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.

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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Tenzi Frenzy!

We are away for the summer at a cottage, with no internet or TV, which I usually love. We’ve already read lots of books, frolicked in the waves, swam to the Big Rock,  sailed to Seagull Island, canoed…dumped the canoe and had lots of good old fashioned fun; but my heart still felt the pitter-patter of excitement when I saw the clouds roll in, because that meant we could drive to the nearest town and spend the morning at Chapters!  

Don’t you love rainy days at Chapters? (Borders would probably be the US equivalent).The boys love playing with Thomas the train in the kids section, and looking at all the books, while I finally get a chance to peruse the latest best-sellers in person, instead of on Amazon! We go to the library every week, but it’s just not the same as a road trip to Chapters. And when we went yesterday, I felt like I hit the jackpot with my new find: TENZI!

Best. Game. Ever!!! Kevin and Steve (the game’s designers), made known by a little piece of paper in the game box with their story on it, may not have thought of the mathematical implications when they came up with the idea for the game, but kudos to them for unwittingly designing a brilliant game suitable for 3-103 year olds!!

Here is the general gist of the game, and I quote: “Everyone gets 10 dice. Then everyone rolls until someone gets all their dice on the same number.” Simple, right? Why am I so excited by this new find? Because of its GINORMOUS educational value! It’s like this game was conceived specifically with the pre-kindergarten to grade 2 curriculum in mind,  yet it’s intended for everyone!

Here’s why I love it:

Subitizing!

Subitizing is the ability to recognize number patterns without counting. Rory quickly grasped what the dot patterns stood for and although he still counted the dots on each new turn, the repetition of looking for the same dot pattern reinforces his learning. I am confident that after a few more rounds, he will quickly and easily know the dot patterns for 1-6 without counting.

Counting on!

If you have 3 of the same number and get one more, now you have 4. Rory was learning and Practicing math skills without even knowing it! He already has developed one to one correspondence and cardinality, but now we’re extending his knowledge. What is 3 and 3 more, or 4 more, or 5 more?! Because each turn is different, he is continually practicing different amounts of counting on.

Decomposition and recomposition of 10 (a very important bench-mark number)!

Because the goal of the game is to get 10 dice all on the same number, you are constantly looking for two numbers that make up 10: those you already have with the same number on them and those you have yet to roll. Rory quickly saw when he needed one more to make 10, and then we looked to see that he already had 9. Or he had 5 of the same number and needed 5 more. And that leads to….

Addition!

Decomposition of 10 is the building block to addition and although we didn’t concentrate on it today (it was our first time after all!), eventually we will use this game to practice our 10 facts. We can easily adapt it to practice our 5 facts first, just play with 5 dice each instead and yell, Fivzi!

Fun!

This game is fun for the whole family!  Oliver got in on the action too but only to yell “Tenzi! “ and steal our dice to make a tower, but I’m sure he’ll see the math value soon!! It was me that finally drew the game to a close after almost an hour; Rory could have kept playing forever!

So Kevin and Steve (fortuitous mathematical master-minds that you are!), thank you for a fun and easy game that everyone can play. It looks like you two have a whole new market to exploit and hopefully I’ve inspired some new fans here!

If you want to know more, check out their website at www.ilovetenzi.com. Thank heavens for rainy days!!

Have other great math games that aren’t actually meant to be math games?

Post them in the comments section below!

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Sorting and grouping

The thing with children is they change so quickly and if you blink, you feel like you missed the moment when a major milestone was achieved. For example, Rory used to say he wanted a hangaber for dinner. Alan and I thought it was adorable, and did nothing to encourage the proper pronunciation of the word! But then one day, we noticed he was asking for hamburgers instead of hangabers, and the moment was gone.

This week, I was playing with Oliver and noticed that he now knows some of his colours! This is a very recent development and one we’ve been anxiously waiting for. Rory knew all his colours by 2 years old, and Oliver is almost 3 and was showing no signs of progress; but then, just like that, he got them all right! This is so exciting for me as a mathematician because it now opens up so many more informal sorting activities!  

“As the children’s vocabularies increase, they will be able to label and describe how and why they are sorting and grouping things.” (Charlesworth, 2012)

Here is an example of Oliver engaged in naturalistic play. Notice how I commented any time he knowingly (although usually unknowingly!) put things into groups. Also notice he learned a new word (rectangle!) and now has additional sorting power for next time!

Because Oliver is now ready for more informal instruction on sorting, I started looking for articles about this important stage of development and was surprised when I couldn’t find many. I couldn’t even find agreement on what strand of math sorting falls into! In some books, classification was stuck under geometry, but the content was directed at a higher age level. For example classifying polygons versus nonpolygons; or triangles with the same area versus different areas.  Another resource I looked at, clumped sorting under data analysis because organizing data into groups is important for graphing. I myself, would have linked classification with logic and pre-algebra, because sorting involves reasoning and logical thought. It is also the precursor to addition (putting groups together) and subtraction (taking groups away).

