# Odds and Evens

I went and asked a few Grade 5 students to tell me about odd and even numbers, and this is what they said: Student A: “3 is odd, 2 is even”  Student B:“anything ending in a 0, 2,4,6,8 is even and anything ending in a 1, 3, 5, 7, 9” is odd. The students told me attributes of even and odd numbers, but not one of them explained to me the mathematical meaning of the terms.

The big understanding is that even numbers can be shared fairly into two groups whereas odd numbers always have a left-over and can not be split into two equal groups. I asked a grade 3 student and she explained it well, “Even numbers can be separated into two parts, like the number 8 produces 2 groups of 4, but odd numbers can’t be separated into two equal parts”. I was impressed that she had retained that understanding! When I asked a different grade 3 student to explain what it means to be even, he said, “I dunno”!

It is very important that we teach for understanding. This understanding starts in kindergarten and there are lots of fun ways to get children enthused. The trick is to find resources that show the meaning of odds and evens first. Then you can supplement with songs and stories that are about even and odd, even if they only talk about the numbers and not the big idea behind them.

My top choices are to give the students a manipulative such as Numicon blocks, which can easily be made out of 10 frames, and ask them to sort them into two piles. Students will usually sort those with bumps and those without. This leads to a deeper discussion about the concept of even and odd.

My second favourite way is to use twins. The twins are happy when they share evenly, but they are mad when they can’t and there is a left-over. Once I explain why they are feeling that way, the students can now discover which numbers are even or odd themselves. Watch as Rory easily conceptualizes the idea of two equal parts or left-overs.

Math is a beautiful subject full of patterns and connections and it is my responsibility as a teacher and a parent to make some of these connections transparent while I teach. At the end of this lesson, we tied it all together, labelled the numbers as even or odd and made some observations about the patterns (skip counting by 2) and connections we found. The fact that even numbers always end in certain numbers never entered our conversation, although it will eventually, when we recognize it as an observable pattern that we can use to predict whether something can be shared equally into 2 groups or not.

The next day Rory and I reviewed the concept by using dot cards. I chose this task as a follow-up for two reasons: he gains practice subitizing while reinforcing his understanding of odd and even. The cards we used come from a game called “Tiny Polka Dot”, but you can easily make your own set.

Students that gain a thorough understanding of the meaning of odd and even, can take their learning further. Now students have a starting place for harder problems that develop fluency such as: what happens when you add two even numbers together? What happens with two odd? Or even further: can you ever get an odd number when you multiply two even numbers together? What about multiplying an even with an odd? If the students have a better understanding of what it means to be odd and even, they can make some observations,  discover some patterns, and further develop on their number sense journey.

# Coding for Kindergarten!

You can hardly get through the week without coding appearing as a headline in the news somewhere. All the big companies are getting involved with coding initiatives and all of the governments are supporting coding with their own endeavours. Here are a few examples:

When I was in school (a long…long time ago!), there was one course in computer programming and it was taught in Grade 11 and I’m pretty sure it was an elective, meaning you didn’t have to take it.  Today, instead of the one high school course, educational systems are trying to come up with a scope and sequence to make coding accessible to all grade levels. There is a push to get everyone to code. There is a push to get girls to code. And there is a push for teachers to incorporate coding into their lessons.

Why is there this big push? Well…

## COMPUTERS ARE EVERYWHERE!

Computers are automating the work-place; so we need to teach people how to programme them. There were as many as 7 million job openings in 2015 in occupations that required coding skills (Click here for source); so we need to ensure our youth are able to apply for these jobs.   Coding is our new literacy! Our national language should be English, French, and reading and writing code! Learning how to code will help kids navigate their future, understand their world and increase their odds of landing a job when they graduate.

But the benefits of learning how to code extend even further than that!

Steve Jobs said,  “I think everybody in this country should learn how to program a computer because it teaches you how to think.”

And I agree! Here are my top 10 ‘mathy’ reasons for teaching kids to code:

1. Coding teaches problem-solving skills
2. Coding teaches logical reasoning
3. Coding teaches spatial reasoning
4. Coding teaches you to check your work
5. Coding teaches you to correct your work (debug)
6. Coding teaches you critical thinking
7. Coding teaches math in an authentic real-world way
8. Coding teaches algorithms
9. Coding teaches coordinate geometry and translations
10. Coding teaches perseverance!

“The computational thinking involved in computer programming involves logic, organizing and analyzing data, and breaking a problem into smaller and more manageable parts. Much of this is also required when solving math problems!” Sri Ramakrishnan of Tynker

So what can you do? Well, at the pre-school level, kids don’t need screen time in order to learn how to code. There are tons of toys and games that support coding these days and you can use them to introduce your kids to coding language.

