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.)

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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

Thinking101 Institutes

Finding Common Ground 2017:

Ways of Knowing: Knowing Ways

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Grade 1         Lethbridge July 3 & 4      or       Edmonton August 14 & 15

Grade 2         Lethbridge July 5 & 6      or      Edmonton  August 16 & 17

Grade 3          Lethbridge July 6 & 7       or     Edmonton August 17 & 18



Monday July 3 &  Tuesday  July 4  or  Monday Aug 14 & Tuesday August 15     one is a snail

Grade One (and pre Grade one)  We will delve deeply into the connections between Literacy and Numeracy and ways of knowing how to build a rich, connected sense of number that does not rely on counting by ones. (Sorting and patterning are purposeful in these tasks). These ideas apply no matter the style of instruction: ‘guided math’, ‘center based’, ‘inquiry, project or problem based’, ‘direct whole class instruction’.  The 2 days will connect literacy to numeracy and numeracy to literacy in a balanced approach to developing ALL the BASICS our students need. (emotional, social, physical and intellectual)

dreaming upOur study will centre on tasks connected to a year at a glance document, designed around the ALBERTA program of studies & organized around the big ideas that connect Number, Space and Shape in a balanced classroom. Reasoning and problem posing and solving are every bit as important as knowing some facts. This spread balances both right from September. It also attends to what we need to know our students CAN do at the end of the year and the evidence we need to gather.  The document has been developed over 8 years of study in Alberta classrooms with Alberta teachers and Alberta students. Children’s Literature, both fiction and non fiction, as well as themes from science, socials and the ARTS are woven in. You will Tinker, Thinker, Make and Shake as we unpack the materials to fit your classroom & community context.



 July 5 &  July 6  (Lethbridge)   or   August 16 &  August 17 (Edmonton)

word FSA

Our study will centre on a Grade 2 year at a glance document designed around the ALBERTA program of studies, developed over 8 years of study and practice in Alberta classrooms, with Alberta students and teachers.  Grade 2 will begin with the important role that shape and space, sorting and patterning play in learning number and number facts and how to connect that to knowing ways to TEACH students how to relate and retain BASIC facts: we must teach students how to learn to recall and how to practice to remember. We cannot just tell them to practice. And games alone are ineffective. Thursday the focus will move to developing 2 digit number and operational fluency.

Grade 2 must develop robust and flexible imagery for 2 digit number. The rest of their mathematics in school depends on it.  Grade 3 teachers will see and hear why How many ways can you catch a flytheir students are stumbling or easily confused when computing with 3 digit numbers and what they can and should do about it. Grade 3 teachers will work with materials that bridge the learning from 2 to 3 digits as well as the most effective tasks for solidifying the “facts” to 20. Filling the numeracy gaps means knowing what connects grade to grade and how to teach so that students REMEMBER, RECALL and APPLY. Measurement, space and shape concepts play a vital role in this learning: bet you’re wondering why? Come find out.

The Grade 2 days will connect literacy to numeracy in order to build comprehension and communication skills as well as critical thinking  in a balanced approach to developing ALL the BASICS our students need to master in order to learn. (emotional, social, physical and intellectual)



 July 6 & July 7 (Lethbridge)  or August 17 &  August 18 (Edmonton)

Grade 3 will join Grade 2 teachers on Thursday working with materials that bridge the learning from 2 to 3 digits as well as the most effective tasks for solidifying the “facts” to how-to-swallow-a-pig20. Filling the numeracy gaps means knowing what connects grade to grade and how to teach so that students REMEMBER, RECALL and APPLY.   Friday we move to the critical connections between place value and multiplication and how important it is that Grade 3 students UNDERSTAND that multiplication and division are more than just a set of “facts”. This is a new way of thinking and if we want to reduce or bridge the numeracy gap, we have to understand the pivotal role that moving from additive to multiplicative reasoning plays in making sense of fractions, divisions and place value.


Ways of knowing and doing mathematics 

Grade 4 teachers concerned about the numeracy gap should attend the 2 Grade 3 days…. find out how to correct the misconceptions.

As with the other grades, our study will centre on a Grade 3 year at a glance document designed around the ALBERTA program of studies, developed over 8 years of study and practice in Alberta classrooms, with Alberta students and teachers. (And if needed a Grade 4 spread to support the bridging.)

2 days:$275      3 days:$395     4 days:$450    5 days:$550

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Registration for Edmonton click here.

Registration for Lethbridge click here.

Registration for Grande Prairie click here    (Grade 1 to 7)

Registration for Whitecourt click here       (Grade 1 to 7)

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.


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Parent letters

I do not and will not support DreamBox or any other on line platform for teaching and learning mathematics but this is one of the components that we really really need to learn from…. Christopher writes a great blog, thanks

Overthinking my teaching

Here is an email that a friend of mine—father of a first grader in the Minneapolis Public Schools, and math-teacher-on-parental-leave—received from DreamBox, an “adaptive learning platform” for K—8 math.

Congratulations! [Child] successfully completed a group of DreamBox Learning lessons.

Are you familiar with the concept of compensation? This is a strategy that can be used to make addition problems “friendlier”. To use it, just subtract a “bit” from one number and add that same “bit” to the other to create two new numbers that are easier to add mentally.

[Child] learned this strategy by completing a series of lessons using the special DreamBox tool, Compensation BucketsTM. For example, when shown the problem 29 + 64, [Child] turned it into 30 + 63. Towards the end of this unit, [Child] was adding 3-digit numbers with sums up to 200!

On the Run: Friendly Numbers

Take turns supplying two numbers to add. The other player has…

View original post 559 more words

Teaching math just as procedures….

This is how too many students respond when we ask them to create their own problems. No understanding, no knowledge and no confidence.Grade 4 student reflection

This is Grade 4… Where will this student be by Grade 6? The research says we never fill the gaps… I say because we still refuse to attend to what we know now to be true: VISUAL, SPATIAL, TEMPORAL REASONING matters….. especially in math. The numbers make sense when you learn to build them, compare them and then image them as related parts and wholes….

If this student had imagery for number and words, she would make sense of the problem, then be able to set up a mental image for what has to happen with the numbers. She still might make a computational error but she would at least understand what the problem is actually asking her to figure out. I cover this in Thinking 101, place value 101…. book a session, workshop or private tutoring session… You will be amazed at how quickly we can turn this student ON, by engaging her in the learning.

I will be all over the province this year, join me for a day of learning…..