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.

sort

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.

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!

book-req

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!

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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!

onesnail

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).

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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.

Click here  for some suggestions!

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

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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.

osmo1

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.

osmo2

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.

3) Addition

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.

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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?

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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!

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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.

 

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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.
  • Addition – subtraction relationships.
  • 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!

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The Facts about Facts!

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!

Addition strategies

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.

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The Importance of Place Value in Primary

The importance of place value

I always start the school year with number sense. Why? Because it gives me a good indication of each student’s comfort with numbers. Number sense is the one strand that is always interwoven into other strands in mathematics. It is also the most telling predictor of a child’s success in math. If they have a good number sense – they can work flexibly and easily with numbers in most situations. If they don’t, I will have an idea of how to construct the rest of my lessons so that those students finish the year feeling more confident with numbers.

So what concept do most teachers start with? Place value. Place value is one of the hardest yet most important skills for primary students to master.

“Place value is the understanding that the same numeral represents different amounts depending on which position it is in.” (Charlesworth, 2012)

“The most critical period in this development occurs in grades pre-K to 2” (Van de Walle, 2014).

The understanding of place value follows this progression:

Level 1:        Initial concept of 10 (see 10 as ten units).

Level 2:        Intermediate concept of 10 (see 10 as a unit of ten ones but rely on
physical or mental models to do so)

Level 3:        Facile concept of 10 (easily work with units of 10 without the use of
                      models)

Place value development carries on through the elementary years by increasing the place value to build bigger numbers. But it is more than just knowing how to read big numbers; it is crucial to the learning of trading rules that underlie whole number operations. As a result, it is the most important building block for number sense.

Sadly, all too many children have a rote understanding of place value without truly understanding what different place values actually mean. They are not able to go back and forth between equivalent representations of the same number (i.e. 31 could be three groups of 10 and one 1or it could be 2 groups of 10 and 11 ones).

Introducing place value

Rory has just started kindergarten (my baby!!!!) and he has no understanding of place value (and nor should he!). He does have a good understanding of conservation of number though, meaning he understands that different arrangements of counters still mean the same number. To introduce him to place value, I did the following:

1)   I asked him which was larger 63 or 36 to see if he has any understanding of what he’s counting or place value. He didn’t. Interviewing children is a great way to gauge their ability and find out what they’re thinking and why.

2)   I gave him a bunch of unifix cubes (a great model to start with!) and asked him to count as high as he could. This is a great test for cardinality and one-to-one correspondence. Both he and I were surprised that he made it to 39 without any assistance!

3)   I asked him to now group the cubes into groups of 10 which he did easily. He quickly noticed if a group of 10 was incomplete (the benefit of a proportional model!).

4)   We finished by noticing that the number of groups of 10 corresponds to the place value in the number. This went way over his head, but I felt better for saying it!

If this were a classroom lesson, I would complete it with a comparison of one ten tower to one cube and engage the students in a discussion of what they notice.

Do’s and Don’ts for Teaching Place Value

So how do you teach something this important? Here are some do’s and don’ts

DO:

  • DO use proportional manipulatives (the hundreds should be bigger than the tens which should be bigger than the ones)
  • DO use many different manipulatives so the child doesn’t think PV can only be represented with one type (i.e. base 10 blocks, unifix cubes, paperclips, counters)
  • DO integrate PV teaching with estimation and computation tasks
  • DO integrate PV teaching with measurement tasks (and use different units to measure!)
  • DO work on PV skills throughout the school year (not just in September!)
  • DO question and assess students constantly to be sure they are really understanding the concepts and not just answering in a rote manner
  • DO engage students in composing and decomposing #’s in a wide variety of ways
  • DO allow children to invent their own computation strategies

DON’T:

  • DON’T give worksheets which encourage a rote understanding not a concrete understanding (i.e 6 tens and 4 ones is 64 in all.)
  • DON’T teach PV abstractly, tie it to a manipulative
  • DON’T rush into operations involving regrouping tasks i.e. borrowing and carrying (mental math rarely involves regrouping as a strategy) and you run the risk of students not conceptualizing place value
  • DON’T start computation with numbers that don’t require regrouping (i.e. 23 +45); otherwise students will think that 56 + 35 = 811
  • DON’T rely on one type of manipulative (i.e. base 10 blocks) or children will memorize that a rod is ten and a small cube is one without actually understanding how many ones make a rod
  • DON’T teach PV in isolation, make it rich by integrating it with other subjects (see the link below for suggestions)

A true understanding of place value doesn’t usually happen until the concrete stage of development (7-8 years old), but it doesn’t mean you shouldn’t provide lots of opportunities for its development. What can you do? COUNT EVERYTHING!!!!

Want more suggestions? Click here!

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Learning to count…baby steps.

Oliver is 2 ½ and quite pleased with his ability to count. He can count forwards and backwards (and sometimes correctly!), but he has no understanding of what the numbers mean, although he does know that numbers relate to quantity. Rory on the other hand is 4 ½ and he can count correctly and accurately up to 10 and sometimes even higher. Oliver can count rotely whereas Rory can count rationally. Here’s a video to show you what I mean:

Learning to count happens in four stages:

Step 1: Number sequence

Between 2-3 years, children are able to recite the numbers in order. This rote counting is done without any understanding of how many things are actually in a set.

Step 2: One to one correspondence

This is the next step where the child is starting to count rationally. They are able to associate a number to an object and therefore count correctly.

Step 3: Cardinality

This usually occurs somewhere between 3-5 years of age. You will know it has happened when your child knows that the last number in a count, is the same as the number of objects counted. In other words, he or she doesn’t need to recount them. Graham Fletchy has a great post of this: https://gfletchy.com/2016/03/05/be-the-teacher-moving-from-counting-to-cardinality/

Step 4: Subitizing

Victory! This is the stage where the child knows the number of a small group without counting. This stage begins in pre-kindergarten and will continue to develop as the child enters school.

The progression between the four stages will happen naturally and your child may show some signs of moving to the next stage with lower numbers, but not with bigger numbers. For example, Oliver can put 4 forks on the table for 4 people (showing signs of one to one correspondence), but he can’t count how many trains are on the train track correctly. Rory can recognize groups of 5 fingers without counting, but he can’t recognize groups of 4 or 6. You can’t push a child into the next stage, but there are things you can do to encourage their development. Your role is to make counting real by pointing out all the real-world applications that you can. I’ve included some great activities on the activities page.

Learning to Count Activities

Have fun!

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