Here are a few of Rory and Oliver’s favourite games:

### Littlecodr (developed right here in B.C.!)

This game comes with specific challenges, although we’ve never used them. Instead, the kids have fun creating their own challenges such as: programme Rory to do a little dance, programme Mom to do a loop around the room, figuring out how far we can get Oliver to go using only 10 codes, and many more!

Oliver programming Rory to dance!

### Code and Go Robot Mouse

There are many different versions of coding robots out there: coding caterpillar, BeeBot, Dash and Dot or ours…the mouse. The mouse also comes with challenges and you can change the board shape around, or add more walls, or even have a rogue mouse and not use the board! Here we are integrating math by teaching the mouse how to count from 1-5. Rory found it challenging and eventually broke it into two parts: code for 1-3 and then code for 3-5.

I was introduced to this game at a pro-d recently and ran out and bought it. I love it for the extensions. You can start with simple commands (so Oliver can play) and you can choose whether or not to include harder challenges as your kids become more comfortable with coding. As a result, your kids can grow with it.

Oliver programming his turtle!

### Code.org

Ready for taking it on-line? Well code.org is amazing. I can’t believe it is free. I can’t believe how well it is laid out. I can’t believe how easy it is to learn how to code! They have lesson plans, tasks, lists of standards etc. Rory and Oliver loved it.

Rory and Oliver programming an angry bird!

Don’t be afraid of coding. I haven’t a clue what I’m doing! Remember, I only have that one computer programming course to back me up and it was in BASIC! But I’m learning along with my kids and loving the process! And more importantly….so are they!

Click here for a list of other coding programs that are good for education.

# Story time in math class!

I love teaching math with stories. Not only do I value the authentic nature of solving math problems from books, I love how quickly they can engage a whole class. I find it so rewarding when a child makes the connection between the story and the math involved. I taught with a book the other day in kindergarten and all of us laughed when one kid yelled out, “hey, this is just like math!”, not making the connection that I was there to teach math!

My pet peeve, is that sometimes it is so difficult to find a book that teaches what I want to teach, when I want to teach it,  in the way that I want to teach it!

I give you Exhibit A!

Even after exhaustive on-line searches, a plethora of librarian requests, asking all my mathy friends on twitter and begging Marilyn Burns to write another book…I still find it hard to get that picture book that would explain the concept I want, perfectly. I’ve even considered asking my friend, who says that she has always wanted to write a kids book, to help me! (Shar expect a phone-call!). I could tell her the concept, she could write the story, and we could get someone else to do the pictures so we don’t scare the children!

I give you Exhibit B!

This is a dog…in case you were wondering!

So when I do find the perfect book, I have to share it with everyone! Last week, I began a kindergarten class with the book called: “One is a snail Ten is a crab – A counting by feet book”, by April and Jeff Sayre and the kids loved it! I was getting so tired of all the books that predictably count up by one or down by one, and although this book does do that, it is a great book to use for showing different ways to compose numbers.

I give you Exhibit C!

One is a snail, ten is a crab by April and Jeff Sayre

I would read a page, and then ask the students so, “What is 6”. Here are the responses I got:

“An insect!”

“ Six snails!”

“ A dog and a person!”

And the really clever individual: “3 people!”

Without knowing it, I had generated a number talk and every time someone offered a different way of making 6, all the kids were amazed! What’s great about the book, is the authors mix up their ways of composing the number too! Sometimes it is just one thing (i.e. a spider), and other times it is a combination of things (i.e. six snails)! This is a great book for those kids who are ready to step beyond the predictable patterns normally found in books.

Here’s Rory and Oliver trying to figure out what comes next. They obviously need more exposure with crabs!

Next week, I am going to use the same book for grade one in order to demonstrate equivalence. “Oh – so a dog and a person (4+2) is the same as six snails (1+1+1+1+1+1).” Since our school just bought a whole bunch of cuisenaire rods, I’m going to have the students use cuisenaire rods to record their responses.

I give you Exhibit D! (In class, we will use the concrete rods and a white board to record our work).

Using cuisenaire rods to demonstrate equivalence.

If the teachers have time, I think we’ll extend it even further by having them make mobiles where one side is equal to the other side! I’m so excited! And what inspired me? A book!

Needless to say, I’m not the only one who appreciates this book. I stopped at page 10 (“ten is a crab”) for my K and 1 classes, but the book continues and could be used for many number concepts in K-2. Even Rory wanted to continue reading and as a result, got to demonstrate his new math milestone: counting by tens to one hundred!

