I love my job ~ educating the children of our future! There’s nothing better than having the opportunity to develop creative lessons and activities that I believe are best for a child’s learning while listening to them discuss new ideas/thoughts and watching their eyes light up when they finally ‘get’ the concept!
I’m always trying to improve what I do in the classroom to better meet the needs of my students. And although I’ve blogged about the idea of “flipped classrooms”, and have even flipped a few lessons here and there, I’ve now gone full force with it. We’ve just finished our second full ‘flipped’ unit and I hope to share some of the things I’ve learned along the way.
I’ve got to admit that it was a little scary at first – letting go of the control and placing it in the students hands. And to be honest, they struggled at first too. I am an extremely organized and well planned teacher, thus had the entire unit outlined and prepared. As a teacher of blocked classes, I provided this simple outline to the students so that they would understand the new process and expectations:
Day 1 
Day 2 
Day 3 
Day 4 
Day 5 

Lesson Objective(s) 
Finish Previous Unit 
NO MATH 
Apply divisibility rules in order to identify a whole number’s factors and write whole numbers as the product of prime factors 
NO MATH 
Students will find the greatest common factor of two or more numbers Students will write equivalent fractions 
What’s Happening in Class? 
Return summative test; students correct mistakes and complete reflections 
Teacher has prepared a quick 5question check for the start of class. Students then break into groups: a focused learning group or an independent learning group (depending on their individual needs). The independent learning group is free to start working through their problem sets. The focused learning group works with the teacher to receive more review. This typically takes the form of an interactive Smart Notebook lesson. These students then also have class time to work on the problem sets. 
Teacher has prepared a quick 5question check for the start of class. Students then break into groups: a focused learning group or an independent learning group (depending on their individual needs). The independent learning group is free to start working through their problem sets. The focused learning group works with the teacher to receive more review. This typically takes the form of an interactive Smart Notebook lesson. These students then also have class time to work on the problem sets. 

HW 
Students finish corrections and reflections; get parent signatures 
Students watch instructional videos, do online practice problems, and optional activities for our next lesson (posted on the HW page). Students complete reflection questions in google form (link available on HW page) so teacher can plan lesson appropriately. 
Students finish the problem sets and correct them. They then complete the Problem Sets Reflection Sheet (google form; linked to our HW page). 
Students watch instructional videos, do online practice problems, and optional activities for our next lesson (posted on the HW page). Students complete questions in google form (link available on HW page) so teacher can plan lesson appropriately. 
Students finish the problem sets and correct them. They then complete the Problem Sets Reflection Sheet (google form; linked to our HW page). 
So, in order to plan accordingly, I started by looking at all of the learning goals for the unit. From there, I began searching for online, instructional videos that were aligned with each learning goal. I tended to use videos from Khan Academy and BrainPop most often; these are perfect for grade 6 math. If I couldn’t find a video that connected, then I would make my own.
Next, I knew I wanted to collect feedback from the students regarding their understanding of the skills/concepts before they came to class so that I could plan appropriately … so for every set of videos, I created a set of reflection questions that they needed to respond to while, or after, watching the videos.
NOTE: I started by essentially asking the students the same questions on every video reflection sheet, but realized that they learned to respond with what I wanted to hear. So, I’ve altered my approach and now create different questions each time, including some that ask them to compare a previous topic to a new one and always one that asks them for questions that they might still have regarding the lesson objective. Most of the questions are posed to get the students to think, not just reverberate what they heard in a video. This has been a positive change.
When the students come into class, I have a sense of what I still need to review and can actually post their misconceptions and/or questions up on the screen while addressing them (I can hide the student’s name since the reflection is done in a google form). I’ve attached a sample reflection form, completed by the students. You can see the highlighting that I’ve done in the spreadsheet – these areas represent things that I want to bring up in class. I actually project this for all students to see so we can discuss the questions or responses together.
NOTE: I didn’t post the students’ responses initially, but found that when I did, everyone saw what the others were writing and this encouraged them to put more thought into their responses.
