## Skip counting or word skipping?

We like games at our house, but there are rules that one has to follow when using them.  And the rules are not just for playing the games (those we actually often modify at will), but there are also rules for putting it away (which are stricter).

One of the rules when putting away the game BLOKUS (a favorite) is that you have to count the pieces and make sure that there are 21 of each color.  Today, after playing with the game, Zoe (4.5 yo) diligently picked up a bag and started putting blue pieces into it.  She counted normally until 7, but then she put in the next piece and said “nine”.

I naturally thought that she just forgot about 8 and pointed out to her that she skipped it.

“I’m counting by 2’s now,” she said.

“But you put in only one piece,” I countered.

Zoe looked at me with surprise.  Without any words, her look said it all:  “Yes. And?”

“When you count by 2’s,” I explained, “you have to put in two pieces.”

“Oh,” said Zoe, and took out the piece that she had just put in.  Without any questions, she picked up a second piece, put them both into the bag and triumphantly said “nine” one more time.

Zoe then proceeded to count the remaining pieces by 2’s, making sure to put in two pieces each time.

I was surprised that Zoe didn’t ask me why she had to put in 2 pieces at a time.  It must have made sense to her.  And yet, she was perfectly happy counting by 2’s and putting in one piece before I pointed this out.

I have actually seen this phenomena before.  Kids understand the concept of counting by 2’s and are good at figuring out the sequence of words, but then when it comes to counting physical objects they don’t connect it to counting 2 of them at a time.

I think that Zoe understood the connection this time, but I won’t be surprised if she makes the mistake again before fully internalizing it.

What similar misconceptions have you noticed children having?

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## Numicon – combining geometry and arithmetic

Translated from this post by Jane Kats.

This year, we are actively using the Numicon tool in our classes.  We have two sets like this one – and it’s plenty for a group of 12. (Translator’s note – here is a slightly smaller set more readily available on amazon).

Sometimes I ask the children to bring me two pieces of different colors and count how many holes there are altogether.

And recently, we drew another set of puzzles for Numicon.

You have to find two different Numicon pieces to make each shape.  (Translator’s note – instructions on the sheet say “Put this shape together using two different pieces of Numicon”).

You need to pick the pieces, see if they fit and count the holes.

It’s interesting that when kids use the same two Numicon pieces for a different shape, they still have to recount the holes.

It’s a nice bonus that many of these problems have 2-3 solutions.

How do you play with Numicon?

## Discussing the meaning of “almost” with an 8 year old

I had previously mentioned that my children are very picky and slow eaters.  This creates much frustration for me, but at the same time has been on multiple occasions inspiration for interesting math conversations.

Yesterday, Katie was eating a cutlet for dinner, and I, as usual, was trying to make her hurry up a bit.

K: I’m almost done with it.

I look at the cutlet and notice that there is still almost half of it left.

Me: What fraction of the cutlet do you think must be remaining for you to be “almost done with it”?

K: I think a third.

Me: Why a third?

K: Because a third feels so much smaller than a half.

Me: Well, you definitely have more than a third left.

K: (taking a big bite) Not any more.

She went back to chewing, very slowly of course, and I thought the conversation was over.  But about 5 minutes later, when she was still not finished with the cutlet, Katie suddenly decided to resume the conversation.

K: Mom, if someone could eat the whole cutlet in just one bite, could they say that they’re almost done with it before they take the bite?

Me: I don’t know.  Can you be almost done with something before you even start?

K: Because if they can’t say it before they take the bite, they can never say it, because after taking the bite they will be fully done with the cutlet!

I couldn’t argue with her there.

So what do you think, does “being almost done with a task” depend on how quickly you do it or just on what fraction of it you have already done?

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## From math-hater to mathematician and computer scientist: The story of Allison Bishop

This is the story of an unlikely mathematician. It is the story of my friend and Funville Adventures coauthor Allison Bishop, who always imagined she’d write a book someday, but would not have guessed it would be about math.

Allison grew up with a strong dislike for math.  She was by no means bad at it and had no trouble getting A’s in all her math classes.  She just found it dull, formulaic, and uncreative.  What she loved was writing, where you could use your imagination and create anything you wanted.

It didn’t help that she never cared for most of her math teachers.  There was even one high school math teacher that felt that girls had no business doing math.  When a girl asked a question in his classroom, his reply was, “You will not need to know that when you are a housewife.”

That’s why when Allison began her undergraduate studies at Princeton, math was not anywhere to be seen on her list of potential majors.  She was leaning towards English, although considering some other subjects in the humanities.

Yet somehow, she found herself being convinced to take a math class her Freshman year.  She didn’t have high hopes for it, but thought that she’d give math one last chance.  The class was an Introduction to proofs through Number Theory taught by Jordan Ellenberg.

It didn’t take long for Allison to fall in love with the class.  But how could it be that this was math?  It was so different from anything she had seen called math before.  It was exciting, thought-provoking, and required quite a bit of creativity.  Was this a fluke?  Allison had to try out a few more math classes to be sure.

By the end of her sophomore year, Allison knew that she wanted to become a math major.  She went on to take many more fascinating math classes and to graduate with high honors in math.

She set out to find compelling uses of math in the wider world, and wandered into computer science research as a result. After obtaining her Phd, Allison became a computer science professor at Columbia University. Though she loves teaching, she is not content to work only with students who have already found their way to math and science. She wants to help more young minds discover the creative side of mathematics, and waiting for college is often too late.

