Numbers on a Line

A few weeks ago, we spent several days exploring and playing with number lines in our Kindergarten math class.  Before introducing the number line, I wanted to surprise the students by telling them about numbers less than zero.  I went around the room, giving the students numbers and telling them to name the number that is one less.  At some point, I pointed at a student and said the number “zero”.  However, instead of the anticipated laughter and comment that there are no numbers less than zero, the kid said “minus one”.  Moreover, none of the other students seemed surprised.  I don’t think that all of them have heard of negative numbers before, but they somehow right away accepted the fact that there is a number called minus one and that it is one less than zero.

And then we moved on to activities involving number lines of various sizes.

We solved addition and subtraction problems by jumping on a giant number line that took up most of the gym!



We played several games of chance on a medium-sized number line:



At some point we played a game that could be won by getting to either 20 or -20.  However, whenever the children rolled a plus and got to move towards bigger numbers they’d get excited and whenever they rolled a minus and had to move towards smaller numbers they’d get disappointed, regardless of where they were standing!  And no matter how many times I’d point out that getting to -20 was a win, I couldn’t change their attitude that “minus is bad” and “plus is good.”  When I later played the same game with the second graders, however, they didn’t have this attitude at all.  Once they were told the object of the game, they were happy to be moving away from zero in any direction.

We finished off the week by using small number lines, drawn on paper, to write down and solve some addition and subtraction problems.  At some point I asked the students why they thought it was called a number line.  They gave me a “are you seriously asking this question?” look and then replied almost in unison, “because there are numbers on a line!”  Not sure what exactly I expected there🙂.  I’m thinking of coming back to the topic of number lines later in the year and having kids make their own.


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Great Mathematicians, Old and Young

Almost every week, I read or tell my second grade class a story about a famous mathematician.  Recently, we have been reading from the book Mathematicians Are People Too: Stories from the Lives of Great Mathematicians and the students have been really enjoying the stories – they ask interesting questions and have lively discussions.

Last week we read about Blaise Pascal.  The students were intrigued by how his father hid math and science books from him, by his invention of the first “calculator” and of course by Pascal’s Triangle.  Exploring the triangle seemed like a very natural follow-up activity.

I printed out a mostly blank Pascal’s Triangle with just the first few rows filled in and with “spots” for the first 15 rows.  The goal was for the students to fill in a few more rows and then I’d give them already filled in triangles for them to look for patterns and color them in.  However, we never made it to this second part because the students got so into filling in the entries that they refused to stop.

I was very impressed with how hard they were willing to work to compute the entries.  I was also pleased to see that most of them noticed the symmetry at some point and used it to their advantage.  I think that for children that are still mastering 2-digit addition, making it into the thousands with minor help is a great feat!





I guess I will have to show them the cool patterns some other time (in fact, I hope to come back to it multiple times because there is so much to explore about this beautiful triangle).  Oh, and at the end they insisted on taking the triangles home with them because they wanted to continue working on them!

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I come before you!

Today I had two really fun lessons on the same topic, one with the kindergartners and the second one with the 2nd and 3rd graders combined.  The topic was “when does order matter?” and I began both classes with a favorite discussion of mine about putting on clothes in the morning.  Does it matter in which order you put on your shirt and your pants?  What about your socks and your shoes?  I’ve noticed that these questions tend to universally put kids into a good mood.

We then moved on to building towers out of blocks.  The kindergartners had to build a tower out of four different colored blocks, draw it, and then draw a diagram that summarized the orders in which the blocks could be placed.  This consisted of first drawing the four pieces in a circle and then drawing arrows according to the following rule: there is an arrow from block A to block B if block A has to be placed before block B.  This represented block A saying to block B, “I come before you!” and was accompanied by a finger-pointing motion that the kids found very amusing.

Here are some pictures from K:




The second and third graders worked in pairs to build towers out of 5 blocks, draw their diagrams, and then exchange those diagrams with other pairs.  They then had to build towers according to the diagrams and draw their solutions.  The towers were often not identical to the originals, and so we got to discuss how different towers can have the same diagram.







It was interesting to see that the hardest diagrams for the students seemed to be the ones with the fewest arrows.  It would take them some time to figure out how one can have several blocks with no arrows pointing at them.  Since the students were working in pairs, there was also much lively discussion going on.  Overall, the students were very engaged and seemed to quite enjoy the activity.  They also got to practice many important skills: working together, reasoning, making diagrams, determining which details are important to the task.

