Author Archives: Golden Key Russian School

Guess and Check

Kids really like guessing games.  However, when they are very little they cannot tell the difference between random guessing and estimation based on partial information.  For example, you might play a game with them where you show them some number of objects and ask them to “guess” how many there are without counting.  When they inevitably then ask to give you a problem in return, they will hide their objects from you and ask you to solve the analogous problem.

As they get older, they will be able to make the distinction, although they may still enjoy games of both types.  Whereas the pure guessing games can be amusing, the games involving estimation are both fun and teach a valuable life skill.

To practice estimation, we played a simple game in our last class. In a large room, a stuffed dog was placed some distance away from where the kids was standing.  The task of each child was to predict how many steps it would take him/her to reach the dog.  Overall, the kids were quite good at estimating the number of steps.  Occasionally, someone would say a really large number, like 100, but this was mostly done for general amusement and did not reflect what the kids were thinking.  It was also amusing to see how some kids really wanted their predictions to be accurate and they would adjust the size of their steps as they were getting closer. (We tried to control the size of the steps that the kids took and suggested that they they put one foot in front of the other but it didn’t always work)

At some point, it took two kids an identical number of steps to reach the dog, even though one of them was standing noticeably closer.  When we asked the kids to explain how this was possible, a few immediately pointed out that one of the walkers had larger feet.



Geometry with Geoboards

Last Sunday we also played with geoboards.

We first asked the kids to make a triangle, and it turned out that they come in many varieties:

ImageImageWe then had them build quadrilaterals.  Here’s a sample:

ImageWe briefly discussed some terminology such as square, rectangle, parallelogram, trapezoid, convex…  The first two were of course familiar terms, and a few kids had heard of a trapezoid but many would struggle to define them if asked to describe the difference.  Some kids would also make comments such as “I made the largest quadrilateral possible” or “I made the smallest one”.  We did not yet discuss the concept of area with them, but I think that we will in the near future.

We then moved on to slightly more challenging assignments.  We would tell the kids to make a figure with exactly one interior point, or exactly five points on the boundary.  For the older kids we eventually would put conditions on both interior and boundary points.

Not surprisingly, geoboards are a great tool for introducing and exploring basic geometry concepts.  In the future, we also hope to use them to continue with the theme of geometry of numbers.

Multiplication before addition?

There are ten people in the room.  How many eyes are there?  How many legs?  What about arms and legs combined?  Fingers?  Our older group didn’t have much trouble with this – they were pretty good at counting by 10’s to get to the answer.

It was next established that out of the 10 people, 7 are kids.  How many kids’ eyes are there?  In theory, this question shouldn’t be much harder than the previous ones, but all of a sudden some kids were trying to count the eyes by 1’s, others by 2’s, and both were making mistakes (probably because everyone was sitting in a circle and the adults are interspersed among the kids).  Finally, someone attempted to add 7+7, and after a mistake or two, the right answer was pronounced.

The actual goal of the lesson was to start informally introducing multiplication and division (for division, the kids worked in small groups and divided various amounts of glass beads amongst themselves).  And, on one hand, it is not necessary to be stellar at addition and subtraction to understand these ideas on a conceptual level.  However, in order to actually do many interesting things with the concepts, it helps when the kids are comfortable with counting at least by every number up to ten.  So does it make sense to introduce multiplication before the kids have mastered addition?

You could go either way on this one.  It turns out that mental math, even within the realm of 20, is a weakness across the board in our groups. And while doing addition drills isn’t exactly in line with our approach to making math fun, getting comfortable with mental math is a critical skill that will enable deeper exploration of many concepts (and will be really impressive to teachers and peers alike). And thus, we’re looking for ideas on how to get comfortable with addition and subtraction for 5-7 year olds in a fun way.  But it’s unclear that we should wait until they do indeed master this skill before introducing more complicated and interesting concepts.  Introducing notions like multiplication and division on a conceptual level may help them better understand the ideas behind these concepts instead of simply memorizing the times tables.  So for now, we’re putting up with counting on fingers and really entertaining mistakes.  Stay tuned for our progress.

A quick update

You may have noticed that after a long hiatus, we have had a few new posts in the past week.  That is because I finally have help in the form of my wonderful sister, who has agreed to not only help me teach my Sunday math classes but also to help me keep up the blog.  The last two posts were authored by her, and I promise there will be more to come, as we describe how our classes are going, the topics we’re covering and how the kids do with them.  The posts will predominantly be in English with an occasional sprinkling of Russian.  And when possible, we’ll post interesting materials that you may find useful (both from this semester, and from an entire fall full of classes when I didn’t have time to keep up the blog).  It’s nice to be back!

Geometry of Numbers

The theme of this week’s lesson was geometry of numbers.  The goal was to get the kids to think about the different ways that one can arrange some fixed number of identical objects.  To that end, we had two activities: one involved the kids recognizing a number from a geometric representation of it, and the second one had the kids arranging the objects themselves.