If you search the net, you will find a lot of resources to help you use this book in your classroom as well.

Meanwhile, do you have other great books for the math classroom? Let me know in the comments section below!

# Digits versus Numbers

Rory was playing the Osmo Numbers Game (he was pretty spoiled by Santa this year!), and he was getting really frustrated because the game wanted him to make the number 12, and he kept doing 1 and 2 (instead of 10 and 2) and not getting it right. This led me to wonder how I could help him understand the difference between digits and their place in numbers.

Oliver learning numbers with Osmo.

Rory is in kindergarten and has no understanding of place value yet, although he should start learning about numbers greater than 10, in school soon. To help him realize that 12 is different from 1 and 2, he needs help conceptualizing the idea that one 10 is different from one 1.

“They must be able to conceptualize place value; the understanding that the same numeral represents different amounts depending on which position it is in.” Charlesworth, 2012.

Rory trying to make 14 with the digits 1 and a 4; then making 14 by adding 10 and some more.

Knowing the difference between digits and numbers is a developmental milestone and comes with a strong understanding of place value. I often have primary teachers ask, if I have any ideas on how to make this difference more apparent, and I do!

There are 3 main strategies you can use to develop this understanding:

### 1) Concrete representation of numbers

Use manipulatives to build the numbers. Showing the difference between 1 and 2 and 10 and 2 using a rekenrek, counters or unifix cubes clearly demonstrates that the digit in the ones position is different from the digit in the tens position. (If this doesn’t make sense, watch the video below!) Grouping or bundling things into groups of 10, or using base 10 blocks, is the best way for students to visualize the difference. Make sure you give lots of time for them to practice counting objects and grouping them into bundles of 10.

“A set of ten should figure prominently in the discussion of the teen numbers” Van de Walle, 2014.

### 2) Patterns

Write the numbers vertically and ask the students to notice any patterns they see. You will be surprised at your student’s or child’s ability to see that numbers repeat from 0-9, or that all the teen numbers have a one in front of it. Bring in the 100 chart! Now you’ve got a starting place for inquiry…why do they all have a one in front? Your goal is to have students come away with the understanding that digits mean different things when they are in different places.

In order to build number sense, you want students to think flexibly about numbers. In other words, can they decompose a number and can they do it in more than one way? When introducing numbers greater than 10, you want your child to decompose them into 10 and some more. This is another reason I love the rekenrek, for its amazing ability to show a number as 10 and some more.

“Mapping the teens number names to a ten and one structure is an important idea.” Van de Walle, 2014.

Counters are great too because they give students practice creating that group of 10: count out 13 ones, but if you group your tens…you have one ten and 3 more. Representing this as an addition sentence is another great connection for your kids to make and is the beginning of learning signs and symbols to represent math problems.

Here is a video of Rory and I learning about numbers greater than 10. Watching it back, I would have done a few things differently (like have my husband look after Oliver!), but I still think it gives you an idea of the 3 strategies in action (concrete manipulatives, patterns and addition)!

Do you have other suggestions of how to build the understanding that digits are different from numbers? I’d love to hear them! Please share them in the comments section below.

# Representing numbers

Rory was writing out numbers the other day and some of them were backward and all I wanted to do was tell him that they were backward and that he should fix them. Instead, I bit my tongue, and silently thought “positive feedback, positive feedback” in my head and then said, “Good job, I see you’ve written the numbers in order from 1-10!”. The experience reminded me of an earlier encounter in a kindergarten classroom, where I was helping a student who was working on the number 3. I forced him to change his backward 3 around, even though he was protesting in frustration. He eventually got it, but from that moment on, I wondered…how important is it for a kindergartener to know how to write numbers symbolically?

Rory writes the numbers 1-10

My conclusion…it’s not! At least not in the early grades! In fact, I don’t even know whether it qualifies as a math skill! I tried my hardest to find evidence to the contrary, but I couldn’t even find one educational math article on how to teach the skill. To further support my claim, if you google ‘how to write numbers’, the first thing that pops up are grammar articles.

So why do we put so much importance into learning how to write numbers? Well, the digits are symbols that are a great way to communicate mathematically. It is a universal language and, once learned, speeds up our representation of problems. As students mature, they graduate from using concrete objects to pictorial representations and eventually to abstract symbols. Knowing how to write numbers is a great way to show one’s thinking in a problem. That being said, I still don’t feel that the ability to write numbers properly, should be assessed as a math skill.

