# Equal Matters

## ASSESSING NUMBER SENSE: Equal Matters

Understanding the concept of equal and the role of the equal sign is one of the 4 foundational pillars upon which  number sense builds.   If our goal is to see students develop and master fluent, accurate and efficient strategies for solving addition and subtraction equations, we must include tasks that attend to developing a rich and robust understanding of what it means to be equal in math.

### When I heard him explain I saw this in my mind

To understand the inverse I can move forward or back between the 2 numbers. If I shift everything to the right I have constant difference. My purple arrows show that. Here is what I thought when he said it was all about difference.  Can students understand the connection between his explanation and my number line?

### How do you know?

Constant difference thinking relates to this type of ‘logic’ problem.

And these problems relate to what are classified as ‘systems of equations’ problems in high school and beyond.

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

## WOULD YOU ACCEPT THESE RESPONSES FROM A STUDENT WHO IS ASKED TO EXPLAIN MULTIPLICATION ?

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

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?

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)

# LET’S BE CLEAR….

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.

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?

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.

## 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?

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?

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?

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

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

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.

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?

# Basic Number Relationships….Is thinking your goal?

### 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?

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?”

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?”

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

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

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

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.

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.

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.

## Teaching early childhood mathematics as the subject of research is undervalued next to literacy.

#### (Linder et al, 2011).

This is a highly mathematical structure built by a Kindergartener. Do you know how to pull the mathematics out of this task in a meaningful way? Do you know how to evaluate his knowledge and skills?

### A body of research supports the realization that mathematical experiences, interactions and investigations in preschool & Kindergarten predict future schooling outcomes. But Edens and Potter (2012) identify a gap in research concerning “ways to provide opportunities to advance children’s early mathematical skills development.”

#### Bobis et al, 2005 go on to state : If preschool is so influential on future success, teachers need to focus on how to promote authentic mathematical learning in a holistic play based environment

Children need encouragement and opportunities to practice with teachers who understand the mathematics children are doing is VITAL. The teachers’ subject knowledge and confidence have influence on the development of children’s mathematical thinking.   Teachers have a huge significance in how we perceive learning outcomes.  (Anthony & Walshaw, 2007; Clements & Sarama, 2003). But the most frequent comments I hear from teachers during working sessions are:” I did not know this is what that outcome meant!”  “I wish I had been taught math for meaning.” “I had no idea this was so important in learning math.” ” I need to teach this much differently than I have been!”

Constance Kamii has written several articles that I have found quite useful. Take a read and see what you think.

#### Kamii Line Up the Fives

Both offer activities that have opened adult eyes to  “see” the assumptions we make about what is easy or hard in math.

Remember what is “first grade” in the Kamii article is based on American curriculum. Expectations in early grades are very different. Her tasks are informative.

Want to make a shift in your attention to numeracy ? Looking for ways to make the mathematics in your “games” and centres authentic? Hoping to make literacy connect to math in more learner friendly ways?

If you are not puzzling, you are not preparing learners for success in mathematics!

Consider joining me for the first 2 days of Summer Institutes 2018. Pre School, Kindergarten & Grade 1 teachers, we will dig deep into the foundations for mathematical thinking that our students are missing at all grades and how to address them in  “play- puzzle-  think”  ways.

Calgary August 7 & 8

Edmonton August 13 & 14

Grande Prairie August 20 & 21

# Effective Instruction is built on COMPARE

### Ways 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.

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.

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

## 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.

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.

## Part whole relationships abound

### Imagery linked to symbols prompts and sustains recall.

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.

BERCS 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.

# Ways of Knowing: Subtraction

## 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 a ChunkitZ puzzle. I see 3 and 2. That’s 5.

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

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 far 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.

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 STUDENTS 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 as 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. Understand 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.