Connecting Area and Perimeter to Art-Piet Mondrian

Whenever possible, I search for ways to integrate the curriculum to create deeper learning opportunities for students and connect to the world around them.  It has always been easy to make connections between geometry and art.  Measurement and art wasn’t something that I had integrated much before.  However, in working with my Teacher Candidate from the Trent University Faculty of Education program, we were excited to see what the students would create.  It only goes to show you that when teachers are able to work collaboratively, wonderful programming ensues for students.

We have been working on perimeter and area for a little while, but students were still having trouble figuring out the difference between the two concepts.  We started by giving the students 9 square tiles.  Students were asked to create a 3 x 3 array of square tiles and determine the perimeter and the area.  The perimeter was determined to be 12 and the area determined to be 9.  From there, students were given a number of different challenges to reduce the area but maintain the perimeter of 12.  The challenges grew increasingly difficult.

1.  Reduce the area by one square unit but maintain the perimeter of 12 units.

2.  Reduce the area to 7 square units while maintaining the perimeter of 12 units.

3.  Reduce the area to 6 square units while maintaining the perimeter of 12 units.

4.  Reduce the area to 5 square units while maintaining the perimeter of 12 units.

5.  Reduce the area to 4 square units while maintaining the perimeter of 12 units.

6.  Reduce the area to 3 square units while maintaining the perimeter of 12 units.

After having the students share their different solutions we thought we would show the students artwork that Ms. Marchiori created inspired by Ellsworth Kelly’s “Colors for a Large Wall”.  In a guided math lesson the students figured out the area and perimeter of different parts of the artwork.  The way in which students figured out the answers to the area demonstrated that they had a much better understanding of the difference between area and perimeter than they had previously.

artworkmath                   artworkmath2

 

At this point, we wanted to get into the artwork and considered the work of Piet Mondrian.  Piet Mondrian is famous for the work that he created using primary colours, horizontal and vertical lines and squares and rectangles.  Perfect for working with area and perimeter and for incorporating the different elements of art.

Ms. Marchiori showed the YouTube video of Piet Mondrian’s artistic life in a nutshell.  Afterwards, the students then created their own Mondrian inspired artwork using chart sized grid paper (6’X6′) and crayon.  To continue our math focus, the students then had to calculate the area of each of the colours that they used and write that on the back of their art “plan”.  From there, the students used acrylic paint on canvas with grids drawn in pencil to recreate their “plan” for their art.

artwork3 artwork 4 artwork 1

A few of the finished artwork samples;

IMG_4067  IMG_4065 IMG_4064

This artwork would also connect to fractions.  Students could express their colour content in a fraction, reducing it to it’s simplest form and then compare which colours covered the largest fraction of the area of the painting.  When the artwork is complete, the students will be adding an artist’s message about what they learned during the process about area and perimeter, about the elements of line, colour and shape and about Piet Mondrian.  This week we will be creating Mondrian inspired artwork while exploring balance and colour in art using much of the same grid technique but with the medium of crayon resist and watercolours.

 

Overcoming Math Phobia

A phobia is defined as an extreme fear or aversion to something. This can often be associated with mathematics both by students and teachers alike. Human nature is such that when we feel we are not good at something, we therefore can’t be successful at it and we tend to avoid that what we will fail at. This self-fulfilling prophecy is often alive and well in a teacher’s or student’s thoughts.

I will be the first to say that at an earlier stage of my career I was very uncomfortable and unsure of myself when teaching mathematics. Sure I knew how to do math, but did I know how to teach something I was not very comfortable with. I had to do something to ensure that my skills and pedagogy were improving. Thus began a voyage of self-learning or self-guided professional development. Now, twenty-five years later I am still on that journey of learning about how to best teach mathematics so that my students learn and are engaged in their world that is so filled with math.

As with anything else you must find the right tool or vehicle for learning. I attended as many workshops as I could on mathematics. The Waterloo Region District School Board offers a wealth of learning opportunities for their teachers as does ETFO and the Ontario Association for Mathematics Education (OAME) (http://www.oame.on.ca/main/index1.phplang=en&code=home).