“Children should eventually be encouraged to connect their drawings to symbols, but they should not be forced to do so too soon.” Van de Walle

What is more important, especially in the early years,  is how a child internalizes the number they are learning. What does 5 represent? Can they show it using different objects? Can they show different ways of making 5 (i.e. 2 and 3 or 1 and 4)? Can they visualize it on a number line?

Here’s a video of Oliver (pre-school) to see what he makes of the number 5.

“Models or representations, whether they are conventional or not, give learning something with which they can explore, reason, and communicate as they engage in problem-based tasks.” Van de Walle

Notice even Van de Walle minimizes the importance of conventional number writing. Instead he places the value on any representation and its use in problem-based tasks. I felt very proud of Oliver in his ability to represent the 5 objects in his own way. He has demonstrated that he can represent 5 using concrete objects and pictorial representations. He is using models and making sense of the number; learning how to write the digit 5 does not need to be rushed.

So what can you do to help develop your child’s understanding of number symbols? For sure, you should still teach them to recognize the numbers orally, and visually. Of course you should encourage them to write it in standard form. But, concentrate on the important stuff: expose students to the number by using problem-based tasks. Encourage them to discover the meaning of each number, by coming up with their own way of representing the number. Allow the use of manipulatives to model the number and allow children to choose their own representation to model their thinking. Surround the learning of numbers with real-life scenarios. Most importantly, include opportunities to problem-solve and decompose the number while they are learning about it. For example, how many ways can 5 people be on a bunk-bed? Or how many girls or boys could we sit at this table to have 5 in total? Always ask, “who found a different way?” to encourage the sharing of ideas.

Here’s an example of Rory learning about the number 5 using a real-life problem. He is still being asked to represent the problem, but I’m allowing him to choose the method (drawing or symbols) to show his thinking. If he had chosen to represent it with pictures instead, I would have been okay with that; however, he is comfortable with writing numbers. I could even extend him by introducing him to equations at this point, but the purpose of this lesson was to think flexibly about the number 5 and so we concentrated on that.

Now…fingers crossed I get that cat so we can use the 5th stocking!

Happy holidays everyone!

# Full STEAM (Science, Technology, Engineering, Arts and Math) ahead!

THE DISCOVERY

This past summer,  I bought Rory a YOXO Helicopter construction toy (for only \$10 at Chapters!). As soon as we got home, he was dying to dig into it, but I said no (…after watching  Shonda Rhimes’ TED Talk  I might have answered differently)! I wanted him to wait until I had time to help him with it; after all, he was only 4 and a half and couldn’t possibly make a helicopter all by himself!

When we finally did open it, I realized I needn’t have bothered; it is designed so that little people can intuitively build things on their own! Rory looked at the pictures and quickly figured out how to build the helicopter all by himself. He dismissed me immediately and insisted that he didn’t need (nor want!) my help. “I can do it myself!!!!”  resonated throughout the house…and the yard…pretty sure even the neighbours up the street heard his proclamations!

After building the helicopter, he proceeded to tear it apart so he could create something else. This continued until we finally had to put the toy away after he pummeled his brother for trying to destroy his new inventions!

YIPPEE FOR YOXO!

If you’re looking for something to do over the holidays, THIS IS IT!! YOXO is an amazing first step to engaging your child (boy or girl) with STEM. Children gain experience engineering their own creation using the material provided, or by improvising with their own. Unlike Lego, that can only fit other Lego pieces, YOXO is built to incorporate many items – boxes, toilet rolls, even Lego bases can fit into the hatch marks on a YOXO piece. As a result, the possibilities are endless and there is no limit to what can be engineered.

It is appropriate for all age groups. Oliver, only 3, was proud of his creations. Rory, almost 5, wanted more challenge. The colour-coding helped when my 3 year old wanted to follow the instructions, while my older son used the numbers to follow the steps. Other age groups may skip the instructions all together (hmmm…sounds like my husband)!

EXPERIENTIAL STEM

STEM (science, technology, engineering and math) or STEAM (includes arts in the acronym) by definition, is experiential; it is taking those subjects, integrating them and then applying them in a meaningful way. If you google STEM, you will find a plethora of lessons or materials that are pushing STEM education right now. The idea of a transdisciplinary curriculum that weaves subjects seamlessly together is not new;  for years, educators have recognized the benefits of an integrated curriculum.

What is new, is the finding that the majority of the jobs in the future will be in STEM fields, so whatever we can do now to prepare our students for then is important.

