Effective Instruction is built on COMPARE

Sitting in chairsWays of Knowing, Knowing Ways: Whitecourt 2017

JULY 10 to 14

Effective instruction encourages students to match, sort, classify and COMPARE. Thinking begins from these simple skills.

As you introduce a new concept be thoughtful. If we only allow one way to build, one way to explain, one way to represent we narrow INTEREST, ENGAGEMENT, UNDERSTANDING and ABILITY.

We make it simple cause we think we are helping. Quite the opposite, we are removing the most important component of learning. COMPARE, describe, explain, discuss, adjust, refine, come to consensus. Once students are willing to engage in learning, you can teach them anything.sorting words for place value

We sorted words. This represents a sort. Do you see any reason for the sort as it sits? Look very closely. Using the words, seeing, saying, spelling them is a critical literacy connection. Students cannot understand a concept they cannot communicate around. Knowing and using the language matters.




CONSIDER AND COMPARE THE STUDENT RESPONSES BELOW. We were identifying the equivalence of one tenth to ten hundredths.  Each representation contributes to part of the understanding.  But what is still missing?

My goal as a teacher is to help students see the connections before we decide how we will as a class represent decimal fractions.


tenths to hundredthstenth hundredthIMG_0774

Spatial Reasoning=Math Success

Study the power of visual spatial reasoning to impact student success in math and science.

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The materials, year plans, lessons and BERCS CARDS  I am sharing at this Summer’s Institutes are designed around visual spatial models that support and sustain attention to the development of reasoning and problem solving skills AS, not after,  students learn, study, & practice in order to efficiently RECALL FACTS.



animate brain

Isolated practice with individual facts does not generate success with reasoning and problem solving. Facts memorized in isolation lack meaning and so are easily mixed up or even forgotten. Students resort to finger counting when under pressure.


As you preview BERCS materials some imagery will pop out at you. Rectangular arrays are everywhere….Student BERCS Sheet

Part whole relationships abound

What covered cardsIf then rods


Imagery linked to symbols prompts and sustains recall.

Cool Fractions eighteenths


Student practice is tailored to meet their current physical, emotional and academic levels of development. The cards provide a range of challenges to allow students to progress toward mastery.

BERCS Cards are supported by a variety of practice pieces. Oral and written.

more WC WSBERCS cards provide you ready made, differentiated independent practice centres. All you do is choose a method or medium through which students will  communicate their learning.









Students can work in pairs or alone. They can use all the cards or just a few.File folder dots


2DM divideparts of set 1


Decimal Numberlines

Bridging the Numeracy/Literacy Gaps

Where are you looking for answers?

Changing Curriculum will not change student engagement, student knowledge, student success.

TEACHER Knowing.

 That is what will change results for students.

Teachers who teach students how to think, reason, remember and recall.

Teachers who teach students how to communicate, connect and critically think.

Teachers who “know” mathematics as a way of thinking, reasoning and explaining the world, not as a set of “rules” to memorize or “answers to try to remember.

Teachers who continue to study and evaluate their own practise. Teachers who understand the human brain seeks always to understand.

This Summer… Join the Evolution!!!

University of Lethbridge: K to 3 July, 2017

Hilltop School, Whitecourt: K to 6  July, 2017

WECA, Edmonton: K to 3 only   August, 2017

Grande Prairie, AB: K to 6  August, 2017

Ways of Knowing: Subtraction

Number One:

If you want to build proficiency with subtraction at any grade, you must not be afraid to move things in your head.

Start by moving things in real time and discuss the affects of the move. So here is aChunkz Fives ChunkitZ puzzle. I see 3 and 2. That’s 5.

How do I know this is still 5? I turned the 2, changed my five(quarter turn left)  flipped it over (reflected) and slid it down to match the left hand side.

8 dots = 8 dots

I see 8 = 8 because I see 6 + 2 in each of them.I turned the 6 (one quarter turn to the left). Then I slid the dot on the far left side of the middle row down below and lined up with the middle bottom dot. I slid the dot on the why does 8 = 8far right side of the middle row up. It is above and lined up with the middle top dot. It is still 6 and 2.

