Filling the Numeracy Gap: Build Recall Skills



Hear the confidence? Hear the enthusiasm for learning?

Distribute 2 waysThey have been working with BERCS materials to build mental fluency with 2 and 3 digit numbers. BERCS materials build multiplicative reasoning, alongside fact recall.

100 grid 77 =They have been working with a BERCS approach to building 2 and 3 digit number.They have a conceptual understanding of “place value” that goes far beyond: ‘line up the digits.’









Hear the thinking, questioning, self correcting?


These students are building recall and memory skills as they practice thinking and talking their way through solutions.The focus is not just get an answer. The focus is know your answer is correct. Know you can think your way through problems. Hold facts at your fingertips and recall them as you need to apply them to solve problems.




Visual spatial reasoning tools develop both RECALL & REASONING skills. BERCS Cards are visual spatial reasoning tools.


AREA MODELS are visual spatial reasoning tools. 

BERCS provides a framework within which teachers can develop concept based approaches to teaching the basics for Number and Number computation.

If you want to fill the Numeracy Gap in your classroom, teach your students to:


       then SELF ASSESS

Teachers interested in learning more about BERCS and using BERCS materials to support the development of reasoning and recall skills for their grade should consider attending a Thinking101 Summer Event.

Parents interested in learning more about using BERCS home materials to support the development of reasoning and recall skills for their children should consider attending a Thinking101 Summer Event.Screen Shot 2019-06-09 at 12.50.20 PM


Teaching division?… Do you know the “basics”?

The trouble with teaching division……


Not multiplication

NOOOOOOOOO !!!!!!!!!! Do not even try to make this make sense. It is wrong thinking.

Not multiplying

Oh dear….. What are groups of numbers and what will you add to solve 3.45 times 6.76 or how will you solve 3 eighths times 4 sevenths?


What is multipl

Telling me a dog is a dog does not explain what a dog is. (We will not even mention the inability to form a statement to explain)


I do not show these examples to have you laugh at, or be amused by, students’ lack of knowledge or communication. I find these examples to be a sad testament to our lack of understanding. These were offered by excited and well intentioned grade 5 students.

They are not funny. They scare me.


Teachers so often ask me to help them with division and my answer is always if students do not understand multiplication and are not able to recall some small multiplication facts you should not be teaching division. It makes no sense.


Division is the inverse of multiplication. That means without multiplication, there is no division or without division, there is no multiplication. They are one and the same.

While you could learn to divide first, then learn to multiply, teachers never have this option because from the moment they start to talk about numbers, well meaning adults are encouraging your children to memorize silly little facts like 2 x 5 = 10 and 10 x 10 = 100. This is not understanding multiplication.

The Alberta curriculum is quite clear in Grade 3. The outcome begins with the statement:

                 Demonstrate an understanding of multiplication as equal groups and arrays. 

“And” in math means they go together so teach them together.


Area models built with square tiles or square grid paper make visible the commutative connection between “basic facts” like 3 *4. By simply viewing the array from a different perspective you can see there are 3 groups of 4 or 4 groups of 3.

3x4 4x3 turned comm

This understanding is a key to division.

The Alberta curriculum is also quite clear in GRADE 3 that students are to

  • represent and explain division using equal sharing and equal grouping


There are multiplication equations to describe each array “and” for each of those 2 multiplication equations, there are 2 ways to express and describe a division.

If the array above is 12 cookies, you could share out in sets of 3. How many groups or sets of 3 can I make?

multiply see 3 in each set?


When I label an array as 3 by 4, the three is labelling the sets of 3 that are running horizontally. There are 4 of them. Therefore 12 ÷ 3 came to be interpreted by many teachers as “how many threes are in 12?” You can see there are 4.

3 in each set multiply


But wait,

that 3 at the top of the array also signifies there are 3 equal groups in 12. See the 3 columns or groups?multiply 3 of 4


If I change my story a little, I have 12 cookies and 3 bags. If I want the bags to be equal I can put 4 in each. Do you see where the 3 bags are in the array? Do you see how I know each bag will have 4 cookies?

multiply 3 groups of 4

The Alberta curriculum make clear in the outcome in Grade 3 that division is to be taught and interpreted in two ways.

What if the array we started with represented 12 cookies and I wanted to give out 4 to each friend? How many friends can I feed?

multiply 4 in each 3

Looks like I can feed 4 friends with 12 cookies because 4 x 3 = 12. Same array, different interpretation.

multiply 4 sets of 3

What if I had 12 cookies and wanted to pack them into 4 boxes. How many would go in each box?

multiply see the 4 sets


We cannot and must not teach division as just set of facts to memorize or a list of rules to follow.  Teaching for understanding means understand FIRST…. not solve and memorize equations and facts FIRST.


