How Do You Measure, Measure A Year?

I have loved the opening song, Seasons of Love, from the musical Rent since the first time I heard it! During a numeracy Masters of Education course we were tasked with finding examples of numeracy and innumeracy. I thought of this song and its mathematical implications. Here are the lyrics and my analysis from my favourite Broadway version:

[see Broadway version:


525 600 minutes
525 000 moments, oh dear
525 600 minutes
How do you measure, measure a year?

In daylights, in sunsets
In midnights, in cups of coffee
In inches, in miles, in laughter, in strife
In 525 600 minutes
How do you measure, a year in the life?

How about love?
How about love?
How about love?
Measure in love

Seasons of love (love)
Seasons of love (love)

525 600 minutes
525 000 journeys to plan
525 600 minutes
How do you measure the life of a woman or a man?

In truths that she learned
Or in times that he cried
In bridges he burned or the way that she died

It’s time now, to sing out
Though the story never ends
Let’s celebrate
Remember a year in the life of friends

Remember the love
(Oh, you got to, you got to remember the love)
Remember the love
(You know that love is a gift from up above)
Remember the love
(Share love, give love, spread love)
Measure in love
(Measure, measure your life in love)

Seasons of love
Seasons of love
(Measure your life, measure your life in love)

By Daniel Noonan


Published by
Lyrics © Universal Music Publishing Group
Read more:


Seasons of Love from Rent Analysis by Fred Harwood

First it is good to note that they describe the 525 600 correctly (without the ‘and’). It is accurate if we ignore the additional 1440 minutes that occur during a leap year once every four years. 24 hrs x 60 min/hr x 365 dy/yr = 525 600 min/yr. Since 75% of the time this is correct, we will allow the poetic license of the song to make this claim. Likewise the second line of “525 000 moments, oh dear” to also be an approximation for the main part of the size of this number being very large with the 600 being only 600/525 000 x 100 or 0.11 % off.

“In daylights,” seems like a strange measure since it refers to when the sun is up and is 365, 75% of the time. It could refer to when you are not sleeping since this is when you are doing something although I have solved many problems subconsciously while sleeping.

“in sunsets” makes us pause to consider how often we attend to the actual setting of the sun and how often we actually are in a position to observe it. Rainy seasons and winters with it being dark when we leave our classes, have a lower frequency than the summer with better weather and longer periods of activity outdoors. Since re-listening to this song, I observed two sunsets per week during this spring season. This might make a good average for the year with less during November to February and more from June to September. If sunsets are metaphors for moments when we pause to wonder, I would increase this total to 5 times per week making a year between 100 and 250 times

“In midnights,” could be metaphorical to those quiet times when the world sleeps but we do not or it could be literally the times when we are still awake. This has greatly reduced since retirement to 3 to 4 times a week making it 150 – 200 times.

“in cups of coffee” would be 0 if it meant ones I have consumed, or 20 to 300 if it refers to my making/buying for friends. (The upper limit occurs when I am a camp cook for 8 to 11 days and make many pots a day.) This could also be metaphorical for time spent relaxing, gathering thoughts or talking with friends.

“In inches,” seems much more like a Fermi question. Does it refer to the actual distance we personally cover in a year? I can gain an estimate on the amount driven in a year as my vehicle passed the six year mark at 84 000 km. There were times when my wife drove without me but these will be roughly balanced by the times I was driven by someone else or was on transit. 84 000/6 = 14 000 km x 1 000 m/km x 100 cm/m = 14 billion centimetres. 14 x 109 ÷ 2.54 cm/in ~ 551 000 000 inches a year. If I owned a ‘fit bit’ to track my steps I could make a better estimate on physical steps. The goal of 10 000 steps a day would give an ideal of 10 000 steps x 30 inches/step x 365 days/yr as 109 500 000 inches per year. If we combined these two, I would have 660 000 000 inches per year.

“in miles,” is likely metaphorical for taking trips, going somewhere out of the usual, or perhaps in the drudgery of commuting. I am grateful to have had 37 years of less than 15 minutes of commuting a day. I am not a big traveller so would not be wracking up air miles. A flight this year and a cruise ship will increase my years total significantly.

“in laughter,” would be a fun measure to attend to. Are they moments when I actually laugh out loud or is smiling okay? For me personally, it would be more of the times I make others smile/laugh as I attend to this often in a day. Joyous times speed by as ‘time flies when we are having fun’.

“in strife”, time seems to move more slowly when under pressure. This would be a depressing measure, but for me, would be a smallish number compared to many other measures like joy. Other depressing measures could be ‘in wasted moments’ or TV watching, solitaire/video game playing.

“In 525 600 minutes” seems like enough time to get things done doesn’t it? But we know there is never enough time.

“How do you measure, a year in the life? IN LOVE!” This is the meaningful measure since love makes the world a better place and gives value to lives and hope for the future. Choosing a measure in love would flow from what your love language is. A look, a touch, a hug, sweet words, thoughtful acts, It also would need to be defined over whether it is the giving of or the receiving of love or both. Should the various measures be weighted? Women might say no, men likely yes due to their dispositions. Research shows that a man might think a dozen long stem red roses would count 12 times what a single red rose would mean. A women sees them as the same expression of love. My wife would prefer one since she doesn’t like to ‘kill’ other things for her pleasure so these could not be universally consistent measures. Like the old king’s foot or forearm changing every time the king died. Measuring our lives by how we love and who loves us is very good for getting our eyes off of ourselves. Healthy relationships do not benefit by ‘keeping score’ on our expressions of love so I would counsel not attaching measures to love but to make loving as essential and automatic as breathing.

