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:  https://www.ted.com/talks/chimamanda_adichie_the_danger_of_a_single_story

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.

Bees and Wasps: Can You Tell the Difference?

European Honey Bees

Notice the wax comb on the wooden frames

European Honey Bee
Notice the large full pollen sacs on her legs (orange)

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, https://commons.wikimedia.org/w/index.php?curid=51253464

Blue Orchard Bee

CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=114919

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, https://commons.wikimedia.org/w/index.php?curid=2577167

Wasp Building a Paper Nest

By Sanjay Acharya - self-made at Sunnyvale, California, USA, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=3952953

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, 
https://commons.wikimedia.org/w/index.php?curid=210595

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.

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

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.

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!).

Coding With Kids

Being able to write code is one way to move passive app and game users into more active learning. Writing your own code puts you in charge of choices and provides basic skill development that could become crucial as technology expands exponentially.

Coding also helps students learn to communicate clearly and with precision, and requires an attention to detail not found in many other areas of communication.

The Hour of Code movement promotes the exploration of code writing by students of all ages in order to introduce coding in an accessible manner. There are many Hour of Code activities that can be done with kids who have no experience with coding. Here is a quick sampler of some of the activities I've tried with my family:

https://scratch.mit.edu/ A family favourite, Scratch is an intuitive drag-and-drop building block style programming platform that helps introduce basic programming logic, yet can be used to create some surprisingly complex programs. It is accessible to even primary students, but provides enough challenge to retain relevance for older students as well. Be careful though, this one is very addictive!

https://code.org/mc Minecraft is a game, yes, but here you can use another drag-and-drop block based programming platform based on Javascript in order to create an adventure for Steve or Alex.

https://www.khanacademy.org/hourofcode Like many other aspects of Khan Academy, the activities here are somewhat more prescribed, but may suit learners who find an abundance of choice to be overwhelming.

https://code.org/learn Here you can find many more activities, apps and also "unplugged" coding activities for Hour of Code that require no electronics whatsoever
. We have tried Rock, Paper, Scissors and enjoyed it without the use of any devices.

http://ai2.appinventor.mit.edu/ For students with some coding experience, MIT's android app maker may be of interest. Students can work with their android device, or use an android emulator on a PC to run their programs.

Grace Hopper, 1952

Where did it all come from?

Along with actual coding, the history of programming is also quite interesting.

In this clip from the 1970's Connections series with James Burke you can see how weaving looms led to punch cards which led to modern coding:

https://www.youtube.com/watch?v=itd-4lMoXgI

Much of modern coding was made possible by the early work of various female pioneers in the field. Grace Hopper create the first compiler, for example, which allowed binary input to be converted into a programming language.

The links below include interesting articles that highlight the contributions of women in the field, and also discuss how the demographic of the "typical programmer" changed over time:

http://www.npr.org/sections/alltechconsidered/2014/10/06/345799830/the-forgotten-female-programmers-who-created-modern-tech

http://www.startupdaily.net/2015/02/women-didnt-just-recently-start-coding-actually-stopped-coding-decades-ago/

http://www.theguardian.com/technology/2016/feb/12/women-considered-better-coders-hide-gender-github

Here is a blog post about how the act of knitting is closely related to coding:

https://www.codecademy.com/blog/70

The history and use of punch cards can be found here:

http://www.computersciencelab.com/ComputerHistory/HistoryPt2.htm

And for those who are particularly interested, here is an odd and long 1st-hand account of learning to program with punch cards in the 70's

http://www.columbia.edu/cu/computinghistory/fisk.pdf

Happy coding!

The Canoe Doesn't Care Who the Leader Is...

If it tips, everyone gets wet. 

~ as quoted by Ela Smith at the You Don't Know What You Don't Know Workshop series.

It's another way of saying "we're all in this together".

Most people are aware of the Truth and Reconciliation Commission Report and its 94 recommendations, but few Canadians are aware of the finer historical details leading up to it. Today I had the privilege of participating in the first two of three sessions aimed at educating professionals who work with children in my area.

I had thought I was relatively well-informed about the facts, but there was much that I learned today-- things that begin to help me make sense of why friends and students I have known have been reluctant to share their heritage.

When I was very young, the family that moved in next door to me had a daughter my age. We played together quite a bit. Her name was Marylee (or Merrilee? I was too young to worry about spelling at the time). There were several milkweed plants behind out houses, which shared a large unfenced yard, and we played with the sap and watched the monarch caterpillars and butterflies come and go together. We dug up worms together and watched them burrow back into the dirt. She tried to teach me how to climb a tree, and I brought out my dolls to play with her. I know it sounds cliche, but we really did make mud pies (and mud cakes, and mud pizzas) together. 

