one another be the best teachers we can be.  




Digital Footprints … Can We Erase Them?

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 Teachers today, who didn’t grow up on the web but are teaching students who did, have quite a lot to wrap their heads around … and I’m one of them. As an educator, I have learned to “go with the flow” and “roll with the punches” over the years. There is always a lot to learn and we must take it in strides and with open arms.   

This question was posed to me lately – “Should teachers be actively blogging and have accounts on Facebook and Twitter? Should we be getting our names out there?”   

I would imagine that most people’s first reaction is “no way”. I truly had to ponder about this for a while because I have mixed feelings, but after doing some reflection, I have come to realize that we should be out there, and here are my reasons why:   

1) Socially, I enjoy being on Facebook because it allows me to keep in touch with friends who live far away, I can share my photos with family and friends, and also see their photos, I can chat with them when they are online at the same time as me, and I can read status updates to see what everyone is up to. All of these reasons are great until I start thinking about what it does to my digital footprint, especially as a teacher. So then I started reflecting on my use of facebook, and I know that I am extremely careful about what I post (I never use profanity, I only post respectful photos, and I have my settings created so I have a decent amount of privacy).   

2) I only started blogging over the last couple of months, and I keep all of my posts connected to education as a way to share ideas with other educators. For this reason, I definitely think teachers should blog.   

3) Finally, I don’t have a twitter account. I don’t think it’s a bad thing for educators to have … I just haven’t found a purpose for it that I feel is worthwhile.   

So, all in all, my answer to those questions is YES. The key to having your name out there in cyberspace is approaching it with caution while also giving yourself a great name.   

I recently googled myself, and several hits came up. This wasn’t the case just three months ago, prior to my blog. It’s interesting to see the changes in such a short time period. As I started to explore the links to my name, they were all related to education, which I was pleased to see.   

Every individual in the world is creating “digital footprints” and “digital shadows” in this day and age, including unborn babies according to this article, and they are here to stay. We need to be careful about what we put out into cyberspace. Prospective employers are starting to use our digital traces to determine if they want to hire someone or not. We need to be creating “a positive and strong online presence that we are proud of”, quoted by a career-builder website, and as teachers we must portray this same idea to our students. They are young and need to be taught this idea, and this link, connecting to an excerpt from Dan Pink’s video on motivation, is a great starting point to get our students to begin thinking about whether or not they are making the best decisions.   

(Click here to calculate your digital footprint; it’s pretty interesting).

Technology to New Heights – Creating MathCasts in Grade 6

I previously blogged about how I have slowly progressed through the adopting and adapting stages that Marc Prensky spoke about in his Shaping Tech for the Classroom article. The final stage (stage 4) in the movement towards “adopting and adapting” is doing new things in new ways. This is the stage that I am attempting to tackle now. I continue to try to find new ways of teaching the grade 6 math concepts so that it reaches the digital natives who are sitting in my classroom. The excerpt below will walk you through the process I went through to get to a Stage 4 activity:

As a math educator, I feel strongly about embedding a challenging problem solving component into my curriculum. This is one area that students need to develop further and deeper, making them better critical thinkers. When I started at TAS (Taipei American School) in 2004, my teaching partner introduced me to the problems he wanted to use for this part of our sixth grade math curriculum; it was completely separate of our textbook. As I looked through the problems, I felt they were extremely challenging and exactly what we needed for our students ~ I loved it. Every year I have taught this part of our curriculum, my approach has been to:     1) provide example problems solved by me, 2) allow the students some time to solve some practice problems on their own or in small groups, and then 3) we discuss certain methods that might be best to use. The students are then left to work on the assignment outside of class individually. I am constantly asked by parents how they can help their child at home, and I don’t really have an answer for them. Additionally, there are some students who struggle all year long with these assignments.

This past year, I felt I needed to do something different in order to help these struggling students find more success … and to give the parents a way to help their child at home. I decided I would make my own “Problem Solving MathCasts”. The math casts that I have created record my voice explaining important rules, step-by-step instructions for how to solve a sample problem, as well as a video showing how I solve a specific problem. (I have included links below for you to access a few of my Problem Solving MathCasts). Because I posted them on the OLC (Online Classroom), the families were able to download all of the videos from home. Students who needed to watch the videos more than once could do so in the privacy of their home. Parents could also watch the videos and then provide assistance to their child. I have received a lot of great feedback from both students and parents in regards to these videos. And in comparison to previous years, I have now noticed that the majority of my students finish their grade 6 math class with a better approach and more self-confidence in solving the challenging problems given to them.

Based on the very positive feedback I received about my math casts, I decided I needed to take these math casts a step further; I needed to get the students making math casts. When creating new lessons or developing new activities, I am constantly utilizing Bloom’s Taxonomy to reach all learners and learning styles, as well as to create a variety of approaches to learning. My ultimate goal is to generate activities that aim for the higher levels of cognitive understanding – that is, getting my students to analyze, evaluate and create. Ironically, it has taken the current technology class that I am enrolled in to learn that “Bloom’s Taxonomy has bloomed digitally”.  In this article, the original Bloom’s Taxonomy Map has been revised by Andrew Churches where he illustrates how it is linked to digital skill (see image to the left). Well, in taking the math casts a step further, I decided that I would have my students create their own math casts. 