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Why learn poetry by heart?

[This is a guest post by Yulia Shpilman.  It is not about math, and therefore a bit unusual for this blog.  However, I feel that many of the reasons and advice that she gives for learning poetry apply to learning math as well.]

We are often asked the question of why we at the Golden Key Russian School ask our students to memorize Russian poetry by heart. Whether you’re a poetry lover or not, the consensus is that it’s something that’s “good for you,” like eating spinach or practicing your multiplication tables. At the same time, it’s challenging and sometimes downright painful for the kids, so is it really worth it?

I believe the answer is a resounding yes, for many reasons (which I believe apply to poetry in English just as well):

Huge sense of pride at accomplishing something difficult 

You may be surprised to learn that, with few exceptions, our students are excited to go in front of the classroom and recite their poems. It actually makes perfect sense – they have worked very hard on something and now have a chance to show it off.

Exposure to vocabulary that they wouldn’t otherwise see or ever say out loud

Chances are, you aren’t using words like чахлая (feeble), лазурный (azure) or праздный (indolent) in everyday conversations with your elementary school age children because you assume they don’t know what it means. That is probably true, but when that word appears in a poem that your child is learning by heart, you suddenly have a reason to explain its meaning. Moreover, after repeating that word fifteen times, your child just might have a better chance of remembering and using it as part of his active vocabulary.

Deeper understanding than you get from just cursory reading of a poem once

The first time a child read a difficult poem with many unfamiliar words, its meaning is likely to go her head. As she begins reading (or hearing) it over and over again to commit it to memory, she starts internalizing it, making connections and coming up with interpretations that would not be possible if she had read the poem once and moved on.

Development of good taste and appreciation for beauty and cleverness of the written word

This isn’t something a child born with – good taste in literature, in music, in art needs to be cultivated and developed through gradual, age-appropriate exposure. Learning poetry by heart helps children better understand and appreciate its beauty, the nuances of the poet’s word choices and its overall meaning, helping them become better and stronger readers and “consumers of literature.”

I don’t subscribe to the argument that one needs to memorize poetry to train her memory. If that was the only objective, there are many other ways to accomplish it and all kinds of things to be memorized, from the multiplication table to historical dates, and I am not sure that poetry is superior to them with regard to memory training.

As for making the experience as enjoyable as possible, here are some suggestions.

  • Allow the child to choose the poem that she’d like to learn – give her several options and let her pick what speaks the most to her (and reject those that aren’t appealing).
  • Work on short bits at a time, so both you and your child can feel like you’re making progress
  • Don’t leave the task to the last minute – memorizing a poem is much harder under the pressure of a time constraint. Instead, make it a part of your daily routine – your drive to and from school or activities is a great time to work on memorizing a poem.
  • Give them background and context on the poem, but let them offer their own interpretation, which may change and evolve over time.

When a child learns a poem by heart, it becomes a part of her treasure, her arsenal against ignorance, her answer to a difficult challenge. We at GKRS are grateful for the opportunity to give this gift to our students.

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Math should be felt with your hands: Interview with Jane Kats

If you ask me whose philosophies on elementary school teaching have influenced me the most, I will answer without pause: Jane Kats.  Jane is a math educator, the author of numerous books and workbooks for educators, parents and children, and a general math enthusiast.  Meeting and interacting with Jane, as well as reading her books, has revolutionized my views on elementary school mathematics and how it should be taught to students.  I’ve had the privilege of having many conversations with Jane and watching her teach.  Recently, I’ve had the chance to interview Jane, and today I’d like to share a bit of her wisdom.

Jane’s views on the main goals and essential components of an elementary school curriculum.

Breadth of topics: In elementary school, everyone usually devotes a lot of time to arithmetic, making sure kids know their digits and how to write them without confusing right and left.  But in addition to that, there is reasoning and topics such as logic and geometry that are just as important.  Many topics in geometry are not given to young children whereas the kids can handle them.  Then the children get to high school geometry and go “Ah, how is this so?”  And the same   problem occurs with a number of other topics.  For example, we first tell kids that we can only subtract smaller numbers from bigger ones.  But then it turns out that you can take away 8 from 5 and get -3.  And the kid goes, “How is this possible?  You can’t take away a bigger number from a smaller one!”  because initially the kid is told one thing and then they are told the exact opposite.”  Young children should get exposed to these things as well.

