Top 10 List of Factors That Made a Difference in Students Learning

Presentation at the 2012 Pacific Northwest Math Conference, Victoria, BC

My Top 10 List – Basis of Success                Fred Harwood, Hugh McRoberts Secondary
I have reflected and decided upon the top 10 things that have made a positive difference in my students’ learning and on my making a positive difference in these last 35 years. My challenge to you will be to consider my list with yours. Will this list change for you? Will your top ten list make me reflect on changing mine? I have arranged these in a more or less chronological order of when I uncovered them in my career. And I am still learning.

  1. Discovery Based Learning:

Early in my career, I stumbled onto this pedagogy by saying to a class that I would prove that their conjecture would not work. In the remaining part of the period, they proved to me that I (and my math professors) could be wrong, that their techniques were adaptable and useful. They engaged fully in the period, owned the math and were motivated by their discoveries. Since that class, I have done discovery based learning often, especially early in the year and in fertile themes. Decimal strings, divisibility and student driven algorithms have been three of the greatest themes for me.

I want the students to see that the study of mathematics is dynamic, exciting and wonder-full. I want them to believe that they can discover their own patterns and to make connections. When they connect questions with answers, questions with questions, concepts to other subjects, math to life, math in life then their growth is more powerful. Students are more energized and thus I am more energized. When I passionately enthuse about a previous classes explorations, questions and discoveries it infects them motivationally to have their names immortalized with their own uncovering of ideas and extensions to the body of learning.

  1. Student Ownership and Student-Named Discoveries:

Giving choices to students, allowing them to pursue their questions and citing them for what they discover have been powerful motivators. Average students have gone on to excel when they began to seek these connections or when the math connected with their interests. Student ownership also makes the math work for them and not on them. Sources for these conditions are in the discovery based learning situations where they generate their own questions and choose what numbers/themes to pursue. Motivational ownership also comes on projects. My favourite involve making equations and inequations draw the pictures that the students choose. I have also seen lives change when they identified with a group that produced something of value. This could be a group generated algorithm for approaching a particular type of problems.

  1. Student Relationships and Caring:

My best classes have been ones in which they support one another, nurture one another and work together. This seems obvious but I needed to come to the realization that it wasn’t just luck or happenstance. It comes from planning and my attention to caring. One of the most powerful tools that I have used has been having all the students’ names on table cards in 48-point bolded font. When they come into my room for the first time, they don’t need to look around to see who they will define themselves to be by who they choose to sit with. I take control from the start and assign them in groups of mixed males and females to give diversity and greatly reduce the amount of work needed to learn who my students are. I can call them directly by name on the first day and often when they enter the room for the first time and sit down. It immediately tells that student that I have cared enough to find out who they are. Attendance takes no time and doesn’t distract from their first activity as we are usually off running right away on a group problem solving activity or table group collaboration. TOC’s, EA’s, student teachers all benefit from also being able to learn the who’s who of the groups when they are in the room. We have started using a new program called Mastergrade which also allows us to arrange the students’ photos in a seating plan. Greeting the students seems like such a simple unnecessary thing but every little act that shows you value them and their own relationships.

Genuine interest in their interests is also a strong relational tool for classroom management and student success. I had one student who went from disrupting every class as a hobby to paying attention just because I noticed the style of lacrosse stick he was carrying and what I thought his position would have been. Another worked harder than she ever had before in math class because I recognized her athletic abilities and was willing to help her develop them further.

Are you paying attention to haircuts, new glasses, dental work disappearing? Does your assessment policy have a humanized component or is it just numbers? Do you teach students or just curriculum?

I would like to think that former students would want to come to my funeral because I had made a difference in their lives. I would like my tombstone to have a line that reads, “A teacher who cared.”

  1. Collaborative Problem Solving:

This has been the powerful tool in making opportunities for students to communicate and reason mathematically. A good problem pulls students into working together, exchanging ideas, challenging one another and to work collaboratively. Good problems expose a lot of different math. One good problem may produce a full understanding of a key topic and generate a family of questions that make concept development more complete. Stiggler and Hiebert’s published The Teaching Gap in 1999 which compared Grade 8 math classes in the US, Germany and Japan. The Japanese model, that has been growing and developing incrementally for decades now, focuses on using a few good challenging problem solving questions for getting students to learn key concepts. This spoke to my heart and has been the birth of many of my best practices.