These are several key areas where you can start your journey of learning. I would like to share three key resources that have helped me become a more efficient and knowledgeable mathematics teachers. The first is the work of Dr. Catherine Twomey Fosnot. Her work and approach to the instruction of mathematics is the number one influence I attribute to my growth in mathematical instruction. I attended several of her sessions as well as visiting her site in Harlem. I would highly recommend her series ‘Young Mathematicians at Work’ as a classroom resource.

The second most useful tool I have come upon is the series entitled Super Source. There are many reasons why I like this resource. The first is the rich problem solving tasks that are in each book. There are a variety of tasks and each task is connected to an area of mathematics where it can be used like number sense or patterning. There is a book written for each type of manipulative (Base 10, Pattern Blocks, Tangrams etc…). The most valuable asset of this resource is that there is a section where the mathematics behind each task is explained to the educator (the big ideas) as well as suggestions on how to bring out the math in your students. As with any resource this provides a jumping on point where a teacher can then adapt the task to meet their needs.

The final resource I would like to share with you is one of the many works of Van de Walle. I used this resource as a teaching tool for myself. It helped me understand the concepts I was teaching and how to bring out both a level of engagement as well as a deeper understanding of mathematics in my students. I hope these resources prove to be as valuable a tool to you as they are for me in my teaching of mathematics.

132

Tech Talk

I would like to share a new initiative my class is starting at our school Ancaster Meadow.

Our class realized the large need for technology in the classroom. We noticed that students are sometimes unfamiliar with apps that are available for completing work and for presenting learning. We are starting a new program called “Tech Talk” where as a class we travel to classrooms and instead of “Reading Buddies” we will assist the class one on one with apps of the teacher’s choice. We came up with a benchmark of five:

  1. Explain Everything
  2. Kahoot
  3. Weebly
  4. Pic Collage
  5. Prezi

These apps will be explored in detail by my class so when we visit our first class after the break, the break off sessions will be smooth and very productive. We will offer this program to grades two-five and speak to teachers before regarding sign up. Students will be able to explore the apps, prepare a small task and then share it with the whole class. The sessions will be fifty minutes long as that is how long a period is in our school. We hope that students will learn from each other and that everyone will be able to use these apps in the new future. I hope it will be a meaningful experience for my class to teach younger students skills that they already have. I cannot wait for our first trial after the break. I encourage everyone to try this  “Tech Buddies” program and share your experiences as well.

Life is for Learning Curves

The view from the halfway mark of my first year teaching senior kindergarten is a lot different than it was just four months ago. Whereas I struggled every day in terms of how to implement the curriculum, now I am so much more comfortable in many, but not all, areas of the program. I hadn’t really taken the time to reflect on how I was feeling about my progress as a teacher and co-learner until, while chatting with my principal the other day, she asked me if I loved kindergarten. I hesitated, because I was thinking back to other grades I have taught, wondering if I had ever said that I loved teaching any of them. I realized, in forming my answer to her, that it is never really the curriculum of any given grade that I love teaching, but rather learning about my learners in order to be able to teach them is what I enjoy the most. I said to her that last term felt like ice-climbing, but now I am on a sort of a plateau – I can take a breath, look back proudly at where I came from, and look forward to how much I still need to learn. I most definitely have not mastered this grade, however, with the loop-de-loop learning curve I have been on this year, I am sure I will be a much better teacher next year.

That is what I was telling myself until recently when our school board confirmed its decision to integrate junior kindergarten with senior kindergarten next year. Even after having lived the senior kindergarten life for six months and feeling they could not be too dissimilar, I could not really tell you what goes on in a junior kindergarten classroom. That is why I am hoping that my principal will give me some coverage time to hang out in one of our JK classrooms to get a feel for the way the program is run and to see how the wee ones go through their day, as a way to help me imagine a blended classroom. Adding French Immersion will be another part of the picture – what will that look like? In-service workshops are to be part of the plan from the school board, and hopefully, so too is having quality time to work it all through with a supportive team of ECEs and teachers. Whereas this year, I was scrambling to make sense of it all as the newbie in the midst of a crew of seasoned kindergarten teachers and ECEs, next year, everyone will be trying to work things through. Life is indeed for learning curves.

