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.