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# I'm helping parents and teachers who do calculus with 5 year olds (and other math adventures). AMA!

Hi Reddit, I am Dr. Maria Droujkova, founder of Natural Math. It is a network of parents, teachers, and authors. Our goal is to help people make their own useful, beautiful, and joyful mathematics. We play and have adventures, like making calculus accessible to young children. Ask me anything!

You might have seen my interview for The Atlantic: http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/ And its discussion here on Reddit: https://www.reddit.com/r/todayilearned/comments/43wby9/til_even_though_calculus_is_often_taught_starting/

My Proof:Photo: https://twitter.com/NaturalMath/status/742874302832381953 Video from SparkCon: https://www.youtube.com/watch?v=eEiG-owLzZs

Update: Thank you for being wonderful. I'm checking in periodically to see that nobody's comment is lost.

MariaDroujkova857 karma

It's a deep question, and I'd like to warn that the answer is somewhat disturbing in its implications. Yes, some of it is WHAT is taught - the number crunching without patterns, the primitive yet tedious topics instead of beautiful adventures of the mind, medieval content not linked to current trends. The "what" part is relatively easy to address: there are wonderful materials out there! Innovative books, cool computer simulations, hands-on construction sets, etc.

And a part of the problem HOW math is taught: we do need to mind what we know about human learning, such as spaced repetition for memory, the power of multiple examples that come from your peer group, the motivation of making mathematics your own.

Yet the most difficult part that tends to stay off-screen is WHY math is taught. Advanced math is taught as a gatekeeper, as a means not to starve. It trickles all the way down - I hear parents of children as young as five or six say that if they don't push math now, the child will fail forever. To quote a presentation: "Why do we need to know multiplication? One reason is that multiplication is on many tests kids take. The story goes like this: if kids don’t know multiplication facts, they will fail tests, which means they won’t get into college, which means no career, which means epic fail of the whole life. For want of a nail, the kingdom is lost."

So people hate math because they learn it out of fear. How can we help kids learn math for meaningful, joyful, loving reasons? That's what it's all about...

cerebral_monkey375 karma

I can definitely say this is why I hated math as a kid. It was always just "memorize this equation" or "if you see a problem that looks like this, follow these steps exactly," without an explanation of how or why. Ironically my dislike of math pushed me towards a career very heavy in math. I started programming my calculator to do the problems for me and I ended up getting a degree in scientific computing.

Thank you for doing what you are doing, it's possible to make math fun and interesting!

MariaDroujkova186 karma

Thank you for making this connection. Computing is one of the awesome ways for children to learn gentle, powerful math. The machine does child's bidding, it takes care of the tedious stuff, and the child has the power to work at the level of concepts, to fly! There are some very neat computer tools I've used with children as young as three - like Scratch from MIT.

Maxpowr929 karma

The application part is what most students, in the USA at least, struggle with. Most can memorize formulas and "plug-and-chug" but manipulating said formulas is a challenge and what leads students to not "getting" maths and the frustration that ensues. Remove the formulas and have numbers with labels in a word problem and having the student figure out what variables go where in the formula is a big problem. I remember in Algebra 2 doing 3-way age problems and so many students struggled with how to solve them.

missjulia92813 karma

I have a learning disability that made it hard for me to apply the word problems in math. But then in my sophomore year of high school, I had a co-teacher in my algebra 2 class and she would sit down with me and explained in depth how to apply the word problems a couple times a week after school. It finally clicked because someone took the time to explain to me rather than just tell me "Put this here and that there and that's how you do math." I ended up getting a 94% on the state math exam that year.

I still have issues with higher math because no one explained how to do it in that fashion ever again but the B+ I got in college stats made me super happy because I was able to apply what that teacher taught me.

MariaDroujkova3 karma

Glad you persevered through your difficulties! Working with a competent, caring person one-on-one is still the best-known way to learn. And as you illustrate, you can stretch that benefit beyond the actual meetings. Check out this classic piece of research: https://en.wikipedia.org/wiki/Bloom%27s_2_Sigma_Problem

The good news is that similar effects can happen with peers, in math circles, and in small study groups. That's one reason why I keep telling parents and teachers to start little playgroups and circles for their children. That way, children learn how to seek partners for learning mathematics - a valuable skill for life.

meatmeatpotato82 karma

Exactly. It never made sense to me. I can easily fix a car's brake system, because i know how it works overall and what each parts is supposed to do so it is obvious to me how to put things together or fix problems. My experience with math was always put X here because the book said to, divide there because that just what you have to do. I never understood the larger picture or what the individual rules were supposed to do so every math problem seemed equally unique and foreign. I still hate math, but wish I could "get it."

