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.

Back to top!

The good, the bad, and the ugly about number bonds

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.

THE BAD….

“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!

Back to top!

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!

Back to top!

Feeling Peppy for Patterning!

The importance of patterns

Pattern

Peppy for patterning!

Young children are naturally drawn to patterns, but as a parent, you may wonder why exploring patterns is such a useful activity.

“Learning to look for, describe and extend patterns are important processes in thinking algebraically.” Van de Walle

Patterning is the process of discovering repetitions or regularities and can be explored used songs, movements, manipulatives, nature or behaviours. Patterns and relations are important for understanding the world around us. Much of our life revolves around a pattern: seasons, days of the week, set the table – clear the table, wash your hands – eat your snack, or my Personal favourite: the 3 P’s: Potty, Pull-ups, P.J.’s!

The stages of pattern development:

In pre-K: Children discover patterns (shapes, colours, routines, nature)
Grade K-1: Children learn about number patterns (odd vs. even, 2’s, 5’s, 0’s, the 100 chart)
Grade 2: Children extend patterns into operations (skip counting, adding 10 each time)
Grade 3: Children use patterns as a strategy for multiplication and division
Grade 4-5: Children use patterns to prepare for expressions, equations and functions
Grade 6: Children see algebra as the study of patterns and relations!

Repeating vs Growing Patterns

Now obviously your 3 year old isn’t about to study functions and relations, but the more exposure he or she has to interpreting patterns the better. Marilyn Burns suggests that even in the kindergarten years, we should be exposing our students to repeating and growth patterns to help students develop flexibility in their thinking. By mixing up the type of pattern you present, you are introducing them to problem-solving experiences that will aid in their development of numerical reasoning.

Having taught growing patterns to Grade 6 for years, I couldn’t imagine a 4 year old identifying a growing pattern. (A growing pattern happens when something is added each time).

I decided to present Rory with a growing pattern that could be represented by x+1. I was quite curious to see whether Rory would see the pattern, and was really impressed at his innate ability to solve it after only a few hints! Notice how I use the key questions listed below to engage him. I also use my voice in a rhythmic way to help him identify the growth pattern (“one fish, one bunny, two fishes, one bunny, etc.). Watch how he does!

How to start

Ideally you want to expose your child to as many different types of patterns as possible. This means use song, movements, nature, the world and of course manipulatives. For teachers, Van de Walle stresses the use of manipulatives instead of work sheets or drawings. Manipulatives allow for trial and error and reduce the fear of being wrong. If you are using worksheets to keep a record, you could always have the students record their work after.

When introducing your child to patterns, there are a few key questions to ask:

  1. Did you see a pattern?
  2. Tell me about this pattern (describe it)
  3. What is the pattern? How do you know?
  4. Can you predict what comes next?
  5. Can you extend the pattern for me?

Here’s a video of Rory being introduced to patterns. I started with an easy one using colours, then I tried a different modality (sound) and showed how it could be related to manipulatives. After some discovery time with repeating and growing patterns, I had him create his own. With each exploration, I kept those 5 questions in mind.

Feeling peppy about patterning? Click here for some activities!

Back to top!

Pattern blocks for pre-schoolers!

Say what?! Melissa and Doug make pattern block toys?! Of course the lead manufacturer of educational toys would make something appropriate for the pre-school generation and I thank them for it!

Math Milestones-007

What are pattern blocks? Pattern blocks are a type of manipulative made up of different shapes (triangle, trapezoid, hexagon, square, parallelogram, and rhombus). As the name implies, they are a great tool for exploring patterns, but they are beneficial for so many more reasons as well!

“ Using pattern blocks … helps students “see” mathematical patterns and differences and develop abstract mathematical strategies.” D. Rigdon, J. Raleigh, S. Goodman

For Oliver, pattern blocks help him develop fine motor control as well as reinforce his knowledge of shapes and colours.  For Rory, the same learning applies, but I can also introduce a level of problem-solving while he’s playing with the patterns. There are 10 key strategies for problem-solving and pattern blocks can be used to develop at least 7 of them:

  • Modelling
  • Guess and check
  • Look for a pattern
  • Use logical thinking
  • Draw pictures
  • Make a list
  • Make a table

Pattern blocks also encourage the investigation of relationships among shapes (how many ways can you cover the hexagon using different pattern blocks?); they introduce children to fractional relationships (how many triangles do you need to make a parallelogram?); and are perfect for discovering algebraic reasoning (2 trapezoids = 1 hexagon).

Young children are naturally drawn to patterns: seeing them, creating them, playing with them. So why not introduce your kids to this fun tool that they will be using often in their elementary years?

Here is a link to the set we used: Melissa and Doug Pattern Blocks

Enjoy!

Back to top!

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!

Back to top!

 

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!

Back to top

Estimation is easy!