Do nothing from selfish ambition or conceit, but in humility count others more significant than yourselves. Let each of you look not only to his own interests, but also to the interests of others.” Philippians 2: 3-4




Problems, Strategies and Games that lead to Math Talk

by Fred Harwood

at the BCAMT Fall Conference-Gladstone, 10/21/16  Twitter: @HarMath Email: and website:

[Red Text throughout this blog are teacher moments that give context to why the workshop is designed as it is. Teaching is like juggling, as you become more adept, you add new objects to juggle, increase the difficulties and applications. Every one is at a different point in their teaching growth so, as you experience various structures in this workshop, which ones feel right to you to explore with your own students? On these first two pages,  I provide the means to research them further with links to people who have been doing this well. You can also contact me for more information or if you have questions about any of these ideas.]

For you to embrace “Math Is Social”, you need to think about how students learn. I believe that learning has both acquisition and participatory aspects. You can be told, shown and discover ‘things’ but much of my greatest learning has occurred in participatory ‘social’ situations with a partner, group or class of individuals working together. Investigating a topic or problem collectively, we bounced ideas off of each other, challenged ideas or we were challenged by someone else’s idea. During the process, the talking and writing clarified our thinking, crystalized our learning and extended our investigations. Achieving success after productive struggles, encountering AHA moments, eliciting peer commendations all built motivation and belief in our abilities to solve other problems and to acquire good dispositions for thinking. I wish for all of you to have these moments and successes.

This workshop will focus on some structures that will support building collaboration, communication, reasoning and understanding socially. If you haven’t learned of a Thinking Classroom from Professor Peter Liljedahl, check out:

We want our students to think, to solve problems and to be able to communicate their solutions (processes and not just answers). The following structures are some of the ways that facilitate these qualities. Some may be already part of your teaching practices and others may be new. Remember, as much as possible, to observe the learning and emotions taking place in yourselves and others while engaging in these structures. Switch your student hat with your teacher hat and researcher hat frequently.

Structures of a socially dynamic classroom include:

  1. Math Talks:

  1. Fraction Talks: and
  1. ImaginEd:
  1. Visual Patterning: and
  1. Chunked Families of Questions: clustering questions in groups with variance for students to seek patterns, connections and to discuss these.
  2. Partnered Games: Taking games, like Ultimate TicTacToe or Mancala, and having students play with a partner against another pair creates reasoning aloud in strategy planning, game observations and pattern recognition.
  1. Good Problems: Rich problems with multiple approaches and good math content lead to students being engaged in the challenge of solving them especially through discussion. Do not fail to realize that you and your students can create your own good problems. They can also be found. Some of my favourite sources are:                              and a myriad of great links through #MTBoS like          and the portal site:

Many great problems also emerge from extensions on any of the problems you might start with. These emergent inquiries are a rich source.

Keep in mind in your planning and execution these dimensions from


[These first two pages are background information and links to pursue any of the ideas the participants find the most intriguing further. I strongly suggest you experience being a student in doing the following activities with others. “Math is Social” so the workshop is designed to model what students would go through in these structures and to then be excited by the amount of mathematics learned, the positive dispositions gained, the lessening of math anxiety and the dynamic and motivating attitude to question further!]

Let us experience some of these structures . . . [groups actually worked through these and we utilized different approaches to structure the opportunities for students to talk, describe, justify, defend and challenge ideas mathematically.]

  1. Math Talks: [In this workshop I used this variation on Number/Math Talks: Give wait time to let students (teachers) think, try the problem on their own for a little while then they worked with a partner or two to add to their techniques. A team was asked to post on the wall one of their ways, then, with each new group, a new method needed to be posted creating an array of around 11 various approaches. The teacher now orchestrates the discussion by having various students explain their posting and then the teacher draws attention to various approaches with questions like: Which ones are similar and why? How is this one different? Which one(s) did you really like? Don’t just leave multiple representations up without students being forced to see more of the mathematics that is there.]

Figure out what 44 x 25 is in multiple ways. Be ready to show others around you how you did each method. You’ll have some think time by yourself and then I’d like you to pair up with someone nearby to share with each other.



2. Fraction Talk: [A variation on Math Talks is Fraction Talks which draw proportional reasoning into their logic skill-set. Proportional Reasoning is one of the really BIG IDEAS in mathematics. I hinted that, after doing some of these with students, they could have solved Fawn’s problem of the missing yellow region’s area very quickly.] 

What fraction of the whole is each region in each of the following diagrams?


Fraction Talk Extensions: [These extensions were an attempt to draw older students into algebraic thinking with clotheslines (rulers) and fraction talks with unknowns but proportionally related.]




4. IMAGINE EDUCATION: Our goal today is to ‘see’ mathematics and to see it talk to us. You will need to imagine how other students are thinking and ‘show’ what they might have done to count how many dots are in an array: [I think I will add some with words instead of arithmetic expressions like #2 which could be “I saw two sixes but missing two corners.”] 


[What mathematics would employed in the following inquiry? What other curricular areas would also be addressed like descriptive writing in the activity? What ownership could be shared if students were allowed to use their camera phones to explore from a ‘lowered or raised perspective’? How much imagination is employed?]