MariaDroujkova126 karma

I am sorry to hear you've been hurt by meaningless math. One of good ways to "get" math, as an adult, is to watch short, recreational videos about topics that weren't spoiled for you as a child. Like this, Infinity Elephants: https://www.youtube.com/watch?v=DK5Z709J2eo

You are probably doing TONS of applied math while fixing cars, by the way. I once took our math circle on a scavenger hunt to a car mechanic, and even little children found many different math ideas.

spinnetrouble7 karma

I dig the idea of math as games--I love logic puzzles so algebra became easy once I started treating them that way--but I still thought of myself as not having adequate math ability for decades until I took a stats class and got to play with numbers and patterns more. Vihart is great at conveying the "play" aspect of math concepts and I wish she'd been around when I was in middle and high school!

Aside from working on children's education now to help them grow into math-competent adults, how can we specifically address the prevailing "oh, I'm just bad at math" mindset so many adults have? (I mean, I had a retiree babysitter when I was a kid who drew the line right after borrowing and carrying.) Have you found any strategies to combat the mental block so many of us have after we've finished school?

MariaDroujkova15 karma

It's a difficult question, because "I'm just bad at math" can mean, "I don't want to do any math and you can't make me, go away!" or, "I've been hurt by math and have post-trauma effects" or...

When I work with people who have been hurt in relation to math, the first thing to do is to find some mathematics that is very different from their past experiences. Something that would not trigger bad memories and pain. For a lot of people, it's games like Set or puzzles like 2048, or maybe beautiful stuff like Origami, or little fun videos Vi Hart makes.

NorthStarZero53 karma

Hey there!

So I was an enthusiastic math student in grade school and high school - I did the annual competitions and did well, adapted well to computer science, etc.

Then, in university, I hit trig and calculus - and it kicked my ass. I did every single calc course (1 through 5) *twice* because I kept failing exams and sup exams.

So fast forward a couple of years, and I discover (to my horror) that I had a problem that required the real-world application of calculus. So I dragged out my old textbooks to review.

They were worthless.

And I had this sudden epiphany as to *why* they were worthless: the whole textbook was a recipe book for techniques for differentiating or integrating various types of functions - "chain rule" etc - but not once (*not once*) did anything in the book discuss *why I should give a shit*. *No* discussions of practical applications. *No* discussion of the history of why calculus was developed. *No* treatment of the types of problems calculus solves, or how to recognise potential applications. *Nothing!*

Just technique after technique for handling weird edge cases in the abstract.

So not only did I come out of that experience hating this weird and arbitrary memorized list of techniques, but I had no idea of what the whole process was actually *for*.

GAAAAAAA!

MariaDroujkova35 karma

Thank you for sharing your story, NorthStarZero. It's sad in itself, but what's really horrible is how OFTEN that happens. And grief stories like that is one of the big reasons why I do what I do - so it would not happen to the children.

One of my co-authors on young calculus materials, Kalid Azad, has a beautiful site on calculus for grown-ups. He starts with one-minute overviews of each topic (that's the part accessible to five-year-olds, by the way). The one-minute version introduces you to WHY. What the topic is all about? Why did people care to develop it? What is it for? Check it out: http://betterexplained.com/guides/calculus/

At Natural Math, we have "math sparks" for each topic we introduce to children - you can check out a couple of calculus sparks on this page, and see how they start with *whys*: http://naturalmath.com/inspired-by-calculus-online/

severoon14 karma

The story goes like this: if kids don’t know multiplication facts, they will fail tests, which means they won’t get into college, which means no career, which means epic fail of the whole life. For want of a nail, the kingdom is lost."

I feel like this gets at something important, but there's another piece of the puzzle.

One reason I've always liked math is that I felt I was graded unfairly in many other subjects. When I analyzed a chapter of *The Sound and the Fury* in English class the interpretation I was supposed to get from the description of Dilsey descending the stairs seemed arbitrary to me. In math class, there's often only one right answer and if you get it, you get the points.

I think this is what terrified other people, though–the completely objective measurement of their understanding of the subject is something many kids did not like because it didn't leave wiggle room. There may be a lot to protest when a child gets a bad math score, but the cold hard reality of it is, however it happened, *they do not understand the material*. When writing a history term paper or an analysis of literature, there's always a way to rationalize, "Underneath it all, I really do get it, I just didn't express myself well," etc.

MariaDroujkova5 karma

There are some parts of math that provide that cold, hard, and beautiful objectivity. And some people find it reassuring, like knowing 2+2=4 in the face of propaganda in "1984" story.