In addition to the controversy over what strand this falls into; sorting and classification only really appears in the pre-k to k curriculum, and as a result it is minimized in the teaching resources or believed to develop naturally. This surprised me because classification is such an important skill not only at school, but also in our daily life. This skill, although it may appear basic, is the basis for further logic and reasoning. It provides an introduction to graphic organizers such as Venn diagrams and to me, it is a life-skill that may even precede executive functioning ability! (New research project?!) Think of the importance of learning how to sort and classify in this day and age, with all the information we have access to.

Now that I have convinced you of the importance of this seemingly natural ability, I want to share with you how to nourish this skill in your child. In the early years, classification activities fall into three categories:

Stage: Your responsibility:       Example:
Naturalistic: Provide free time, material and space
  • Blocks, cars, farm animals, nature things
Informal instruction Provide comments or tasks that identify or encourage sorting
  • Your picture has lots of red.
  • Can you separate the forks from the knives?
  • Could you put your cars in the car bin and your balls in the ball bin?
  • I see you’ve arranged your dolls from smallest to largest.
Guided instruction Give specific objects and guide classification strategies
  • Find some things that are___.
  • Tell me why these belong together.
  • Sort these into groups, how did you decide?
  • Is there another way to sort these?

 Rory has a larger vocabulary than Oliver and a larger understanding of the universe. For his sorting activity, I used guided instruction. You’ll notice he came up with interesting ways to sort things: by function (button, sticker), by colour (red, blue, yellow, green) and by category (animal, vehicle, shape).  I guided him by encouraging him to think of different ways to sort his materials; however, it was ultimately his decision.

Next time, I might choose different objects that force him to make different decisions. For example, choosing all cars but different sizes, or choosing all art mediums (canvas, paper, felt etc.) and let him sort by texture, or all natural objects and have him sort by common features. I would also provide objects that relate to different content areas. For example, objects that float or sink (science), pictures of workers and different materials (social studies), or sorting plants into edible and non-edible. The possibilities are endless! The only thing to keep in mind is that classification activities should follow the same progression as manipulatives (see my post on this here), so start with 3-D objects and then move to cut-outs and then to pictures.

Although I couldn’t find much on how to teach classification, I found bucket-loads of activities that involve sorting.

Click here  and sort through these for starters!

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Making Math Meaningful with Manipulatives!

If you’re not a teacher, you may not be familiar with the term manipulatives, but you can probably infer what they are. Manipulatives are models that help children think and reflect on new ideas in math. They include resources that allow children to explore, question, guess and check, but more importantly, to play with the problem. Counters, toys, linking cubes, abacuses are just a few examples.

Why use manipulatives?

We all know the old adage: we learn better by doing and math is no different! Manipulatives give students, of all ages, opportunities to have a hands-on approach and develop deeper understanding of concepts. Research has shown benefits to using manipulatives all through life! That means, don’t be in a rush to move your child into more abstract ways of solving problems. There is a natural progression to manipulatives and you need to assess your child’s readiness before pushing them to a more abstract level. When choosing materials, they should be sequenced from concrete to abstract and from 3-D to 2-D. See the chart below for more information.

Transitioning from concrete to abstract manipulatives (Charlesworth, 2000)

1) Start with real objects. Sensorimotor stage.
2) Move to real objects supplemented by pictures. Pre-operational stage
3) Once the first two are mastered, you can use cutouts of real objects. This is the transition from 3-D to 2-D, but the objects can still be manipulated. Pre-operational stage.
4) Now move to pictures. Transitional stage
5) Finally (and much later!) use paper and pencil. Concrete operations stage

So where do virtual manipulatives fit on this spectrum? Good question! I’m not sure! My guess is that they act like real objects because you can move them, but because they are 2-D, they might be more on par with the cut-outs level, in terms of concreteness (see step 3 above). Let’s see what Rory thinks. I’m going to get Rory to do a task with real objects and then do the same task with on-line manipulatives. Then we’ll see what he has to say! This task is an introduction to addition but it would also be great to use for lessons on one:one correspondence, decomposing numbers, counting on and  cardinality.

Well it looks like Rory prefers virtual manipulatives. It may have been the novelty of it or the fact that the computer images acted more life-like than the real objects! He claims that the boat was more real compared to my egg carton version and he liked that the bears kept looking at him (in case you couldn’t tell)! The important thing is that children are given the freedom to choose their own manipulative so that they aren’t restricted to one method. That way, they can discover their own way to reach a solution that makes sense for them. If he likes the on-line tool, on-line tool it is! But I’ll make sure he has the real objects on stand-by in case he’d like to use them as well.

Looking for manipulatives? Look no farther!

Click here for a list of manipulatives that teachers often use with this age group!

Are you a parent? The great thing is that anything can be a manipulative! You don’t need to run to a teacher supply store in order to help your child.

Click here for a list of great things to use at home!

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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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