Understanding, not memorization: But the most significant thing is for children to realize that mathematics is not something that can be memorized but something that needs to be understood.  If we’re talking about arithmetic, then math is something that should be felt with your hands, so it is important not to rush as quickly as possible to writing down the number 21 using digits, even if the kid is capable of doing it.  It’s better for the kid to see it visually first, say as two sticks of 10 and 1 little cube. Then the kid understands it better. The longer a kid spends with concrete counting materials, the stronger a foundation they’ll have.

The same goes for addition and multiplication facts – they do not need to be memorized.  Because when a kid knows that 6+4=10, she does not recognize 3+3+4 as 10.  Because it is a different fact, one she did not learn.  And 3+4+3 is yet another, completely different fact.  In games like Turboschet children can see that it’s the same end result.  When you memorize something, then it’s a different kind of knowing; one that works in math class but not in life.  Various games are better at teaching how it is in life.

Kids can come up with the problems too: It is also important to show children that math isn’t something where the adult always knows the answer – that there exist problems that a kid can make up for an adult and they will be interesting.  For example, “Here, I drew for you the view from the front, what can the tower look like from the side?  Now let’s take turns making up problems for each other: you give me one, I give you one.” So that there’s not this impression that the adult is the only one who can come up with problems and the kid is the only one who can solve them.  

Varied problems with multiple solutions: It is wrong when a child associates math with columns of exercises.  Plus, minus, plus, minus.  Not only that, but they are all exercises where only two numbers are being added.  Then when such a child sees a problem with three addends they go, “How is this?  This can’t be.” It is also is important for a child to encounter problems that have several correct answers and problems that do not have an answer. For example, “List all numbers that are divisible by 5 that are more than 6 and less than 30.  Now list all numbers divisible by 5 that are bigger than 3 and smaller than 5.  Those don’t exist?  You are correct, those don’t exist.”

Hands-on exploration of math: One of Jane’s motto’s is, “Everything needs to be explored in a hands-on way.” In elementary school, students need to learn how to count and understand the base 10 system. But in addition to that, a program should involve exploring these concepts on fingers, cubes, beads, and any other concrete materials longer than is customary in most school programs. There needs to be a lot of repetition, but it needs to be in varied forms.  For example Jane likes to do with kids things like when she says “Kar” you wave your hands, when she says woof you jump and when she says meow you wipe your mouth like a cat.  “Now let’s do kar, kar, woof, woof, meow.  How many total movements did we do?”  Math needs to be explored with the whole body and not just be all abstract and virtual.  In another activity, for each meow the children take a red stick, for each woof a yellow stick and for each kar a blue stick. Then they may have to create the sequence meow, meow, woof, woof, kar again and exhibit it as a picture. Children need to learn to translate between these different “languages”

Taught by math enthusiasts: Math worksheets with redundant exercises induce nausea and not the love of math.  The goal should not be to force-feed the kids mathematics.  The goal should be to find some areas of math that will make them go “wow”.  Different teachers love different topics, and that is normal.  If a teacher loves a topic they will present it to the kids in exciting ways and the kids will say, “that sounds cool, I want to do that too.” For example, Jane can spend hours cutting up a Möbius strip with the kids and won’t get bored.  Every adult has their own favorite ages to work with and their own favorite topics, and that’s okay.  If the adult isn’t fascinated by what she’s teaching, the kid won’t get interested either. If a child talks to different teachers, then someone will tell him about the Möbius strip, another one about geometry, a third about graphs and a fourth will show arithmetic tricks.  And that is good. It is rare that a kid is equally good at all areas of math.  But if they find something that they take pleasure in, that’s already good.

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What is a pattern?

This week in Kindergarten our focus was on patterns.  I began by asking the kids whether they knew what a pattern was.  All said “yes” in unison.  So I called on someone to tell me.  “It’s when you have one color and then another color, and then the one color, and then another.”  Does it have to be color?  The student said that it does.  But other students disagreed.  “It could also be shapes,” a second student said.  Everyone agreed that you could alternate shapes instead of colors.  Anything else?  No, just colors and shapes.

I called on a kid and asked her to continue the pattern: woof, meow, woof, meow.  Not surprisingly, she had no trouble continuing.  The next kid was asked to continue the pattern clap, stomp, stomp, clap, stomp, stomp…  Again, this was pretty easy.  We continued with a few more variations on the theme, mostly using sounds and physical motions.  Sometimes the students would get confused, but for the most part they were able to continue.

