MY PREFERRED MODELS FOR DECIMALS include areas that can be folded. That includes strips and rectangles. Number lines that can be partitioned into smaller and smaller equal parts.
Decimals are fractions. ****Read them as fractions. ****Build them by dividing wholes into equal parts. *****Hundredths are in every tenth. Thousandths are in hundredths are in tenths.
NUMBER IS IN NUMBER
This can be quite evident if students are expected to divide tenths into hundredths in area models and on number lines. This can be quicly lost if students are allowed to create a new set of models for the next unit of decimals. For example in the pictures below. The pink balls that represent tenths are separate from the pink balls that represent hundredths. Whereas in an area model or linear model, the decimal parts can be seen to be inside each other. DENSITY
The models we use to represent concepts in mathematics exist in space. They prompt visual spatial reasoning. For ease of discussion, I loosely sort models into the following categories:
Linear Area Volume
The key word here is loosely. I am simply trying to begin a conversation with others that will help us build some common understandings.
One of the things I study is the impact that a specific category of models may have on the development of the idea at hand. For example, if I am talking 2 dimensional shapes: quadrilaterals, triangles, multi-sided, regular and irregular polygons but I am holding a prism, what affect will my model have on the thinking and reasoning my students are able to generate from the object?
How will/do their perceptions and understandings change if they hold the object instead of me? How will/do their perceptions and understandings change if they place the object on a surface and trace the face I am describing since I am discussing 2 dimensional shapes?
The answer, significantly!
If we want our students to experience higher success, higher achievement in mathematics, then we must engage them physically in the learning.
Stop showing and telling and expect them to show and tell.
NO one model on its own is ever enough. All models break down somewhere. That is why we must COMPARE, COMPARE, COMPARE. Teach students to analyze: How are these the same? How are they different?
I love that it was all on Zoom. When I had to miss a session, I could still watch it later. I have been pulling out the tapes and re watching them this year. Lots of quick ideas, right when I need them. I will definitely be back for Summer 2021.
Number Sense frees students to think as they learn. Number facts become automatic, not rote chants. A student with number sense does not resort to counting by ones.
Measurement Sense underlies estimation and reasoning skills. Measuring tasks are hands on-minds on and develop quick recall that links to sucess with multiplication & division.
Literacy & Numeracy are linked. That is why this summer we will focus on planning regular practice with math tasks that also build reading and comprehension fluency. This is not just about word problems. This is about building vocabulary, reading and writing skills alongside key math skills.
Here is an interesting visual I found for a problem posted in the Balanced Assessment Project.
Consider how your students might view and understand this repeating pattern. From Kindergarten to Grade 3, how could you use this pattern to develop an understanding of elements, core, repetition.
Given dot paper are they able to reproduce a section of the pattern? All of the pattern?
Are they able to complete the missing section of the pattern?
Are they able to extend the pattern? both forward and backwards?
In the original problem students are asked to work with $25, $100, $200. These are nice mental mathematics numbers for working on fluency in Grades 3 and up. For Grade 3 we might wait until mid year, but for Grade 4 and up this would make a nice opener to building your approach to problem solving alongside checking on mental fluency with these common sense numbers at the start of the year.
HOW DO YOU BUILD RECALL SKILLS WITH YOUR STUDENTS??
Hear the confidence? Hear the enthusiasm for learning?
They have been working with BERCS materials to build mental fluency with 2 and 3 digit numbers. BERCS materials build multiplicative reasoning, alongside fact recall.
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.
CONFIDENCE BUILDS ENGAGEMENT.
ENGAGEMENT facilitates MORE LEARNING.
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:
BUILD EXPLAIN REPRESENT COMPARE
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.
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.
Why do I want this skill? Understanding equal is a skill that forms the foundation for future success in school math. The equal sign does not mean ‘here comes the answer’. We need the equal sign to trigger thinking about relationships so that when students meet equations like 0.200 = 0.2 or
2x + y = y + 2x they are not looking to complete an operation or get an answer.
There are 2 ideas embedded in the kind of thinking this young student is practicing. Automatic recognition of quantity: I just know the quantity each expression represents so I know this is equal.
Balance thinking: I see how the parts are representing the same quantity (but not thinking about what they add to).
The act of explaining out loud develops working memory & recall skills. She is practicing holding parts in her mind. She is developing confidence by explaining out loud.
This next clip shows fluency. He just knows what is missing. He was going to draw it out somehow but I interrupted by asking him if he just knew it. If we want fluency, we must prompt for it. But I am not sure what he understands about equal. He refers to balance but does he really mean balance on each side or does he mean just solve the missing piece?
The boys in this next clip demonstrate another aspect of understanding equal.
The boy talking “balances” the equation by adding to get to “friendlier” numbers. He knows that you need to change both numbers by the same amount and automatically knows what that same amount is (53). Watch
He says take- away is just a difference. I would prefer to hear subtraction. All subtractions are about differences. ‘Take away’ creates a faulty imagery. All subtractions are not taking away. But all subtractions are comparisons.
Does he understand how inverse and differences are connected? Does he actually demonstrate an understanding of difference on the number line? or is he more focused on the computation than on the relationship?
I like what I hear and see but is it robust enough? Does he understand the addition & subtraction relationship or is he only thinking in one direction. I ask because it will matter once we move to algebra. So I want to see and hear what he has to say if I ask him to prove with the inverse.
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?
I would like to ask:
What are 2 more numbers that have a difference of 488?
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.
The final check to know if students truly understand a concept:
Write a similar problem.
How are these problems similar? Are they similar to the original?
Are they similar to each other? Are they correct? Do they both work?
What do your students think?
CONSIDERING WAYS TO ASSESS NUMBER SENSE AND HOW TO USE THOSE ASSESSMENTS TO FURTHER STUDENT LEARNING is the goal of my work.
Find out more by participating in a Summer Numeracy Event. Studying student thinking will form a big part of our study.
WOULD YOU ACCEPT THESE RESPONSES FROM A STUDENT WHO IS ASKED TO EXPLAIN MULTIPLICATION ?
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.
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 shareout 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.
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?
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.
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.”
“And what about this?”
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 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.
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).
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.
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?
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.