Following our discussion regarding their responses and questions, I project 5 basic questions that the students should be able to respond to in less than 7 minutes. We grade them together and this becomes the design of that class period. For any student who answers them all correct, they are considered independent learners. They gather with other students and begin to work through a set of assigned problems. For any student who misses 3 or more, they must work in a small learning group with me. We use an interactive Smart Notebook that I have prepared. We spend the 10 minutes discussing the concept, creating connections to real life, and finally getting the students up to the board, manipulating various things to try and understand the math better. At the end of our short lesson, they are now independent learners. And finally, for those students who missed 1 or 2 on the 5question check, they need to do some reflection and determine the best place for them to work.
NOTE: At the start of flipping my class, I had not incorporated the 5Question Checks, but have found that the students are held more accountable for ‘trying’ the understand the concepts before class with them in place.
Once the students are all working in their independent learning groups, I wander the room. I constantly remind the students that only mathematical conversations should be taking place, and it’s great to hear them. The environment in the classroom is one of open questioning, helping one another, conferencing with one another (including me), and lots of mathematical conversations.
With this model, the students are learning to become responsible individuals for their own learning, advocates for themselves when they don’t understand something, reflective learners, motivated to do more if needed, and resourceful.
This is what I believe powerful learning is all about!
As teachers, how do we get students to show their true understanding of a mathematical concept without just solving a math problem or explaining the steps? I spend a lot of time pondering this question and would love to get to this point, on a daily basis, in my classroom. What a shift in paradigm that would be … imagine it … a math class that didn’t focus on just skills and solving paper/pencil math problems! One, that instead, requires students to think and stretch their minds and produce work that is connected to real life.
At the start of my “Statistics and Graphing” unit in grade 6 math this year, I asked students to identify similarities and differences in a variety of graphs that they had found. They shared their findings with the class and actually described the different graphs much better than I would have imagined. Immediately I knew that the entire introduction to the unit was a powerful one because the students had done the thinking, connecting, and verbalizing without much guidance from me. This inspired me to continue along this same path … so the next day the students came into class, they saw six different questions on the board, ranging from “How many siblings do you have?” to “What’s your favorite pizza?” to “What month were you born?”. They were set up so that the data collection was done through tally marks.
As we began to enter the data into a spreadsheet, the students needed to figure out the best way to record it. Most students chose a ttable demonstrating the frequency as a whole number. A few students actually figured out how to enter the data as tally marks (I am still not sure how they did that :)). Once again, I didn’t guide them much other than to explain what we use spreadsheets for and what purpose a spreadsheet cell serves. I wandered around the room for the next 20 or so minutes listening to the little fingers tapping the keys of the laptops, and an occasional student asking someone else for help. Once all the data was entered, we were ready to move onto the application (‘fun’) part of the unit.
Students would come in each day with a new task written on the board. For example, they might see these questions:
Guiding Questions:
1) Look at the different data sets. Which ones could be used to create a simple bar graph? Why?
2) Which ones could be used to create a circle graph? Why?
Students would be required to answer these questions on their own, based on the intro lesson and class discussions that had led up to this point. Once they developed an answer and the mathematical reasoning to support it, we would do the task at hand. Of course I needed to teach the students how to graph using ‘Numbers’ since this entire program was new (and I also created an instructional video for those students who needed to view it a second time)
Click here to view the embedded video.
~ but once they got the hang of it, they were able to do most of the graphs on their own. As they would produce the graphs, I would slowly walk around the room and ask openended questions like “so what does your graph show the reader?” or “is that the best choice of graph to display the data from your data table? how do you know?”. I would hear students say “Ms. Nave, this doesn’t look right” or “Oops, I didn’t chose the correct graph”. I love the moments when students are able to identify their own mistakes and work through them! Eventually, each student completed the process of making their graphs. Once each graph was completed, they had some “higherlevel thinking questions” to respond to based on the graph they completed. (See the attached thoughtprovoking questions) The unit continued in this manner until all data was graphed and all graphs that the students needed to learn, based on our standards and benchmarks, were made.We then moved onto the idea of statistics (mean, median, and mode) … and continued to use the laptops to teach and practice these mathematical concepts. We taught our students how to write formulas and sort data within the Numbers program. We provided our students with data that we had collected from the students since the beginning of the year (diagnostic test results, prediction of a minute, MAP test scores). This made it exciting for the students because it was real data! See the attached activities, used within Numbers for Mac.