One day I heard my mom and grandmother talking about how that "Indian family next door from the reserve" had a problem with their son lighting fires around the neighbourhood, and after some deliberation, I was told I could play with Marylee, but not her older brother. Fair enough--who would want to play with a stinky older brother!

After we moved, I attended a new school and was instantly branded a troublemaker because during recess, I became very upset and agitated when the kids played "cowboys and Indians". The goal was to kill off the Indians! Marylee was an Indian! Why on earth would anyone want to kill her or her family?! Unfortunately, neither the other students nor the teacher on duty saw it my way.

We moved again, and soon after another family moved in beside us. It was a family with 4 kids. and the two girls were close to my age. We played together often, but I was a few months older than the eldest girl, and as result, I tended to become "the leader" in our play. I remember one day I crossed a line in my bossiness, and the girls went home and told their older brothers. They all came out and the brothers gave me a stern talking to. Being an only child, and having lots of toys, they saw me as spoiled. I saw them as a true family, united and strong, and I doubt they ever understood just how envious I was of them. I remember in that moment realizing that no toy I could ever own would ever match the bond they had as siblings, and it was perhaps one of the loneliest points in my childhood. I learned a couple of years after they moved in that their father was First Nations. I probably should have clued into this when they took me to Hagersville for ice cream and to visit to the uncle and grandfather on the reserve, but I was 8 at the time, so perhaps that is to be forgiven. The part I do remember is the way that whenever this came up in discussion, it was always spoken in hushed tones, very seriously, and very secretively. 

I grew up and became a teacher. More than once in my experience in teaching, I found out at the end of the year that a student of mine identified as First Nations. This was generally only revealed to me hesitantly or after knowing the student for several months, and still happens in this way even now. It saddens me that there is a silence and fear of revealing having such a rich and beautiful culture, and only after learning more, am I beginning to understand the role history has had in what can only be honestly described as cultural genocide. Generations of western immigrants and ignorant political and social policies and attitudes have made the original peoples of Canada become, in many ways, invisible and voiceless.

In Canada, in 2016, we have numerous communities without access to clean drinking water. We have housing issues, issues with mouldy, decaying schools and homes, high suicide rates, MMIW, and whole communities of students who have to face moving away from their homes and communities just to attend high school.

I did not know:

- about the "pass" system

- that there are 634 First Nations in Canada
- that homes on reserves are not eligible for mortgages and can only be purchased outright

- that "status" rules are in many ways discriminatory regarding gender

- that "status" "benefits" most often do not materialize--for example, although dentistry is supposed to be covered for status Indians, since the Federal government is less than punctual and reliable in paying for this, few dentists will honour it

- that not only were kids taken to residential schools and abused, but that they were often made the subjects of medical experimentation

- that kids were sent not to the closest residential school, but instead were sent an average of 6 hours away so that they would not be able to easily run away or see their families or communities

...and the list goes on.

Today, in discussing the history in more detail than I'd ever known before, and participating in the blanket activity developed by KAIROS, I am left reflecting on what I didn't know, what I still don't know, and my responsibility as a parent, teacher and Canadian to learn more in order to do my part in the reconciliation process. It took seven generations to get us here, and it will take seven to recover, and it is my responsibility and yours to help make that happen.

Pieces of the Whole

In working through activities for basic numeracy skills, the "pieces of the whole" idea keeps surfacing, regardless of what manipulatives or operations are being used. It started when I was looking through sock matching and sorting activities, in which two matched socks make a pair, with the "pair" representing a whole. It continued through Lego brick building, pattern block mosaics, and looking at everyday grid patterns such as are found in trays of cans, chocolate bars, golf balls, etc.

Whether you are working with addition and subtraction, multiplication and division, fractions or percentages, the concept of "part" forms a vital part of the math lesson.

Some examples:

In this chocolate bar, the whole is "5", so one piece is 1/5 of the whole. Fractions work when the size of the parts is the same for each piece. 

We can also share a single piece with 4 friends and have one left over to keep. 5 ÷ 1 = 5

We can break off two pieces to make the subtraction sentence: 5 - 2 = 3

and then add them together again to make the whole bar: 3 + 2 = 5

In this lasagne, there are 5 columns, of which one is missing. Therefore, 5 - 1 = 4, or, there are 4/5 of a lasagne left. We can also say that 20% of the lasagne has been eaten.