NUMBER TWO: Subtraction is an action that emerges when you want to make things equal or maintain equality. These are not equal. One way to make them equal is to remove from one side.Not equal dots

IT IS CALLED SUBTRACTING not take away. first I did a quarter turn to the left. Then I removed one dot and pushed the other to the right. Yes, the action was a removing action but the mathematical term is subtract and the sign we use is a subtraction or minus sign. WHEN make equalSTUDENTS HAVE THE VOCABULARY, THEY CAN JOIN THE MATH COMMUNITY. Vocabulary is a part of community. You have to speak the language to join the club. we correct DA DA to become daddy or dad or even father but we leave take away??? I do not agree. You do not take away to solve every subtraction problem.

Number 3: Addition and Subtraction are RELATED. Teach them together, first as a relationship.

If you can add, you can subtract. Since 1997, the Western Protocol for Mathematics and the Alberta Program of Studies for mathematics which emerged from it, has stated this What's missingas an outcome in mathematics. Students need to know and understand that addition and subtraction are related. In the 2004 revision, the statement think addition for subtraction was used as a strategy. HOWEVER, it is not just a strategy it is an actual property of our number system. The focus is relationships before equations. WC If thenUnderstand the relationship, talk, describe explain the relationship. Then introduce the notation.  Not the same day, but once they can explain…..

INVERSE OPERATIONS, is a critical understanding students need to be able to apply when they work with integers, rational numbers and algebra.

Number 4: Subtraction is not just a “taking away”.

It is a comparison. Taking part of the whole away can reveal the answer but the comparison might be phrased as difference between. In that case you are looking at 2 lengths or heights, or weights or quantities and comparing them.

My sunflower is 4 meters tall. My mom’s is 3 metres tall. How much taller is my mother’s sunflower. (You do not see taking a sunflower away in your head. You see the space between the sunflowers.)

inverse to 1002 digit inversNL 3 digitScreen Shot 2017-05-10 at 3.39.09 PM

B.E.R.C.S. cards and tasks were designed to prompt and promote visual spatial reasoning. They focus on the relationships that matter. They prompt communication, problem solving and critical thinking. Use them to support your Guided Math approach. Use them to open lessons as warm-ups or as the introduction to a lesson. Use them with small groups or individuals for added practice. Use them to differentiate so that every student builds his or her full potential.

Want to learn more? Join me at a Summer Institute:Lethbridge, Whitecourt, Edmonton, Grande Prairie  2017

Join the Evolution: Infographics

Summer Institutes 2017:

Finding Common Ground:


Thinking101 Institutes are dedicated to fuelling the evolution of learning that is teaching. Every summer, every session deepens the connections between Numeracy, Literacy, Science and Math. The ARTS are INFUSED throughout, a little or a lot. An approach to teaching and assessing that values- no prizes visual/temporal spatial reasoning is an approach that includes EVERY LEARNER, EVERY CULTURE, EVERY BACKGROUND.

It embodies and encompasses a holistic world view, a holistic learning view… we all share the gift of coming to know as a body mind connection… Visual models can bridge any gap be it:  Literacy, Numeracy, Remembering or Recall.

Infographics is more than reading graphs and organizing data. It is an approach that makes connections and patterns burst forward as students discuss and COMPARE.

The focus is thinking and comparing: the teacher role is to prompt, probe, prod…. immerse students in the vocabulary, shape and mould the pathway from seeing and experiencing through thinking and doing to understanding the conventions of the discipline. We do not start from memorizing symbols and rules… but we can certainly end up with understandings of the symbols and rules.

How does this:knot 12

Link to this:


And to this?:Screen Shot 2017-04-26 at 10.22.38 AM


And to this?robotic-assembly-line


Or to this:Screen Shot 2013-09-04 at 10.12.16 AM


The only way to face the future is together, another one jumps the cavern

the only way to survive the future is to evolve… Join the EVOLUTION…..


Summer Institutes 2017:

Finding Common Ground:


Thinking101 Institutes are dedicated to fuelling the evolution. Every summer, every session deepens the connections between Numeracy, Literacy, Science and Math. The ARTS are INFUSED throughout, a little or a lot. An approach teaching and assessing that values- no prizes visual/temporal spatial reasoning is the BASIC SKILL for learning, anything even MATH FACTS!

This summer my new word is INFOGRAPHICS. The topic is huge and all encompassing. Making sense of the mathematics that is all around us, often used to influence us, often used to bias our decisions. Infographics refers to an approach to actively, deliberately teaching students to connect  and communicate the development of their literacy, numeracy and critical thinking skills through the use of graphics based artifacts. STEM, STEAM with a focus on learning to think, not a focus on making products.

