Get ready for Funville Adventures!

Funville Adventures, our book with Allison Bishop, will be starting its crowdfunding campaign later this month and the plan is for it to be published by Natural Math this summer!

Here is a sneak preview of our awesome book cover:

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Imagine a world where functions are not confined to a blackboard, but rather come to life as magical beings! This is Funville, a world where each inhabitant has a special power to transform objects.

Stay tuned for more updates on the campaign and more sneak previews.  And please help us spread the word!

 

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Games with the multiplication table

The following was translated from this post by Jane Kats.  Translation and translator’s notes are by Yulia Shpilman.

(Translator’s note: I taught a lesson inspired by this post to my math circle for 1st and 2nd graders on Sunday and we had a blast. We spent about 20 minutes filling out the multiplication table (some kids finished the whole thing and some about half but everyone was able to get through something) and spend another 20 minutes looking for various numbers and patterns.  

Everyone had their own strategy of filling it out (just as Jane describes) and, to my great surprise, everyone really enjoyed doing it. One of my favorite by-products of this activity is that it allowed the students to find and correct their own mistakes – if someone were filling out the 4s row, for example, and goofed up, she saw that something didn’t work when she was filling out the 8s column because the numbers didn’t make sense! It’s a rare activity where the mistakes are so obvious and easy to fix – it was a great lesson for the kids!  

And last note – if you’re going to give your kids a visual multiplication table, please use this version – not the version that lists out all the equalities such as 2×2=4, 2×3=6 and so on – that is NOT a table and it’s not at all useful for seeing the beautiful patterns in the actual multiplication table!)

And now on to Jane’s post.

At our math circle, we don’t do many arithmetic exercises, but we try to show our students some rules and patterns.  For example, last week, we explored the multiplication table.

We gave the kids a blank table and discussed that the first row is numbers 1, 2, 3, 4 and so on – we count by 1s and can keep doing it indefinitely.  In the second row, the numbers are 2, 4, 6, 8 – we are counting by 2s.  In the third row, we count by 3s and so on.

So how do we will out the table? In which order?

Someone was filling it out by rows. Someone else preferred columns.

Many were actively using their fingers, adding by 3s, 4s, 6s, 7s.

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Some kids filled out the the 10s row and column, leaving the 7s, 8s and 9s for later.

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Some kids filled out the same row and column together

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Someone started with squares

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Everyone had their own strategy.

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Everyone chose the “most convenient” cells and filled them out first.

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Actually, from my experience, different people find different multiplication problems challenging and everyone finds their own way of remembering the answer.

Let’s say, for example, someone memorized the correct answer to 7+8.  Someone else might check himself, adding 7+7+1, and yet another person might get to 10 first, 7+3+5.  And everyone finds their own way most convenient and natural!

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The tables are filled out.  And, might I point out, they go up to 14, not up to 10 (translator’s note – ours went up to 15 😊).  That’s because I know a lot of people who think that the multiplication table only “works” through 10!


Now we can play with it:

Color the number whose neighbors are 5, 8, 12 and 15.  How many numbers like that did you find?

Now color the number, whose neighbors are 8, 9, 15 and 16.  (Translator’s note: this was a challenging activity for our students, but they slowly got the hang of it with practice.  I also showed them a “magic trick” – they gave me the neighbors and I guessed the number without looking at the table).

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Now I give out a filled-out table and ask the kids to color in all the 12s, all the 16s, all the 72s – and to understand how many there are of each number.


Then I ask them to cover a part of the cells in the top left corner with a small square sheet of paper – and I guess, how many are hidden.

After that, I write the numbers 3, 6, 9, 12, 15 and 18 on the board. The challenge is this – find all the numbers the sum of whose digits is one of the numbers on the board, and circle them.

everyone has a different strategy

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many start by looking for the same numbers and circling them first

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I walk around, admiring their work, and wait till someone notices a pattern.

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but it won’t happen right away.

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not yet

jane_mult13this is getting closer

jane_mult14even even closer

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Got it! Someone figured it out!

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I ask him not to tell the others and we quietly discuss the found pattern.  

I think this is a much more interesting way to discover the rule about divisibility by 3, rather than just feeding it to the kids as something to be memorized.

Do you remember when you learned the multiplication table?  Which problems were the hardest to remember?

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Logical Fun, Part I

Once a week, I teach a joint math class at my school which combines children of ages 6-10.  For these lessons, I generally try to pick an activity that I can present as a series of “problems” of increasing difficulty and depth.  Thus, the more advanced students (generally the older ones, but this really varies by topic) can go further and deeper whereas the ones who struggle with the topic can spend more time on the introductory problems.  Often, there is also a hands-on element to the activity.

This week, however, I taught a joint lesson in which there were no hands-on materials and no worksheets with problems or puzzles.  The whole class consisted of discussions, where students of all ages contributed greatly.

The topic was logic.  Most of the problems/puzzles we discussed came from Raymond Smullyan’s wonderful book What is the name of this book?