5. Visual Patterning: [Visual patterning in growing patterns is a fantastic way to engage students in mathematical thinking, patterning, generalizing, verifying, and much more high quality mathematics. I used stars in these ones so I could have flip chip manipulatives available for participants to make their pattern’s growth obvious and to defend their generalizations. I found my teacher/students had 3 different working formulae for each of the patterns which led to a good discussion on equivalent expressions and on ‘seeing’ how each could be represented in the chips powerfully.] 

1.) Work with partners to solve this growing pattern’s questions. Remember you are trying to ‘show’ how you thought through the problem clearly enough for others to understand your method and the process you did to solve the problems.


  1. What would Figure 4 look like?
  2. How many stars in Figure 8?
  3. How many stars in Figure 12
  4. Can you generalize a formula or formulae to predict the number of stars in any Figure?
  5. If there was a Figure 0, what would it look like and why?


Bonus Pattern: Answer the same questions as above:


5. Chunked Families of Questions: Figure out each and connect them with a larger pattern of understanding. Create your own algorithm. Can they all be done in your head after you’ve discovered the larger pattern? [By working out the answers but knowing someone can do all of them in their heads faster than they could be entered in a calculator engages the students to connect questions with answers, questions with questions and to discover larger patterns of understanding. Visual patterns did the same thing in connecting Figure Numbers with their shapes to generalize a more powerful pattern. These families of problems also provoke students to ask their own questions to pursue the inquiry further. The movement of ‘nixing the tricks’ and ‘banning the worksheets’ misses how much mathematics can be uncovered/learned in well designed sheets which draw students into inquiry (much like a science lab does). These create life skills, attitudes and dispositions that will lead to much success.] 




6. Partnered Games: playing a variety of games especially as two against two to create the structure for math talk with each other (also called strategy, strategic planning, observing the enemy, seeking patterns . . . ) [100 minutes left us just mentioning these last two sections but these are very powerful as structures for mathematical discussions while being engaged in the activities.]


Ultimate Tic Tac Toe

Rules can be found here:

Online version:



7. Good Problems: These are single problems that are challenging enough to make students want to persevere in solving them but need to do lots of talk with their groups to get to their understanding of the problem to solve it.




Square Peg in a Round Hole

What is a better fit, a square peg in a round hole or a round peg in a square hole?


How many squares on a weird 6x 10 chessboard? How many rectangles?




A Prequel on Divisibility as Inquiry

Reaping a Landslide of Student Excitement

By Fred Harwood when I was Mathematics/IT Coordinator Hugh McRoberts Secondary School in Richmond, BC and students/Math Club members:Mark Funston, Suzanne Lee, Philip Gao, Steve Welsh, Thomas Mickie & Ekaterina Daviel


Many exciting student discoveries have transformed my classes into learning communities. These past discoveries provide a valuable set of “anchor papers” or standards that current students can strive to surpass. This search for meaning and for extending the science of patterns, create a vibrant sense of the dynamics of mathematics that motivate students (and teachers!) to strive for understanding and excellence as mathematical researchers.

This fall (2004) has been my most exciting of the past fifteen years with Mark Funston’s breakthrough unifying theory of factor finding creating connections to other uses at a tremendous pace.

This discovery-based learning is an invigorating process that I highly recommend for teachers. I’m employing this approach in my regular Mathematics 9 classes and in my accelerated and enriched classes in grades 9, 10, and 11. I’ve also employed it with grade 8 classes in the past and know of elementary teachers doing work like this as well.

This article will deal with the Funston Factor Finder written up my my Math 11 Enriched student, Mark Funston. Suzanne Lee interviewed him on the creative process and then Philip Gao and I extended it and applied it to new processes. Two weeks later, the math club was working on a separate concept and we saw the application of Mark’s theory’s underlying concept to exponentially increase our understanding of fraction conversions.


I had given the students in my Enriched Math 11 Principles class an overview of divisibility tests as a mathematical tool. I taught it to them in an organized big picture approach.

Divisibility Tests

Test for divisibility by 10 (and 2 & 5 since they are factors of 10)

Last digit is divisible by 10 ( or 2 or 5)

 Test for any power of 10 (or power of 2 or 5)       

Check the same number of last digits as the power for divisibility by the number

Eg. Test for 8: check last 3 digits for ÷ by 8 because 8 = 23.

 Test for 9 note: it is almost our base value of 10 (and 3 since it is a factor of 9)      

       Add up the digits to see if divisible by 9 (or by 3)

 Test for 11 note: it is almost our base value of 10 (one of many tests but this one is similar to 9’s)

       Alternate the digits + – + – + – + etc until you are out of digits

                   and then add up the digits to see if ÷ by 11. Remember that 0 can be ÷ by 11.

Test for 7 (a student discovery that works for 3 digit and larger numbers)

       Starting on the right, break the number up into groups of 2 digits, the first digit is normal (x1), the second group is doubled (x2), the third group is x4, 4th is x8 etc until you are out of groups. Add up the products of the groups to see if divisible by 7. If the total is still too large to tell, repeat the process until it is a two digit total. (Note: 91 & 98 are ÷ by 7)

       Ex: 1572354: breaks up into 1 57 23 54: The total = 8×1 + 57×4 + 23×2 + 54×1 = 8+228+46+54 = 336 and then 336 breaks into 3   36 so 3×2+36×1 = 42 which is ÷ by 7 so 1572354 is also ÷ by 7. [Note: this is especially useful for 3 & 4 digit numbers.]