The evidence is clearly established that once you think you “know how” to do something you are much less likely to care how or why it works. When we focus on memorizing meaningless facts and multi step algorithms like “long division”  before we develop with students an understanding of how multiplication and division are related both visually and spatially students do not come to understand. That is what makes teaching division so difficult. They have no way to make sense of what they are doing or why.

They “get” answers that have no meaning.

Long division means what?

My BERCS cards are set up to be puzzles. Puzzling attracts the brain. Puzzling takes the pressure off “know the answer” quick as you can or fill in the blanks by moving the focus to hmmmm I wonder what is going on here? Puzzling engages students in thinking, talking, comparing, connecting all the components of LEARNING.

Is this solved with multiplication? with division? or with understanding both?Division cards

It is just the basics, folks….. THINKING 101.

Basic Number Relationships….Is thinking your goal?


As number sense is foundational (Baroody et al, 2009) and predicts later skills (Edens & Potter, 2013), guiding the child to know and enjoy numbers is critical. Other research into the connection between body, mind and emotions (Alcock & Haggerty, 2013) points to mathematics strategies that are tactile, fun and shared. Peters and Rameka (2010) explain that as ECE teachers we need to be confident that practices and resources will foster learning and enjoyment, not simply temporary gains in particular skills.


Reading the paragraph above inspired me to write today’s post. Chunk-itZ are a thinking tool that have emerged and evolved across my work with learning to teach mathematics through thinking and reasoning.  The Chunk-itZ hook students because they fit together in puzzles. 






As students puzzle, teachers observe & listen for opportunities to prompt lessons around number, space and shape.


Do you see 3 and 2? How about 5?

Do you see a staircase?

Do you see a decagon?



                                 Ideas about number emerge.

The best way to learn mathematics is to follow the road that the human race originally followed: Do things, make things, notice things, arrange things, and only then reason about things.

    ***WW Sawyer

Ideas that form the foundations for multiplicative reasoning and what teachers refer to as  ‘place value’ emerge.

As he was exploring the ChunkZ, a Grade one boy called me over to ask: “Is this 1 “two?” Then turned it over and finished his question: “or two “ones?”

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I said: “hmmmm, interesting. How much is each of them worth?”

He responded, “they both are worth 2 but one is a 1 two and the other is 2.”

I asked, “What would this be?”Screen Shot 2018-07-25 at 8.33.57 AM

He responded: “Two ‘twos’. That’s four.

I turned the pieces over and asked: “And now what do I have?”

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He responded: ” Two ‘ones’, and that is still four.”

“And what about this?”

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He responded: “three ones, that’s 6 cause each one is a 2, or if you turn them over three twos and that is still 6.”


He is demonstrating the ability to unitize. He recognizes that we can talk about numbers as single units or as one unit of… this idea is critical to understanding place value which is based on multiplicative reasoning.


Ideas about what it means to be equal emerge.

5 equals 5





Can you “see” 4 and 1 inside 2 and 3?







Do you see how 5 + 2 and 4 + 3 are related in this puzzle?


ChunkitZ engage learners through visual spatial reasoning.  As they turn and trace the pieces to make their puzzles, learners develop intuitions and insights into how parts are related into wholes. In the hands of a skilled teacher, those wholes can be identified and related using numbers and number expressions.  Those wholes can be described and related using the attributes and properties of shape and the vocabulary of space.

This three has 6 sides. What do we name any 6 sided shape? Are any of your puzzles hexagons? Can you use ChunkitZ to build a hexagon shaped puzzle?



This student counted the sides of the three Chunk and said it has 6 sides. He then built a ’12’ puzzle by putting 2 three ChunkZ together and was wondering how many sides it would have. Will it be twice as many sides?  






Nearly a century of research confirms the close connection between spatial thinking and mathematics performance. The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related. 

***Mix & Cheng, 2012

      The connection does not appear to be limited to any one strand of mathematics. It plays a role in arithmetic, word problems, measurement, geometry, algebra and calculus. Researchers in mathematics education, psychology and even neuroscience are attempting to map these relationships.

Combined with dot collections, Chunk-it tasks differentiate to challenge ALL learners as they develop, adapt, practice and refine key skills and attitudes that are critical to developing more than just RECALL of facts. ChunkitZ engage young learners in mathematical REASONING that will make the difference to their success across all the strands and all future grades. In the hands of a skilled teacher, the discussions and drawings form the basis for teaching children to explain their thinking through oral and written communication. Reading, writing, symbolizing the mathematical relationships, those are the goals. Automatic recall of facts are just a given, it comes and it lasts.

ChunkitZ help develop proportional reasoning which forms the foundation for making sense of multiplication, division, fractions and decimals. Outcomes related to proportional reasoning include comparing size of units to number of units needed when measuring (Grade 2), understanding the relationship between minutes and hours, days and weeks, months and years (these are all ratios), cm to m and mm to cm and m (more ratios) making sense of how multiplication facts are related and how area grows, making sense of how fractions are related and how equivalent fractions are related… the list goes on.