In this carton of eggs, there are 12 eggs. Two of the eggs are white, one is blue, and the rest are brown. 2/12 or 1/6 of the eggs are white. 1/12 is blue. 9/12 or 3/4 are brown.
2 + 1 + 9 = 12
We can also say there are 6 pairs of eggs, or 6 x 2 eggs in the carton.
If we only want to use the brown eggs, we can remove the others: 12 - 3 = 9
We can divide the carton by rows, columns, pairs of columns, or into two equal columns 3 eggs wide. In doing so, we can investigate factors of 12, and experiment with various potential common denominators when exploring related fractions, and explore equivalent fractions.

Parts of the whole form a basis from which we can build on many mathematical concepts. We can extend this for use when speaking of angles, while referring to the circle (360 degrees) as the "whole" from which other angles are compared. This is, in fact, exactly how pie charts work.

We can even take this into polynomials by calculating the area of a deck for a pool:

If the pool is 8 m x 15 m, what is the area of a deck that surrounds it if the width of the deck has a universal width of 4 m?

The pool and the deck together become the "whole" combining the area of the pool and the area of the surrounding deck. 

This concept, of parts making a whole, is also a vital part of integral calculus in which the area under an irregular curve is calculated.

Encouraging students to explore these concepts using mathematical terminology and sharing their discoveries can help in relating previous knowledge with new concepts.

On Class Discussions and Participation

In nearly every classroom we see it--a handful of students who always raise their hands to participate, and a majority who do not. For some, it may seem "uncool" to appear keen in class. Some may be naturally shy or introverted. Others may simply not know the answers or have ideas they wish to share. Others still may be off-topic or not engaged mentally in the topic of discussion.

How do we, as teachers, engage everyone in discussion while remaining respectful of our students and their individual needs?

I recently read an article by Alfie Kohn about this very problem. You can find the article at this link: http://www.alfiekohn.org/blogs/hands . 

We've all encountered teachers who call on students they think are distracted or not engaged in the lesson, whether through film, television or real life. I would imagine that most of us can agree that putting students on the spot in this way can cause fear and anxiety. Personally, this is not the way I wish to approach my students. I want my students to feel safe and respected. I want them to feel like they have room to take chances, make mistakes and learn and grow without being subjected to humiliation for their efforts.

So then, how do we engage all students in class discussion?

The students themselves understand these patterns

Since it is their learning that is at stake, inviting students to come up with solutions for class discussions as well as questions related to the concepts being studied is a meaningful and respectful way to ensure that they are invested in the process. However, finding solutions may take time, especially with students who are used to more typical classroom discussion. In the interim, as students are left with the challenge to think of solutions, some of the following ideas can be worth trying out.

In the meantime...

One method is taken from Susan Winebrenner's book, Teaching Gifted Kids in the Regular Classroom. Winebrenner proposes a name card method. The benefits of this method are that each student has a chance to confer with a partner before answering a question, and that the random shuffling of cards helps the teacher to avoid bias in choosing students to participate. The method works like this: each student's name is written on an index card. When the teacher asks the whole class a question, he or she gives the whole class 30 seconds to discuss it with their elbow partner. Then the teacher shuffles the deck of index cards and calls on the student whose name is on top. The student can then answer or consult further with their elbow partner if they are at a loss for an answer. The teacher repeats this for several students, and then the class looks at the answers and discusses the relative merit of each.

Some teachers I know have developed a system in which students have a signal that they wish to talk, such as might be shown by cards or cups on their desks. Rather than raising their hand while someone else is talking, they show their signal and wait their turn. It is a little bit like using a talking feather in a sense. A talking feather is a First Nations tradition in which a feather is held by the speaker; while the speaker has the feather, they may speak. When they are finished, they pass it on to the next person such that each person has a chance to speak without risk of interruption.

For a class discussion, rather than specific questions, the idea of the talking feather or a similar method may cause less interruptions in the flow of the conversation. 

Some further ideas to engage students that I've seen have included having a question box as well as a comment box where students who feel their ideas were not yet heard can write them down and have them brought up in the next class. This can also act as a review of the previous day, and can preserve anonymity for students who lack confidence. The students can choose whether they let the teacher know who the question or comment came from, but the teacher promises to preserve their anonymity from the class as a whole. The teacher can then pull out the questions and comments and the class can address them. Ideally, as less confident students hear that their input is valued, they will gain enough confidence to begin to contribute without using the boxes.