I think for children who aren't into that, there are parts of mathematics that aren't like that, as well. For example, problem-solving is more closed-ended than problem-*posing*. When people submit sculptures to the math art conference called Bridges, they are judged on criteria that are open and somewhat subjective. Mathematical modeling is a very open field, because you often need to apply the model to the realities of the situations, and those can get messy - especially in sociology or political science.

In general, I wish math activities were more tagged along the dimension you brought up: subjective/interpretive to objective/closed-ended.

MariaDroujkova8 karma

It's a very broad question, because no person can suck at ALL math - there's too much of it! I am yet to see anyone who hates math art, programming, hands-on modeling, problem-solving, social justice statistics, the sixty-three major math subject areas - all at the same time.

So, what do you like, what are your hopes and dreams? What books do you like? Or maybe it's not a book, but a game or a movie? Let me know. Meanwhile, here's a site I respect for book reviews: http://livingmath.net/Readers/tabid/268/Default.aspx

FourOfFiveDentists30 karma

Because it is the most demoralizing experience ever. It makes you angry, depressed and feel like a loser. My experience is that math teachers are beings of pure evil devoid of any emotion.

Every math class I have ever had was basically "Figure it out on your own, and if you don't you'll get left behind and fail. You'll ask a question, then get made fun of by your peers for not knowing. But remember kids: LEARNING IS FUN!"

I am going to be 31 years old and any time I have to do any kind of math I *still* get the same scared/panic feeling I got as a kid.

Oh well. Those mother fuckers told me I'd never have a computer in my pocket and look at me now!

MariaDroujkova7 karma

"Grief stories" is what I see a lot - people share their hurt when mathematics is mentioned. Sorry to hear that. I hope we can help children better than that!

mcflufferbits8 karma

I believe the main reason is that they don't see why they need to learn it. If parents/teachers gave examples on how crucial it is to learn math then it'll be easier to encourage the kids.

MariaDroujkova27 karma

Good reasons are important, but please be careful! It's very easy to send the message of fear to children: "Learning math is crucial, because bad things happen without it." Bad things can mean being poor, or not saving the world from some tech disasters, etc. - the point being, if we try to scare children into learning math, it does not motivate them well.

notforlily2 karma

Why do so many people hate History, or Science, or Foreign Langauges?

I think there is no deep meaning in it, but rather different people like different things. I hate math. I have a calculator for math.

But I enjoyed the hell out of diagramming sentences. Different strokes, ya know.

MariaDroujkova4 karma

It's interesting that you mention diagramming sentences - a very particular (and very neat) activity, as something to love. But things you mention people hate are large and abstract.

Maybe instead of trying to embrace all of math (too large!) we should seek particular small and lovable parts of it. Like fractal art, or funny paradoxes ("This sentence is a lie") or that thing about .999...=1

seriouslyliterally177 karma

Ok, I get how a 5 year old could probably understand limits but please explain to me how the average 5 year old can understand differentiation and integration?

MariaDroujkova166 karma

First of all, let's be clear that a 5yo can understand the concept and idea of limits or integrals, but probably not computations of them from something like AP Calculus test. It's such a FUN task to ask yourself - how can I help a little kid to play with derivatives? And to figure it out! A lot of it is in storytelling and hands-on play.

Zeno's paradoxes or Hotel Infinity stories are perennial favorites for finite and infinite limits: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

Here are our hands-on activities for integrals and derivatives, from the Inspired by Calculus course. They have particular examples.

Integration as 3D printing, layers on an onion, LEGO building - "What Would 3D Printer Do?": https://drive.google.com/file/d/0B6enMfoYXJb3RjRzUks0bVpQaHc/view?usp=sharing

Integration as mosaic, beads, pixel art: https://drive.google.com/file/d/0B6enMfoYXJb3N0Z5SXFFdEdBeXM/view?usp=sharing

Integrals and derivatives in "X-ray vision" metaphor - making flipbooks! https://drive.google.com/file/d/0B6enMfoYXJb3cWxhUTB3SGlIM3M/view?usp=sharing

I hope these examples help to get started. There are more, like this interactive toy for beautiful rotations - kids love it: http://www.zefrank.com/string_spinv2/

Wombcorps37 karma

I'm 30 and I have no idea what integral a and the other stuff in this post are :/

Failed GCSE maths 4 times before giving up...still 6 6 a levels, a degree and never been out of work. Maths really really confused the hell out of me at school and more so now :/ thank fuck for phones and calculators

OMG Rip inbox! Thank you all for explaining, I will try and reply to you all. Most of it has gone over my head, and the bits I can understand leave me with the question 'why do you need to know that?' I think one of the big issues with school (certainly my experience of it anyway) is that they failed to make things *relevant* or *useful* it was just 'hey learn this shit to get a C grade'