After weeks of practice, not only in Mac Numbers, but also with stretching their thinking in a different way, the students were ready to show us what they learned and understood about statistics and graphing … so the assessment was created in Numbers. And you guessed it – they completed the test on their computers and submitted it to us digitally. I have included a completed test here: Completed Student Assessment. The true test came about 4 weeks ago when a grade 5 teacher asked if my 6th grade students could teach her students how to graph within Numbers. My first reaction was … yeah, that sounds great … but as I thought it about more, I wondered if I was going to have to reteach everything again. I decided to spend the next math class doing a quick checkin to see what they remembered. I asked my students to open up their laptops and go into Numbers. (Yeah, of course I heard a few grumbles, but who doesn’t when you teach sixth grade?) I posted some make believe test scores on the screen and asked the students to create the best graph to display the data. I would say that about 80% of my students could complete this task individually, and then they were ready and willing to help others in the class. I was definitely pleased with that outcome 5 months after we completed the graphing unit. When they heard that they were going to get to teach this to the 5th graders, they actually cheered!
]]>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 stepbystep 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:
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.
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During this past summer, I took an online class called “How the Brain Learn Mathematics.” The class was based on David A. Sousa’s book and was offered through Learner’s Edge (a program with several classes, geared towards teachers, taught by teachers). I have recently been extremely interested in how our students’ brains are wired and this class has now elevated my interest to learn even more. As educators, it is important to recognize that students retain knowledge when they have moved information from their working memory to their long term memory. There are several techniques we can use in our teaching to help students with this transition, but the one that stands out most to me is that we need to allow our students to establish meaning of their learning, relate to it, and make reallife connections. As my interest in brain research continues to peak, I am really hoping to attend the Learning & the Brain conference. If my school approves this PD opportunity for me, I could possibly see and listen to one or more of the following presenters: David Sousa, Howard Gardner, and Marc Prensky. WOW!
So, I’m now curious to know what the big brain researchers say about flipping classrooms. Would they agree or disagree? I’m sure there would be people who stand on both sides of the fence with the idea of flipping a classroom. Some students may absolutely excel with a flipped classroom, while others may struggle. With almost any new approach that comes through education, I believe we must find a nice balance … and in this case, a bit of flipping and a bit of facetoface teaching. The pendulum is constantly swinging from one idea to another, rather than taking several ideas/approaches into account. I feel like we need to slow the pace of this pendulum down a bit, and provide our students with a variety of learning experiences (some including the ideas from a flipped classroom, of course!)
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In the MS mathematics department at JIS, we believe in tiered instruction, challenge by choice, and tiered assessments .. and that is exactly what we offer to every student that walks through our classrooms. We offer one math class at each grade level, but within that class, we offer our students three differentiated levels (green, blue, black) of instruction, homework, and assessments. The idea is that we teach a gradelevel concept/skill (green level) which all students need to understand and grasp by the end of the unit. For those students who tend to be more advanced with their understanding of a specific concept/skill, they may choose to work at one of the higher levels. The higher level will still focus on that skill, but ask the students to think and apply their understanding at a deeper level.
Here’s an example of some questions from our most recent test:
Tell which power has a greater value. Explain your decision by showing your work.
Green (standard) Level: 3^{4} or 4^{3}
Blue (advanced) Level: 0^{8} or 8^{0}
Black (highly advanced) Level:
We teach our students to selfreflect EVERY DAY on their own learning and choose the level that is best for them based on their understanding. We tend to base our wholeclass instruction towards the green level (standard; grade level expectations), but do make our way around to all groups and work with those students as well. Students who choose to work at the blue (advanced) or black (highly advanced) level are typically selfmotivated to learn something on their own that they don’t already know. This approach allows every student to feel ownership of their own learning. The students spend half of our 90minute block working through problem sets, while I walk around and work with small groups. It is truly amazing seeing these grade 6 students (11year olds) working to their maximum capacity, and being highly motivated by the program. And not only do I see the advanced and highly advanced students extend themselves, but I have also witnessed the standard level students raise their expectations, provide amazing support/assistance to one another, and work at a higher level than I anticipated. What I observe every day in my classroom is empowering!
