Exploring the illuminations website: An Activity Analysis

The illuminations NCTM website is a great resource for teachers who are looking for lesson ideas. There is A LOT on this website! In addition to lessons sorted by approximate grade levels, there are also interactive tools.

The activity I chose to critique is called Hopping on the Number Line. I chose this lesson because I was interested to try the activity and it fit with my current teaching topic. I felt that my students had some prior knowledge with addition facts to 18; however, had not started to look closely at the number line as a tool. Also, we have just begun exploring and discussing addition facts, so I considered this as a way to provide some extra practice and exploration.  My students had not used the number line yet as a tool to show ideas during problem-based learning. Breyfogle and Williams (2009) state that, “the teacher must consider how well the task provides the opportunity for students to investigate the mathematics content in an open but structured way and how well the task connects with students’ existing knowledge while pushing them deeper (p. 277). I considered this task to provide that open but structured way to connect existing knowledge with this task.

When using the Pisa test levels, I would level this activity as a level 2 or 3.  Students were using an understanding of addition and the decisions they were making were fairly sequential.  This activity was engaging and provided students with practice and exploration.  However, I did not consider this a problem that had my students delving into deeper learning.

The students liked the frog jumps and it was an easy “leap” (pun intended) to connect. The students also were able to use the dots on the dominoes as quick identification and they had some prior knowledge/experience with subitizing DSCN0059(seeing an amount under 10 quickly without counting. This blog, Teaching Math,  has resources too. Check out dreambox for interactive 10 frames that could be used).  This activity afforded students the ability to see numbers in addition sentences as whole parts. Students who were having more difficulty with addition sentences were challenged to leap at least the first number, rather than counting all by one, starting at one. This encouraged students to use the count-on strategy.

We also discussed the communative property. It is hoped that students who are counting by one, may be more able to grDSCN0057asp the idea of counting up from the larger number. They would explore and understand the communative property through play.

My more experienced students were able to add the numbers quickly and make the necessary leaps according to the domino. It was interesting to watch and listen to students who were able to leap the first number, and then break apart the second number. To record this, I adapted the activity to allow the students to show their small leaps under the larger leap that was indicated on the domino.  Sharon Friesen (2010) states, “teaching practices that help students build mathematical proficiency combine concept formation with procedural fluency. In this way students gain a deep understanding of what they are doing and how to do it (p.52).”  This activity and this adaptation helped my students combine concept formation with procedural fluency. Albeit, that some students are not quite proficient yet; however, I did not expect them to be after this one experience.DSCN0051There were some limitations to the activity. I felt that using the number line from 0-10 was a good way to begin; however, it may lead some students to think that a number line must include 0. As we delve more into addition and subtraction of larger numbers, the open number line will be used to demonstrate student thinking so that it too becomes a tool students can choose to use. In ways, this is not a limitation, but definitely something to think about when using the lesson. As for possibilities, this lesson could easily be adapted to create additional lessons to explore larger numbers, maybe by using dice (6-10sided) or cards. Another possibility could be to explore more than 2 addend addition sentences, again using the dice and rolling three leaps. Students could also be given an open number line to record on.


Interesting to explore what happens past 10 on the number line

I enjoyed this lesson; it was easy to implement and had the students engaged using the number line. It provided me an opportunity to observe students who had challenges with the number line and possibly number concepts and understanding amount. The activity page in the picture shows a student I will be observing and listening to closely in the lessons to follow.  This activity afforded formative assessment into knowledge about number lines and number concepts.


This student gives me something to watch for…possibly he is unsure about the concept of an amount of 5, confused about the number line, or confused about the concept of addition.

Friesen, S. (2010). Raising the floor and lifting the ceiling: Math for all. Education Canada, 48(5), 50—54.

Breyfogle, M.L,  & Williams, L. (2009). Designing and Implementing Worthwhile Tasks.  Teaching Children Mathematics, 15(5), 276—280.