Junius_Bonney31 karma

A derivative of a graph at a certain point will give you its slope at that point. An integral will give you the area under the curve from point a to point b. Both use the concepts of limits, which are just what they sound like. Find the slope between to points on the graph and then move the points closer togetger until the distance between them is effectively 0 - you get a derivative. Use the height of the graph at a point and make a rectangle with that height and a width of 1. Do that for every part of the area you want to measure, and add them all up. Then make rectangles at every 0.5, 0.1, 0.05, etc until you get to rectangles with 0 width. Add them all up and you get the integral. Hope this works as an ELI5

nivekevil10 karma

I appreciate the effort! But no. Still dumb here.

Edit: Some excellent responses below! Thanks!

MariaDroujkova97 karma

Imagine an interesting shape, like the Millennium Falcon. How much space does it occupy? Or maybe, how much plastic would you need to 3D print it? Now, imagine building that spaceship out of LEGO blocks. You can then count the blocks to estimate the volume. This, in a nutshell, is integration: building a shape out of easy, simple little shapes.

samsonizzle21 karma

This is an explanation of integration from someone who actually intuitively understands what integration is. I don't think I met anyone until I was in my masters that actually understood it well enough to explain it in simple terms like this.

Crazypyro15 karma

Isn't that just what the Riemann sum definition basically explicitly states? Definite integrals can be approximated by a sum of smaller shapes in a region.

MariaDroujkova11 karma

The definition does say the same things, but *too explicitly* (as you said) - so it's not a relatable story.

MariaDroujkova7 karma

We often need to know volumes to estimate how much material making an object will take. Also: you have an awesome username.

lukesvader4 karma

Thanks :) I'm Afrikaans-speaking (*father* = *vader*) and my son's name is Luke

Wombcorps9 karma

Because 6 a levels in different topics...the only gcse I failed was maths! The point I was trying to make is that it's not a disaster if maths doesn't click with you, you can still be successful elsewhere and build knowledge without it

MariaDroujkova12 karma

Your message should give people hope, Wombcorps. Some kids fail to learn *because* they are told math makes or breaks a person. The pressure to pass the test (or else fail at life) is too much for many kids.

So, I do not condone frightening kids into math. Yet I think much of mathematics can be gentle, and playful, and useful, and beautiful. THAT is the kind of math I want to offer children.

randomguy1861 karma

One of the most challenging aspects of mathematics is that every level builds on the previous ones: addition / subtraction ==> multiplication / division ==> exponents / roots ==> logarithms and algebra

geometry and algebra ==> trigonometry and analytic geometry ==> calculus

Oversimplified a bit, but if you fail to fully grasp any single topic, you cannot fully understand any topic that follows.

MariaDroujkova3 karma

This is true - within curricula that are built that way, linearly. The good news is that you can build mathematics more like a net than a tower.

For example, if I play with a three-year-old, we can have some toys out (I'll use symbols): @ and # and *. I ask the kid to close eyes, then hide a toy under my hand (which has letter X on it), so only two are left: @ and *. What is hiding? Eventually we go into using multiple toys of the same kind, and arrive at rather formal equations. You can play with the IDEA of an unknown before learning numbers.

Likewise, you can play with the idea of limit or infinity or function before (or after) learning addition and subtraction.

Here's a presentation I like on the topic: https://prezi.com/aww2hjfyil0u/math-is-not-linear/

We have it as a part of the large anthology called Playing with Math: http://naturalmath.com/playingwithmath/

maneo21 karma

Oh! My Dad (former physics major turned computer scientist) used to talk about "Hotel Infinity" all the time. I never fully understood the implications of the story, and it kind of just fell to the back of my memory by the time I was learning the math concepts it related to.

But if things like that were a normal part of our culture where everyone is familiar with those stories, puzzles, etc then teachers could reference them when finally teaching the details of the math, making it a lot easier to wrap our head around tough concepts,and providing some meaning and context to what otherwise seems like random gibberish processing of numbers.

MariaDroujkova18 karma

That is my hope and dream:

if things like that were a normal part of our culture where everyone is familiar with those stories, puzzles, etc then teachers could reference them when finally teaching the details of the math

Hotel Infinity, Zeno's Paradoxes, playing with zoom on your camera - all these stories and hands-on experiences can make easy bridges into formal understanding.

MariaDroujkova19 karma

If you zoom close to a graph, you can see it turn into a straight line - a slope. That's the essence of taking a derivative.