Here is a task for proportional thinking: I show this puzzle on the overhead or on a task card. Students work with the full size Chunk pieces to reproduce it without coming up and direct comparing.

Chunk pages 1

He is directly comparing. He is fitting the pieces in to make the puzzle. The beginning of proportional reasoning.


The task above is very different from the task below. I show this puzzle on the Smartboard or give it to students on a small task card and they build it with the Chunk pieces which are 3 tior four times the size.  The outline they create will be similar (the same shape, angles are same) but not congruent (same side lengths). This is a much more sophisticated version of the Grade one matching above. And Grade 5 to 8 students love it.

Once they get good at it they are able to also double and triple the dimensions of the shapes, using chunk “ones” as their referent.

Make my Chunk puzzle





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

Spatializing math “facts” builds executive functioning skills….

Math facts are much easier to learn and remember and more rapidly recalled from memory when we embed them within physical, visual investigative tasks. The intent of working with blocks, tiles, grid paper and the like, in math class, is to ENGAGE the senses in the learning. The mistake teachers often make is to turn using the manipulative into just another set of rules. The materials should be used flexibly. COMPARE!




Screen Shot 2016-07-29 at 12.49.09 PM

These are all ways to see 7. What do they have in common?

Students should never be afraid to test out a turn, or a flip, or a slide…  What changed, what stayed the same.

The COMMUTATIVE PROPERTY emerges when we encourage students to move materials and images:

Screen Shot 2016-07-29 at 12.14.12 PM

I store blocks in fives of one colour. I want students to trust the five from mid Grade one on…  I can move the image, I can move the symbols…


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Understanding is grounded in the physical experiences. Language is a symbol system for communicating understanding. When we start in the real world of objects and materials, every student can begin.


The imagery continues with models into 2 and 3 digit numbers. The blocks become cumbersome so we gently but intentionally move to hundred grids and numberlines.

Screen Shot 2016-07-29 at 11.33.28 AM


B.E.R.C.S. cards and tasks are designed with spatial reasoning in mind. But any tools for thinking can be proceduralized into meaningless rules if we do not learn how to use them. Teachers have no choice. We must continually learn……

do do birdEVOLVE or GO EXTINCT…








The imagery continues into multiplication…


Students build multiplication models with grid paper or tiles, with cusienaire rods & diagrams. Flips (reflections), slides (translations) turns (rotations) do not change the product.

Screen Shot 2016-07-29 at 12.35.46 PM


Students physically experience the commutative property. Link the images to the equations and practice mentally transforming both imagery and symbolism.




Screen Shot 2016-07-29 at 12.36.40 PM

The Commutative Property for multiplication is spatial.


So what’s up with xy or yx?  x + 4 or 4 + x?

Algebra becomes a puzzle not a chore!

EDMONTON AUGUST 8 to 12, 2016

What do these books have in common?

dreaming up  a seed is sleepyOne Plastic Bag

What’s the basic math in the samples below?

another buttons more

Screen Shot 2016-04-18 at 5.36.52 PMpirsquare

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Registration will be live this week.. Mark the dates on your calendar. August 8 to 12.

Register for 3 days or all 5. August 8, 9 , 10 the math focus is on making sense of early numeracy (Pre K to Grade 3) and filling the gaps for students who are still struggling with the basics by Grades 4 and 5.

August 10, 11, 12 the math focus moves to how multiplicative reasoning emerges and can be nurtured beginning as early as Kindergarten. Then studying the links to division, fractions, place value and beyond.(Specific focus Grades 3 to 7).

School leaders, numeracy and math coaches and consultants, teachers who cover more than one grade need to participate in the whole week.

Registration will be online this week, meantime mark your calendar.. or email





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Can you “see” where to make the two folds that will leave you with only yellow (only blue) showing on one side of the paper?


PAPER FOLDING Tasks have huge potential for engaging ALL students in critically important



Intellectual activities

that lead them to success with learning MULTIPLICATION and MULTIPLICATION FACTS.

Some food for thought as you prepare for 2016 and making the DIFFERENCE that will CHANGE the FUTURE for you and your students. Include time to mentally puzzle in your everyday classroom routines- EVERYDAY.



your brain

Teach your BRAIN TO SEE: Without touching paper or speaking any words, THINK through and create a set of gestures to teach someone else to successfully fold a piece of 8.5 x 11 paper into 3 equal pieces.. (You may need to resort to speaking as you build your plan. But be prepared to teach the skills without using any oral language.)




PAPER FOLDING links to learning to think in ways that are opening the door to solutions to complex social, scientific, economic and political problems .