Our beliefs about learning shape the environments we wish to create

My previous post blends with this topic. My argument about the importance of the environment being interconnected is of great importance.  The teacher, the students, the community and the disciplines are interconnected.  “Knowledge is assumed to be the dynamic by-product of unique relationships between an individual and the environment; learning, then, is a natural by-product of individuals engaged within contexts in which knowledge is embedded naturally” (Choi & Hannafin, 1995, p.53).

Within these elements a structure for learning and knowledge building is grounded. Scardamalia and Bereiter (2010) state that “a supportive environment and teacher effort and artistry are involved in creating and maintaining a community devoted to ideas and their improvement (p. 8).  Classroom environments that are interconnected, safe, and welcoming cannot be taken for granted, diminished, or understated.  It appears seamless, but is ever so important in supporting deeper learning. When visitors or guests walk into a classroom that is conducive to knowledge building, they comment on how “nice” the room is, maybe it’s the lighting, maybe it’s the shape of the room—but most likely it’s more than that.

As an educator, passionate about functionality and about aesthetics, I have planned my environment carefully. At times, (as I have learned it is not always apparent) I must highlight what I believe makes it a productive learning environment, carefully structured to support the collective and the individual.  Here are a few things I have thoughtfully and carefully structured in my environment.

My room has tables. IMG_7191Not desks in rows. When people enter, they can almost feel a sense of collaboration.I have also planned for many adaptable learning spaces. There does not appear to be a “front” to the room and I have often been asked “you only use that small board?” Knowledge and ideas are shared by the collective, not just from the teacher. Areas around the room allow students to IMG_7787sprawl out on the floor, pull up chairs, or huddle together around supplies or tools when they are working.



Students know, expect, and appreciate that we will work with each other, share our ideas and thoughts, question, or agree with each other. I provide opportunities for these OLYMPUS DIGITAL CAMERAexperiences, but we also define what learning together is and agree on what we believe will work. We also discuss what it looks and sounds like, and I model productive working with a partner. At times, it is necessary to review this criterion, but at no time do I think (as the teacher) that we will no longer be collaborative IMG_2777because the students can’t handle it.  I’ll be the first to admit, some years seem more challenging than others, but I believe collaboration, learning to work together, and sharing thoughts and approaches, are valuable life skills and necessary to build knowledge.

Scardamalia and Bereiter (2010) developed 12  “knowledge building” principles (p. 9).  When reflecting on my classroom environment and learning, many of these principles are present.  As my students collaborate, they are also building knowledge as a collective and improving ideas as they share, present, and listen to each other. In addition, through collaboration and sharing of their ideas, during exploration of authentic problems and presenting of findings, idea diversity emerges almost naturally.

All learning tools are available to students. Students can use the tools whenever they feel it is necessary.  Students decide what tool would best suit the task and their learning. IMG_2809 In order to build knowledge, knowing the destination is not enough; teachers need to consider the possible steps in-between or the possible strategies students might try, but also listen, observe and allow students to share ideas. By having tools available, students are able to use tools to help explain or work through their ideas. It also helps to “democratize knowledge” and give “epistemic agency” by allowing students to share their individual knowledge and value each others’ ideas as legitimate. These two principles are also part of the 12 knowledge building principles.

How we structure our learning environments, what our expectations for learning are, what we see and hear when we observe our students, and the way we present lessons and investigations, are all connected to what each individual teacher believes about teaching and learning.  I know what I’ve done in my classroom has been firmly grounded in what I believe.

What do you believe? How does that shape your classroom environment and structure?

I learn

I learn….many of these qualities apply to my students, but also to me. The words on the bottom could also read “I teach”


Choi, J. & Hannifin, M. (1995). Situated cognition and learning environments: Roles, structures, and implications for design. Educational Technology Research and Development. 43 (2), 53-69.

Scardamalia, M. & Bereiter, C. (2010). A Brief History of Knowledge Building. Canadian Journal of Learning and Technology,36(1). Retrieved from http://www.cjlt.ca/index.php/cjlt/article/view/574/276


What Makes a Good Mathematical Task? Is there a secret?

As teachers we are charged with the task of creating, planning, sharing, and structuring our classrooms, our environments, and our programs. Wow, what a rich task that is! But what is a “good” or “rich” mathematical task?