The Music of Math

I secretly want to be a conductor of a large musical ensemble. My visits to the Toronto Symphony Orchestra always reignites my passion for music and I will always catch myself using my hands and body to dance with the beautiful arrangements of the melodious notes. However, one small obstacle has barred me from realizing this dream: I have not yet been able to master any instruments much less lead a body of musicians. Sigh! Nonetheless, being a teacher gives me the opportunity to integrate many of my passions into my classroom practice. As such, my love of teaching and my dream of conducting an orchestra have a place to collide. My class call signal – teacher: “Bah dah bump bah dah dah…” students: “Hey!” – can seem to go on for hours as I engage my students in various melodious versions of the common tune. Needless to say, I can totally get lost in the call and response clap patterns that allow me to use my hands as if they were conducting an orchestra, pulling out the very beat of each note value as students respond with the exact arrangement of claps. Each time I am left invigorated and excited and my students love it. These sessions often end up with smiles and laughter in our classroom, opposite to the quiet tone I originally was trying to achieve.

This year I teach math to Primary, Junior and Intermediate students. Math and music have long enjoyed a harmonious marriage with the plethora of interconnected concepts that can be explored simultaneously. This year in my Grade 3/4 math class, we had the opportunity to embrace this fusion. We have been exploring geometry concepts by naming, sorting and identifying characteristics of polygons. This aspect of geometry is extremely language-rich and I needed a way to solidify the students’ conceptual understanding of the principles of geometry while not losing my students in the barrage of its discourse. So we decided to accent our learning by capturing it in a song.

I invited my class to decide on a familiar tune that we could arrange our lyrics to fit. Hot Line Bling and Watch Me (Whip/Nae Nae) were among the tunes that were suggested, but didn’t make the cut. We opted to use the infamous Mary Had A Little Lamb as our melody base since it offered a simple structure for the type of song that we wanted to construct. We then brainstormed all of the ideas we had learned up to that point in our polygon journey. These were written on the board and were the ‘meat and potatoes’ of our song. Then the creative juices began to flow. Each aspect of our learning was written as a verse in our song. We had to be creative with our word choices and syncopation, as we had more to say than the structure of our tune would allow. So once the lyrics were written, the music teacher and I collaborated to hash out the note values of our song. Ta-dah! Music literally met math. Coupled with this, Mary Had A Little Lamb just happened to be the song that my students just finished learning to play on their recorders. Bazinga! Music met math again. And guess who got to direct this beautiful masterpiece?… ME!

File_000 (1)

So what’s my point? Learning is awesome! No really – when a passion for teaching meets willingness to embrace creativity, amazingness happens. Children learn and teachers have fun. Being authentically excited about what you do is an important ingredient to any meaningful experience and the polygon song is such an experience which I hope my students will never forget. Because of this, neither will the concepts be lost. As for me, my dream will continue to grow. Who knows what else I might try? I’ve secretly wanted to be an opera singer…

Brain Shifts

Have you ever been on a very long car ride and just as you think you are not able to continue you come upon a service center? Within 10 minutes of having a change of state, you eagerly hop back into your vehicle and are ready to go for several more hours. The same is true for students. By allowing short (1-2 minute breaks) you help energize the students to continue on with their learning, fully engaged.

I call these breaks ‘Brain Shifts’. For example my students just finished working on a problem-solving task in Mathematics and prior to starting Science (which is the next item on our daily agenda) I ask them to do a Museum Walk around our displays. The first benefit is that they are physically moving and thus increasing blood flow to all body parts, including the brain. Secondly they are now focussed on our science topic and have started to activate prior knowledge even before the lesson begins. The final benefit is that it allows socializing to occur amongst my students.

