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.”

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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

 

 

 

 

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.

 

 

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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.

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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

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

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!

 

 

 

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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:

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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.

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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.

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

 

 

 

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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

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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 glorway@thinking101.ca

STAY TUNED FOR UPDATES AND THE UNVEILING OF OUR SPECIAL GUEST SPEAKER….

 

 

Maker, Shaker, Thinker, Tinker

M.ath, M.usic, A.rt, D.rama. Dance

the MMADD mathematician wonders constantly about pattern. It is everywhere……So is Art. Art creates curiosity and curiosity drives CONSTRUCTIONINVENTION. This is how the human brain takes control of its own learning.

Compare 6

6 = 2 + 2 + 2 6 = 2 + 4

I was thinking about ways to play with 6 when I wove these.

I specifically had 4, 5 and 6 year olds in mind. I wondered if this might be an interesting centre task. Today’s focus is 6 and only 2 colours are available to use.

I made looms that had a warp of 5 (no particular reason except I wanted an odd number).  I was thinking of ways to see 6 in 2 colours. I see 2 + 2 + 2 and I see 2 + 4.  But then I thought I also can see 3 + 3 in the one on the left. So 2 + 2 + 2 = 3 + 3

still 6   Then I made these 2,  and I saw 3 + 1 + 2 but that is really just 3 + 3 or is it 5 + 1.

Playing in the numbers I see 3 + 1 + 2 = 3 + (1+2) so 3 + 1 + 2 = 3 + 3 I see 3 + 1 + 2 = (3+2) + 1 so 3 + 1 + 2 = 5 + 1.

Changing the order or arrangement of 6 does not change there are six. And the nice thing with paper weaving is that I can pull out and trade off the colours if I do not worry about gluing down edges…

See my section on WEAVING for more on this.

Why would I weave with 6? I was thinking about little fingers working on fine motor skills. I was thinking about the attraction of colour. I was thinking about the focus of remembering to weave in and out while you stick to the constraint of only 6 but in 2 colours. I was thinking about translating from colours to numbers, is any understanding lost or is new understanding gained.

A hugely important thinking skill is the ability to sort by 2 or more attributes. I link that idea to what I was doing here. I have to keep several different criteria in mind as I work. 

I was also thinking, it is good for students to “play within” a number. So this task is about 6. Ways to describe 6. Ways to arrange 2 colours or 2 parts so that 6 is clearly evident. That is a pre requisite skill to developing number sense.

But then I wondered what would happen if I had a larger loom and I could keep going? Could I still see in sixes? What kind of pattern would emerge? What would the core of each be? Same or different?

I find children and some adults are easily fooled into thinking the core ends when they decide it ends, rather than it ends when it repeats….What is the core of my two patterns in the weaving above?

What would come “before” if I extended them backwards? What would come “after” if I extended them forwards? What would 6 repetitions look like in each core?

Could you translate these patterns to music? What would they sound like?

Could you translate these patterns to movement? What would they clap like? Step like?

To really embody patterning as a way of organizing data, seeing relationships, creating thought-provoking ideas, students need to EXPERIENCE pattern with their whole bodies.  So we look for ways to BUILD.

As I begin to EXplAIN my experiencing, the complexity that is embedded in the task comes through.. I see more connections and begin to spread my thinking in other directions. Finding pattern in the data I collect helps me FOCUS the task.

What I choose to REPRESENT as my understanding of the task will direct where my learning goes. At the represent stage, in a classroom, the teacher may begin to take a more direct role to guide students toward specific goals.

For more about Weaving and Math go up to the menu and look for the drop down box for Math, Art  under the section for Support. You will find a section on Weaving. Newly updated.

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?

sweater

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.

Grade 4, 5, 6 :     NUMBER IS IN NUMBER      RELATIONSHIPS MATTER

 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

HE PUZZLED AND PUZZLED TILL HIS PUZZLER WAS SORE

ARE YOU EXPECTING, ENCOURAGING CHALLENGING YOUR STUDENTS TO PUZZLE?

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. 

Multiplication Fact Practice in the 21st Century Classroom

Multiplication Fact Practice in the 21st Century Classroom

A problem solving approach engages the senses to inspire visualization and engagement. When students have to puzzle and think, they build conceptual memories that last. Finger Fold Puzzles are a way to engage in daily practice that focuses on Distributive Property, Commutative Property and building Mental FLUENCY with facts.

My newest set of BERCS cards are built around MULTIPLICATION FACT PUZZLES.. Wow what fun! We tried them out at Spirit of the North School this week… Really engaged in some having to think hard with a follow up practice sheet to maintain daily records.

We are practicing every day and the results speak for themselves. Students are confident in their knowledge and understandings of multiplication facts, connections to division and their ability to demonstrate and apply the DISTRIBUTIVE PROPERTY.

 

Image 2

 

This Vimeo will give you a feel for how we are working in Grade 4. It’s raw, because it’s real kids in real classrooms.

<p><a href=”http://vimeo.com/91841558″>Multiplication</a&gt; from <a href=”http://vimeo.com/user15812216″>Geri Lorway</a> on <a href=”https://vimeo.com”>Vimeo</a&gt;.</p> 

 

YOU WON’T GET THERE WITHOUT PRACTICE

BUT WHAT CONSTITUTES EFFECTIVE PRACTICE  ?

ENGAGE THE WHOLE BRAIN: VISUAL SPATIAL REASONING

READ ME Picture This Spatial Learning in math and science

These Grade 4 students have been studying multiplication. They build, explain and compare arrays to build a conceptual base that includes imagery for thinking in “units” not singles. Multiplying by 3 or 4 or 7 is a growth by a factor or unit of 3 or 4 or 7. Cuisenaire rods are an effective model for demonstrating multiplication because they are built as units. As student build with them they are challenged to see the growth in Units, not by ones.

Students build and match FINGERFOLDS to equation cards (answer included) so that they MATCH the correct FACT to the IMAGE.

FINGERFOLDS ARE a hand held reference tool to support understanding.

Once students “know” some of the facts they challenge each other to remember them.  The two students you hear are explaining to me how they understand the task of practicing.

PRACTICE SHOULD ENGAGE THE BODY, MIND AND SPIRIT. If you have to think, reason and manipulate things you are more likely to remember long term. That includes students actually being able to explain how the practice is set up and what it is accomplishing.

VISUALIZATION IN LEARNING MATHEMATICS HAS BEEN IGNORED FOR MUCH TOO LONG. Why do we make it so hard to learn when imagery makes it so easy to ENGAGE and then to REMEMBER.

Image

THIS IS PRACTICE that I have playfully termed: FingerFolds…. because you can manipulate the distributive property at your fingertips.

THERE IS not a lot of benefit in practicing something you do not understand.. You waste brain power trying to remember what to do… if you were involved in building the understanding of what to do…. then you are practicing something you understand over and over. Now if you practice that thinking in collaboration with a partner you have company and a little competition and a space for innovation and personal mastery.

Use visual artifacts,  PUZZLES and FINGERFOLDS to help students stay engaged in their practice. Students control the level of Differentiation

BERCS CARDS are all about VISUAL SPATIAL REASONING being embedded in the learning, experiencing and assessing.

 

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The cards are starter tools, they open the door to student driven investigations. Or can be used as REALTIME ASSESSMENTS

 

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Layer in the complexity.