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!)!

Click here for activities!

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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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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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It All Starts At Home

Getting a child inspired and curious about math begins at home. If you are a teacher, you can relay these messages to your parents; and if you are a parent, you can begin right now!

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Here are five things you can do:

1) BE ENTHUSIASTIC!

I can’t stress enough the importance of positive math messages at home. I’m still surprised at the number of parent-teacher meetings I’ve had, where the parent has told me things like, “well, we were never good at math”, or “she must get that from me because I never got the right answer”, or “I’m not a math person”, or even, “I hate math” (I know, I was shocked too…who hates math?!).

If they are telling me these things, then I’m pretty sure their children are hearing the same messages at home. Sadly, these are the worst messages to pass on to your kids. They teach kids that math is a do or don’t subject; you either like it or you hate it; you either inherited the ability or you didn’t; you either understand it or you don’t. As a result, children who hear this develop a fixed mindset and believe that they will be good or bad at math or they will love or hate it, when in fact there may be some aspects they love and some they don’t love as much.

So what can you do? BE ENTHUSIASTIC! Show your child you love problem-solving and puzzles and math problems…even if you don’t! Help your child develop a growth mindset! I think you can, I think you can, I think you can!

2) PROVIDE A MATHEMATICAL SETTING

Have you ever read the Math Curse by Jon Scieszka? It’s a book about a boy whose teacher tells him that everything is a math problem….that teacher was right!!! Everything can be turned into a math problem! When children are young, they love investigating the world around them and they love math! You will find them naturally gravitating to things they can order; you turn your back and all their trucks are in a line or they’ve matched up all their dolls with pink dresses followed by all their dolls with yellow dresses or they have ordered their cars from smallest to largest. Take advantage of their innate urge to do math and develop it further.

3) ASK QUESTIONS ABOUT EVERYTHING

Provide your children with opportunities to do math at every corner: “can you help me measure ⅓ cup of water for the muffins?  How many steps will it take you to get to the car, what if you hopped? How fast can you get upstairs for bed? Let’s estimate how many minutes Dad will complain about the commute home tonight!” Just like the Math Curse says, everything can be turned into a math problem, but you shouldn’t push it. If your child isn’t interested, don’t force them. If they get the wrong answer, point out what they got right. Your role is to introduce math so that it is fun and playful. Your goal is to help them develop confidence in their efforts. If your child isn’t interested, try again another day or change your questions to ones that intrigue them. Most importantly, have fun!

4) PROVIDE BUILDING BLOCKS

By this I don’t mean scaffold learning, I mean actual building blocks!

Children’s play with building and LEGO blocks in the early years has been identified as one of the key reasons for success in mathematics all through school”, Jo Boaler (2015).

Anything you can provide that encourages patterns and spatial reasoning is beneficial. And there are so many to choose from!

·       Coloured wood block set

·       LEGO

·       Magnetic 3D tiles

·       Jenga blocks

·       Magic Bricks

·       Keva Blocks

·       Train tracks

·       Robots

·       Puzzles

·       Cogs and Gears

·       Race-tracks

·       Perler beads

·       Light-brights

·       Geoboards

·       Pentominos

·       Tangrams

·       Pattern blocks

·       Unix cubes

5) NUMBER TALKS

Children who do well at math have great number sense. That doesn’t mean they have memorized their times-tables or are the fastest at their math facts. It means that they can decompose and recompose numbers easily. How do you encourage this? Number talks. Number talks encourage children to think flexibly about numbers and can be started as early as you want. It helps children realize there is more than one way of doing something and also helps them develop critical and creative thinking skills. Here is a video of Rory and I discovering that you can make 4 in more than one way.

Have fun!

Ann

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