Test for composites: break into mutually exclusive factors (they must not have a factor in common other than 1)

6 = 2 x 3 so the tests for 2 and 3 must work. Eg. 2 must ÷ last digit & digits add up to ÷ by 3

12 = 3 x 4 so tests for 3 and 4 must work. Eg. 4 must ÷ last two digit s & digits add up to ÷ by 3

15 = 3 x 5 so the tests for 5 and 3 must work. Eg. 5 must ÷ last digit & digits add up to ÷ by 3

18 = 2 x 9 (not 3 x 6)                                 etc.

Some of the pieces were not in place by the end of Thursday’s period so I assigned the students to discover their own tests for 13 and for 17. Half the class was working on 13 and the rest on 17. On Monday, Mark Funston greeted me dancing and with a huge smile and a couple of pages of much more than a test for 13. The class itself had 5 of their own discoveries but Mark’s dwarfed them all and, in fact, generated many of the students other discoveries. I originally called it the Fun Key Funston but Mark has renamed it the “Funston Factor Finder”.

Here is Mark’s writeup

The Funston Factor Finder by Mark Funston

            Ever since the beginning of civilized math, we humans have tried to find little tricks to help us solve problems more quickly and efficiently. One of the major problems that we have is finding whether a number is a factor of another one, like whether 7 is a factor of 140. We have developed methods that work for different numbers, but there are so many that it’s hard to remember them all. What if there is a universal method for finding the factors of numbers?

            There is one that works for all numbers and it is called the “Funston Factor Finder” or “FCubed” for short. Let’s use the example of finding whether 7 is a factor of 140. You will be able to execute this method in 3 easy steps.

  1. Find the multiple of 7 that is closest to 100 without surpassing it

(7 x 14 = 98). Then find the difference between 100 and that number (100 – 98 = 2 = d).

  1. Separate 140 into groups of 2 digits starting from the right and continuing to the left.

II     I

1 | 40

The numbers on top are the group numbers. This will become more important with more digits.

  1. Multiply the number left of the first separator (1) by 2 (the difference between 98 and 100) and add it to the numbers on the right of the first separator.

1 x 2 + 40 = 42

Notice that 42 is a multiple of 7. This tells us that our number is also a multiple of 7. We’re finished this simple example, but this method works for all numbers so let’s try it again.

Once the number in question surpasses 4 digits (greater than 10,000), there will be more groups of 2 digits. Let’s try another example:

 Eg. Is 12 a factor of 10404?

  1. Closest multiple of 12 to 100 without surpassing it.

12 x 8 = 96 (difference from 100 is 4)

  1. Separation

III   II     I

   1 | 04 | 04

  1. We multiply group II by 4 and group III by 42 because it is in a different group. (If the number got big enough, group IV would be multiplied by 43 and so on for each of the consecutive groups. Note that Group I is multiplied by 40or 1)

(1 x 42) + (4 x 4) + 4

1(16) + 4(4) + 4

= 36 (which is a multiple of 12)

Again, this works for ALL numbers.

However complicated this method may look at first, it is really quite simple and efficient. I believe it will revolutionize the way we think about factors and it may replace the current methods of determining whether one number is the factor of another. Try it out, it really does work!



I demonstrated this method to the class for 7 and it generated the “Attridge Attributed divisibility test” I’ve included in the Workshop materials. Allan Attridge was a Grade 9 whose original discovery fifteen years ago excited me about divisibility test discoveries. We then tried it for 13 and 17 and several of the students’ own discoveries were generated. A test for 11 could be that it goes into 100 nine times with a remainder of 1. So we can check for 11 by adding up pairs of digits starting from the right and checking the sum for divisibility by 11 (or by repeating the process until it is small enough to tell.) For example, 1286472 can be split into 1+28+64+72 = 165 and 165 can be split into 1+65 = 66, a multiple of 11 so 1286472 is a multiple of 11 as well. Try it on other numbers.

I demonstrated it to some of the math club members at lunch that day and Philip Gao and I thought that it could work for a factor of 10 as well. Take a number like 7. How many times does it go into 10 without going over? It goes once with 3 left over. The Funston Factor Finder extension would then say that the units digit of a two digit number would be multiplied by 30 or 1 and the ten’s digit by 31. Think of the multiples of 7 that we have stared at for years. 14 à 1×3+4=7; 21à 2×3+1=7; 28 à 2×3+8=14 and we could repeat the process to 14 to get 7. How could such an obvious pattern escape me for so long? Now the practicality of this test is not high since we’ve learned our multiplication table for 7 already but look at some of the other single digit numbers. Trying 9 (or 3) we see it goes into 10 once (or three times) with a remainder of 1. This means that each digit of a number gets multiplied by a power of one thus generating the presently practiced “add up the digits to see if the sum is divisible by 9”! Applying the theory for 10 to a factor of 10 yields the test of checking the last digit for divisibility by 2 or 5 if we loosely claim that 00 is 1 then only the last digit counts as the rest are multiplied by a power of 0.

We tried the test with the concept of 1000. For example, 11 divides into 1000 ninety times with 10 as a remainder. Progressive groups of three digits would be multiplied by a power of 10 before summing. 1286472 would split into 1×102 + 286×101 + 472×100 =100+2860+472 = 3432 which can be split into 3×10 + 432 = 462. Then if 462 is a multiple of 11 then 1286472 is too.

In writing this last paragraph I thought to try the concept of 11 into 10 with a remainder of (-1). This would generate the test of “alternating digits plus, minus, plus, minus etc and then adding to see if 11 is a factor of the sum.”