To enhance our math program, I have just started using Khan Academy with my students for the first time. Not only do I love it, but my students do too! Any chance they get, they ask if they can “practice” on Khan Academy. They are highly motivated to earn badges; something so minor in our adult world, but so important to a 6th grader. And then there are those intrinsically motivated students who don’t really care too much about the badges, but watch the videos to teach themselves the next level of math so that they may advance in their own mathematical knowledge, as well as the levels of Khan Academy. I believe it is Khan Academy that got educators talking about how powerful this type of learning can be, and what classrooms would look like if we used this same model.
There is so much published about ‘flipped classrooms’ now, and how they can definitely benefit each and every student. According to the material I have been reading, there are many reasons that teachers should flip their classrooms. The best list of reasons that I stumbled across while perusing the web were the following, taken from from this edublog:
Although I flip my classroom at random and various times throughout the year, it’s now my turn to take that next giant step towards totally flipping my classroom. It’s exciting, but scary at the same time!
]]>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.
]]>Education has had a complete makeover since I was a student … and even as a teacher, I still feel that I’m a learner in my own classroom. Everyday, I learn something new, usually related to technology. And often times, I am learning the skill from one of my students. I have learned that I must be openminded, flexible, and approach teaching as a twoway street in my classroom.
As educators, we know that our goal is to prepare our students to be functional citizens in the global world. We do this through teaching academic skills, as well as life skills. We know that our students must be able to ‘decode, comprehend, interpret, and develop a new understanding of the various materials they read. We also know that we must develop independent learners who can gain the necessary skills to survive and succeed. Through the acquisition of these skills, we need to prepare our students to use the skills, understand their own responsibilities, and take time to selfassess – the key ingredients to thriving in a complex environment’ (adapted from the AASL standards). These standards are written in a fashion that leaves them somewhat openended.
In looking at the NET standards, the underlying themes are that students will:
These two documents truly overlap one another. The bottom line is that when we put all of it together, we should be utilizing a variety of resources (digital, visual, textual, and technological) to create students who are competent with multiple literacies. In putting this into practice with my grade 6 students, I attempt to teach by utilizing the various literacies. I start by preparing an interactive, digital lesson (using Smart Notebooks) every day. The students are active participants and learners throughout the lesson. I provide the students with a prepared, paper copy note page where they must record notes from the lesson (i.e. vocabulary, practice problems, real life connections). At times, I ask students to watch a BrainPop video on their own, complete the review quiz, and send me their results; or we may even watch a video together as a class. (In fact, here’s such a fantastic one that I had so many students wanting the link so that they could go home and watch it again.)
Click here to view the embedded video.
And this link, somewhat related, is a great one too! Other times, I may ask the students to log into a simulation or game website to practice math problems. I teach my students how to properly read a math textbook, find key words and/or ideas, and choose problems that relate to our learning objectives, while also challenging them at an appropriate level. My students are asked to log how much time they spend on homework, whether or not they choose problems that related to our learning objectives for the day, and if they were properly challenged from their homework. At the conclusion of each unit, the students reflect on their learning from the unit and share their reflection with me.I strongly believe that 6th grade students (aged 11 & 12) still require a lot of guidance, but my hopes from all of this is that they will learn to function independently by the time they are in grade 7.
]]>*For more info on ‘flipped classrooms’, check out this blog post. In fact, a flipped classroom book will be coming out this year and seems to be the new way of teaching — a great way to get “increased student interaction.”
For the Math Cast projects, each student chooses a topic or skill that they have learned throughout the year in Math 6. They then teach their specific concept to a 6th grade audience via a digital presentation. During the digital presentation, students will explain the concept stepbystep, using visual examples and problems to demonstrate each step. Additionally, students are required to show how their mathematical concept is applicable in real life.