Zoom is useful for several other math things, like orders of magnitude, but the slope is my favorite calculus example.

yumyumgivemesome2 karma

Saved for when I visit my young cousins. My cousins at ages 7 and 8 are struggling with math. Do you recommend any learning tools that will help get them excited about math? I feel like the excitement is what will drive the interest and focus in school.

MariaDroujkova2 karma

Board games and computer games is what gets a lot of children excited, because children already like to play games. But different children like different things. Try to find out what your cousins are into - and then search the web for math activities with that. Like math and art, math and sports, math and programming.

r1zz8 karma

Was also wondering this, but in the article it says 5 yr olds wouldn't be able to do the actual equations, but they would be able to sort of understand the concept and it just lays the foundation (I think).

She doesn’t expect children to be able to solve formal equations at age five, but that’s okay. “There are levels of understanding,” she says. “You don’t want to shackle people into a formal understanding too early.” After the informal level comes the level where students discuss ideas and notice patterns. Then comes the formal level, where students can use abstract words, graphs, and formulas. But ideally, a playful aspect is retained along the entire journey.

brett968 karma

I imagine it's something like "ok, for the derivative of x^{2} take the tiny 2 at the top and bring it to the bottom and put it in back of everything already there. Then minus the tiny 2 at the top by one. Ok good, now what's the derivative of sec^{3} x + ln(3)?"

MariaDroujkova5 karma

With young children, you can invite them to make a graph of x^{2} and x^{3} in a beautiful computer tool like Desmos, and then zoom in close enough to compare slopes. https://www.desmos.com/calculator/xknso1kdan

jespicy75 karma

I have a 4 month old daughter now. What would you suggest I do now until she is five to help her enjoy math. Do you have any books or classes on teaching young kids?

MariaDroujkova82 karma

Moebius Noodles is our book of activities adapted all the way to babies. There are 18 chapters of activities there, such as "live mirror" games where you repeat what the baby does. You can play "hide and seek equations" with toys under a hat, or tape two mirrors together for hours of symmetry fun. If you share some more of what your baby already likes, I'll try to suggest math games to match her interests. Meanwhile, here's the book: http://naturalmath.com/moebius-noodles/

jespicy18 karma

She loves playing with her owl toy that has a bell on it. It swings in her seat. We have also introduced a rainbow abacus that she moves back and forth with surprising control

MariaDroujkova27 karma

One of the topics in the book I mentioned is gradients - like Goldilocks and the Three Bears. In baby language, it means making several versions of the same toy. Say, small, medium, large, giant bells, together, so that the baby can compare, contrast, and select what she likes. The sounds will also change - and add easy complexity to it. Like a xylophone. Rainbow is a gradient already (of color). You can make a lot of easy toys that way: several pots to bang on (small, medium, large), or containers you fill with cold, warm, and hot water - to explore in the bath.

cerebral_monkey39 karma

How do you feel about teaching young children proofs? I sometimes feel that my early math education would have been easier if I had seen proof of the methods/equations we were using, but I'm not sure if that's just some kind of bias since I now have a degree in a subfield of applied math.

MariaDroujkova41 karma

I am all for helping children see WHYS. Why does the quadratic formula give us solutions? Why have people bothered to invent limits? Why does the Pythagorean Theorem always work? I love to work out child-friendly whys behind more advanced topics, like complex numbers (carousel around the axes) or integration (assembling blocks in Minecraft).

And also, logic and rhetoric and paradoxes (topics related to proofs) are both accessible and fun for young children. We have a book out called Camp Logic, and people report good results with it: http://naturalmath.com/camplogic/

You are right about bias, though. Some people, including children, are much more interested in building, modeling, or artistry in mathematics - the HOW of it rather than the why. So, the proofs and whys should be there, and accessible, but as one venue of exploration among many children can choose.

SolvaySlovak18 karma

I was one of those kids who always asked 'why?' to everything. In order to truly understand HOW it worked, I had to first know WHY it worked. Once I know why it works I can easily rearrange the formula to solve for any variable. Too many people only know how to solve for 'x' because that was the only way the formula was taught.

I can recall one lesson in Physics where we learned how to apply a formula to solve for the pressure exerted at any given 'x' depth on a dam wall. For some reason, this one clicked in a way I'll never forget. It made me feel like I could wield Thor's hammer. (Plus, I had a great Physics teacher!)

Once I understood that this applied math is what allows us to precisely determine exactly how much concrete is needed to prevent a dam from catastrophic failure, the excitement of applying math in other areas came naturally.

