# Posts tagged early math

## Last-Minute Christmas Decorations and Tangled Lights

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Days are flying by and everyone is busy preparing for the holidays. While you might not know what gift(s) you will be getting this year or whether the kids will catch the sniffles right before the party, one thing you can count on is tangled lights. Seriously, no matter how carefully we pack them, when we open the boxes next year, here they are, all knotted up. Turns out, it’s all about math and physics as opposed to our carelessness. Check out this NPR article for details. Here’s the best part – next time you have knotted up strings of lights, give them to your little one to undo and proudly tell your friends he was busy solving a knot theory problem.

In addition to tangled lights, another thing you can count on is a broken ornament or two or three. If your tree looks a bit bare, don’t rush to the store. Instead, consider some really cool last-minute DIY ornaments. Bonus is they are all about math. The easiest ones are paper chain garlands. This year, add a twist, (ok, half a twist) and turn them into Moebius chain garlands. Or bend pipe cleaners into shapes and grow Borax crystals on them. What shapes you choose is totally up to you. Inspired by Vi Hart’s Borromean Onion Rings video, we made this Borromean Rings ornament.

Or how about turning cardboard boxes you were about to throw away into star ornaments just like Malke and her daughter did on the Map is Not the Territory blog.

Finally, we’d like to share a holiday printable we found thanks to the link from one of the readers, Elena T. She printed it for her daughter to color days ago, but we just got around to it. This Christmas tree might seem like just a giant cute coloring page, but don’t let it fool you. It’s got lots of things going on, including some great math. Can you find examples of gradients, tessellations, pattern, rotational symmetry, radial symmetry, iconic quantities and a lovely Sierpinski triangle?

Share your holiday math with us on the blog and on our Facebook page.

## Discovering Signs and Symbols

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Lately my 5-year old has been very interested in signs – road signs, signs at the entrances to parks, museums, office buildings, etc. Which led to some really interesting conversations about how rules (and ideas in general) can be represented as symbols.

Unlike written words or letters (which are symbols as well), well-designed signs are much more intuitive and easier for pre-readers and early readers to interpret independently. By the way, have you ever noticed how many of the signs we encounter are the ones that prohibit something rather than inform or encourage? I never did until my son pointed it out saying “see, this sign says no smoking, this – no drinking, this – no music, this – no guns. Signs are for saying “no” to things.”

So back to the conversation that we, my son (S) and I (M for Mama) had a few days ago:

S: Mama, when my tree house is finished, I’m going to invite all my friends and put a big sign “no girls allowed”

M: How would you make a sign like that?

S: Easy, I’ll just make a big red circle with a thick line across like that (draws in the air) and there will be a girl on it, like on bathroom doors.

M: Ok, but what if your cousin A comes to visit? Can she play in your tree house? (My son loves playing with his oldest cousin)

S: (after some thinking about it) Sure!

M: But then you need to make a different sign. What would it look like?

S: (after some more thinking) Ok, I’ll just put her picture next to the other sign. It has to be a smiling picture.

M: What if (names of a couple of girls he knows well from playdates) want to come play? Will you let them into the treehouse?

S: (after even more thinking) Yes. All girls I know can come and play. Only girls who are strangers can’t come. And if they are not very little.

M: Ok, but then you have to change the sign.

S: (sounding a bit weary) I dunno. Put more pictures. (runs away)

We had a few more conversations about signs that were similar to this one. My son would come up with a very broad rule and a sign for it. I would then suggest scenarios that did not fit the rule and he’d adjust the rule. And we’d try to figure out how to create a sign that would accurately reflect the new rule.

Since all these conversations were completely “on the fly”, usually while walking or right after reading a bedtime story. Which, I figured out, is not the best time since we don’t get to put any of the sign designs on paper.

But now I’m thinking what kind of a sign-making game can I put together (something that wouldn’t take too long). Any suggestions? Please share!

## Different Perspective

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Yesterday I chatted with a friend whom I haven’t seen in a while. Her child, a bright and energetic 8-year old, participates in quite a few extracurricular activities – ballet, gymnastics, tae kwan do, and art. Next year, music lessons might be added to the mix.

I asked my friend how they choose the activities for their daughter. Well, ballet was Mom’s choice since it was something Mom always wanted to do herself, gymnastics – Dad’s, since he was in gymnastics as a boy. Martial arts was a joint decision because it is known to improve child’s discipline. And art was the girl’s own choice.

Then I asked this question: if there was a math club or a math circle near you, would you consider signing up your daughter?

Her answer was “Absolutely! We are actually considering some extra math drills for her since she has some problems in school.”

