Thursday, 9 March 2017

Cuisenaire Rods for Intermediate Math

One of the larger goals of math in the intermediate grades is to learn about proportional reasoning (multiplicative thinking) as applied to fractions, percentages, proportions, ratios and patterning. This provides a solid foundation from which higher levels of mathematics can be explored, including geometry, linear and quadratic functions, and data management.

In the primary and junior grades, among other mathematical constructs, students work with decomposing larger numbers into smaller quantities, primarily as addends, and eventually working towards factors as used in multiplication.

Cuisenaire rods are a useful tool in math, allowing students to easily compose and decompose small quantities while showing the proportional relationships. For example,

Image result for cuisenaire rods

Each block can represent an even proportion of units. For example, if we call the white rod "1" and the red rod "2", then it follows that light green is "3" and so on up to 10.

We can combine rods to create larger numbers, such as 12 by using red and orange, 20 by using 2 oranges etc.

We can also assign a larger value than "1" as our base (white) unit. If we assign it as "5", then we can count by 5's up to 50 without the need to add additional rods.

Using these relationships, intermediate students can use these rods to explore the concepts of proportion, ratios, fractions, and the distributive property.

Factoring and the Distributive Property With Cuisenaire Rods

Using the principles listed above, students can use the rods to factor a larger number. For example,
in the following photo, 12 is factored in a variety of ways:

 In the dark green row, we see 2 rods of 6 or 2 x 6. Likewise, in the red row, we see the reversed version where there are 6 rods of 2, or 6 x 2. Through the use of many examples, students can determine through exploration the commutative property of multiplication, and also the relationship between multiplication and division.

Factor trees can also be built in this way, by further reducing the larger factors. Eventually, each component of each row can be factored down to the prime number components.

If we decided to use a base unit other than one, we could compare similar numbers based on proportional relationships. For example, if we chose "3" as our base for the above example, the top row would be worth 36. The second row would be 2 rods of 18. The third row would be 3 rods of 12, and so on. We could also build the 36 by using more orange and red rods and keeping the base unit as 1.

In the following picture, "9" has been divided into various rows in a somewhat different way:

Moving down from the top, the second row shows 3 rods of 3, or 3 x 3. In the third row down, we see something new. Suddenly there are two different colours in the row. Can we write a multiplication expression for this row?
Similarly, the 4th row also has two different colours.
In the 5th row, however, we see a pattern emerge. There are 3 instances of red-white, or 2, 1 in this row. How might we express this mathematically?

This is an example of the distributive principle of multiplication. How else might we use cuisenaire rods to show this?

So far we've been looking at rows of rods to compare relative values. How might we move into an extra dimension where we can compare both rows and columns?

Eventually, algebra tiles will help with these visualizations, however, the challenge of combining grids to visualize proportions allows for concrete abstraction that will eventually lead into linear and quadratic applications.

Tuesday, 22 November 2016

Why I Will Not Teach Tolerance

Coco0612, Educ 323, CC BY-SA 4.0
We've all heard much over the past several weeks about "political correctness". Quite frankly, I suspect all of us have grown tired of that phrase.

I cringe when I hear that phrase, as well as its close relative, tolerance, particularly when they are touted as societal ideals. I believe we can do much better. I believe we must do better if we are to create and maintain a safe, inclusive and globally aware community of learners.

To me, political correctness is much like putting on a mask and pretending not to hold stereotyped prejudices (or downright bigotry), or that it's OK to not learn more about the differences of others. To be fair here, sometimes it might be a matter of time and exposure--perhaps there just hasn't been a chance to do better; particularly for an individual who has never met a person with said difference. However, much can be said about the attitude with which people approach those with differences. In most cases, I believe that being politically correct simply allows people to use the benefits of someone who is different than them without showing them respect beyond the very basic etiquette required of public or business dealings. It is like covering one's eyes and waiting for something undesirable to go away--there is a part of you who knows it exists, but you really wish it would just go away or be done with so you can get on to more interesting or important things. I believe this sort of thought pattern is dangerous, and can lead to the phenomenon we see in which white people fail to see racism as a problem in North America (yes, Canada, we too have a racism problem, and it involves our indigenous people as well as other visible minorities). If political correctness were represented as such in books and movies, it would involve tokenism and stereotypes, with all of the main roles being taken by the dominant group.

One step better than political correctness is tolerance, in which people agree to ignore or overlook differences in order to get by when they must interact. It might contain an inner message such as "I like you despite this one part of your identity". It doesn't necessarily imply that someone is using the other for their own benefit, but it also requires no more than the basic recognition that there are differences in race, religion, socio-economic factors, politics, gender identity, gender expression, sexuality, sexual orientation, age, etc. It requires no discourse, conversation or understanding beyond what is immediate to the circumstances of necessary interaction. If tolerance were to be represented as books and movies, there would be some effort made for representation of a variety of groups, but those would usually play minor roles with the main roles being taken by the dominant group. Some stereotypes would remain.