In our changing 21st century world, what is considered to be a good mathematical task has also changed. There is a lot of information about the history of our education system and the move toward change. We see in history a need for procedures and a setting of standardization. I believe what we need now is creative problem solvers and critical thinkers who are engaged in learning. Sharon Frieson spoke of the history of our education system at a CEA conference in Calgary.

Good mathematical tasks need to be motivating and engaging. As teachers we look for ways to connect math to activities students are already engaged in. calvin and hobbesSome may argue that not all can be motivated, like Calvin; however, if connections are made students are more readily willing to attempt and continue with the task. Tasks need to provide students with connections between prior knowledge and also prior experiences. I think the beauty of prior experiences is not only the students’ personal prior experiences, but also the collective classroom prior experiences from previous math tasks and explorations.  I see this in my classroom when I am coaching a student and ask them to think about what was discovered during a previous investigation. You might hear me saying to a student “do you remember what we did when wIMG_5572e collected and counted all those items in our classroom?” Students then are directed to connect to a shared experience, they are able to remember that experience and they are able to remember what they learned from that experience.

If students are to remember an experience, then they must actually “do” mathematics. Good tasks engage students to actually be involved in the learning. They are the ones to decide what tools would be needed, what approaches they will try, what to do when they get stuck, etc.  This type of learning is engaging, there is not one correct approach or one way the IMG_5598teacher is looking for. The students use what they know and their own approaches to solve the task or problem. A good task would allow for multiple entry points and multiple end points. There is a sense of play and engagement in the task.

This little video/story highlights in an endearing way how thinking “outside” of the box and working collaboratively our students can attempt problems in different ways and soar!

It is important to identify, acknowledge, and appreciate the need for a learning environment that supports good mathematical tasks.  The physical learning environment needs to allow for space, movement, and availability of tools. However, the learning environment is also based on interdependent relationships between teachers, students, the community, and the subject disciplines (Friesen, 2009, p. 6). These relationships allow tasks to create and unfold in ways that appear seamless. The interconnected learning environment creates and promotes a culture of learning. “Over time, as students experience these relationships and learning environments that support caring, risk-taking and trust, students’ confidence in themselves as learners grows” (Friesen, 2009, p. 6).

Good mathematical tasks also need to involve communication between learners working together on the tasks and sharing ideas, approaches and questions. There is also a need to share findings together as a whole community of learners and allow questions, thoughts, and ideas to be invoked through the communication and sharing of approaches used. An interconnected learning environment is what good mathematical tasks need to nurture the mathematical tasks as well as the learners.

So is there a secret?

I believe the secret to good mathematical tasks lies not only in the task itself, but in the teacher, who with students creates the environment, looks creatively at shared experiences, and attempts risks. It is the teacher who realizes the potential in mathematical tasks introduced to learners and connects that, to the knowledge of learners in her care. It is the teacher that realizes the potential in what may appear as an “ordinary” moment and leverages that potential.

I teach


Friesen, S. (2009).  What did you do in school today?:  Teaching effectiveness and rubric.  Toronto: Canadian Education Association.

Math Fair: Creativity in problem solving

A math fair is a defined on the SNAP website as a fun, recreational way to share and learn through a variety of different math problems.  The SNAP website and the Galileo Educational network have many examples of problems that could be used in a math fair.

Whoa, wait, math—fun and recreational?

During our math fair at Telus Spark, I saw firsthand how fun and recreational a math fair can be, from our experiences (my graduate class) to the public, who approached our booths with hesitancy and curiosity and left feeling excited having enjoyed the recreational math they just participated in.

Well I would hesitate to call creating our display and coming up with a theme recreational.  I will be the first to admit that that stuff was a lot, but exploring the math problems (my own and my classmates) was definitely fun and recreational!  We were also extremely creative. I was blown away by how interesting, thought provoking, and fun each of my colleagues made their math problems.

So how does this relate to teaching mathematics in a math fair vs. a typical math lesson? Well there are the obvious time constraints involved. Planning and creating such displays would not be manageable especially because our teaching jobs and daily teaching duties may look something like this:jobs of a teacher And coming up with all those amazing different themes would be daunting. I’m creative but come on THAT would be taxing and possibly stifling in the end. Take a look at some of these displays!

