“The Purpose of Education is to Have an Interesting Life. Now.”: conversation with Maria Droujkova

Maria Droujkova is the founder of Natural Math, a curriculum developer and a mathematics education consultant.

When asked what she thinks should be the main goals of an elementary school curriculum, Maria answers with a quote from a favorite book, The Diamond Age, “The purpose of education is to have an interesting life.”  And then she pauses and adds just one more word to it, “Now.”  When educating young children, people think of the interesting lives those kids will have in the future, but for Maria the purpose is to have “the adventures and deep intellectual experiences, to exercise their curiosity and make connections, now, as they go through that education.”

So how should we evaluate a math curriculum and the math education our children are getting?   Maria believes there should be three components to the assessment: mastery, adventure, and growth.  More specifically, we should evaluate the growth of pattern thinking, as well as mastery of values such as precision, reuse of what works, search for algorithms, things of that nature.  “And you evaluate these values by how they are expressed in whatever the child does.  And what the child does should be individually owned by the child – their own examples, their own problems, their own adventures.”

And what about memorization?  Do certain things in math absolutely need to be memorized?  Maria believes that memorization can be joyful and can help understanding if done right.  She believes that we need to offer children different ways to memorize, have them work heavily on patterns, problem solving, and projects, and in the context of that, let them do memorization that they find useful.  She herself recalls having some issues with memorization and building her own systems to go around it.  However, she feels that if someone had forced her to memorize, she would have likely quit the field because it was always about patterns and problem solving for her.  

I asked Maria why she thinks there is so much math anxiety these days.  The reply, “There is math anxiety and fear because there is violence.  Mathematics is a strong influence on your mind that could be good or bad.  What happens is that people have no way to limit how much they do it, with whom, and when to stop it, and that is cruel. We, educators and professionals, need to stop that, we can’t be that cruel to people.”  She imagines that when we give children the freedom to do as much or as little mathematics as they want, and let them choose when to do it and when to stop, then we will see much less of the anxiety.  She sees that happening, for example, in math circles, “when people are there for love, they get together and they do it because they choose to and it’s a celebration and festival of joy.  Or at least it’s ok, it doesn’t have to create a wonderful thing, it just has to be fine enough.”

On the bright side, Maria sees progress.   She sees many more people, in proportion to the whole population, being mathematically literate.  There is much more conversation about gentle and joyful ways of doing mathematics than ever before.  There is much more awareness.   “And we have a female Fields medalist, yay!”  

Maria’s advice for someone designing an elementary school math curriculum?   Get children involved in your endeavor.  She promises that you’ll see miraculous things.  “Have a mechanism for them planning the next lesson,” she says, “Children are such good designers in a group with adults and they have very divergent thinking.  Every design team should have a 5 year old on it!”

As for great activities to do with children, Maria singles out one of her all time favorites, The Mirror Book: two mirrors put together at an angle.  “It’s so rich, leads into group theory easily because of the wallpaper groups it produces, and it leads to infinity – you can see the infinity, and the symmetries.  It is low floor and high ceiling.  It has all the aspects that I value in an activity, people like it, and it’s playful.”  

People sometimes laugh, and ask what math can you learn from children?  “What you learn,” Maria responds to them, “is to change your definition of mathematics, what it is.  That is really valuable, and we should keep doing it.  We should keep changing our mathematics and we can do it better with children.”

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Math education, here I come (full time!)

In short: In the fall I will become the Head of Math at a small private school in its second year of existence.  I will be developing their elementary school math curriculum, starting with the younger grades and working up.  I will also be trying it out on some guinea pig Kindergartens and second graders (this year), and other grades in the future.  I am super excited!

A bit of context: For the past five years I have been working as a research mathematician, and for the last three of them I have also been teaching enrichment math, first just to my daughter and a few friends and then to several larger groups of kids at the Golden Key Russian School.  Both were very exciting and fulfilling endeavors, but recently I had been feeling more and more of a pull from the math education side of things.

To make a long story short, I was presented with an exciting and unique opportunity to develop an innovative elementary school curriculum and teach it to small groups of students. So next year I will be Head of Math at the Main Line Classical Academy (website in progress).  I will make sure to blog regularly about my adventures!

Meanwhile, stay tuned for a series of interviews with some people doing amazing things in math education and their views on building curricula and teaching math.  First up: Maria Droujkova!



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Is three minutes a long time?

I was putting Zoe to bed today and the process was moving rather slowly.  She kept coming up with excuses not to change into pajamas or brush her teeth.  Finally, she sensed that I was getting frustrated and that it was time to start giving in.  The following wonderful conversation followed:

Zoe: Okay, I’ll go sleep for three minutes and then come back to play.
Me: No, three minutes is a very short time.
Zoe: But three minutes is a very long time for warming up milk – it’ll boil!

Zoe sure has a way of saying things seemingly out of the blue that are yet not completely crazy.

Me: That is true, three minutes is a very long time for warming up milk, but it is a very short time for sleeping.
Zoe: Three minutes is a short time for sleeping?

I would have been certain that she was playing with me if this had not been said with such a straight face. Or perhaps she is a much better actress than I give her credit for🙂. Still, I had what I thought was a pretty good come back.

Me: Is three minutes a long time for a cartoon?
Zoe: No! Three minutes is a very short cartoon!
Me: Just like three minutes is a very short nap.

And somehow, after that, Zoe dutifully went to change into her pajamas without asking any further questions.

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What makes a triangle?

Last weekend we went to a wonderful family camp with an Ancient Greece theme. I was tasked with coming up with a fun, hands-on geometry lesson for 5-7 year olds.