“Employment in occupations related to STEM—science, technology, engineering, and mathematics—is projected to grow to more than 9 million between 2012 and 2022. That’s an increase of about 1 million jobs over 2012 employment levels.” Dennis Viloria, 2014

To me, STEM is more than the sum of its parts, it’s a way of thinking.  At a STEM conference, my colleagues and I quickly realized that we wished the acronym STEM didn’t exist because it turns away all those kids with aversions to science or math. We thought a more appropriate name should be DESIGN THINKING. Regardless of what you want to call it, there are lots of things you can do to aid in its development and get children excited about it, even in the early years. YOXO is just one way.

Here are a few other suggestions, so you too can move: Full STEAM ahead!

1. Nature!!!!
2. Bloco
3. Lego
4. Little bits
5. Transport toys from Battat
6. Magnatiles
7. Maker space
8. Design Challenges
9. Tinker toys
10. Clipo by Playskool or Funskool

Have more suggestions? Leave them in the comments section!

# All things are not created equal!

All things are not created equal…including the understanding of the equals sign! Did you know that Americans have a very poor understanding of the equals sign, especially when compared to students in other countries?

“Ninety-eight percent of the Chinese sample correctly answered 4 items indicating conceptions of equality and provided conceptually accurate explanations. In contrast, only 28% of the U.S. sample performed at this level.” (Li, Ding, Capraro, & Capraro 2008)

Students on our continent tend to see the equals sign as meaning ‘the answer to’. In other words, the answer to 2+3 is 5.

What’s wrong with that you ask? The true meaning of the equals sign is as a symbol that indicates equivalence; it does not mean ‘an answer’. It means that one side is equivalent to the other. The equals sign acts as a balance in that the two sides must balance; it represents sameness as in the same amount, but not necessarily using the same things.

Still don’t see the difference? You’re not alone! Most of our students don’t understand the difference either. When students think that the equals sign means ‘the answer’, this is what happens:

Want another example? How many of you have written a problem like this on the black-board: 10 = 6+4 and had your kids tell you “that’s wrong!” because you wrote it backwards? If so…you have some work to do!

But don’t fret! You can quickly transform your students’ thinking by incorporating a few new habits into your routine.

Experiencing equivalence using a balance scale!

Rory quickly and easily conceptualized the true meaning of the equals sign using this home-made balance scale. I was surprised at how accurately it worked too! It was also a great model to demonstrate equivalence such as: 4+4 is the same as 5+3; and Rory was very able to prove their equivalence to me, by rearranging the towers of 4 +4 to look like 5+3, with unifix cubes. In later lessons, I would use the balance scale to ensure concrete understanding of the other relationship symbols, ‘less than’ and ‘greater than’, as well.

Here’s what you can do:

• Most importantly: Use your words carefully! When reading equations, reinforce the idea of equivalence by reading the equal sign as ‘is the same as’ or ‘is equivalent to’. Don’t ask for the answer to a number sentence, ask for what it is equivalent to.
• Write number sentences backwards and forwards. In other words, alternate which side has the operation to be performed. For example, write 9 = 5+4; don’t always show just 5+4=9.
• Give questions that ask children to find equivalent expressions, not just questions with one number answers. For example,  5 + 4 = ? + 1 instead of 5+4 = ?.
• Reinforce the commutative property: 3 + 2 = 2 + 3.
• Cuisinairre rods! These are great for finding and showing equivalent representations of expressions.
• Use a balance scale! The more concrete and real you can make the understanding, the better. And not just for early grades, upper levels appreciate the visual as well.
• Give experiences with true AND false number sentences.
• Watch for textbooks and worksheets that don’t promote this way of thinking. The study mentioned earlier, blames a lot of the misinterpretation of the equals sign on North American textbooks.

Building the understanding of the equals sign, as a relationship symbol, starts in kindergarten and therefore the earlier you can promote the proper understanding of it, the better. The equal sign is the primary symbol used to understand relationships in our number system. Understanding its meaning promotes algebraic reasoning and gives students access to powerful relationships for working with numbers. With a thorough comprehension of the equal sign, other representations, such as the symbols for  ‘less than’ and ‘greater than’, make more sense. And that’s our goal! To make more sense!

Stay balanced!

# What the heck is a rekenrek?

Never heard of a rekenrek? You’re not alone, but I’m here to tell you, it is one of the best manipulatives for developing number sense in early primary, and you can make it yourself! Originally developed in Holland, it looks like a mini-abacus, but functions quite differently. Although you can have many rows in a rekenrek, it usually comes with just 2 as seen below.

There’s an app for that!

Why do I think it is the best manipulative out there? Well, let’s start with pre-school. It is a great instrument for working on one-to-one correspondence, counting and cardinality. Watch how Oliver (3 years old) easily subitizes a group of 5 on his first try with a rekenrek!