MariaDroujkova10 karma

Thor's hammer < 333 Thank you for the story - you captured that feeling so well! We talk about "superpowers" as a good metaphor for what happens when you learn conceptual math. For example, understanding integration is like *time-lapse vision* that shows you how objects are made of slices: https://drive.google.com/file/d/0B6enMfoYXJb3cWxhUTB3SGlIM3M/view?usp=sharing

A student I know described it thus: "Suddenly I went from it being wizard magic to it being wizard magic and I was the wizard."

thinkhuman31 karma

Overwhelming amounts of research that show the last thing pre-7-year-olds need is math training. In several countries--like Finland--math and reading come later, yet they score higher than most every other country in the world in math and reading.

Given this, why in the world introduce calculus at an age when it's convincingly wrong for brain and emotional development?

Edit: For angry-responding folks that want "proof", a few quick examples:

http://www.scholastic.com/teachers/article/what-happened-kindergarten

MariaDroujkova49 karma

Indeed, we have to be careful here! That research looked at *formal* math training, such as teaching children computational algorithms. I am talking about something very different: play and hands-on activities inspired by rich math ideas. For example, building your own fractals out of toys, or pretend-playing Zeno's paradoxes.

Samseaster30 karma

Can any of your research and methods be used to help adult learners gain confidence and skill in calculus or is it something that would work only for young children? Thank you!

MariaDroujkova36 karma

Secretly, all Natural Math methods are for adults too. (Well, it's not really a secret, just something my colleagues like to joke about.) People feel that if a method is gentle, playful, accessible enough for five-year-olds, they might give it a try! Children are like pied pipers, leading their grown-ups to give math another chance.

Some people even "borrow" children for the purpose: a niece, a friend's child, or a group at a math circle.

What grown-ups say: "I've never thought of it that way - never linked integration and topo maps, but now that we made them out of foam, it's so obvious!" "I come to the math circle for my child, but it's so fun and therapeutic for me. I play for my own sake."

If activities are deep, easy, and playful, grown-ups find them valuable too. Math anxious people find it therapeutic, and professionals in math-rich fields find new inspirations.

tehrabbitt20 karma

do you find they learn the concepts quickly?

I wish they would teach calculus at a younger age main-stream. I took calc in college, and then so much of it made sense WHY I had to memorize all that stuff in HS. . Honestly Calculus made more sense to me overall than half the stuff they taught us in geometry in HS.

MariaDroujkova27 karma

I am with you here: ideas from more advanced math make life, the universe, and everything (including math!) EASIER to understand. These ideas give us power to make sense of the world.

Children are geniuses about learning new ideas, concepts, patterns. They learn new languages that way, if they are exposed. Now, growing deep, connected, fluent understanding of math ideas isn't quick. But you can easily begin the journey, for example, children love it that you can zoom into a fractal and see *exact same fractal* - it invites them to think differently, to have new adventures of the mind.

LeighmurBJJ21 karma

I used to coach brazilian jiu jitsu to kids. As far as sports go, bjj is fairly difficult to learn: it takes balance, coordination, spatial awareness, core strength, cardiovascular and muscular conditioning, all in addition to what can be often be fairly complicated techniques. It's most definitely a thinking man's sport.

The kids learned it *freakishly* fast. The kids who have trouble with usually aren't able to pick it up because they physically aren't strong/coordinated enough to do whatever technique. The hardest part about teaching wasn't having kids understand the techniques; it was making sure the more advanced kids didn't get bored. I eventually had to start teaching one technique to everyone, then branch it together with two or three more just to keep the more advanced kids stimulated.

I really think if we just allowed kids to progress at their own rate while giving them some guidance, a lot of them would be able to learn things much faster than how we currently teach them in a structured environment.

MariaDroujkova14 karma

This story is *amazingly* similar to what happens when you invite children into complex math. Young children can't do heavy lifting (in math: long tedious calculations), but they can deal with the complexity of ideas!

DrHappyFunTime2 karma

Ever try to help them learn about gradients? The fact that everything happens because of or can be modeled with them is pretty cool.

MariaDroujkova2 karma

Yes, I love gradient-based games for everybody from babies to grown-ups. Montessori developed some good gradient games, for color, weight, temperature, sounds... You can do fun work with two-dimensional and three-dimensional gradients, too: http://naturalmath.com/2014/02/multiplicationtowers/

CandysaurusRex15 karma

I work as a nanny for two 2.5 year olds. What can I do to help them get ready for and excited about math and numbers?

They can count to 20 with help and like to count with me, but have no concept of proper counting on their own--they point to objects at random and start saying whatever numbers come to mind, not always starting with 1.

MariaDroujkova21 karma

The world needs more thoughtful nannies like you. Kudos!