Obviously, there is no limit to the applications for factor testing but some will be less helpful than others. The important thing is that they do exist and this method can generate a wide variety of relationships between the digits of a number.


Crossover Extension:

Our math club met two weeks later and I mentioned again the excitement generated by Mark’s discovery and how a magazine article was being written. I thought they might discover something to form the foundation of their own fame by developing some new patterns for the conversion of fractions to repeating decimals. I had shown Steve, Thomas and Ekaterina the conversion of one seventh to a decimal by doubling 7 to 14, 28, 56, 112, 224, 448, 896, 1792 etc. By placing these into two digit registers with carryovers going into previous registers, the sum of the columns generate the six-digit repeating pattern for 1/7 as 0.142857 repeating.

These are examples of where inquiry, discovery-based learning, thinking classroom communities make a tremendous difference in students learning, being engaged and motivated to further investigations and to generate new ideas.




Algebraic Clotheslines & Fraction Talks

An Andrew Stadel (@mr_stadel ) tweet on parallel number lines with one being numerical and the other algebraic got me thinking. “What if I combined proportional distances on number lines with algebraic clues or areas on @fractiontalk diagrams with algebraic clues?” I started playing with a few ideas. Seeing how too much information led to straight forward thinking, I wanted just enough to be doable but still ensure mathematical reasoning. Here is the first iteration awaiting testing with students. If you find it fits with your classes, please try them out. They might make good review of several concepts blended.

Clothesline algebratalks 1

Clothesline algebratalks 2

Link to Word handout:  Algebra Clothesline to fractiontalks




Teatherboard Inquiry Activity

Several months ago, I saw a colourized version of a teatherboard at I had been attracted to it by a tweet that led me to their “Prime Climb” imaging/game.

Prime Climb Factor Poster

When I saw a similar ‘factors’ & prime factoring image on twitter that used symbols, I knew that this would be a great way to have students look for patterns, do inquiry, practice multiplication and division facts while building on the structure of numbers. It was posted by @MarkChubb3 at

Teatherboard Activity

I chose to give them a swath of known symbols to see patterns in and pose questions. I also wanted some ambiguity to seek out what could be known and what could not be known from the given data. I sought a balance between too little and too much known information to start. Too little would raise the bar for entry into the problem. Too much would make it mechanical and boring. It seemed important for inquiry to keep the opening question simple and open-ended. I also wanted to lose colour considerations so that it could be photocopied and not challenge someone who has some colour-blindness. [A brail version might be really intriguing. Anyone interested?]

TEATHERBOARD ACTIVITY: The numbers inside of the bold black boxes have a symbol or symbols shown. The rest of the squares do not have any symbols yet. Which ones can you figure out and what would be their symbol(s)?” Part way into the activity of testing this on six grade 6 students and two teachers I added, “Which ones cannot be figured out from the given information?”

Teatherboard Inquiry shot
Before continuing to read this blog, I recommend you consider the puzzle yourself first. Once you feel you’ve mastered the thinking, then continue to read as the next section will have spoilers.
The previous paragraph is in reaction to what the two teachers said after several other teachers had dropped in to see what math club was doing this week. I gave them the puzzle and one of the club sponsors enthused over it to them. After they left she said, “I wish I didn’t tell them it was about multiplying. I stole the joy I had in figuring that out from them!”
It was interesting to see engagement that grew as they started to uncover its secrets. I encouraged students to work in pairs to keep mathematical reasoning and communicating happening. I watched for strategies they used. A couple of students demonstrated exponential thinking by filling in 2 – 4 – 8 – 16 sequentially. Another pair worked horizontally with factors like 20-40-80. There was a positive energy throughout the activity and one even commented 18 minutes in, “This is surprisingly fun!”
Guess & check-with-teacher is still being fought in this group as not an acceptable heuristic for problem solving. “You know I don’t respond to is-this-right? questions.” When we did a gallery share of various squares that they figured out or ones they couldn’t figure out. I asked ‘K’ for 84 which he posted as 2×47 symbols. I let several students disagree then asked, “K, why did I ask for this one?” He replied, “because this was the one I made an error on. I should have checked my own work.”
Envision how you can utilize an activity like this and what students would gain from it.

Link to document: Teatherboard Activity



Professional Development: my story & beliefs

written: February 27, 2014

Fred G. Harwood’s Philosophy of Professional Development

Teaching is not my job, it’s my passion; getting better at it – that’s my job!                     ~Jose Popoff from his website and twitter chats

This motto is a vision statement for my view of personal professional development. I have always been fascinated with learning since first observing a flaming bolide meteorite’s path through the sky at age 10. This led me on a path towards a PhD in astronomy. In my second year of university I was diverted from my path into a pursuit of a physics degree with a math minor in order to be a teacher. Beginning my career with a majority of mathematics courses, I utilized my training as a scientist to explore my role as a teacher and of students’ learning. Always curious, my focus was on learning about the craft and science of teaching while exploring mathematical ideas. Discovery-based learning is a major part of my teaching and assessment research is another major focus.

Professional development (Pro-D), in the beginning, was attending workshops and ‘receiving’ from the experts, the experienced, and the gurus. But I was learning so much from my own students through explorations, questioning and reflective observing. The learning was so rich, I felt called to share with others, not as an expert but as a passionate learner full of wonderment and joy that I wanted others to experience.