Not convinced yet? This math model offers teaching opportunities ranging from pre-school all the way to grade 2! Here’s a list of the other math milestones you can use it for:

• Counting strategies: count all, count on, see groups.
• Visualize numbers 1-20.
• Develop benchmarks of 5 and 10.
• Commutative property.
• Number conservation.
• Decompose and recompose numbers.
• Subitizing.
• Math facts to 20.
• Visualize doubles and doubles plus one, one or two more and one or two less .

Here is a video of Rory and I exploring some of these things on our home-made rekenrek.

Are you still thinking to yourself: “I can do all of this using 10 frames or unifix cubes” so what’s the big deal?” I’m definitely not saying: ditch the others; children should be exposed to many different manipulatives, but here’s a few reasons why rekenreks should be one of your first choices:

Rekenreks are concrete. There is often this push to move students to think abstractly too soon despite Piaget’s theory which states: children may remain in the pre-operational stage, and thus not able to mentally manipulate information, until the age of 7! Rekenreks are a concrete model that children can use to do their work, to communicate about their work, and to assess their work. Worksheets should be a method to RECORD their work. If you move too quickly to worksheets without giving the developmentally-appropriate materials to complete them, you may be hindering the development of true understanding. Rekenreks help develop the true understanding at a developmentally appropriate level.

Rekenreks have a 5 structure instead of a 10 structure; as a result this is a useful first step to learning place value. Young children find unitizing a group of 10 (seeing a group of 10 as one ten) difficult and this is what place value is. A rekenrek helps children unitize a group of 5 first, and then 10. In other words, it establishes those major benchmarks of 5 and 10, which will aid in subitizing skills (knowing the number without counting) and later in unitizing skills (seeing a group of 10 as one 10).

Rekenreks allow students to count in groups. When using unifix cubes or counters, children are manipulating one block or counter at a time in order to put them together. With the rekenrek, children can move a group of objects (i.e. 4) at one time. This means moving from a manipulative that requires counting all (counters), to one that encourages counting groups.

Rekenreks help in the development of pattern recognition. Pattern recognition is the basis of number sense and its development begins in the early years. Identifying patterns in pre-school leads to pattern recognition with numbers, which ultimately leads to greater number sense all around. By seeing what 5 is on a rekenrek, and then 5 and 1 more, and then 5 and 2 more, children begin to ‘see’ these patterns in their mind and thus master math facts. This leads to the next point…

Rekenreks help develop automaticity with basic math facts. This is the ability to produce answers in a few seconds by relying on thinking of the relationships among the operations rather than recalling answers. Because rekenreks support subitizing skills, they help kids achieve mastery of addition facts. Students begin to ‘see’ the answer, and don’t have to calculate the answer.

Rekenreks reinforce  the different relationships between addition and subtraction. There are three categories of problem structures for these relationships: change problems (join and separate), part-part-whole problems and compare problems (how many more or less). The rekenrek is an excellent manipulative to use to model all of those structures using rich real-world problems to support them.

Rekenreks allow students to think flexibly about numbers and construct their own strategies. They are an ideal tool for number talks. Starting a class off by saying: I made the number 11, can you guess how I did it, opens up the floor to hear and discuss alternate ways of making 11. For example, I made doubles of 5 and 5 and then added one more or I used 10 on the top and added one more on the bottom. “Not all children invent their own strategies. So strategies invented by class members are shared, explored and tried out by others.” Van de Walle.

Still a non-believer? Well believe this: evidence abounds!

Well…actually it doesn’t… but what little research I could find showed that there definitely was a difference between children taught with a rekenrek and those taught without.

“Results indicated that students in Group 1 (who were taught with the rekenrek) scored significantly higher on an addition and subtraction test with numbers from zero to 20 than did students in either Group 2 (taught without the rekenrek) or 3 (not taught).” Tournaki, N., Bae, Y., & Kerekes, J. (2008).

So what are you waiting for?! Start using rekenreks in your home or classroom today!

#### Why bother?!

In this age of technology, do we still need to learn number facts? Absolutely! Thinking flexibly about numbers opens up thinking for more complex mathematical problems. Mental math is also a survival mechanism when you are out in the real-world! Think of how often you use it: How many km am I going over the speed limit?! Do I have enough money to pay for the groceries? How many points is Hillary Clinton up by?!

#### How do you teach mental math?

Traditionally learning math facts has always been focussed on drill, drill and more drill. In recent years, however; teachers are straying from this model and realizing that drill is not as effective as other methods for ensuring that deep understanding and number sense.