How much math would my students learn if all they ever did was watch me solving a question my way . . . perfectly. . . every time? How much of this will they remember a week later? How much more will they remember when they are actively engaged in collaborative problem solving, pushing each other with different perspectives, backgrounds and ideas? How powerful would it be for them to be verifying each other’s ideas and having peers validating their ideas? Have you learned that many students have better ideas at solving some problems than we do? If not, then perhaps your practice doesn’t allow for students to risk, question and present their own ideas for comparison.

Our text books/math class cultures have been guilty about fragmenting learning. We have bought into a world view of easy, medium, hard questions in this order and if it’s Tuesday then it must be Sine Law and every question on this page is involving Sine Law – each page’s title even reminds us of this fact.

We SHOW them some easy questions and then assign questions for them to practice what they have seen (easy, medium, hard) and then expect them to extend this to the harder questions. Why do so many students return the next day with many of these not done? How many are just waiting for us to SHOW them the Way the next day!

Why do we always ask measurement questions about circles, quarter-circles, semi-circles, and three-quarter-circles? By fragmenting these away the real concept of “the fraction of the circle we are utilizing” we encourage memorization and incomplete understanding. Dr. Walter Whitely shared with us about Australian publishers of children’s literature always showing a triangle with point up and either equilateral or isosceles. Children were not seeing the real features that define a triangle and it was impeding their geometric understanding.

I think it has made a big difference in student learning when I reflect upon my own practice and culture and look for these weaknesses. For example, I am very good at drawing 3-D objects from one particular view (an isometric mental grid that I have practiced on) and not so good rotating them to a different orientation/vanishing point. Try it yourself. Immediately draw a rectangular prism (box) and then compare it to your neighbour’s view. How many different styles have been drawn around you? Was your perspective viewing from above and to the right? How flexible a drawer are you? How easily can you change to viewing the same drawing from below left?

  1. What’s the Big Idea about Chunking:

This has been one of the richest sources of new ideas, connection-making and growth for my students, my peers and myself. Chunking is different to me than fragmenting since it involves grouping problems into families for the question/answer combinations to raise questions about patterns/connections and to formulate conjectures. Chunking forces us to connect questions with answers and to compare these pairs with other pairs for a bigger idea. If students just complete one question at a time, fill in a blank in a workbook, write out answers only without the questions how active are they at making connections?

After all my years of school and twenty years of teaching math I was caught in the middle of a class ‘chunking’ some common fraction operations and a question popped into my mind. Recreating the process here:

How do you multiply fractions? -> (multiply tops)/(multiply bottoms)

Powering is a type of multiplying so how do we power fractions?

-> (power top)/(power bottom)

[Stop moment!] But dividing is a related operation so shouldn’t dividing fractions also use a similar skill?     -> (divide tops)/(divide bottoms)?

My class then explored this and found it to be quite useful. Why were we never taught or allowed to explore this simple, obvious idea? Why did I not see it earlier? Intentional chunking revealed it. We want students to bump ideas into one another and to use one body of learning, one type of problem to help us extend to others to expose the bigger ideas involved.

Philosopher David Appelbaum speaks of what he calls a “stop”, a moment of risk, a moment of opportunity. A stop, he tells us, occurs when a traveler (teacher, researcher, child encounters an obstacle, and is momentarily paused in action. A stop is a moment of hesitation, a moment that calls our attention to what is hidden–a vulnerability, an intimacy, a longing. A stop invites us to question our habits of practice and to engage anew. A stop is an invitation to understand things, events, experiences and/or relationships from a new perspective” (Fels, 2010, p. 2).

  1. Manipulatives – Thinker Toys:

When my eldest daughter was six, she watched me mark some grade 9 algebra papers. She then started doodling 6x + 2x = 8x. This wasn’t on the test. When I asked her about 5x + 4x she told me 9x. I tried 2x2 + 4x2 on her. She screwed up her face and said 6x2? Excited about her being a genius prodigy, I asked her what an x-squared was? “I don’t know. Is it something like kilometres?” AHA MOMENT! It was a week later that I built my own class set of algebra tiles from cardboard cutouts I made and my two little girls coloured one side – the positive side of course!

How many students get “what is a negative plus a negative” wrong but do not fail at seeing that a debt and a debt make a bigger debt? Memorization without a physical, relational, visual anchoring in the concrete is doomed to failure in all but a few. Four red tiles added with six red tiles do not make a pile of ten blue tiles. Drawing fraction division question visualizations made me realize that I had only learned one division concept growing up. 12 ÷ 4 meant dividing 12 into 4 equal groups with 3 in each group. This also meant that I did not know how to visualize . What is two-thirds of a group? 12 ÷ 4 also meant dividing 12 into groups of size 4 with there being 3 of these groups. So now I visualize as 12 divided into groups of size 2/3 and can see there are 18 of these groups.