Rather than promote the above superficial methods of dealing with people who are different in some way, my goal as a parent and teacher is to promote dialogue leading to understanding, acceptance and compassion. I will never be an Hispanic man. I will never be an Inuit child. I will never be a Syrian refugee. I will never be a black transgender woman. I will never fully grasp the experiences of those who are. But I can listen, learn and, albeit to a limited degree, empathize with the experiences and perspectives of those who are, and I can teach my children and my students to do the same. This doesn't mean that we have to believe what others believe, or follow their cultural traditions, or in any other way emulate them; it simply means that we need to look a little closer with our eyes, our ears, our brains and our hearts to better understand their world view. By connecting as humans with others and sharing a larger group membership, be it as a classroom, a local community or a national identity, we all become richer, stronger, and learn to understand and communicate effectively for the benefit of all. If understanding, acceptance and compassion were represented as books and movies, there would be a rich mosaic of representation, with individual characterization and situation replacing any stereotypes.

Some related thoughts on this topic are well articulated in the Ted Talk, "The Danger of a Single Story" by Chimamanda Adichie:

Since this is a particularly volatile topic at the present moment, I will not be allowing comments on this particular post. Yes, I do see some irony in that decision, but I will stand by it nonetheless.

Sunday, 25 September 2016

Bees and Wasps: Can You Tell the Difference?

European Honey Bees
Notice the wax comb on the wooden frames

Bees, particularly honey bees, have been in the news often over the past few years, mainly due to their decline as a result of colony collapse disorder.

Honey Bee Workers
Honey Comb in the Making
Photo Credit: Gordon Fountain, 2016
Worker Bee Filling Comb
Photo Credit: Gordon Fountain, 2016

Although honey bees are not native to North America, we have come to depend on them as important pollinators for a large portion of our crops. They are beneficial mainly for their pollination services, and more famously for their honey and wax.

Other bee species that are native to North America include bumble bees and many species of solitary bees. These are also important pollinators, although they do not live in hives the way honey bees do, and do not produce honey.
Common Bumble Bee
Large and "furry"
By Paul Stein from New Jersey, USA - Azalia Blossoms, CC BY-SA 2.0,

Blue Orchard Bee
CC BY-SA 3.0,

At this time of year, there is an abundance of wasps and hornets. These are not bees, but are often mistaken for them. They do not produce honey, are considered less effective than bees at pollination due to the lack of a hairy body, and are predatory. Among their prey are bees. They are aggressive, and are attracted to garbage bins, pop and fruit juices and any other sweet smells.

This is a Yellow Jacket WaspBy Richard Bartz, Munich aka Makro Freak - Own work, CC BY-SA 2.5,
Wasp Building a Paper Nest
By Sanjay Acharya - self-made at Sunnyvale, California, USA, CC BY-SA 3.0,

Notice that the body of the wasps are completely devoid of "fur" or hair. The connection between the thorax and the abdomen tends to be much thinner than in bees. The wings are elongated. Their legs have no pollen sacs.

Bees and wasps are also different in their behaviour. Since wasps are predators, they tend to be much more aggressive than bees. The exception is Africanized honey bees, which were introduced to North America in an effort to increase honey production, They are know for their aggression, but they also require a warmer climate than is found in Canada. Honey bees and other bees generally only sting when their safety or the safety of the hive is in question. Drones (male bees) cannot sting. If you are not opening up or tearing apart their hive, blocking their flight path to the hive or inadvertently crushing them, you are unlikely to be stung by a bee.

In the off chance you do get stung, the pain can be reduced by adding baking soda and/or crushed plantain (the common "weed", not the banana-like fruit) to the area. Watch for symptoms of anaphylaxis.

There is a small proportion of the population that is allergic to stings, and these stings could include hornet, wasp or bee stings. Learning to distinguish between these could become a matter of life and death for those affected.

Common PlantainBy Rasbak - Own work, CC BY-SA 3.0,

Friday, 1 July 2016

Feedback and Finals: The Problem with Culminating Tasks

In the field of education, one of the big buzzwords right now is "growth mindset". In essence, it asks students and teachers to visualize themselves on a learning journey in which mistakes can lead us to interesting places and new opportunities for learning; where a positive attitude is vital and reflection and improvement are key.

And yet, we still have final exams and unit tests.

Of course we can't spend forever on a specific topic, but to end the learning with a judgement activity that gives no feedback other than a mark does little to inform a student or contribute to further learning.

Perhaps I am still fixated on my high school's motto from many years ago, which was "Know the Reasons". I certainly live with that idea firmly implanted in much of what I do. However, it seems to me to be an important concept for continued learning, reflection and critical thinking which are sometimes found lacking in wider society (not to name any recent political campaigns or such, but I'm sure we can all find a real-life example in which critical thinking was found to be lacking).

So how can we approach summative assessment while encouraging students to continue to reflect on their understanding beyond "the test"?