The Brain Shifts are not just random activities but rather have a very specific purpose. That purpose is dependent on what is occurring or not occurring in the class at that particular time. For example, if the energy level seems low, the activity will be chosen to increase the energy in the room (find 3 people and share with them the strategy you used to solve this task). If the students are bordering on being hyper, the task is designed to bring a calm to the room (look around the room and show with your fingers a score out of 10 that you would give this latest chapter in our book). There are so many ways that Brain Shifts can occur. The key is that you develop activities that are a good fit for your teaching style and accomplish what they are intended to be used for.

Number Lines: A Game Changer in Math Class

This year, as in times past, planning my math program was filled with the usual excitement of clustering expectations and imagining ways in which I could creatively address the curriculum. Teaching a grade 3/4 combined grade for the first time, I was eager to be on top of my planning and ensure that I was creating parallel learning experiences by simultaneously addressing the curriculum expectations for both grades.

When planning the learning experiences, I colour code the strands, cut out each expectation and cluster them according to big ideas. This results in cross-strand learning experiences.

For my first series of learning experiences, I really wanted to invite my scholars to engage with number lines as a model that would help to make their thinking visible, but also as a tool for exploring many math concepts such as quantity and measurement. A number line is a line segment that can either be vertical or horizontal that represent a series of numbers that are marked at intervals. Re(introducing) number lines to my scholars early on in the year was important for me because I view number lines as an invaluable math model. Every day numbers lines are used to measure time – a clock; distance – a ruler; capacity – a measuring cup; temperature – a thermometer and mass – a scale. Number lines also allow students to model the strategies they use when adding and subtracting, as well as composing and decomposing numbers. Starting with the exploration of number lines proved to be an important first learning for my grade 3/4 math class.

Students were able to identify a variety of tools that have number lines embedded in their function.

Beginning the year this way was exciting but it was even more invigorating when I realized how many authentic, cross-strand connections that actually took place by centring the learning around investigating the use of a simple tool. The following are a few reflections from the journey:

1. Once the orientation of a number line was solidified, scholars were able to make connections between the number line drawn on the board with the number lines embedded in everyday objects.
The most prominent feature in my class at the start of the school year were the quantity number lines hanging from my classroom ceiling. These were made by cutting pool noodles into two-inch disk and placing them on rope. One was counting by twos (two-coloured pool noodles) and grouped by twenties to hang on the ceiling. The other was counting by fives with alternating colours and grouped by 25s. Students were invited to inquire about the quantity of pool noodles, the groupings, the skip counting, etc. and this became the introduction to number lines for the year. Using the quantity number line along with an actual number line that marked the perimeter of the front of my class, we explored the nuances of number lines by describing their directionality of increasing and decreasing quantity. We also explore the ways in which intervals could be marked on an open number line so that users didn’t necessarily have to always count by ones, but skip counting was made possible by using appropriate quantity intervals. We then named all the number lines that were present in the classroom. They noticed the clock on the wall to be a circular number line and the “How Much are Your Growing” height chart as a vertical number line. Excitement bubbled as scholars scrutinized the number lines we named by the elements of number lines that we had previously defined. They were set and ready to continue their investigation of how number lines could truly enhance their learning experiences as young mathematicians.

File_000 File_001

Quantity number lines hanging from the ceiling of my classroom.

2. Counting on a number line allowed number patterns and relationships to jump out at scholars.
Once scholars were comfortable describing the orientation and use of number lines, we began exploring quantity by representing skip counting on an open number line. We counted forward by tens by modeling jumps of +10 on an open number line and then later by nines. Scholars were able to name the relationships between counting by tens and nines by creating expressions that named the pattern (i.e. 10-1). We then explored counting forward by 100s, 50s, 20, and 11 to explore the other number patterns and relationships. Once scholars we comfortable counting forwards, we then counted backwards. They noticed relationships around place value, odd, and even numbers, what happens when counting by numbers with specific digits, and the nuances of increasing and decreasing values. The grade 4 scholars were then invited to apply their understanding of skip counting by whole numbers and moved deeper into counting by fractional amounts and decimal numbers up to tenths. Students were taught to use the number line to keep track of their thinking by using it as a thinking tool and not a picture. Hence, numbers and jumps were added in sequence as the counting progressed and not merely by drawing the elements of the number line and labelling them afterwards.