These were my criteria for the activity:

1) Easy prep of materials.
2) Easy mechanics of working with materials.
3) Opportunities for kids to make interesting observations.
4) An opportunity for the kids to discover some cool math fact.
5) At least a loose connection to Ancient Greek geometry.

So, here it goes: Little kids discovering the triangle inequality using pipe cleaners.

Prep: Cut up pipe cleaners into pieces of various sizes (our range was probably from 1 to about 7 inches).

The lesson:

I began the lesson by asking the kids how many sides a triangle has. Of course they all knew that it was 3. I then proceeded to take 3 pipe cleaner pieces of about equal lengths and make a triangle out of them. And then came the key question: will I always be able to make a triangle no matter which three pieces I take? If some sets of three pieces cannot make a triangle, can you come up with a rule for when triangle can and cannot be made?

There were six kids and we split them up into two groups of three. Each group was given plenty of pipe cleaner pieces. The kids were encouraged to experiment with the pieces, make observations, and discuss them among each other. Each group was also assigned an adult scribe to write down the kids’ observations.

Here are some pictures of the process:


IMG_2793 (1)





Pretty soon I heard kids in both groups saying things like, “If there is one long piece and two short ones then it won’t work.” Naturally, at this point, I encouraged them to try to formulate more precisely how short is “short” and how long is “long.” I hinted that a good way to do this could be to fix the long piece and then try out longer and longer “short” pieces until you can make a triangle.

And then I saw this and heard the following wonderful formulation:


“If the two short pieces are placed along the long one like this and they don’t reach each other, then you can’t make a triangle.”

And there you have it, a 7 year old’s formulation of one of the most fundamental theorems in geometry!


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M&Ms: Munch and Math

It all started with a bag of peanut M&Ms. There were three of us and we would take the M&Ms out of the bag three at a time; Zoe would pick her favorite color out of the three, then Katie would do the same, and I would get the one that was left over. Finally, we took out the last three and there were none left in the bag. It worked out perfectly. And all of a sudden, Zoe said that she wasn’t feeling too well and she did not want her last M&M.

“So what do we do now?” I asked Katie.
“I know!” she replied immediately. She took one of the M&Ms, bit off half, and then gave me the remaining half.
“Now there is one left for each of us and it’s all fair,” she continued.
“So how many did each of us get in total?” I asked.
K: One and a half.
Me: And if there were 5 of them, how many would each of us get?
K: Two and a half. And if there were 7 of them, then we’d each get three and a half.

At this point Katie paused and thought for a moment.
K: You know, I am 7 right now and Zoe is three and a half. That means that Zoe is half my age!
Me: That’s a nice observation, but it’s not fully true because you are actually seven and a half.
K: I guess that’s true… Wait, but next year she will be 4 and I will be 8 so then she’ll be half my age.
Me: Nice. Will she ever be half your age again after that?
K: (confidently) No.
Me: Why is that?
K: (less confidently) Well…when she is 5 I’ll be 9 and twice 5 is 10, then when she’s 6 I’ll be 10, and twice 6 is 12, when she’s 7 I’ll be 11, and twice 7 is 14. You see, twice her age is getting further away from my age.
Me: So what is special about 4? Why will she be half your age only when she is 4?
K: Because then I’ll be 8.
Me: Right, but why does it only happen with 4 and 8. Why will you not be twice her age when she is 5, 6, or 7?

Katie thinks about this for a moment, then replies, “Oh, it’s because I am 4 years older than she is!”

And it all started with just a bag of M&Ms…

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Tough as an Eggshell

Math is not the only thing we do in our household!  This is a guest post by my husband about a fun science experiment he did with the kids:

Recently I stumbled onto The Everything Kids’ Science Experiments Book  on one of our bookshelves. It was a present from someone on Katie’s birthday a few years ago, and at that time I thought her too little for it. And, to be honest, I didn’t consider myself sufficiently ‘handy’ for this.

After flipping through the pages, I decided some of the experiments even I should be able to do. And that is how, one spring evening, our whole family found itself contemplating the hardness of eggshells and the significance of domes!

This simple experiment starts with taking 4 egg half-shells, trimming them down to about the same size and arranging them in a rectangle.
Then we … pile books on them! The kids were very excited and started making predictions about how many books it’d take to crush the shells.
To their, and our, surprise these predictions fell short. The books kept piling.
Then, with a slow crunching sound, one shell  gave way.
The rest held longer, but one-by-one they too succumbed. And this was all that was left of the eggshells, aside from pictures, fun memories and some vague plans to do more hand-on experiments with the kids on our part.
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Diagonal is not straight

Zoe and I often play games where one of us makes a figure with some objects (stones, chips, snap-cubes, etc.) and the other one has to make an identical figure.  Today, for this purpose, we were using an old Russian game with pegs (really a generalization of tic-tac-toe where you have to make 5 in a row, but that didn’t matter to us).   At some point, Zoe made the following figure:


(Her final figure actually had the yellow pegs all the way across.)

She then said,”Mommy, now you copy my long straight line.”

What interested me most was not the figure that she made but that she was so explicit about what it was that she made so that there would be no mistake about it.  This led me to an idea.  I made the following picture:


Then I asked Zoe whether this was a straight line.  As I had suspected, the answer was no (accompanied by a “why are you asking me a silly question” look).  And then I asked Zoe what the figure was.  Her answer: it’s like a bishop moves, it’s a diagonal!

At this point it was time for her to go take her bath and so we had to stop.  But now I wish that I had taken it a little bit further.  What if I had rotated the board a bit, so that the line looked horizontal from her perspective, and asked the same question?  Would the answer have changed?  I guess there is always tomorrow!

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