I loved how the students did not just look for “the one right answer” but looked for different interpretations, explored different definitions, and questioned “intended” answers.

Here is an example problem:

A train leaves from Boston to New York.  An hour later, a second train leaves from New York to Boston.  The trains are moving at equal speeds.  Which train will be closer to Boston when they meet?

The answer given in the book, and the one that I thought of right away with my experience with such problems, is that of course they are the same distance from Boston when they meet.

The students, however, started asking questions such as, What does it mean to meet?  Does it mean that the fronts are next to each other or the whole lengths of the trains are next to each other?  And how do we measure closeness?  Do we use the front of the train or the end of it?  The trains are not dots after all.

Different answers to these questions lead to different solutions.  We had truly wonderful and very mathematical discussions about how “it depends on many factors.”

And then one student asked whether it takes more or less than an hour to get from Boston to New York, so we discussed whether this would affect the answer to the problem.  Again, we decided that it depends on where the second train is before it starts off on its route.

I truly did not know what to expect from this lesson.  I was a little worried that the only response I’d get would be silence and confused looks.  However, I must say that this was one of the most enjoyable lessons that I ever taught.  And I was in absolute awe of the students, the freedom and depth of their thinking and their complete lack of fear in expressing their thoughts or of being wrong.

I wonder what would happen if I tried a similar lesson with much older students, or even adults.  Would they have the same freedom of thought or would they focus on looking for the one right answer?

Here’s another problem that we discussed with the students that perhaps requires a bit more brain twisting and that is a favorite of mine:

A man is looking at a portrait.  A passerby asks him whose picture is he looking at, and this is his response: “Brothers and sisters have I none, but this man’s father is my father’s son.” Who is in the portrait?

Stay tuned for a separate post about the students’ discussions of this one.

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Arithmetic games – is that boring?

Translated from this post by Jane Kats.  I very much subscribe to this philosophy when teaching math to kindergartners.  As for the games mentioned, I love playing many of them throughout elementary school.

In our math classes, we do a lot of geometric activities, play logic games, move around, but you can’t say that we don’t do arithmetic at all.

What we actually do very little of is calligraphy with numbers. We don’t write much – instead, we draw diagrams of various structures.

But we do some counting during every class.

We don’t write numbers in our notebooks – we count on our fingers, count steps, count animals on cards or holes in Numicon frames.

To master counting within ten we actively use our fingers,

and card games such as HurriCount.

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The condition: there must be fewer hedgehogs than frogs. What cards need to be hidden, what needs to change in the open cards to meet the condition?

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We actively use Numicon – counting frames with holes

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you can clearly see that two odds add up to an even number!

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snap cubes

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the game “Many-Many”

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we also like base 10 blocks

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Or we make batches of 10 matches or 10 craft sticks (using rubber bands).  Or we build towers of 10.  It’s much easier for children to understand tens and hundreds that way!

Number twister for the fingers.

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You can practice addition or multiplication.

Put your fingers on two circles so that the sum is 19.  And now, same question, but you can’t use the number 9!

We also use dice a lot: roll a die and glue that many stickers onto a page, or draw that many petals on a flower, or color in that many cars on a train.

Our favorite card games include “Fruktazh” and “Kotosovy”, where you have to quickly count and recognize quanities (these games, like HurriCount and Many-Many are made by the Russian company “The Brainy Band” but are not currently available in English).

We also practice counting things that you can’t touch, for example “meow-meow” means touching your cheek twice like a kitty and “kar-kar-kar” means wave your arms three times like a bird.

In many of our active games, we like to start by saying that the kids have to stop (or hide) when the leader says, for example, “eight”. Then, when the leader says “twelve”. Then, after the leader says 26, but counts by twos. And only then will we choose one of the kids as a leader, and suggest that those who know how, count by twos.  
What are your favorite games with counting and mental math?

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Playing with symmetry in kindergarten

In our kindergarten classroom, there is a big mirror that is usually covered by a giant map.  And for good reason!  As soon as the students find themselves anywhere near the mirror, it attracts them like a magnet.  They make faces at it, wave at it, and just stare at their reflections.  They are fascinated by it.

This week, however, I began my math lessons by taking down the map and thus exposing the mirror.  I gave the students 5 minutes of free time in front of the mirror, but I told them to make lots of different motions and observe their reflection carefully.  I was going to ask them questions after the five minutes were over.

Here are some of the questions I asked:

  1. What does your reflection do when you wave your right hand at it?
  2. What does it do when you take a step forward?
  3. What does it do when you jump to the left?
  4. What does it do when you stick your tongue out at it?

The students had no trouble answering the last question and had a good laugh when I asked it.  However, for all the other questions, the answers varied.  We had to all stand in front of the mirror again and do the motions together.  There was still a bit of discussion about which hand the reflection waves when you wave your right hand, but quickly everyone was converted to the view that it was the left hand.