Who would have believed, when we were folding snowflakes in Grade one, the potential for lifelong success in learning mathematics to the highest levels was right on our fingertips.

SO WHAT HAS THIS TOPIC GOT TO DO WITH MULTIPLICATION AND BASIC FACTS???? I will start my answer tomorrow…. today, try my puzzle. Video yourself teaching someone else to fold in 3, using only gesture and send it to me:

It’s Not about Coding, It’s not about iPads, Smartboards or Apps. WE MUST TEACH STUDENTS TO THINK…

WHEN I spend too long at my computer, working on materials to help teachers work with students, to help build understanding in mathematics and science, I start to have “panic” type attacks. The world is moving at such a pace, how are we going to keep up. I guarantee it will not be by memorizing flashcards or remembering the 13 or more steps to long division. I agree there are some things I would like kids to have as tools to aid their calculations BUT NOT AT THE EXPENSE OF LEARNING, of knowing how to learn, what to believe, trust, question, how to challenge yourself to learn as much as you can, to pursue more and more knowledge, to love to think, puzzle, problem solve, to want to solve the puzzles in this world.

The sad and simple truth I have learned, is that the majority of our kids do not learn to love learning by memorizing facts and filling in worksheets. Do not confuse learning with compliance. And do not get me started on how much of the day the majority of our students in the majority of classrooms are doing just that!!! No matter what nonsense the media might choose to splash across their headlines and blog pages, very few classrooms across North America look any different than they did in the 50’s. I am not criticizing teachers, just reporting the facts m’am.

When I watch the latest buzz around how we will make the DIGITAL GENERATION MORE attentive, interested, engaged , when I see the latest book, app, gadget supposed to teach them some key skill,  I shake my head…. One of the latest crazes is programming and coding… Folks get with it… if I want to learn to code and I have a flexible, critical and well-oiled brain, I will learn. But will I need to learn to code, within a few months, days, minutes, seconds, why will it matter? I can do anything I need to do with computers without knowing coding… and that will include getting some pretty high paying and exciting jobs.

(this excerpt from a new product at Spark Fun prompted this thinking…)

By interfacing the Sandbox to your computer via a USB cable, the Sandbox can be programmed using the popular Arduino programming environment. To further simplify the learning experience, we’ve designed the Sandbox and its guide around using a simple, “blocky”, programming add-on to Arduino called, Ardublock. Using ArduBlock – a simple, graphical version of the popular Arduino programming language – you will be able to program all of the experiments with a simple graphical interface instead of writing code.

INSTEAD OF WRITING CODE is what caught me.

We are already out of date and gosh I just heard that “coding” is the new “skill.” That’s why I continue to rant and rail. Stop being sidetracked by the lure of quick fixes….TEACH KIDS HOW TO LEARN, HOW TO THINK CREATIVELY, CRITICALLY AND CONSTRUCTIVELY. Yes, you might be able to do that through some coding activities but TELL them the point is to learn how you learn… to be a better, more effective and efficient LEARNER.

Let them in on the learning we are doing around thinking… engage them in problems and contexts that require thinking and talk about the thinking NOT THE ANSWERS. Teach them to pay attention to thinking, organize their thinking, share their thinking, communicate about their thinking, evaluate their own thinking.

If the thinking is SOLID, the Answer will be correct but this is not a two way street.

But IF THE ANSWER IS CORRECT does not necessarily mean the thinking was solid.!!!!

I love this visual. a grade one thinking about 7. What is my job as a teacher? He has done his job, he solved the problem… now what do I do?


How can I help him capture, refine and improve his mathematical reasoning. How can I help him “KNOW” what he obviously knows and move on to new learning?

I am greatly disturbed by this visual from a Grade 7… How did this student get to Grade 7 without any idea whatsoever of how to image a fraction of a number?




cherries 3

This is a “picture” to explain her thinking for adding fractions. 1/3 + 1/4 = 2/7 = 25. The 25 is a direct literal interpretation of the picture of 7 dots as her answer. Two are black and 5 are not so she wrote 25.


 How are they related is THE QUESTION. It is THE question, in every discipline…..

Imagery opens the door to imagining and imaging means you are actively engaging your brain. Take the time to let students build their understandings….  Compare FractionsPuzzle with kids. Teach them to puzzle and think and puzzle some more….

puzzled copy copy



No Need to Count N.N.C.

ten folder

Students use dot cards to see that numbers are in numbers.

If you look into 10 you will see 5 + 5 and 3 + 2 + 3 + 2. How are they related?

You might see 3 + 3 + 2 + 2 or you might see 6 + 4. How are they related?

Relating the facts, increases the likelihood of retrieval because things are linked. More important relating the facts allows students to build meaning and make sense. Number properties become more evident and transfer more readily to algebra.