Think of some concepts beyond counting. How about symmetry - kids love to play with mirrors, or you can fold paper and cut out a doodle and it will open symmetric. If you fold a paper and trace a hand, the unfolded paper will have 2 hands. But if you fold a paper twice, you'll make - 4 hands! What if you fold the paper again?!!

We have games like that for toddlers in two of our books (which are, by the way, Creative Commons and available at name-your-price, up to free): Moebius Noodles http://naturalmath.com/moebius-noodles/ Socks Are Like Pants, Cats Are Like Dogs: http://naturalmath.com/socksarelikepants/

kiwiJONES6 karma

I feel like by making kids this young struggle with math before their brains are really even developed enough to understand more abstract calculations could be more detrimental to them in the long run. It teaches them that math is frustratingly hard from a young age, whereas later in life they could learn the same math MUCH faster and learn to enjoy it rather than despise it. What do you think?

MariaDroujkova7 karma

making kids

We can remove the rest, and stop there. Making kids do *any* kinds of math is problematic. I suggest gently inviting children to do mathematics. If they don't accept the invite, offer something else.

Just imagine what learning would be like if children chose what to do, all the time, with gentle help but nobody *making* them, ever...

MPREVE5 karma

Hi Maria,

I'm a student finishing up my bachelor's in math, and I have a 12-year-old sister who's interested in math as well. She enjoys interesting concepts, but doesn't like actually solving all the algorithmic problems she's given in school. How can I best support her and show her the parts of math that she'll enjoy?

MariaDroujkova8 karma

Hi MPREVE - there are several things to do that will help. One easy thing is to find cute, profound, cool math pieces that are easy to share, and send them your sister's way, maybe when you chat. Like these videos (just start with one you like): https://www.youtube.com/user/Vihart

You can also help your sister find a math circle, a robotics group, a hackerspace - any group that does math, science, and engineering for love, and that would treat your sister well. There are girl programmer groups and so on - look around, ask your sister what she loves, keep her involved in all decisions. If she connects with such people and enjoys the experience, it's a big huge step toward keeping your own values in learning (in spite of other, less meaningful experiences).

drrrraaaaiiiinnnnage4 karma

I'm afraid this will get buried, but i want to give it a shot anyway.

For as long as i can remember I've been terrible or, at the least, pretty bad at mathematics. I really started falling behind when i switched to public school and got into the more abstract mathematics such as algebra I, geometry, and precalc. It got to the point where I was spending 4 hours in tutoring after school and still not finishing the homework and still failing the exams. This was troubling to me as I was always good in science and english. I seem to be a strong abstract and philosophical thinker, but it just seems that standard methods in teaching math just do not compute. I've always been tempted to work the problems my own way, which to the teachers seemed far more arduous than it needed to be, and as it turned out would generally be the wrong way or would be too hard when it came to more advanced problems. Now that I'm going into my senior year of college, I realize that the only careers I'm very interested in require a certain level of math beyond which i can really do well (specifically statistics, some higher level algebra/programming, and perhaps some calculus as well). I'd really like to learn mathematics and feel comfortable with it. I recognize that I will never be an expert, but I'd at least like to become skillful enough that I don't start hyperventilating when I see too many symbols in a problem. What would you recommend that someone like me do to become more competent and more comfortable with the subject?

MariaDroujkova3 karma

Thank you for sharing your story. I would like to applaud your determination and grit. There are a couple of things you can do. First, it looks like you have math anxiety. When you are anxious, higher order thinking turns itself off. So, if you are hyperventilating, learning is probably very hard or futile. There are books on the subject, like "Overcoming math anxiety" - and your college may have counselors and therapists and learning specialists who can help. But some of it is basic. If you get that shot of fear when you see too many symbols, get up and take a break. Do a short breathing exercise. Lift some weights. Some people have good success doing math outside on the grass, or while petting a cat, or listening to music. Anxiety is a physical reaction, and needs bodily measures.

Second, it may help you if a learning specialist looked at some patterns in how you learn. Most people know what dyslexia is, but there are lots of quirky things like that that require different approaches to learning. For example, some people need to highlight symbols in several colors, because of how their attention works (there's even software that does it for you). See if someone can identify what would help you more.

If you know what courses you plan to take, you may want to audit them ahead of time online. No pressure, no grades, just exposure. It does make a difference. Your mind habituates to the new words, new ideas.

From what you describe, it looks like you should be able to pursue your dreams, with some support and planning. I hope you make it happen!

The_Potato_God994 karma

Do you think that some kids are pushed by there parents to do your class against there will? Because I know people who would argue that 5 years old are meant to play and enjoy their youth. Do you think that your class might be bad for some of the kids, because their probably is a ton of stress associated with following a math class at 5 year old?