And then we moved on to some acting.  The students went up in front of the class, two at a time, and one person had to act as the other one’s mirror image.  Overall, they did a great job, but the hardest part was getting the non-reflection person to make their movements slowly.

Here are the students in action:

And then we made symmetric pictures on a checker board

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drew second halves of symmetric images by folding them and holding them up to the window

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colored them in, symmetrically

played “reflect my picture” using pattern blocks

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Sometimes, when the students weren’t sure whether they made the reflection correctly

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we checked it by using a mirror

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and that helped them correct their mistakes.

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Some students got creative and stacked their pattern blocks vertically.

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Symmetry is one of my favorite topics to explore with young children.  One other activity on this topic that I did not do this time around but a number of these children saw in the past is mirror books, described, for example, here.

What are your favorite symmetry activities?

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The joys of peas and toothpicks for all ages!

Translation of this post by Jane Kats.

A building set from peas and toothpicks is a simple and winning proposition. I can keep building from it myself again and again, inventing new structures every time.

You can make platonic solids, or you build a house or a rocket, a bird or an animal.

This is my favorite setup – where a free-style creative activity provides “development” on its own, without additional directions or instructions.

If you decide to build a cube, you will figure out the angles at which you have to put in the toothpick into the pea in the process of building. If you want to build an octahedron like the kid next to you, you will have to study it carefully, understand the shape of its faces and count its edges, figure out the angles at which the toothpicks connect.

For our math festivals, we soak 9-11 pounds of peas and buy more than 20 thousand toothpicks!

And like with any other building material or construction set, if there is a lot of it and many builders, then the buildings turn out more interesting than when there are only one or two builders.

When one person starts building a multistory building, her neighbors also get inspired to build something similar.

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the buildings are growing

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building houses is a good place to start!

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some windmills appear

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and rockets

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and dinosaurs!

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a tetrahedron and an octahedron

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a crown

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Parents are also excited to build with the kids.

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The structures aren’t fitting on the dedicated tables and shelves anymore!

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We have had this station at many of our math festivals and we aren’t tired of it! Some children stay with the toothpicks and chickpeas for over an hour. The parents get tired and try to move on, but the child continues his experiments with shapes and angles. Don’t rush them, they are learning math!

 

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Math enrichment – what is the value?

Another guest post by Yulia Shpilman on why we teach enrichment math and what it means to us.

When you look for additional math classes for your child, what do you look for?  We (and by we I mean Golden Key Russian School teachers) are often asked about what exactly we offer in our unique math classes and what skills we teach our students.  I think that we have not always been successful in articulating clearly and concisely what we actually do and why we do it.

Questions we get most often are: (1) Are we like Kumon? (we’re pretty much the opposite) (2) Do we teach “Russian math”? (not even sure what that is, though starting to realize people are probably referring to RSM) (3) Do we tutor kids who need help with their school math homework? (not really) (4) Do we train students for Kangaroo Math? (somewhat, in that we solve interesting problems, but that’s not the main goal of the program).

Well, what is it that we do, then? At GKRS, we invite the students to join us for enrichment math classes, and we take the word “enrichment” seriously in that we truly want the activities in the class to enrich their lives and be an enjoyable experience. That said, in our epoch of over-scheduled kids, any extracurricular activity must provide additional value beyond just pure entertainment (if you can even believe that math classes can be entertaining, which ours very much are 😊).

What we strive to teach our students, first and foremost, is the skill (and art) of problem solving.  

Why do we focus on problem solving?  Well, it’s a skill that is very broadly applicable to modern everyday life, regardless of whether you’re a mathematician, an engineer, a scientist or a writer.  All of us, on a daily basis, are faced with problems the answers to which are unknown and the approaches to solving which are undefined.  Both big societal problems like building the driver-less car, finding a cure for cancer or improving our education system, or smaller problems you might face in your job like figuring out a way to improve a production process or market something effectively on social media, don’t have a defined path to an answer.  It takes creativity, grit and a lot of trial and error to figure them out.  

If you think about it, traditional math classes in school teach the exact opposite lesson! There is always one (and only one!) correct answer, and you arrive at it by a set of predefined steps that you must memorize and then apply at just the right moment. This couldn’t be further from reality of most of today’s careers in just about any field.

But how do you become good at these skills? How do you develop creativity? How do you learn to persevere after trying several failed approaches? How do you know if your result makes sense? The answer is, of course, practice, and a lot of it, over long, continuous stretches of time.

And so, we will not guarantee that your child will win a math Olympiad or will manage to do mental math very quickly in her head (although over time, there will be improvements on these fronts also). That said, we can promise that she will learn to be excited, not scared, when she sees a new, unfamiliar problem, in math or other areas of her life. She will learn to analyze the information that she is given, think about what she might already know about it from previous experience (pattern recognition), try an approach and course-correct based on the results.  And we will argue that there’s tremendous value in that.

So back to my original question: what do you look for when you look for additional math classes for your child? I am most curious to hear from parents of elementary school children, but also from parents of older kids.

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