In this project, students must assume the role of “digital educator” and think about how to best engage and communicate their lesson to their audience (the other 6th grade students). They need to be well prepared before recording their Math Cast, which necessitates the creation of a “lesson plan” that incorporates an introduction, a body, and a conclusion. Furthermore, they have to be discerning in their selection and explanation of proper math vocabulary and academic language within their presentation. And finally, the students are taught to choose only legal graphics/music and cite their sources, when needed.
Students need to evaluate the suitability of the technology tools they will utilize throughout the project. Being that this is the first year the students come to class with their own computers, I do provide a lot of guidance towards the programs. For the visual presentation portion of the project, students can choose from the following software programs: Smart Notebook, One Note, Power Point, Word, and DyKnow Panels. For the audio portion of the project, students can opt to use Smart Recorder or Cam Studio. Each program offers something a little different from the others, thus requiring students to examine their outline/plan to guide the selection of the most appropriate tool.
Two very positive aspects of this project are that it offers students the opportunity to work in an area of interest or in an area of mathematics they feel most comfortable with, and secondly the authentic nature of the tasks enables them to truly make meaning of their topic.
Here are some final MathCasts:
Click here to view the embedded video.
Click here to view the embedded video.
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Once the students understand the golden ratio, we will begin to look at architecture, pieces of art, ratios of body parts, and objects in nature that all are related to the golden ratio. I may even extend it further, to those students who have a good understanding of the topic, and ask them to create a piece of artwork that incorporates the golden ratio.
I would love to hear some feedback on this lesson:
I will keep you posted on how this actually turns out in my class!
At the most recent EARCOS conference 2011, I attended a session on just this topic. Unfortunately the session was a quick 45 minutes, and we only touched the surface. In those 45 minutes, of course I was reminded that the various regions of the brain have different functions, but I’ve never thought about it in terms of teaching my students math. I learned that neurons fire to both the motor cortex and also to the left parietal lobe. The motor cortex is the region used for controlling fine motor skills (i.e. moving fingers – thus the reason students start counting by using their fingers). The left parietal lobe is the region used for controlling symbolic function in language and math (number symbols). And a totally different region, called Broca’s area, is the region that processes language vocabulary (i.e. numbers written as words). When we begin to think about this in academic terms, we realize that there are several parts of the brain that students must utilize in order to truly understand a math concept.
NOTE: Most of the data that was presented that day came from a book titled “How the Brain Learns Mathematics” by David A. Souza, and I have just signed up for a summer course on just this topic. I can’t wait!
Aside from the conference session, I have also read an article written by several doctors from the Harvard’s Children Hospital titled Trigger for Brain Plasticity. The article explains that neuroscientists from this hospital have “identified a protein called Otx 2 which may trigger the brain’s ability to learn. They have discovered that this protein helps a key type of cell in the cortex to mature, initiating a critical period — a window of heightened brain plasticity, when the brain can readily make new connections.” In essence, the eye is telling the brain to become plastic rather than the brain functioning on its own.
So, put all of this together and we see that both the brain and eye function together in the learning process. This all leads to the reasoning of why students learn best visually. As educators, we now need to be focusing on developing visuallyliterate students. “We need to develop critical thinking skills in relation to visual images, enhance verbal and written literacy skills and vocabulary to be able to talk and write about images, and encourage students to critically investigate images and to analyze and evaluate the values inherently contained in images” (taken from The Visual Literacy White Paper written by Dr. Anne Bamford).
This year, I set a goal to redesign my geometry unit so that it was taught predominantly through the use of visuals. Since geometry is a topic that is represented by so many natural objects in the real world, I figured that was a great start to creating lessons focused towards developing visual literate students. In comparison to last year, the students performed better this year on the final unit assessment. I would like to believe it’s because I made positive changes in the way I delivered the curriculum – through visual means. I have included a few snapshots of some of the lessons throughout the unit, as well as a student’s final project (using GeoGebra – free Mathematics software).
Snapshots of visual images from lessons:
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