File_000 (2)

Counting forward by 9 using a quantity number line. Student notice and described patterns.

3. Using number lines as a tool for measurement invited scholars to apply their understanding of number concepts.
Measuring mass using a triple balance scale, capacity with graduated cylinders and time with a clock were all the more meaningful when scholars were able to apply their understandings of using number lines as a tool for mathematical thinking. This is how our Number Sense learning experiences married Measurement. When scholars measured mass using the triple balance scales, they applied their understanding of decomposing numbers into place value (including tenths for grade 4s) in order to calculate the mass of the item they were measuring. Similarly, when measuring capacity using a graduated cylinder, scholars needed to apply their understanding of skip counting in order to accurately represent the capacity using a variety of differently shaped and sized graduated cylinders that each increased by differently valued intervals. Given their prior knowledge of skip counting, and composing and decomposing numbers using number line, measurement became a context to continue to expand their understanding of quantity.

4. Counting on a number line open up the way to modelling mental math strategies for addition and subtraction.
Once scholars were comfortable skip counting, composing numbers and measuring different units of measure using number lines, adding and subtracting was a natural next step and application of their previous understanding. Using addition and subtraction strings (intentionally crafted equations to be solved using mental math strategies), I modelled scholars’ mental math strategies visibly on a number line. Scholars articulated strategies such as adding on, counting, back, compensation and even decomposing numbers in order to add and subtract. The application went through the roof when grade 4 scholars were able to demonstrate their understanding of elapsed time using these same strategies. This was especially helpful because scholars could be fluid with the units of time they were adding or subtracting (i.e. hours and minutes) and could avoid the complexities of the base-60 system of time (i.e. 60 seconds in 1 minute, 60 minutes in 1 hour) when it came to calculating time that involved “regrouping” which uses a base-10 number system.

File_000 (1)

Students articulated mental strategies while I modelled them on an open number line.


File_000 (3)

Student calculating elapsed time using strategies for addition and subtraction.  

During these first months of school,not only was I able to introduce scholars to an invaluable learning tool, the number line, but I was also able to seamlessly integrate two strands of math and a multiplicity of essential number concepts. The number line has truly been a game changer in my class. I am already seeing the connections for moving forward with experiences that explore data management (scales on a graph) and geometry (side length attributes of polygons). …the journey continues…

Tortoise Brained Learning and Students

In my last post I focussed on the philosophical belief that quality vs quantity of professional learning is a more effective way of enhancing pedagogical practice. What does that mean for my classroom instruction? As I grow to understand the presence of different learning styles in my class, the presence of multiple intelligences and the wide variety of learning rates it forces me to re-examine both the long term and short term planning that I set up.

In the earlier part of my career my long-range plans were reflective of an efficient way to ensure that all of the curricula was covered. This I now refer to as curriculum planning and not student centred planning. As my understanding of differentiated learning and assessment grew, so did the need to adjust the way my planning unfolded. What I had experienced was a short-term understanding of content and when that topic was revisited months later there seemed to be a regression in the level of understanding of my students. That forced me to ask myself as to how well they had really learned the content in the first place.

Through years of experimenting with both my long range planning and unit design there arose two aha moments for me. The first was the need to revisit big ideas (overall expectations) through a spiralling curriculum. This means that I would chunk the content into more manageable pieces and revisit the content several times over the course of the year (quality vs quantity).