Allison joined the Funville Adventures project because she believes that mathematics needs more writers, and writers need more mathematics. The kind of thinking muscles that mathematical studies develop are sorely needed in today’s world, and the creativity that lies behind the shape of a theorem is not so different from that behind the shape of a story. And every little girl, and little boy, should get to grow up loving math. Because in addition to enriching their lives, they will need to know it after all.

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## A number talk that turned into an investigation

It all started out with

## 12+12+12

a problem that was not particularly difficult for any of my 1st/2nd grade students.  The students took turns suggesting strategies for doing this computation and I recorded them on the board (this was an example of a “number talk”, something that we do on a regular basis at the beginning of class).

However, one of the younger student’s in the class, made a typical mistake for his age of getting the digits reversed in his head.  So instead, he solved the problem

## 21+21+21,

correctly, I must add.  Little did he know that his mistake would spur a discussion that would continue over several days (he was very proud of this afterwards).

Here is what the board looked like at the end of this first number talk.

Except that it wasn’t the end at all!  After looking at the board for a moment, one student raised a hand and voiced an observation – he noticed that 63=21+21+21 is the reverse of 36=12+12+12, just like 21 is the reverse of 12.  “Does that always happen?” he wanted to know.

Some students immediately said “yes”, while others said “no”.  And so the investigation began.

I asked them what other number they wanted to try it out on first.  Someone suggested 15.  We tried it, and it didn’t work!  Then someone suggested that the number had to be smaller than 15, and we tried 14.  That didn’t work either.

Now that it was clear that the property doesn’t always hold, the students started hypothesizing when it does hold.

“It works only for numbers smaller than 14,” one student suggested.

“But it worked for 21,” another one countered.

They tried a few more numbers, and the property did not hold.  At this point, most of the students had the sense that in order for the property to hold, the numbers couldn’t be too large.  But it was time to move on to a different topic.  I promised that we’d come back to this problem soon.

The next day, we had the following number talk

It wasn’t long before someone commented that the number 23 satisfied the property from the day before.  So the students did a bit more investigating.

And now the story should end with them figuring out exactly for which 2-digit numbers the reverse of three times the number equals three time the reverse of the number.  But they didn’t – not quite.

I thought I heard someone say something about the digits at some point, but when I asked them to repeat it, they weren’t quite sure what they had just said.

I told them that we would come back to this problem again soon.  Some students wanted to know the answer, while others really wanted to figure it out for themselves.

To be continued…perhaps.

What is your earliest memory of working on a problem for more than one day?

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## Funville Adventures live on Kickstarter!

Funville Adventures, our math-inspired children’s fantasy with Allison Bishop, had a great first day on Kickstarter!  We are very excited by the interest and support that it is getting!  You can check it out here.

Our goal with this project is to introduce children to mathematical concepts in a story-based form, appealing to their natural creativity and love of stories. We believe this will help reach not only children who already love math, but even those who may have struggled with typical classroom presentations.

And here’s a small preview of the story with some pictures!

After 9-year old Emmy and her 5-year old brother Leo go down an abandoned dilapidated slide, they are magically transported into Funville: a land inhabited by ordinary looking beings, each with a unique power to transform objects. The siblings discover that Funville is full of surprises; one never knows when something will be shrunk

flipped upside down

turned into an elephant

or erased!

Find out more about the story in the book and the story behind the book by vising our campaign.  If you like what you see, please consider backing the campaign and helping us spread the word by telling/emailing your friends and sharing on social media.

And don’t miss out on watching our video – a lot of love and effort went into making it!

## Logical Fun, Part II

Here is the long overdue follow-up post to Logical Fun, Part I.

The previous post left off with the following logic problem being posed to the students:

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?

As soon as I read the problem, a number of students (about a third of the class) started saying that they know the answer because they heard the problem before.  I encouraged them to still think through it and make sure that what they thought was the answer made sense.

Interestingly, everyone who thought they knew the answer from before, remembered it incorrectly (or perhaps they had heard a different, similar problem).

The students were then told to discuss their thoughts in small groups.  As with several previous problems, some students immediately started thinking outside the box and asking insightful questions.

One student brought up, “The problem says brothers and sisters have I none.  What if he had them and they passed away?” We had to agree that in this case we are going to take “have” to mean “never had”.

Another student asked about half-siblings and step-siblings.  After some discussion, we decided that half-siblings count as siblings and step-siblings do not affect the answer.

When we finally started discussing the problem as a class, we had three proposed answers: the man himself, his father, and his son.  Only one person voted for each of the last 2, and the rest voted for the man himself (a few abstained, saying they weren’t sure).

I then asked them who “my father’s son” is.  They all agreed that it must be the man himself.  So the problem became:

“Brothers and sisters have I none, but this man’s father is me.”

At this point, most of the class said that they wanted to change their vote to “the man’s son”.  However, we still had a few that insisted that the answer must be “the man himself”.

I decided not to say anything else myself, but let the students discuss it among themselves for a bit longer and convince each other.

In the end, we had one non-convert whose reasoning was “I heard this problem before and I’m sure that the answer was the man himself, so I am sticking with that.”

It was hard to argue with that, but I did tell him to try to reason it out for himself and not to rely on his memory or what he was told by others.

I already mentioned in the previous post how impressed and happy I was with the students’ freedom of thought and lack of fear.  I can’t say that the discussions were perfect, without any yelling or hurt feelings.  But overall, the class was a very enjoyable experience, definitely for me, and I believe for the students as well.

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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:

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.

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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.

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

Some kids filled out the same row and column together

Someone started with squares

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

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!

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).

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

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

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

but it won’t happen right away.

not yet

this is getting closer

even even closer

Got it! Someone figured it out!

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?

## 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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