In attempting to hit the highest level of Bloom’s Taxonomy for my digital learners, having my students create their own MathCasts would be the perfect task. I introduced this assignment to them at the end of the year and, of course, I heard a few groans, but for the most part, the students seemed excited. We brainstormed all of the topics we had studied and learned about over the course of the year on a google doc, and then I shared the document with everyone. The students needed to choose one specific skill/concept to “teach” through a MathCast. The only directions I gave them were:

  • Their presentation needed to be Grade 6 appropriate (proper language, vocabulary, problem choice)
  • They needed to use a software program that included video and audio functions
  • Each student needed to cite sources (i.e. graphics)
  • Only legal music could be used

I gave them a few options for different programs they might use, but they had to figure out how to use them by exploring and helping each other. I gave the students class time to work on these, although several of them spent hours perfecting them at home. They were inspired!

I have to be honest with you – I was a bit nervous about this assignment because I didn’t know what to expect. There were very loose guidelines and little direction. In the end, all of the students submitted their projects on time … and that then meant it was time to grade them. I opened the first file and began to listen to and watch my 12-year old student explain how to balance an algebraic equation. Wow ~ It was amazing! And to think about the transfer of knowledge it takes to go from understanding a topic to teaching a topic. Another Wow! The students’ MathCasts were so fantastic that I decided to upload them to YouTube for other math teachers to use in their classrooms. When I mentioned this to the students, they wore the biggest smiles and sat tall in their seats – they were so proud of themselves! These same videos are also being used as resources for my students this year. I can’t wait to do this assignment again this year with my current students.

YouTube Links to Ms. Nave’s Problem Solving MathCasts:

Draw a Diagram:

YouTube Preview Image

Systematic Lists:

Eliminate Possibilities:

 YouTube Links to Grade 6 Student-Created MathCasts:

Finding the Mean of a Set of Numbers:

Dividing Integers: 

Reading Stacked Bar Graphs:


Our current students, the digital natives of the web generation, have definitely grown up quite a bit different than my generation did.

  • As a kid, I played outside; kids today play computer games in the house.
  • As a kid, I spoke to my friends on the phone; kids today text one another.
  • As a kid, I had to go home to listen to music on my cassette player or record player; kids today pull their iPod out of their bag and put on their headphones.
  • As a kid, I played games on my Colecovision console every so often (I didn’t play it often because it took so long to pull all of the necessary components and parts out of the cabinet in order to set it up); kids today play games on their travel PSP/gameboys, cell phones, or i-Touch.

  • As a kid, I had to research academic information from “real” books or encyclopedias; kids today hop on the internet and find everything they need within minutes, sometimes seconds.
  • As a kid, I took pictures with my camera and then had to wait a week while they were getting developed before I could see them; kids today see their pictures immediately in their digital camera screen.


When you compare the two generations, it’s quite obvious to see that our expectations of daily life have been quite different. And as a teacher, it is my job to reach these students.

I just recently finished reading an article titled World Without Walls: Learning Well with Others by Will Richardson. He speaks about our jobs as educators, and what does it mean to be teaching the digital natives. There were some great points made, which I have quoted below:

“Anyone with a passion for something can connect to others with that same passion — and begin to co-create and colearn the same way many of our students already do.

I believe that is what educators must do now. We must engage with these new technologies and their potential to expand our own understanding and methods in this vastly different landscape. We must know for ourselves how to create, grow, and navigate these collaborative spaces in safe, effective, and ethical ways. And we must be able to model those shifts for our students and counsel them effectively when they run across problems with these tools.

Anything less is unacceptable for our kids.”

What a task … but I find it to be an exciting one! As a teacher in this day and age, we must grasp what is in front of us and run with it openmindedly. Reading articles such as these inspire me to look for new ways of doing things in the classroom that will better meet the needs of my web generation students and ultimately be a better teacher!

Wanna join this journey with me?

Reaching out …

As I begin my very first blog, I am hoping it will become a portal to communicate with several other math teachers out there.

As a middle school math educator, I love the challenge of creating lessons/activities that will test the students to think outside the box and find connections to the real world.

A few years ago, our school had the vision to go 1-to-1 … where every student in grades 6-12 would come to school with their own laptop. In order to prepare for this shift, the school purchased a tablet for all teachers to experiment with for an entire school year before the students would come to school with their own computers. We worked in teams to research, explore, and create.

Students have now entered my grade 6 math class daily with a laptop  for two years. I can honestly say that my approach to teaching hasn’t changed all that much. Listen to this sound clip I created in response to the question: “What impact has technology had on teaching styles?” Technology Shaping My Classroom Teaching

My goal, through this blog, is to share how I am currently integrating technology within my math classroom … and I hope to get ideas from you too.