“Posing a story problem…followed by a brief discussion of the strategies that children used, can improve children’s accuracy and efficiency with basic facts.” Rathmell, Leutzinger, & Gabriele, 2000

In fact (no pun intended!), it has been found that these children actually master more facts than children in a drill program! In addition to story problems, number talks that expose children to multiple strategies goes much further than drill alone. A number talk is when you pose a question (ideally in a story problem) and engage the students in a dialogue about how they solved it.

Playing games has also been shown to increase students’ recall and accelerate their understanding of facts. This is only true, however; if the use of reasoning strategies is explicitly built into the games and reinforced through interactions with others. So, if you are a fan of worksheets, drills, timed-tests or Kumon – hopefully this post will change your mind!

It may interest you to know that there is a huge difference between the way we present facts in North America, compared to China. In China they teach fact tables (for example, the 6 fact tables are all the facts with 6 as an addend: 6+1, 6+2, 6+3 etc.). In America, we teach fact families (for example all the ways to make the sum of 12: 1+11, 2+10, 3+9 etc.). The result? Using fact tables, there are 81 facts to learn (although if you know the commutative property that can be reduced to only 45!!!); using fact families, there are a whooping 153!! (W. Sun and J. Zhang, 2001) So I encourage you to rethink the way you present facts in the early years. Your child may thank you 153 minus 45 times!

Although students should be exposed to multiple strategies, it is important that they are allowed to invent and use their own strategy and not be pigeon-holed into a teacher-chosen one. That being said, there is usually one strategy that is more preferable to use for the given scenario. Discussions about efficiency should take up a lot of your time. You’ll find children are quick to realize that one strategy is better than another and your job is to make the opportunities for those discussions to happen.

The main mental math strategies are:

• Counting all
• Counting on
• One more or one less
• Two more or two less
• Making 10
• Doubles and near doubles

Does order matter? Kind of….the first few methods are counting skills so children should find those mental math skills easier to master. The next few are reasoning methods which require a lot more higher thinking. In order to master those, students must be comfortable counting on and back as well as familiar with composing and decomposing numbers. At school, we just started a unit with making 10’s, but only because they had just finished place value and it seemed like a natural progression. But even within that strategy, we can make it easier or more difficult for students that need it. To make it easier – start with the 9 fact table (easy to make 10 by counting one more). To make it harder – start with the 6 fact table (how many more to bridge 10?).

The important things to remember? Ensure repeat exposure with each strategy and have patience! “It can take between 2-4 lessons before most students really internalize the reasoning strategies discussed in class.” Steinberg 1985

#### Isn’t using manipulatives with mental math cheating?

I like to call it reinforcing! Child development naturally progresses from concrete to pictorial to abstract, so do that! Help build the mental picture in their minds by using concrete manipulatives. Here are some good ones to use for facts to 18:

• Five and Ten frames
• Rekenreks!
• Abacus
• Base 10 Blocks
• Number lines
• 100 chart
• And one of the favourites! Fingers!  Read an interesting article HERE on the importance of fingers and mapping within the brain!

Here’s a video of me introducing addition facts to Rory using 5 frames. My first check is to see whether he knows what addition is; then I follow to see whether he has the ability to count on. If he doesn’t, then the only mental math strategy accessible to him at the moment will be counting all. I was super surprised and excited when Rory automatically made groups of 5 with his 5-frames and even more impressed when he made the group with the larger number (i.e. 4 instead of 3), especially since he has never seen this manipulative before. This just shows the power of a manipulative in learning addition strategies.

#### Keep in mind...

Kids need continual practice throughout the year. Look for ways to incorporate mental math into your day naturally. Even exceptional students benefit from conversations about the efficiency of each strategy. For example, when should we use bridging 10 instead of making doubles?

Can you expect your 4-year-old to master the facts? Not usually and not with full understanding. The ability to reason and achieve full mastery for addition and subtraction facts up to 18, usually occurs in Grade 2, but it is different for each child.

What can I do with Rory at the kindergarten level? I can expose him to problems of joining and separating and teach him the meaning of the words add (join) and subtract (separate). I cannot teach him reasoning strategies until he has the ability to count on and is developmentally ready, but I can see where he’s at. In Grade 1, I can use 5 frames and rekenreks to help him develop his facts up to 10 and help him achieve mastery of those before entering grade 2. And then in grade 2, expose him to different strategies so that he can think flexibly and easily, mentally.