Of course, much classroom work is also done in groups. The challenge here is to encourage students to ensure that quieter students are allowed to contribute as much as the more outspoken ones. It is a lifelong lesson that is important to hone. Society works best when all voices are heard. Some teachers actively assign roles for group work, and these roles change from one assignment to the next. Encouraging students to find their own ways to address this challenge can help them build collaborative skills that will last a lifetime.

Multiplication Woes

Multiplication and multiplicative reasoning can make a world of difference in a student's mathematical development, however, for many it becomes a stumbling block that slows them down, sometimes to the point of hindering their studies in high school and limiting their post-secondary options.

There are many different ways to understand multiplicative expansion (the word "growth" can become confusing when students begin to multiply fractions and integers). One way is to see it as a series of addition problems, such as:

2
2+2
2+2+2
2+2+2+2
2+2+2+2+2

and so on. As we move down each row, the multiplier increases by 1 so that:

1x2 is 2
2x2 is 2+2
3x2 is 2+2+2
4x2 is 2+2+2+2
5x2 is 2+2+2+2+2

 You can also say it as "3 two's make 6" or, "3 groups of 2 equals 6".

Another way to represent this is to use grouping. Students can make piles of manipulatives such that each pile has the same number of items in it. This time, let's use multiples of 7. In this case, each pile would have 7 pieces. Let's say they wanted to know how many 4 groups of 7 are, or, 4 x 7. Students can use various strategies to determine the total.
They can:

• count up the total number of pieces by either counting each individual item in all of the piles
• start counting-on using their understanding that the first pile has 7, then continuing counting the remaining piles from 8 onward
• count each group "by 7's", also known in some circles as "skip counting"
• count the number of groups and use their remembered answer for 4x7

Each of these stages show a different level of mastery of the concept.

However, piles of manipulatives, or circled pictures of groups of objects on a worksheet have a limited usefulness when it comes to visualizing the patterns that are common to multiplication.

For this reason, we can try and move to an area model as shown below. It is called the area model, because the solution to the multiplication problem also represents the value of the area of the rectangle. Area is another way in which multiplication can be visualized, and it also shows a practical application of the concept.

In our example the number of units in each row is 7, while the number in each column is 6. We have 6 rows of 7, or 6 x 7 units in the rectangle.

We can look at this model in two ways. We can look at the columns (7 columns of 6 units each, as shown on the left), or we can look at rows (6 rows of 7 units each, as shown on the right). We have simply lined up the groups into columns or rows to make counting, as well as visual representation, easier.

The grid lines in these pictures don't have to be there for the model to work. Simply knowing the base and the height of the rectangle gives enough information so that the multiplication problem can be solved, and the area found.

Adding the grid lines helps us see the groupings involved, and makes the visual representation of the problem clearer, particularly for students who have not yet reached mastery. Approaching the same problem using groups of rows and repeating it using groups of columns helps reinforce the key principle of commutativity, in which the order of the numbers multiplied does not change the final product.

The grid model can be used with manipulatives that allow for columns of units to be connected, such as unifix cubes, multi links, Lego, or square tiles. These columns can be put together to form the rectangle that represents the problem.

Once the multiplication concept has been explored, students will eventually need to learn to access those facts quickly. There are a number of options that can help with this including:

• learning to skip-count (counting by a number, such as 3-6-9-12-15-18-21-24-27-30-33-36-39-42-45 etc.
• classroom games
• finding number patterns to follow (even numbers for 2's, ends in 0 for 10's, digits add to 9 for 9's etc.)
• recognizing patterns in daily life (eggs come in 2x6=12; a case of canned vegetables has 4x5=20 cans, etc.)
• intensive answering, such as with regular timed tests, Mad Minutes, etc.
• rote repetition
• written tables such as the one to the right, which are given blank for students to complete and mark for patterns
• musical chants /songs
• classroom charts
• calculators 

Each of these has its place at various times, however, students who can spend less effort to retrieve these facts do better as more concepts are introduced in higher grades.

The over-use of rote methods, and the dawn of Bloom's Taxonomy, have made the task of having students memorize their times-tables unpopular in the classroom. This is slowly changing.

The problem is not so much that students spend time memorizing these facts, which is admittedly a lower-level task, but that if they do so without an understanding of how multiplication actually works, the knowledge of "facts" will have limited value as more complex mathematics are introduced.  Students who understand the language, the commutative property, the groupings, and that multiplication is an advanced form of addition, can show how multiplication patterns continue, and likewise, how using the inverse operation of division causes the pattern to reverse, will be well equipped to apply it to fractions, decimals, integers, algebra, etc. They will also be better able to handle related concepts such as area and volume.