I joined the Pro-D committee and for the next 26 years was always part of the team and, on occasion, the chair. I had also discovered the joy of learning with others and being challenged and enriched by their various backgrounds and viewpoints. When we entered into curricular pods and integration, my English/Socials partner made me read Nancy Atwell’s “In the Middle” on reading and writing workshops. It was the best educational read I had had as I was constantly viewing it with thoughts of how this approach could happen in my math and science classes. The sharing together of our students’ journeys was so rich. We would also hold regular meetings to plan and pursue educational ideas together. I had been blessed with cross-curricular learning. Teachers from other disciplines helped me better understand teaching and learning. I learned to teach students and not just curriculum. I sought out other teacher teams as professional learning communities. After 16 years at London Jr. High, I moved to McRoberts as the Math & Science coordinator because I was inspired by their vice principal who raved about their young teachers passion for learning together. They had a Wednesday Morning Study Group that began as a survival group for 17 young teachers. There was a core of us veterans that knew the power of a PLC and cross-curricular learning so we joined in and grew together. It ran for 13 years and I tried to never miss a morning. The last 8 years I acted as a lead teacher/organizer. It drew to a close when many of the core teachers moved on to administrative positions and had young families.

I sought out other PLCs by joining district study groups, focus groups and book studies. I joined a series of Lesson Study investigations out at UBC hosted by the Pacific Institute of Mathematical Sciences (PIMS). At my second meeting, I was recognized as a leader and joined the team organizing it. It ran successfully for 6 years. Lesson study has taught me that professional growth is a gradual, incremental process and not a quick fix or an application of a single idea. Learning is a complex process involving many factors. Our lesson study teams were powerful groups of multi-level teachers of math, math educators and mathematicians and we learned much together. Solitary learning is inefficient.

The SFU Field Studies Grad Diploma program was offering grad classes in our district and seemed to be a perfect fit for my desires in cross-curricular and multi-age learning. Our 2007 cohort was Today’s Classrooms, Tomorrow’s Future (TCTF). We joined a cohort entering their second year, Teachers as Learners and Mentors (TLM) and then had the Diversified Learner cohort join us a year later. This was professional development at its finest. I thrived in the professional learning atmosphere where we ‘bumped up against’ a rich assortment of ideas, distant thinkers and our in-class investigations. It is talking through, observing together, and challenging one another in our learning that growth powerfully happens. It energizes us and exhorts us to learn more and to recognize the power of the participatory metaphor for learning (Sfard, 1998). Even though I was a few years away from retirement age, I knew that this was going to be a valuable direction for my own professional development. I have chosen my three references from these groups who can speak to my participation and passion for learning in the field studies program.

Another powerful source of professional development for me has been utilizing technology to access other resources, thinkers and practitioners. I love reading and hearing about theories and then synthesizing them and adapting them for my own classrooms. I have a connective intelligence that can see the efficacy of ideas. My own metaphor for learning has been weaving a tapestry. Each new idea is held up to the collective learning of my past to see how the idea fits in with the bigger picture. Sometimes the idea forces me to pull out threads from the tapestry that no longer work. The complexity of learning is like the messy jumble on the backside of a tapestry but the clarity of the front side can only come from the complexity of pulling together many threads. I joined several on-line conferences on education and have been a very active contributor to listserves like the BCAMT’s. I joined the Canadian Assessment for Learning Network (CAfLN) as another learning network. In laddering to my Masters in Education Practices, I was a major contributor to our class moodle because I knew the power of participatory learning and wanted to stimulate others and to be stimulated by their ideas and explorations. In the last few months, I have greatly expanded my professional learning network (PLN) with twitter. My twitter community is diverse, passionate and reflective. Technology has opened doors to learners for professional development and the improvement of student learning. I have used Google Docs, blogging, and EdCamps as other ways to grow as an educator and lifelong learner.

The following quote is an excellent summation of my professional development model that the Field Studies program provides.

Recent research on how people learn points to another reason why the current structure of our schools is dysfunctional. This research has demonstrated that robust, fluid, and usable knowledge must be grown by learners through highly active engagement with ideas and their interconnections. Knowledge is neither acquired nor applied mechanically or in piecemeal fashion. It evolves into ever more complex, integrated bodies of thought and skill. Knowledge does not just sit there, waiting to be retrieved; it must be tended, fed, and used. In fact, the way people learn anything — from the ABCs to cooking to astrophysics — is by energetically connecting ideas with action. . . Learning those kinds of skills is not a solitary endeavor; rather, it needs to be a highly social one.

   It depends on continual discussion and demonstration. People learn by watching one another, seeing various ways of solving a single problem, sharing their different “takes” on a concept or struggle, and developing a common language with which to talk about their goals, their work, and their ways of monitoring their progress or diagnosing their difficulties. When teachers publicly display what they are thinking, they learn from one another, but they also learn through articulating their ideas, justifying their views, and making valid arguments.

[Deanna Burney, Craft Knowledge: The Road to Transforming Schools, 2004 including a quote from (Bradford, Brown and Cocking’s)]

UPDATE: March 23, 2016  [I wrote this in 2014. I publish here as a move from my old website that mysteriously disappeared a year after retiring. Since publishing thi, I retired in June of 2014, went back to grad school at SFU for another Masters, became a mentor teacher in my district. I  volunteer mostly in elementary schools; I continue as a education consultant doing workshops, research, and other educational pursuits. I utilize twitter (@HarMath) as my main source of PD both in learning and contributing. I also attend edcamps in the area and joint proD collaborations in my district. I continue to be active on my mathematics association’s conferences and listserve. So I’m still working on getting better at my passion and sharing my passion with others to bless them as I have been blessed.]