This reminded me of another conversation, months ago, when another friend said that she wasn’t interested in a math club for her daughter because “she was doing well enough without it”. Yet another friend, this one with a preschooler, said that it was simply too early for her child to learn math.

I find it very interesting why there’s such a difference between attitudes towards, say, a dance class or an art class and a math club. Why the reactive attitude? I mean, why wait until a child falls behind in math class? Why math clubs are thought of as places for remedial math? Why sending a 3-year old to ballet classes is perfectly “normal” while playing advanced math games with her is not (“are you trying to raise a genius?”, “why are you torturing a child?”, “why don’t you let her enjoy childhood for now?”, and similar questions).

What if we looked at art, music, dance, gymnastics, LEGO and other children’s activities from a different perspective. Each and every one of these offers so many opportunities for mathematical discoveries! We just need to help our kids recognize and explore these opportunities. How can we do it? Where do we start?

## Playing with Platonic Solids

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There’s a lot of talk about how playing with building blocks helps children develop math skills. But what about children that are too young to even “tote and carry” blocks? Have you thought about introducing them to Platonic solids? Ok, here’s a little refresher about Platonic solids:

A Platonic solid is a 3D shape where each face is the same regular polygon and the same number of polygons meet at each corner.

If the idea of introducing this concept to a small child sounds a bit over the top, here’s a surprise – your infant might already be enjoying one. After all, a cube is a Platonic solid. But why leave out the other four – tetrahedron, octahedron, dodecahedron, and icosahedron? That’s exactly what British mathematician Richard Elwes and his wife Haruka have done. Here’s Richard’s story:

When some friends told us they were having a baby, Haruka set to work making a soft cubic toy to give the child, by sewing together square patches of colourful cotton cloth left over from other projects, and stuffing it with cushion-filler. Being a mathematician, Richard immediately suggested the set should be expanded to include all five Platonic solids. (One challenge was to make sure that no two adjoining faces were made of the same cloth.)

These toys are intended for very young children, so it cannot be expected that they will ‘learn geometry’ in the usual sense. Instead, what we hope is that they will begin to foster a geometrical aesthetic, enjoying the symmetries of the toys, and developing a familiarity with these five solids, which will remain throughout their lives.

As the children grow older, we hope they will keep revisiting the Platonic solids in other forms, perhaps as wooden or plastic toys, maybe as dice or puzzles, later making them themselves out of paper or card. But there is no need to stop with the Platonic solids! As soon as practical, why not introduce shapes like prisms, antisprisms, and Archimedean solids (along with their duals: bipyramids, trapezohedra, and Catalan solids)?

For a small child meeting the Platonic solids for the first time, there is is one potential problem: apart from the cube, the names of these shapes fail to reflect their elegant simplicity. For a toddler, the word “icosahedron” is surely a bridge too far. So why not reduce them to their initial syllables: tet, cube, ock, dode, & ike? This will allow the child to have fun identifying and comparing the shapes, without getting bogged down in unnecessary Greek verbiage.

## Peas and Carrots Math

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A few days ago my 5-year old and I were busy picking peas in our vegetable garden. The 30 or so pea pods looked so delicious, that we decided to eat them right away. And since shelling pea pods takes some time, we had a moment or two for the all-about-peas math:

• Each pod snaps into two halves length-wise. Let’s count how many peas are in each half?
• How many peas are altogether in each pea pod? Let’s count them to make sure.
• Can you see without counting how many peas are in each half?
• Can you tell how many peas are in a pod without counting? (this can be done either with subitizing or by adding peas from the two halves)
• Which half has more peas in it?
• Does this pea pod have more peas in it than the one before?
• Can you divide peas from this pod between the two of us so we both get the same number of peas? Why? Why not?
• How many peas do you think will be in this pod? (keep track of this data; we found out that most of the time we had pea pods with 7 peas in it; 5 was also pretty common; only a few pods had 3 peas in them; just one had 8 peas; there were several pods that appeared to have 6 peas, but on closer examination we would always fine the 7th tiny pea at the tip of the pod)
• Do you think we will get a pea pod with no peas in it? With 100 peas in it?
• What do we find more often – pea pods with odd or even number of peas?

Now summer carrots are almost ready for picking. I’m thinking we might explore gradients (length, thickness, weight, taste), fractals (carrot leaves), measurements (including how tall are you measured in carrots).

## Silly Robot Math

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Robots are cool! Just ask any 5-year old (I just did a quick survey of 3 5-year olds and they confirmed it). So let’s play a game that is all about robots, math and silliness.

Objectifying people is dangerous, but math entities – operations, functions, algorithms – love to be objectified! “Function machine” metaphor helps kids to see actions as objects or things. This allows kids to progress to the next level: that is, to perform math actions on functions. When functions are objects, kids can act on them: sort, improve, analyze, compose, reverse, loop and so on.