Anyone who has had me as a teacher in Grade 6 or 7 has heard me echo, “Estimation is Easy!”…but is it?! Estimation can be a difficult skill to learn because it involves number sense, spatial sense, measurement sense and lots of mental computation. This important skill is often left out of the curriculum, or is inserted as an insignificant add-on because approximate answers are not valued as much as correct answers; but they should be!

that was easy

That was easy!

Why is it important?

Estimation may not be easy, but it is essential! We use estimation every day, whether estimating how much sand to buy to fill the new sandbox, or guessing whether our suitcase is overweight, or figuring out if we have enough money to buy something. Estimation helps develop number sense and fluency and is a great way to get children to visualize amounts mentally.

“The emphasis on learning in math must always be on thinking, reasoning and making sense.” Marilyn Burns

And what better way to emphasize these skills than to begin problems with estimation!

What does it accomplish?

Mathematicians agree on these four things:

  1. It helps children focus on the attribute being measured (length, time, volume etc.)
  2. It provides intrinsic motivation for measurement because children want to see how close their guesses are.
  3. It helps develop familiarity with standard units, if that is what is being used to estimate.
  4. It develops referents or benchmarks for important units and as a result, lays the groundwork for multiplication.

When should you start?

My belief is that once your child can count with meaning and he or she can comprehend the language of comparison (more, less, the same), you can start estimating with small amounts. Why not start when the kiddos are small and not yet pre-programmed to believe that right answers are more valued than close answers?

That being said, Rosalind Charlesworth suggests that children can’t make rational estimates until they have entered the concrete operations stage (ages 7-11 yr). This is because she believes children should have already developed number, spatial and measurement sense before they can make educated guesses.

Well, let’s see what Rory thinks; will he make a wild guess or a rational estimation? He’s 4 1/2 years old and in the pre-operational stage of development, but I believe he can make good estimates through motivating activities, coupled with appropriate phrasing of questions.  Let’s see how he does.

The Problem

We are going to our new house on the weekend to measure the rooms so we can plan where to put our furniture, but oh-oh…Daddy forgot the measuring tape! What could we use to measure instead? Oliver! And he is a very willing helper…at least in the beginning!

As you can see from Rory’s first attempt, he made a rational estimate even though his estimate wasn’t that close. He thought it would take 10 Olivers to line the wall, but it only took 5.  I knew his guess was rational because he explained it to me and it made sense. His mistake was that he counted 10 steps initially, instead of 10 Olivers.

Notice how he has already improved his understanding and his next estimate was much closer; he guessed it would take 1 Daddy and it actually took 2. I give him a thumbs-up for his first estimate activity and look forward to doing more with him to see how he improves.

Teaching Tips

Here are some tips so your pre-school children meet with success also:

To start, provide numbers for the children to choose from so they don’t have to pull numbers out of thin air.
Stick with numbers they can count up to.
Begin using benchmarks (5 and 10).
Use and teach proper words: about, around, estimate.
Ask good questions that encourage comparisons:

·         Will it be longer, shorter or the same as _____________?

·         Will it be more or less than ____________?

·         Will it be closer to 5 or 10?

Ask good questions to ensure understanding:

·         How did you come up with your estimate?

·         How can we find out whether your estimate is reasonable?

Start with length, weight, and time.
Let the estimate stand on its own; do not always follow with the measurement.
Develop the idea that all measurements are approximations (thus estimates!), the smaller the units – the more precise but still approximate.
Incorporate estimation activities into your every-day life so it becomes second nature.

It is easy to incorporate estimation activities into your daily life and the more your child practices, the better they will become! Need help getting started?

Click here for activities!

Back to top

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!

Back to Top

C is not just for cookie!

Although some would argue that I am a cookie monster and thus not able to think outside of the “C is for Cookie” box, I say: have you met my friend Mahsa?! Hee hee.

Today I will prove that I can distance myself from cookies long enough to tell you about the 5 C’s of mathematical engagement. (That being said, I should probably admit that as I write this, I am waiting for the oven to pre-heat so I can make cookies! Coincidence? Aha! Another C word! But I digress…)

How do you get children inspired about learning math? You get them excited about solving problems! And how do you get them excited? Get them engaged! And how do you get them engaged? With the 5 C’s of mathematical engagement (Jo Boaler, 2015)!

Curiousity!

Find a problem that they want to find the answer to.

Math Milestones-005

CURIOUSITY!

Connection making!

Find a problem that has connections with other subjects as well as connections with other strands of mathematics.

Math Milestones-006

CONNECTION MAKING!

Challenge!

A problem is not a problem unless it poses a challenge to the learner. Find a problem that will lead to a productive struggle.

Math Milestones-003

CHALLENGE!

Creativity!

Find a problem that is open-ended in its methods or in its solutions, or even better, in both! A problem that encourages them to think critically AND creatively is ideal.

Math Milestones-004

CREATIVITY!

Collaboration!

Find a problem that requires group work to solve. Encourage conversations. Make math social!

Math Milestones-002

COLLABORATION!

and if all else fails….EAT COOKIES!

Back to Top