Thinking Competencies: Creativity Through Math

THINKING COMPETENCIES: Creativity through Math with Fred Harwood April 24, 2015

It is the supreme art of the teacher to awaken joy in creative expression and knowledge. Albert Einstein

What are cross-curricular competencies? At the heart of the definition of the cross-curricular competencies is the principle that education should lead to the development of the whole child—intellectually, personally, and socially. In a world of growing diversity and challenge, schools must do more than help students master the sets of knowledge and skills acquired through the standard subject areas. They must prepare students fully for their lives as individuals and as members of society, with the capacity to achieve their goals, contribute to their communities and continue learning throughout their lives.

The cross-curricular competencies are the set of intellectual, personal, and social skills that all students need to develop in order to engage in deeper learning—learning that encourages students to look at things from different perspectives, to see the relationships between their learning in different subjects, and to make connections to their previous learning and to their own experiences, as members of their families, communities, and the larger society.

The conceptual framework described here envisions three broad cross-curricular competencies: thinking competency; personal and social competency; and communication competency.

Thinking competency, which encompasses critical, creative, and reflective thinking, represents the cognitive abilities that students develop through their studies. Personal and social competency represents the personal, social and cultural abilities that students develop as individuals and members of society. Communication competency represents the abilities students need to interact and learn effectively in their world. Together, these three cross-curricular competencies represent a holistic and unifying approach to learning, spanning all courses and grades in the common purpose of enriching students’ learning experience and preparing students for the future.

These cross-curricular competencies are interconnected; and they are not three linear and discrete entities . . .

Creative thinking is the act of generating and implementing ideas that are novel and innovative to the context in which they are generated. A creative thinker is curious and open-minded, has a sense of wonder and joy in learning, and demonstrates a willingness to think divergently and tolerate complexity. A creative thinker uses imagination, inventiveness, resourcefulness and flexibility and is willing to take risks to imagine beyond existing knowledge in order to generate and implement innovative ideas.

Students are enabled to think creatively through opportunities that allow them to take initiative, exercise choice, explore ideas and options, question and challenge, make connections, and imagine and visualize the possibilities. Teachers can foster creative thinking by welcoming students’ unexpected answers, questions, and suggestions; delaying judgment until students’ ideas have been thoroughly explored and expressed; offering students opportunities to work with diverse materials in various ways; and supporting and scaffolding students as they explore new and unusual ideas.

‘It’s never enough to just tell people about some new insight. Rather, you have to get them to experience it a way that evokes its power and possibility. Instead of pouring knowledge into people’s heads, you need to help them grind anew set of eyeglasses so they can see the world in a new way.’— John Seely Brown


3 components of creativityFigure 2.  Adapted from Adams, K. “Sources of innovation and creativity: A summary of the research.”

Some helpful links for future research:      Roger von Oeck’s “Whack on the Side of the Head”

Chic Thompson’s “teaching children to be creative first and critical second”          “What a Great Idea”

John Spencer Blog on “Why Consuming is Necessary for Creating”

“The truth is that consuming well is a part of how we develop a taste for what we like. It’s part of how we gain information. It’s part of how we fall in love with an art or a science or a craft.”

On the Edge of Chaos Where Creativity Flourishes by Katrina Schwartz

Ken Robinson’s drive for getting more of the creative competencies in education:

“Creativity is the process of having original ideas that have value.”

“Creativity is not an option, it is an absolute necessity!”

Revolutionary Ted Talk:

From: “Sir Ken Robinson: Creativity Is In Everything, Especially Teaching”

“Creativity is putting your imagination to work. It is applied imagination. Innovation is putting new ideas into practice. There are various myths about creativity. One is that only special people are creative, another is that creativity is only about the arts, a third is that creativity cannot be taught, and a fourth is that it’s all to do with uninhibited “self-expression.” None of these is true.

Creativity draws from many powers that we all have by virtue of being human. Creativity is possible in all areas of human life, in science, the arts, mathematics, technology, cuisine, teaching, politics, business, you name it. And like many human capacities, our creative powers can be cultivated and refined. Doing that involves an increasing mastery of skills, knowledge, and ideas. Creativity is about fresh thinking. It doesn’t have to be new to the whole of humanity— though that’s always a bonus— but certainly to the person whose work it is.

Creativity also involves making critical judgments about whether what you’re working on is any good, be it a theorem, a design, or a poem. Creative work often passes through typical phases. Sometimes what you end up with is not what you had in mind when you started. It’s a dynamic process that often involves making new connections, crossing disciplines, and using metaphors and analogies. Being creative is not just about having off-the-wall ideas and letting your imagination run free. It may involve all of that, but it also involves refining, testing, and focusing what you’re doing. It’s about original thinking on the part of the individual, and it’s also about judging critically whether the work in process is taking the right shape and is worthwhile, at least for the person producing it.”

Competency: Creative Practice


Innovation is a key component of creativity: if there is no innovation, then there is no creativity.  To be innovative is to have ideas and contribute to, or lead, activities that have not been tried before. As a creative practitioner, innovation should be the constant driver for your work; while originality is your ultimate goal.  Consistent innovation requires a strong disciplinary base of knowledge and skills.

There are two main ways in which something can be new: a new method or a new context.  Secondly, something could be new to you, new to your client or client group, or completely new to the sector, or even the world – as far as you know!

The more of these boxes it ticks, the closer to ‘original’ your work will be.  However, behaving innovatively is not about achieving an ideal; it is about demonstrating, in all your creative practice, that you are constantly striving to be innovative and that you understand where and how your action is innovative. 