Jumping Jamboree 1sword of knowledgeOursSo I turn away from the logistics of time, classroom constraints, gathering supplies, planning etc., (all the things that swirl in a teacher’s mind and best described as this)

a teachers mindtowards the math problems themselves.

These problems are engaging, interesting, and require creativity on the learner’s part! These problems definitely incorporated the skills of the 21st century learner:  critical thinking and problem solving, effective oral communication, but also grit, agility, adaptability, and at times, hope and optimism.

skills and attributes(This graphic came from Jackie Gerstein’s blog)

face blurred 2As I watched young learners come towards our display, they were curious and almost instantly engaged in our math problem.  I noticed that when Tina and I were able to explain and show our participants what was meant by equivalent addition sentences, especially younger learners, felt more at ease with the problem. Tina and I modeled this by having an example attached to our shirts and then another equivalent addition sentence on our backs. Similar to classroom lessons, demonstration and explanation helped students feel more at ease face blurredwith the problem and ready to attempt it.

I also observed a variety of approaches. I watched students stumble with the addition and just make a guess. I watched students adding using their fingers, using strategies like count-three-times, counting on from the first number, counting on from the largest number. I watched others who seemed to have a much more efficient strategy, possibly using a double or knowing their combinations to make 10. Similar to a classroom, there was a variety of strategies being used and a variety of levels to approach the problem.

Similar to classroom experiences, I also watched parents engage with their children. I’ve watched similar engagements during activities in student-led conferences. There were parents who wanted to jump in to give the answer, not giving their child a lot of time to attempt the problem in their own way.  Or who were quick to provide a strategy.  I also noticed parents who looked a little more annoyed that their child didn’t know the addition as quickly as they wanted them to. Possibly not knowing how to help, they simply just kept repeating the addition question: “what’s 5+4?”

In many ways, coachingthe best teachers the math fair problem was also similar to coaching in a typical mathematics class. I looked for ways to support without giving away the answer and ways to honor their first attempts without diminishing it by encouraging the student to look for other possible approaches.

What I really noticed during the math fair problem was a sense of creativity. The participants showed creativity in their approach to the problem and ways they tried out an idea. Each participant used creativity differently.  Creativity is the “secret sauce” of learning, especially in math and other subject areas too. Play is important for productive thought. Play is also important to build a sense of freedom from having only one correct approach. Math fairs, but also engaging math problems within our classrooms, incorporates that play with concepts and nurtures creativity.

Higginson uses an ecological image to describe school mathematics. He describes paths through school curricula as being “well marked, heavily used, and very straight” (1981, p. 505). He continues by stating that it is easy to get “lost” when moving away from these straight paths. However, the “occasional cross-country ramble can give the mathematics student insights into the nature of the discipline that are much harder to gain from the express lane” (Higginson, 1981, p. 505). When engaging in these recreational problems, taking a pathstudents explore off the path. They may get “lost” a little, but the creativity, excitement, and exploration may have made the problem enjoyable and will call them back to explore further another day.  Isn’t that what we hope and strive for when trying to engage our mathematics learners?

The Amazing Mathematical Clown Coaches!

The Amazing Mathematical Clown Coaches!



Higginson, W. (1981) Mathematizing “frogs”: Heuristics, proof, and generalization in the context of a recreational problem.  National Council of Teachers of Mathematics, 74 (7), 505—515.


Moving from the illusion of falsely exclusive pedagogical choices to sophisticated hybrids

As a classroom teacher of mathematics, I have heard of, seen, and taught during the “new” math reforms.  Of course, the most current reform and now the “new” math wars debate. As test results emerge from countries around the world in mathematics, I like to remember this: what tests don't measureWhat is worrying about the debate about which direction school mathematics should take, is the idea that simply imposing a “back to basics” or whatever other push towards teaching, will provide an absolute solution to much greater issues. It seems that in our changing world of 21st century learners even the definition of what being a numerate citizen means has changed. I am left to wonder if a resurgence of procedure practice and rote memorizing is the approach we should be taking to support our learners in the world outside of school. This leads me to ponder which one approach is deemed most valued? And by whom is it given value? Is there only one approach? I will not be delving into these questions during this blog, perhaps another topic for another post.