Providing grade nine students with some blocks and string allowed several groups of them to solve a locus problem that took the calculus class twenty minutes to solve. Manipulatives (or thinker toys as van der Walle characterized them), visualizing, making physical models and creating analogies or metaphors allow students to grasp concepts with deeper understanding.

  1. Attention to Diversity:

Another source of influence and success has been in paying attention to diversity in my classes. Have you ever heard something to the effect that ‘there is more diversity in my Essentials of math class of 16 kids than in my regular class of 30 students’? Reflect for a moment on how possible this is. Why would we believe this to be true when the mathematics says it must be the other way? Is it because we teach the Apprenticeship & Workplace students differently and perceive their needs as great enough to utilize a more student-focused pedagogy?

The biggest mistake of past centuries in teaching has been to treat all children as if they were variants of the same individual, and thus to feel justified in teaching them the same subjects in the same ways. (Howard Gardner in Siegel & Shaughnessy, 1994; Phi Delta Kappan)

  1. Alternative assessments–Assessments FOR, AS, and OF Learning:

This area has preoccupied me for almost twenty years as my reflections and changes in practice kept exposing how poorly my assessment practices connected with my goals and accurately reflected my students’ learning. I was also convinced through many situations that I was either saving or destroying students with how they were assessed or evaluated. Following is the summation of the Assessment Reform Group in Great Britain. Black & Wiliam and Shirley Clarke are distant thinkers of mine that have had major impacts on my understanding.

Assessment for learning

  1. is part of effective planning
  2. focuses on how students learn
  3. is central to classroom practice
  4. is a key professional skill
  5. has an emotional impact
  6. affects learner motivation
  7. promotes commitment to learning goals and assessment criteria
  8. helps learners know how to improve
  9. encourages self-assessment
  10. recognises all achievements


  1. Professional Learning Communities, inspiring peers, distant thinkers:

The wide variety of opportunities I have had to reflect on and develop my craft of teaching within a community of practice has radically changed me in a positive way and has made a lasting impact on who I am and what I believe. The BCAMT Listserve is one of these communities and all the conference workshops and sessions they’ve arranged. Others include our PIMS Lesson Study Group based at UBC, our Wednesday Morning Study group at McRoberts, which I am missing greatly, our Richmond District’s “Teaching Gap” book study and assessment study groups. The four varied and passionate cohorts of post grads at SFU have been amazing examples of how various perspectives and experiences strengthen and encourage growth in learning. Even as I write this, I see going back through the years that there has been an amazing number of ‘groups’ and individual’s who have influenced me. I will pause here to just encourage everyone to seek these out to enhance your own educational lives and to reflect on ones you have already experienced.

Individuals also have impacted me greatly whenever we collaborate or just reflect together on teaching and learning. I have also been greatly affected and enriched by members of my school community and beyond who were not ‘mathematics’ teachers but educators. I have partnered with some amazing people who shared the same students as I but taught them in different subject areas.

These great educators also connected me to a wider range of thinkers. Reading books/papers from English, art, drama, science educators and researchers. Generalists, poets, philosophers, radicals and scientists have all played a role of who I am and what I see and respond to. Who would you like to invite to your dinner table to discuss learning with? Comparing guest lists with others opens the door to interesting paths to pursue.

And the # 1 reason: The freedom of not having to know everything!

            My best lessons have turned out to be ones when I asked a question that I didn’t know the answer to. I became a student/problem solver with my students. They need to see us modeling the process – the messiness. If students are more willing to risk on non-permanent vertical spaces, then how damaging is it for us to ‘show’ a perfect solution . . . every time! They need to see that problem solving can be messy–starts, stops, retracing, verifying, question posing and ‘aha’ inducing.

My life and the lives of my students have been so much richer when I let go of having to be the expert and to only ‘explore’ known ground.

Antonio Machado Poem:

Traveller, There Is No Path

Traveller, your footprints
Are the path and nothing more;
Traveller, there is no path,
The path is made by walking.

By walking the path is made
And when you look back
You’ll see a road
Never to be trodden again.

Traveller, there is no path,
Only trails across the sea…”