Algorithm is a very curious thing indeed: it’s an objectified sequence of many action, or a Rube Goldberg machine composed of many functions. The usual developmental wisdom is that kids can deal with algorithms with as many steps as their age, for example, a two-year-old can (1) take socks (2) from top drawer but forgets to (3) put them on. However, repeated, fun, meaningful algorithms kids care about work well at earlier ages.

Algorithm is one of the BIG Concepts that children can learn in the Silly Robot game. Other BIG Concepts include

• Command
• Function
• Loop
• Cycle

So let’s play! Here’s HOW

“Silly robot” turns familiar, everyday task into funny and quirky “Wonderland” games. Choose a simple task, such as filling a glass with water, or putting on shoes. “The silly robot” should be someone who knows the game well. Robots only understand simple, one-step commands, such as “move forward” or “pick up the glass.” They make silly noises if the command is wrong or they can’t do it.

The robot is trying as hard as he can to mess up the task without actually disobeying directions. For example, kids say, “Put the shoe on” and the robot puts it on his head. Kids say, “Put the glass down” and the robot does, except the glass is sideways and all the water spills out! The goal for the robot is to find funny loopholes, and the goal for the team is to give commands without loopholes. Only the robot (the game leader) needs to know this at first – kids discover how the system works! Then they want to play the role of silly robots – analyzing actions as they try to mess up on purpose.

Reusing and improving algorithms is a huge value of mathematicians. Another value is the precision of language and action. One way to be a good “silly robot” is to take commands literally, which helps kids to pay attention to details of actions and their descriptions. The task of playing the robot is difficult, because the robot needs to be slightly annoying, for laughs, but not too annoying.

Infants – Use “mix-up” gags about routine tasks. I think every parent of a baby I know has at least one photo with underpants on the head, done for the baby’s amusement. Tell the baby the dog goes “Meow” and point at the belly button after asking, “Where is the baby’s nose?”

Toddlers – Even before kids are verbal, they use gestures to tell parents what to do and what they want. You can play silly robot games around these commands. Just make sure to laugh with the kid, not at the kid.

Older Children - Progress to more complex tasks and those that involve repetition, such as cutting several tomatoes for a salad, or decorating a room with several pictures. This promotes reuse of algorithms and their parts (cycles or loops). Use computer algorithms.

Other ways to explore algorithms are

• Story ideas: A storyboard is an algorithm for a movie or a game. Look at favorite books, movies, games, jokes to analyze their algorithms and their parts, such as plots or motifs. For example, knock-knock jokes and “Why did the chicken cross the road?” jokes have distinctive algorithms. Fairy tales and stand-up gags often feature two repetitions and the third action breaking the pattern, for example, “Three little pigs” or “Three billy goats gruff.” Check out this huge depository of story algorithms.
• Show kids computer tools for capturing algorithms, such as concept mapping or diagramming software.
• Play “Silly Robot” with math computations. How many mistakes can you make in one problem? Making mistakes on purpose is hugely therapeutic!
• The game can help with the annoying, persistent issue: kids not remembering some steps in everyday tasks, such as putting lids back on food. After playing for a while, the algorithms are not only laughed at, but also debugged, and everybody is in a better mood.
• The clearest, easiest programming software, accessible even to toddlers (with parents helping) is Scratch from MIT. It presents steps in algorithms as Lego pieces you put together.

Do you like robots? Does your child? Share your experience playing this game.

## Press Here

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Have you read this book? Maybe “read” is not exactly the word here. Have you played this book yet? If not, go ahead and give it a try. The idea is simple – each page of the book tells you what to do, but doesn’t tell you why you are doing it or what to expect. You discover what has happened only after you flip the page.

This is not a math book. Yet there is a lot of fun math hidden in it. In just a few pages you play with iterative functions, several function machines and a fun pattern.

Even better, the book ends with an invitation to play it again, looping back to the first “press here” page.

This book gives a child a perfect opportunity to figure out patterns of action and to predict outcomes. “What do you think is going to happen now?” is a question that is ingrained into the book.

After going through it a few times, children might feel inspired to create their own books similar to “Press Here”. All it takes is a few sheets of paper, stickers and markers.

One thing I haven’t tried yet, but that sounds intriguing is to see what happens if you stop in the middle of the book and go back to beginning and repeat the instructions. For example, what if we stop on this page with a pattern and then go back to the beginning, follow the instructions and press yellow dots only. I think it would be a fun way to explore algorithms with nothing but a large piece of paper and a few Dot Dot markers (or regular markers or stickers).

Have you played this book? Share your ideas and experience with us!

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