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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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!

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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!

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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!

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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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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!

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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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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.

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CURIOUSITY!

Connection making!

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

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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.

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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.

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CREATIVITY!

Collaboration!

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

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COLLABORATION!

and if all else fails….EAT COOKIES!

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Houston, we have a problem!

…Granny is coming for dinner! Now I’m not just saying this is a problem because she is my mother in law (seriously, I love my MIL!). I’m saying this because this is an example of an appropriate math problem for Oliver! We normally have 4 people at the dinner table, but we just found out, Gran is coming! Can he set the table for one more? Turns out he can!

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What is a problem?

“A problem is any task or activity for which children have no prescribed or memorized rules or methods, and for which they do not have a perception that there is a specific “correct” solution method” (Hiebert et al., 1997)

“There must be some block preventing immediate resolution. If there isn’t a block, then the situation is not a problem for that student.” (Burns, 2015)

If you are a classroom teacher, this is a critical concept and it is the reason why differentiated instruction is so important. Gran coming for dinner was a challenging problem for Oliver. He struggled with it and could not solve it correctly until we put down the visual cues of 5 placemats. This was not an example of a problem for Rory though. Rory can count easily and adding one more to the table wouldn’t phase him; there would be no block to challenge him. In fact, he kept trying to confuse Oliver by saying he wouldn’t be home for dinner tonight (“he had sports”!) and that Gran was sick and couldn’t come! Trouble-maker!

If this had been a classroom setting, I would have needed to provide extension activities to allow Rory to experience a productive struggle; while also ensuring I had supports available to ensure other students are not engaged in an unproductive one.

The importance of problem solving

Problem solving has always been an important part of mathematics, but in the last few years, it has become even more so. Why? Because our society is changing…fast….and in order to be successful in it, you need to be able to reason, think critically and creatively, use logic, and apply your knowledge to new situations where you may be missing some or all of the needed information… in other words: PROBLEM SOLVE!

Now you may be thinking, at this stage of development, shouldn’t I concentrate on teaching my child the arithmetic first, like how to add or subtract? The answer is an emphatic no! Research shows that students who are given problems to work on before they are shown methods to solve it, actually perform at higher levels (Boaler, 2015). But there is a caveat, the type of problem you choose is important.

Choosing Rich Tasks

There are good problems, and bad problems, even at the pre-school age. Below are the important criteria that should be applied when choosing rich tasks for any age. I’ve also shown how my two problems measure up.

RICH TASK CRITERIA FOR PROBLEM SOLVING Oliver’s Task Rory’s Task
A familiar context X X
The outcomes should matter to them X X
Involves math they are confident with X
Low floor (lots of ways to enter problem) X X
High ceiling (can extend problem) X X
Have appropriate materials to solve it with X X
A perplexing problem that the child understands X X
Have more than one solution X
Be interesting for them so they want to solve it X X
Challenging but accessible, provoke productive struggle X X
Encourage open thinking X X
Allows for connections to be made X X

I chose a rich task for Rory, based on a problem from Van de Walle  (2014), that has all the characteristics of a good problem. Rory will be turning 5 in a few months, so I’ve asked him to figure out how many different ways he could put 5 yellow or blue candles on his cake. This problem is great for introducing the following concepts: decomposition of 5, exploring 0, the commutative property, exploring part-to-whole relationships, counting, and cardinality…just to name a few! Let’s see how he does:

Well, Rory surprised me by taking this problem on a totally different tangent, and I let him! It’s important to allow children to try different strategies, develop their own solutions, and make mistakes and Rory did all three! Notice I did not provide him with answers or lead him to the ‘correct’ way to solve it. If I was in a class, I would get other students to share their strategies and have a number talk about the decomposition of 5, or have them work collaboratively to find different combinations. In this case, Rory worked independently (with a little help from his brother!). I haven’t told Rory the final answer; instead, we’ll re-visit the problem later. Perhaps we’ll see how he does arranging 3 candles on Oliver’s cake…stay tuned!

Your role in all this?

BEFORE the task, you should activate prior knowledge and be sure that the problem is understood by the child.

DURING the task, you need to step back and allow the struggle to ensue. Resist the temptation to lead them to the answer! Ask appropriate questions so the child experiences a productive struggle and not an unproductive one.

AFTER the task: talk with them about alternate methods and solutions so that they see there is more than one way to solve a problem.

 

Need help coming up with problems? No problem (ha!)!

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My American Idol (although she’s British)

I once had a friend who was obsessed with U2 to the point of ridiculousnous. If she saw Bono on TV, she would scream and tell everyone to be quiet so we wouldn’t ruin the opportunity for her to hear him sweat, like that was even possible! When her husband bought her concert tickets for her birthday, she collapsed on the floor in a gigantic heap, bawling. I never understood how anyone could be that obsessed with someone, especially someone they had never met and didn’t really know.

And then the MTBoS (math-twitter-blog-o-sphere) introduced me to JO BOALER. I became obsessed! I followed her on twitter @JoBoaler, questioned the appropriateness of friending her on Facebook, stalked her on Youcubed.org, bought all her books, and competed in the #withmathican contest in an attempt to win a virtual professional development session with her!

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Me and My American Idol!

Dreams of doing my PhD at Stanford, under her confident leadership, ensued (until I priced out said-degree and realized I may not be able to afford the $47,000 USD per year price tag), but dare to dream! Unlike my friend, I didn’t collapse on the floor bawling when I won the #withmathican contest (here’s my lesson), even though I had just won a chance to impress JO BOALER virtually. Instead, I envisioned a skype session with just me and Jo, hanging out, engaged in deep stimulating discourse about mathematical pedagogy. Two women, two moms, two inspiring mathematicians (well, maybe one inspiring and one aspiring!).

I nearly did break into tears when I realized that I wouldn’t be virtually face-to-face with my idol at all, and that the prize was enrollment in a MOOC (massive open on-line course) instead. Although I am super-excited to take the course (it starts in June), how would I impress my mentor and woo my way into Standford now?!

Now you’re probably asking, why am I obsessed with JO BOALER?

Well, the best way to explain it is to quote youcubed’s goal:

“Our main goal is to inspire, educate and empower teachers of mathematics, transforming the latest research on math learning into accessible and practical forms.” https://www.youcubed.org/ourmission/

WELL JO BOALER…MISSION ACCOMPLISHED!! I AM INSPIRED! I AM MORE EDUCATED! AND I FEEL EMPOWERED!

You can get to know Jo Boaler yourself with these links below. I hope you are as inspired as I am…or maybe not…it is kind of creepy!

Jo Boaler on wikipedia: Wiki Jo Boaler

Math mindsets by Jo Boaler: Math Mindsets

What’s math got to do with it by Jo Boaler: What’s Math…

Youcubed at Stanford University: www.youcubed.org

Jo Boaler’s on-line course for teachers: How to learn math

 

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