What are number bonds? Number bonds are just another way to visualize addition facts and are a huge component of Singapore math. What is Singapore math? It is a collection of computation strategies that arose out of Singapore, where students are supposedly ranked among the best in the world in math achievement. So are number bonds all they are cracked up to be? I think they might be! But I’ll let you decide. First, here is a video of Rory discovering number bonds for the first time so you’ll know what I’m talking about.

#### THE GOOD…

“Emphasizing number relationships is key to helping children fully develop number sense.” Van de Walle

And the most important relationship to develop? Part-part-whole. Number bonds are a great visual to see the part-part-whole relationship.  Focusing on a quantity in terms of its parts is a major milestone for young children and number bonds can help them get there. Just make sure you keep the big idea in mind by consistently using the “part-part-whole” terminology.

The building block of number sense is to think of numbers flexibly. Number bonds help develop number sense by showing different ways to decompose and recompose numbers. By showing different number bonds for one whole number, children see multiple ways of making (or unmaking!) a number. 5 can be made from 2 and 3, or by 4 and 1. Number bonds show this connection well.

However, what is my favourite attribute of number bonds? Number bonds are a great way to teach addition and subtraction at the same time. Rory and I continued our lesson with questions such as: “With 5 whole cars, if you had 4, how many did Philian get?” (one). Number bonds allow children to link the inverse operations easily so that they develop the two skills together. This creates fluency in both operations at the same time, and not a weakness in subtraction, which we often see when the two operations are taught separately.

“Primary teachers have the tendency to rely too heavily on textbooks, workbooks and photocopied support materials.” Charlesworth

Number bonds lend too easily to this. It is too easy to photocopy a bunch of circles and have the students fill it in with little attention to problem solving. As a result, students are not given the chance to discover the meaning of the relationships on their own. Instead, why not have the students choose their number and discover the number bonds that connect it? Or why not use number bonds as a method to record their work for a problem, but not as the problem itself?

Number bonds shouldn’t be presented independently; use manipulatives to make it real. Adding is putting together groups of objects to find out how many, and students need practice actually doing this with concrete objects. Notice with Rory, I don’t start with the number bond visual – I start with the concrete manipulative to mimic a real life situation. I use that to build the number bond and whenever Rory got stuck, where did he go? He referred back to his concrete manipulatives: the cars. Number bonds help you to see the two parts, but manipulatives make the two parts real.

“Research has demonstrated that when kindergarten and first-grade children are regularly asked to solve word problems, not only do they develop a collection of number relationships, but they also learn addition and subtraction facts based on these relationships.” Van de Walle

Number bonds should not be presented first and then problem-solving second. Instead, allow students to discover the facts for themselves and then use number bonds to make sense of their work. Rory was able to solve the problem for himself and will have a deeper understanding of ways to make 5. Using real problems makes the learning more engaging for the child, especially if the problem involves them. And as the research shows, it also helps them achieve mastery as well!

#### THE UGLY…

You might be tempted to use number bonds as flashcards for a drill or a timed test or prolonged practice. Number bonds used as drill creates anxiety and stress and doesn’t encourage an understanding of the part-part-whole relationship. Children should learn facts through discovering patterns and relationships. By focusing on families of facts and their relationships, in a problem-solving environment, you are encouraging mastery of facts through exposure. Rory quickly noticed the patterns that make 5 – as one part gets smaller, the other gets bigger. He giggled when he saw that 3+2 is the same as 2+3, but he was discovering these relationships under the guise of a real problem. There was no anxiety or pressure for him to memorize number bonds.

Some kids aren’t going to be ready for number bonds because they are too abstract. To aid with this, start with 5 or 10 frames and concrete manipulatives to help them see the facts more easily. Keep the manipulatives going and don’t switch to the abstract (pencil and paper) until much later. Allow the child to determine the pace of the learning….as long as you are providing opportunities for them to engage in problems that reinforce the facts, you’re good!, You will know mastery is achieved when they don’t have to count how many are in each group, they just know and this typically doesn’t happen until 3rd grade!

#### THE BOTTOM LINE…

So are number bonds good, bad or ugly? Well, you always need to keep in mind your big idea…you are trying to build number fluency which involves efficiency, accuracy and flexibility. Number bonds are another tool in your toolkit to help students visualize part-part-whole relationships. As long as you remember to focus on those relationships and surround the students learning in problem-solving and not drill, then number bonds are a great resource. Just beware of what could make them bad or ugly as well.

What does Rory think? He loved creating them so much, he carried on and made his own number bonds independently afterwards…some more abstract than others!

Have you had any experience with number bonds: good, bad or ugly?! I’d love to hear about it! Tell me in the comments section below!