The props for the first activity were an empty candy box (with holes) and Mancala stones.  One of the adults would arrange some number of stones in a nice pattern (without the kids looking) and then open up the lid in front of the kids for about a second.  The kids would then have to identify the number of stones in the pattern.


It was interesting to see how the kids had different approaches towards solving the problem.  Some would try to count the stones during the brief moment that they had for looking at them.  However, if the number of stones was greater than 3 or 4, this approach generally did not work too well.  Others would try to remember the pattern, such as three rows with four stones each.  Once the lid was closed, they would then do the calculation in their heads and come up with the answer.    Some kids would give the correct answers with fairly large numbers of stones (8-10), but when asked how they came up with the answer they would say “I just knew” or “I just guessed.” 

Another interesting thing to note is that success in this activity was not always correlated with age.  One four year old boy did better than most of the six year olds. 

For the second activity, the kids each got 8 snap cubes and had to make a flat figure out of them.  They then had to draw (by coloring in squares) an identical figure on graph paper.

ImageThis proved to be a bit more challenging for the younger kids.  However, with some guidance, everyone was able to draw at least one such figure successfully.  One 6 year old girl drew a scaled-up version of her figure: each cube became four squares on paper.  A few other kids tried to do the same but were unsuccessful.  Similarity and scaling seem like good future topics.


Creative Classification

Last Sunday started a new semester at our school.  We now have three groups: littlest kids (around 4 years old), middle kids (around 5 y/o) and oldest kids (around 6).  This post is about an activity that we did with all three groups, although the approach varied slightly.

The goal of the activity was to separate 5 or 6 picture cards into two groups so that all of the cards in one group shared a common characteristic that was not present in any card of the other group.  With the little kids this usually took the form of the adults separating and the kids guessing the rule, whereas with the older kids we would have one of them separate.  There were usually multiple solutions, and in general, we encouraged the kids to be creative.

Here are two examples with 6 cards:

1) apple, banana, kiwi, orange, pear, strawberry

The two groupings that came up during the lesson were: (apple, kiwi, pear) vs. (orange,banana,strawberry) and (apple,kiwi,orange) vs. (banana,pear,strawberry).  Can you come up with others?  The explanations are also left as exercises to the reader.

IMG_07722) fish, octopus, cricket, worm, ladybug, dolphin

Here we tried to guide the little kids to come up with the grouping (fish,octopus,dolphin) vs. (cricket,worm,ladybug) on their own by asking where each creature lived.  However, we came to a stumbling block when we asked “Where does a fish live?”  The first answer was “in an aquarium!” and the second was “at the beach.”  The other grouping that came up had to do with the presence or absence of legs (for some reason in the pictures we had, the octupus, cricket, and ladybug each had 6 legs, and the kids were quick to point that out.  We had to explain that the two remaining legs of the octopus were hidden.)

IMG_0781 IMG_0782

The examples with 5 cards were a bit more challenging.  Here is one:

3) drinking glass, belt, window, necklace, glasses

Initially, the older kids separated them as (glass,window) vs. (necklace,belt,glasses), with the reasoning that in the first group the objects were made out of glass and the second one contained objects people wear.  We then pointed out that glasses were also made out of glass.  The kids were then faced with a dilemma of how to make the glasses belong to the circles for both groups.  The older kids figured out pretty quickly to intersect the two circles (some of them have seen similar things before), the little kids had to have it shown to them, whereas the middle kids got it with some guidance.


Помоги мне?! Help me?!

English text follows Russian.

У меня такое впечатление, что эта фраза у нашего почти 4хлетнего Ари идет рефреном когда рядом кто-то из взрослых и распострoняется на все аспекты его жизни.  Одеть ботинки? Сходить в туалет? Составить из кубиков узор? Нарисовать картинку? Поесть кашу? “Помоги мне!” – сопровождаемое очень несчастным взглядом, и занудным подвыванием.  Причем все эти вещи он прекрасно умеет делать сам (и иногда на него даже находят приступы независимости и он таки их делает, и приходит весь гордый показать результат).  У нас в семье нет однозначного подхода к таким ситуациям – иногда, особенно когда мы куда-нибудь торопимся, то кто-нибудь помогает, но часто мы раздражаемся, а Ари обижается.  А как вы поступаете в подобных ситуациях – когда ребенок просит помочь с вещами, которые он явно умеет делать сам?

I sometimes get the feeling that this phrase, “help me”, is all I hear from our almost four year old son Ari when there are any adults around and it applies to all aspects of his life.  Put on his shoes? Go to the bathroom? Solve a puzzle? Draw a picture? Eat his oatmeal? “Help me!” – followed by sad puppy eyes and less sad, more annoying whining. And the kicker is – he knows how to do all those things by himself really well (sometimes he is hit by bouts of independence, goes and dresses himself, or finishes breakfast, or does a page in a workbook and comes running proudly to show us the results).  We don’t have a consistent approach to handing the requests for help – sometimes, especially if we’re in a hurry to get somewhere, someone will help him, but oftentimes, we get frustrated with him and he gets upset.  What do you do in situations like that – when a child consistently asks for help with things he can clearly do on his own?