Some teachers allow for re-tests, although this is becoming unpopular in some circles. Many teachers opt for summative projects rather than tests or exams. Some incorporate a self-evaluation into the culminating task. Are there other ways to approach this?

We've all seen it: student x gets back the test, looks at the mark at the top, then tosses it in the recycling bin never to be read or seen again. All of the notations and feedback on the paper are ignored, and an opportunity to learn from the assignment is lost.

This happens at all stages of a particular strand or unit, but is most common after the summative, perhaps because the student knows it will be a long time before they see that concept revisited. However, this is a valuable time in which to receive feedback because it is likely to address the student's highest level of understanding yet for this topic, skill or subject.

How do we get students to buy into such reflection?

The learning cycle is a lot like using a lint roller: every time you go over an area, a little
more stuff sticks. That also applies to reviewing all tasks. The mentality of "I'm so glad it's over, I never want to see x again!" is hard to overcome. Part of the issue may also be the idea that since the course has been passed (or not), the learning on that subject is no longer necessary or worth pursuing. When students see the report card or credit as the goal rather than the learning itself, the idea of review after the last task may seem pointless to them.

When students see the report card or credit as the goal rather than the learning itself, the idea of review after the last task may seem pointless to them.

Some schools have a period of time in which students can go over their final exam or assignment with the teacher. Other ideas might include a class discussion over the classes website, Facebook page or other online discussion venues, or a scheduled drop-in time where students can meet with their teachers. Some teachers build a reflection component into the final task itself. Although it can also be argued that a little more time and feedback could aid in reflection, taking time to do so at all is still a positive step.

I am curious about how students in classes wherein a growth mindset is strongly and embraced by the students might respond to such opportunities to receive more feedback and review/reflection when compared with classes who continue on a more traditional path. I welcome your insight in the comments below.

Sunday, 26 June 2016

Life with Bees

My son and I have become bee havers this spring. I use the word "havers" rather than "keepers" since the bees pretty much take care of themselves. We've only been doing this for a couple of weeks so far, but I've already learned much along the way. 

It's an interesting and calming activity to share with a kid who has always had a fascination with nature and small creatures.

Honey bees, for the most part, are docile and will only sting when the hive is under attack or they perceive it to be. There are Africanized honey bees that are more aggressive, but those only survive in warmer climates much further south than Canada. Our bees are a mixture of various European breeds that have been bred locally for several decades.

Honey bees are not native to North America, but since their introduction, they have become very important pollinators, and much of our food is the result of pollination of honey and other bee species. They may be known for their honey and wax, but their true importance to humans lies in their pollination abilities.

Many people mistakenly call wasps and hornets bees, but they are very distinct species with different habits.

Wasp nest--not that it is paper, not wax
Above are a wasp and wasp nest. Note that the wasp's nest is paper-based, not made of wax, and that while it has hexagonal cells and rows of paper comb, the overall shape looks like a round paper lantern. Below is a picture of a hornet.

Below are some honey bees on comb, and below them is a bumble bee.

Honey bees, on wax comb
Bumble bee
Wasps and hornets can be aggressive and most stings people receive come from these insects. Bees tend to be more docile, although they will defend their hives.

If you have bees on your property, do not use insecticides or call a pesticide company. Instead, contact your local beekeeping association and they will very happily remove the bees. Everyone wins as the beekeeper gets free bees and you get the safe removal without the use of toxic products.

Most honey bees are female. There is the queen, who is central to the hive and the only bee that reproduces in most cases (occasionally worker bees can become fertile, but can only lay unfertilized eggs which become drones). The queen is an egg-laying machine who depends on several workers to feed her.

The majority of bees in a hive are worker bees, who take on different roles during their short lives, including nurse bees who feed and care for larvae, foragers, builders of comb, defenders, etc. All worker bees are female.

A small number of bees are males, called drones. Drones are slightly larger than workers, have no stingers, and their only job is to mate with a distant queen. 

On our recent visit to the apiary (bee yard), we noticed that the bees at the entry were being groomed by other bees from the hive. This is an important activity as it helps the bees keep down the number of mites in the hive. Varroa mites are a serious threat to honey bees, but healthy hives are able to keep down their numbers through various activities, including grooming.

Some of the bees watched us as we took out frames for inspection to determine the health of the queen. I would guess that they were trying to determine if we were a threat to the brood. Since we moved slowly and carefully and did not threaten the hive, we became more of a curiosity than a threat.

Since these are new hives, it is important for us to know that the queen is healthy and laying new eggs. If she stops, the hive will need to build queen cells and start making a queen. They do this by choosing young larvae to feed royal jelly and bee bread, which is a mixture of pollen, honey and various enzymes. This feeding difference is the key to determining whether a bee will become a queen or a worker bee. Several queens are created. Upon emergence from the cell, the quickest and strongest one fights off the others to become queen of the hive.

Those who have not visited a bee hive may think this sounds absurd, but watching the bees go about their business has a very calming effect on people. It certainly does for me (and I was very hesitant about the whole idea not very long ago!).