Alberta learning’s poster on teaching and learning mathematics states the vision of the educated Albertan as: Engaged thinker and ethical citizen with an entrepreneurial spirit.(math_poster_eng_web_print) The necessary question for all educators continues to be how to create an environment that encourages, promotes, and entices students to learn, explore and become engaged thinkers, ethical citizens and develop that entrepreneurial spirit. I am sure it will not come solely from memorization and rote learning.

As Sharon Friesen states in her article Math: Teaching it Better, the debate over “new” reforms is not a new one. When looking at the historical contexts, one that Friesen notes as occurring in the wake of the launch of Sputnik in the 1950s, we can take comfort in the fact that this is a long, complicated debate, with people falling on either side, fierce to defend what they believe in mathematics education (2006, p. 7).  To compound this issue, is that the span of people this debates includes is far reaching; educators to parents, politicians to mathematics, doctors to lawyers—it seems everyone has an opinion about what mathematics is and how it should be taught.

And so we are where we are, and have been here in the past, people willing to debate and battle over their ideas fiercely, without end and without pause to consider the positions of the “other side” assuming of course there are two sides.  As Sharon Friesen so eloquently states that all of this battling “exacts a considerable toll on those willing to wade into the fray. Nowhere, it seems, is the status quo defended with such fervor” (2006, p. 2). It is us teachers, left to wade into the fray of such heated debate, with considerable stakes at hand; parents, principals, fellow teachers, and most importantly our students.

David Clarke wrote an interesting article about oppositional dichotomies. I begin to ponder the dichotomy of a back-to-basics approach with a problem-based constructivist approach. Of course, this is assuming that these are the two sides in the math wars debate. I would venture to guess that there are many dichotomies involved in the debate and a useful first step may be to clearly define what these sides are, a task that may be impossible considering the diverse nature of education in general and the variance in classroom teaching in particular.

In Clarke’s article he states that “constructing such dichotomies as oppositional creates a set of false choices, sanctifying one alternative, while demonizing the other”. I think this is where the math wars debate falls. However, possibly turning towards what is complementary in these issues is one area we as educators could turn to as we venture into the fray. In Clarke’s article, he addresses tensions between “telling” vs. “not-telling” and “student-centered” vs. “teacher-centered”. It is important to remember and consider the teacher’s role in these dichotomies and also realize that they do not need to be at odds. Could this be the same for memorization: Look where it falls in this graphic… new math and old mathnot completely gone…not at odds with the “new” math

Another such example, is “telling” vs. “not-telling”, the simplistic notion of “not-telling” undermines the incidences when mathematical knowledge shared explicitly would benefit student learning (Clarke, 2006, p.383). A teacher can initiate new items into discussion, as these teacher-told initiated ideas are inserted into discussions in hopes that the students will interpret them in different ways rather than passively accept them, as a way to further and deepen a classroom discussion.

In identifying the notion of “telling” (a part of the back-to-basics approach called for in the current debate) vs. “not-telling”, there is a way to view these as complementary rather than a dichotomy.  I feel that if we look for incidences that embrace ways to view these teaching practices as complementary, we may begin to get somewhere, at the very least, begin to realize that mathematics education is not made up of these falsely exclusive choices. I would argue that for the sake of our students, (with the understanding that classrooms are not numbers and data to be collected in order to provide a one-way approach) we need to look for ways that these practices might prove to be complementary.  In doing so, educators will be developing more inclusive constructs. As Clarke states, “educational theories that employ such constructs will be more likely to support the development of more sophisticated, hybrid pedagogies, adapted to the demands of each classroom setting, but less constrained by culture or convention” (2006, p. 385).




Clarke, D. (2006). Using international research to contest prevalent oppositional dichotomies. ZDM, 38 (5), 376–387.

Friesen, S. (2006). Math: Teaching it Better. Education Canada, 46 (1), 6-10.