Teaching Creativity

A helpful website exhibiting a creativity structure called SCAMPER is found at

Art of Creativity


Fraction Talk Compilations from has a wonderful website devoted to visual fraction puzzles that build proportional reasoning while embedding in geometric shapes, real life and whimsy. Here are some teasers. Their website have explanations on how to use them as individual fractions talks. These three pages are designed to lure teachers into seeing their power to increase fractional understanding and logical and geometric thinking.

Enjoy the pdf  Link:      Triangle Fraction Setups

Also follow on twitter  @FractionTalks & the author @NatBanting

And check out their websites:     and


Visualizing As A Tool For Building Number Sense

Below are the materials that I utilized at the Northwest Math Conference 2015 in Whistler. In the presentation, we looked at various aspects of visualization and how they empower a greater sense of number, problem solving ability and enhances communication and reasoning. Student visualizations give us access into their thinking which reveal their strength of grasp or the misconceptions that are interfering with their progress.

Visualizing as a number sense tool


Visualizing Divisibility Tests to Strengthen Numeracy

Visualizing Divisibility to Strengthen Numeracy

By Fred Harwood, SFU grad student, educational consultant, and (semi)-retired math educator

I am currently studying elementary and middle school numeracy at SFU. As a recently retired secondary math teacher, these studies give me new perspectives to look at how students develop their mathematical understanding. In secondary schools, we often lose sight of the underpinnings of numeracy and utilize patterns or rote memorization.

Marc Garneau provided me with a beautiful perspective that was very empowering. In a 2014 NCTM conference presentation he showed how the divisibility test for nine could be visualized. Picture the number 3213. For decades I taught the test of adding the digits to see if the total is divisible by 3 or 9 as this is what works. Here 3+2+1+3 = 9 so both 3 and 9 would divide into 3213. Marc said to look at the place values of 1000’s, 100’s, 10’s and 1’s.


You can think of 10x10x10 cubes, 10×10 squares, 10 strips and leftover 1’s. One less than any power of 10 is obviously divisible by 9.

Taking the one cube away from the 1000 cubes leaves 999 cubes. If students don’t see this is divisible by 9 then look at the fact there are 99 tens and 1 nine. Taking one from each of the 99 tens give us 99 nines and the 99 ones can be made into 11 sets of 9. This gives us 111 sets of 9.

If we take 1 piece from each massed set we would end up with groups that are all products of 9. In our number we would see three sets of 999, two sets of 99, one set of 9 and then all the leftover pieces loosely gathered that total 9 pieces. Every set of pieces is divisible by 9 so the whole number 3213 is also divisible by 9.

In the example 64 236, we could take 6 singles, one from each of the ten-thousands, a single from each of the 4 thousands, 2 singles from the hundreds, 3 singles from the tens and amass them with the 6 leftover pieces. There are now six sets of 9999, four sets of 999, two sets of 99, three sets of 9 and 6 + 4 + 2 + 3 + 6 = 21 left over pieces. All the complete sets are divisible by 9 and also 3 except for the 21 leftovers. Therefore 64 236 is not divisible by 9 but would be divisible by 3.

Now consider other divisibility tests. Powers of 10 are divisible by 2, 5 and 10. For example, 100 is 50 twos, 20 fives or 10 tens. Since every place value is based on a power of 10 then every place above the ones place is already divisible by 2, 5 and 10. Only the leftover 1’s place is left to check for divisibility. In 2 352, the two thousands, three hundreds and five tens are each divisible by 2, 5 and 10. So we check the two leftover 1’s. Only 2 goes into this so 5 and 10 won’t go into 2 352 but 2 will.

Which is numerically more powerful: Even numbers end in 0, 2, 4, 6, or 8 or seeing these numbers as one 2, two 2’s, three 2’s, four 2’s or five 2’s by rearranging little blocks?

Testing for 4 or 25 (or 100) we can have students memorize looking at the last two digits because 4 is 2 squared and 25 is 5 squared OR we can have them visualize that all powers of 10 greater than one are divisible by 4 and 25. For example, thousands and hundreds are both divisible by 4 and 25 so we only need to check the tens and ones to see if they too are divisible.

Eleven has many divisibility tests. Thinking about what numbers close to a power of 10 is divisible by 11, we see that 11 is 1 more than 10. 99 is one less than 100, 1 001 is one more than 1 000 and 9 999 is one less than 10 000 etc. Therefore the scenario alternates between having a place value needing one more for each holder or one less. Taking 12 837 as an example, we need to subtract 1 from the 10000, add 2 ones o the two thousands, subtract eight 1’s from the hundreds, add three 1’s to the tens to make each of these groups divisible by 11. If we were using flip chip colouring, we would have 1 + 8 + 7 whites and 2 + 3 reds making a total of 11 once the zero pairs are deducted. The alternating of digits positive and negative and summing to test for 11 is now obvious that it works. 872 146 822 would have 8 + 2 + 4 + 8 + 2 of one colour and 7 + 1 + 6 + 2 of the other. 24 – 16 = 8 so this big number could not be divisible by 11.

Having student teams develop their own tests for divisibility using their understanding of powers of 10 place values would be very powerful and involve much good mathematical reasoning together. Manipulatives could aid this reasoning and visualizing for numbers under 10 000. Given an existing divisibility test, trying to prove why they work instead of just demonstrating that they do would also be a valuable number theory activity for students from intermediate to post-secondary.