Next, we moved on to doing some patterns on paper, and I was a bit surprised to see that these were much harder for the kids.  The “patterns” were no harder than what we had done with the sounds, mostly ABAB, ABCABC, and ABBABB, but half of the kids had to say the names of the objects out loud to spot the pattern and figure out what should come next.

I began day 2 by asking the students whether a pattern always has to repeat.  I was not surprised to hear a unanimous “yes”.  I then asked the kids one by one to continue the following patterns: 1, 2, 3, 4…;   2, 4, 6, 8…;   10, 20, 30,…;   7, 6, 5, 4.  They had no trouble with any of these.

Then came the most fun part: letting kids make patterns of their own using snap cubes.


We finished off the week with tessellations.  We read Emily Grosvenor’s book Tessellation!   The children loved the pictures of tessellating nature.  This one was voted as the favorite:


Another image that they explored for a very long time, even before we started reading the story, was this one:


They said that it looked like a beehive and also like flowers (a hexagon and all the ones that touch it make a flower).  They also wanted to count all the hexagons and had a discussion about whether it is easier to count them left to right or top to bottom.

Next week I plan on having them look for patterns on the big hundreds chart that is hanging in their classroom.  I am also looking for other ideas.  How would you explore patterns with Kindergartners?





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As many opinions as there are shapes: fun with “Which one doesn’t belong – A shapes book”

Last week, the main theme of both my Kindergarten and my second grade math classes was geometry.  We did a number of activities that explored various shapes and their properties, but the biggest hit in both classes was Which one doesn’t belong: A shapes book by Christopher Danielson.

Each page of the book consists of four shapes and a repeating question: which one does not belong?   The answer is also always the same: all of them, but for different reasons.  Doesn’t sound very interesting?  That’s because the real answers are the reasons and observations that the kids come up with, and these are truly wonderful and creative.

Here are some highlights from the discussions around two of the pages from both age groups.

Page I:


On this page, the Kindergartners right away noticed the big square and pointed it out as different, whereas with the second graders this was the last shape that they found a reason for. In both classes someone pointed to the rectangle and said that it has two short sides and two long sides.  In the second grade group, one kid also pointed out that the rectangle is the only non-square.  

It was interesting that it took both groups some time to notice that one shape was a different color (perhaps because they weren’t thinking about color as a “geometric” reason?).  In K, someone even pointed to it and said “the blue one is different because…” and then they couldn’t articulate a reason.  After a pause I couldn’t help myself and suggested, “maybe because it’s blue?”  The kid got very excited and agreed.  When one of the 2nd graders pointed out that the shape was blue another one said, “Color doesn’t count.”  I had to gently point out that every reason counts and all observations are welcome.  When the kid kept insisting that he won’t count that, I told him that he was welcome to come up with a different reason as well.

The bottom left figure didn’t belong because it was a diamond (same formulation in both age groups).  I asked the kids what makes it a diamond and the reply was, “it’s standing on it’s corner.”  (Here the kid that didn’t want to count color as a difference said that this shape was just another square that was turned around, so he wasn’t going to count that either.  I told him that he was making interesting observations but that he has to be respectful of other people’s reasons.)

Page II:



This one was a little more complicated for the Kindergartners, but they were truly fascinated by the shapes.  I also loved one kid’s observation that the top two are essentially the same shape but the heart is more curvy.  The other reasons they came up with were: the bottom left is different because it has lots of pointy ends and the bottom right is different because it has circles.  

The second graders had a ton of fun with this one.  Some of their reasons:

  1. The bottom left is the only one without something pointing in (what better way to introduce convexity?).  
  2.  It also looks like a cupcake.
  3.  The bottom right is the only one with 4 identical sides.  
  4.  It is also the only one with circles for its corners.  
  5. The top right is the only one with all straight lines.  
  6. The top two are the only ones that have very pointy corners.  
  7. The bottom left is the only one not with 4 sides.

There was a fun discussion about how many sides the heart and the cupcake have. Someone argued that they both have 4 (one kid said, “I know it’s not a straight line, but it’s like a side”).  Others said that the heart had none.  Still others thought that you couldn’t tell.  

When we work on problems with the second graders, the most common question I hear from them is “Is this right?”  One of the many things that I loved about this activity was that I did not hear this question a single time.

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