Over the next few weeks I will share samples to demonstrate the breadth and scope of using INFOGRAPHICS as the organizer for curriculum. Here is one:

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As a class we looked for connections between the problem told orally and the diagram this student created to share her solution. Here is what we came up with through that discussion. We can now use the information on the diagram to re write the problem or to improve our own solutions.we did

or to start building the number relationships that will support the kind of number practice Grade twos need to constantly engage in. Do you understand this table that we created from information in the picture?the table

Can you make a t table? Here is a different problem. Build a t table to show some solutions.

My dad gave me 12 nails. He said I had to let my sister hammer more than 6 into the wood.

How many could I let her hammer? How many will I get to hammer?

Coming to Know, so much more than “Math”.

This past weekend a phenomenal groups of mathematics educators, researchers and learners met at the BIRS research station, Banff, Alberta for the annual Math Fair Retreat. I share here thoughts and reflections the weekend spawned…..

Traditional cultures, traditional arts, traditional ways of “knowing” ourselves and the world are highly spatial, highly visual and can be used to build, adapt, refine spatial reasoning skills that improve our abilities to reason and think. Why are we not immersing ourselves in the experiences, then noting the “math” connections as they become apparent. Why do we start from “math” first as though the knowing of mathematics is more important than the Coming to Know…. 

Read the following excerpt that describes the traditional art of bark biting and tell me again about what mathematics and knowing mathematics entails???

As I was when I first heard an interview with a traditional bark biter, I am left breathless in wonder as I consider the incredible skill and knowledge involved in birch bark biting. This is mathematics in our world.

Another of my favourite topics is Knot Theory and for just a few brief moments I got to immerse myself back into the world of knots. Knots are opening the door to math, science pursuits never before dreamed possible. The world awaits and all you need literally is a shoestring!!

This summer we will re enter the world of spatial reasoning with a powerful look at INFOGRAPHICS and how we can use the appeal of the visual spatial temporal world to connect ALL students and ALL cultures to Literacy as Numeracy and Numeracy as Literacy. Ways of Knowing, Knowing Ways….


They will learn arithmetic faster, more fluently, with confidence, speed and accuracy.


Here’s an article to consider: screen-shot-2016-10-23-at-9-46-26-am



The evolution has begun, WE WILL CHANGE THE WORLD… Join me for SUMMER INSTITUTES 2017…. Multiplication is not repeated ADDITION… It is much, much more and screen-shot-2016-10-11-at-10-39-00-amour KIDS DESERVE TO KNOW “why?” AND “Why DO they have to learn this?” Make the mathematics they learn meaningful in a way that leaves them wanting to LEARN MORE….

LITERACY IS NUMERACY, NUMERACY IS LITERACY and let’s not leave out SCIENCE.  Comprehension, reading and writing skills do not grow in a vacuum. flying frogsWe use the content of math and science in engaging, authentic and challenging explorations and experiments to develop the skills that link them all.

IT IS WELL PAST TIME FOR TEACHING TO EVOLVE OR GO EXTINCT…. THE  world demands it, our kids deserve it… do do bird


JULY 10 to 14:   WHITECOURT, ALBERTA, CANADA      Finding Common Ground: Making the Connections: Literacy, Numeracy, Critical Thinking…. K to 6  Registration is now open



TEACHERS, COACHES, LEADERS who are engaged in change at any grade level… This is the continuing journey. All you need is a mindset not willing to accept the MEDIOCRITY that education contains to MIRE DOWN IN…. and a passion for EXCELLENCE. Every learner deserves to learn…..

Creativity in Mathematics



How do you think about the link between 21st Century Competencies and the learning of mathematics?  The list below sounds suspiciously like the reasons I teach mathematics ergo mathematics is a creative endeavour…. More evidence for why our teaching must EVOLVE or go EXTINCT…. mathematics should make students curious, joyful and excited to discuss, compare and create…..



I would add:

Students composing and decomposing number, space and shape in multiple ways through various mediums.


response to build a number (1)screen-shot-2016-10-01-at-11-12-18-amScreen Shot 2016-09-03 at 7.44.46 PMscreen-shot-2016-10-01-at-11-17-37-am