The second profound understanding was in time management and how do I accomplish the ability to revisit overall expectations with so many demands on the school day. Thus came the desire to increase my skill set in integrating learning across a variety of curricula. The following is a direct reference from the 2006 Ontario Language Curriculum:

In cross-curricular learning, students are provided with opportunities to learn and use related content and/or skills in two or more subjects. For example, teachers can use social studies reading material in their language lessons, and incorporate instruction in how to read non-fiction materials into their social studies lessons. In mathematics, students learn to identify the relevant information in a word problem in order to clarify what is being asked. In science and technology, they build subject-specific vocabulary, interpret diagrams and charts, and read instructions relating to investigations and procedures. All subjects require that students communicate what they have learned, orally and in writing. Their studies in the different subject areas help students develop their language skills, providing them with authentic purposes for reading, writing, listening, speaking, viewing, and representing.

Needless to say, this is a spiralling learning experience for me as I continue to help my students consolidate the learning that they are a part of each and every day.

Yikes! I’m Teaching A “Split” Grade Class

Meet Linda. The start of the school year has come and gone and she has finally been successful at landing her first Long Term Occasional teaching assignment in October. It’s an FDK/Grade 1 class, and given the dynamics of the FDK program, she struggles to even make sense of what that might actually look like.

Meet Lester. After finally feeling settled with his Grade 5 class, he learns that as a result of a school wide reorganization, his neatly packaged single-grade class has now become a “split-grade” class.

Do either of these situations sound familiar? Do these narratives hold true for your current experience? Before you pull out your hair or scream, here are some practical strategies that will help you navigate this new experience. Consider the following 3Cs, Context, Community and Curriculum, for teaching in a multigrade classroom.

Context: Although a multigrade classroom is not the ideal setting for an optimum teaching and learning experience, it is worth re-imagining the situation as a combined class rather than a “split grade.” The idea of a “split” gives the impression of a type of separateness that does not accurately describe the context of learning in a class of students who generally are within the same developmental stages. The great news is that combined grade class have of two consecutive grades in one class as opposed to grades that are years apart. In this way, a combined grade class maintains a similar diversity of students, range of learning styles, social skills and academic needs as in a single grade classroom.

Community: Rethinking the notion of a “split” grade can do wonders for the type of classroom environment you nurture. As in any class dynamic, students learn best in an environment that is inclusive and fosters a of sense community and belonging. As such, the ways in which the class environment is setup should work to unite students rather than separate them based on grade. Students should have opportunities to work in flexible groupings using a range of collaborative learning structures as often as possible to invite positive peer interactions through both collaborative and independent learning situations. A combined grades should never be thought of as housing two classes in one room. Rather, students should know that they are one class engaging in similar learning opportunities. Students should be taught how to navigate their unique learning situation by explicitly teaching the importance of cooperation, and how to engage in learning that takes on a variety of groupings; such as whole class, small group, partners or even independent learning. Inviting leadership opportunities and encourage students to view their peers as resources will help to establish a community based on cooperative interdependence.

Curriculum: The idea of covering a combination of two curricula is one of the primary concerns for teachers who have been assigned combined grades. In Ontario, thankfully, many of the content areas align in ways that make tackling both grades more manageable. These content areas include Mathematics, Language, Health and Physical Education and The Arts. In these curricular areas, differentiating instruction is an essential tool for ensuring that the overall expectations of each grade is addressed and assessed. Differentiating content, process and product will allow for a seamless flow between the different grade expectations. At the same time, using the gradual release of responsibility will set the tone for independence that is often essential in a combined grade classroom. For content areas that are less similar such as Social Studies and Science, teaching through an inquiry framework allows teachers to focus on essential skills and invite students to apply those skills in different contexts. By focusing whole-class instruction on big ideas, thinking processes and strategies, teachers are able to engage students in parallel learning experiences while differentiating content.

Teaching a combined grade is never easy. But with the heart for learning and the art of teaching with finesse, even the least of ideal situations can be one that is done successfully.

For more information about teaching in a combined grade class, check out the resource Learning Together: A Teacher’s Guide to Combined Grades at Shop ETFO.