MariaDroujkova3 karma

We aim to play and enjoy. Having said that, not all children like all types of play. Please don't push children!

Shinvo4 karma

What do you think would be the best components and concepts of an interactive math learning game targeting both children and adults?

MariaDroujkova8 karma

This is a large question. I am so tempted to just spend the next hour on this :-) Any concept in math is someone's baby - some people worked it out with love and care. So, any math topic or problem - Euler's formula, geometric series, knights and knaves puzzle - can be INTERESTING. However, some are more POPULAR than others, and some are more CENTRAL to other topics.

If you pick a popular topic, it will be more memetic and easier to share with others. For example, fractals. Everybody loves fractals, it seems. They are in the main song in a Disney movie (Frozen). They go around a lot on social media. So if you make a game about fractals, people will probably relate to it - both children and adults.

You can also study what hang-ups people have in math, and target those. If you make a game that teaches people the deep calculus issues related to .999...=1, I bet both children and adults will be thankful! We worked out a hands-on model for this one, where children cut squares or triangles (into smaller and smaller pieces) and then reassemble them back into the whole thing.

tweeks113 karma

Hey there,

Being 24 I remember graduating high school when technology was starting to enter the classroom. Everyone had cellphones, the Ti-89 calculators could solve derivatives and integrations for us; basically, technology was in the hands of all students and we knew how to use it to our advantage.

What technology are these 5 year olds or other students using? Do they incorporate tech into mathematics or are you strictly teaching math without technology?

If you are not using tech, why?

MariaDroujkova7 karma

I love tech and use computer-based math a lot. Tech can give children the power to explore without the burden of doing tedious little things every step of the way.

Sometimes we use (or make) apps that go with particular topics, such as Universcale zoom that goes with fractals, infinity, and multiplication: http://www.nikon.com/about/feelnikon/universcale/

One of my favorite general-use child-friendly tools is Scratch from MIT (for algorithms, programming, and making your own apps): https://scratch.mit.edu/

And this beautiful grapher: https://www.desmos.com/

The decision on which activities are best "unplugged" and which are great with tech is always interesting. For example, integration with LEGO is a lot of fun in physical space - but Minecraft is great for the purpose, too.

seafood103 karma

I have a 5 year old and feel like he is being left behind now! What can a Dad in SoCal do? I realize that is a vague question but I would love to start him with this.

MariaDroujkova5 karma

You can have a math board game night with some friends, or another type of little math circle (puzzles, hands-on models, robotics - whatever your son likes). Also, you can do some math activities and games with your child at home: special time with dad. Check out Moebius Noodles and see if any activities catch your eye: http://naturalmath.com/moebius-noodles/

If you tell me what your son likes, I can offer more particular suggestions.

s4ltydog2 karma

Thank you for this AMA!! I was wondering as an adult who in the next few years will be helping his kids with advanced math, what would a good way be for me to start brushing up on algebra and trig and start learning calculus? I've always found math fascinating but was always frustrated with my teachers teaching so I could pass tests, not so that I could understand the WHY an equation works. What, in your opinion, would be the best way for me to get started? Thank you 😊

MariaDroujkova2 karma

Before I answer your question, here is an extension. Beside brushing up, find things of beauty and joy. Recreational, fun math that may not be very utilitarian, but keeps you going. Something you want to share with friends and family, just because.

Now for resources. Try these free high-quality collections: http://betterexplained.com/guides/calculus http://www.jamestanton.com/?category_name=curriculum-tidbits https://www.youtube.com/user/numberphile

jumpinjiminy3212 karma

One question, not trying to be obnoxious, out of genuine curiosity, Why?

MariaDroujkova5 karma

I really like pretend-play games that children play around math models. For example, we were making topo maps out of foam (integration), and children started to play pirate islands and tell all the stories about their islands and characters: http://naturalmath.com/2014/04/inspired-by-calculus-math-circle-week-4/

One of my favorite math computer game is the very old Zoombinis (soon to be re-issued after a successful crowdfunding), because it invites children to experiment, to explore, and to make interesting choices.

A couple of mobile games I like: 2048 (a number sense puzzle), Dragonbox (concept-centered game of early algebra) and Geom-e-tree (a fractal maker) - but these aren't my top picks among many fine little apps, just something that came up recently so it's on my mind.

The game of Set is an example of a good board game. Making your own set of cards with different shapes and properties is a good activity for a family or a math circle.

SigmaEpsilonChi2 karma

Oh well, I was fishing for SineRider but those are all excellent games too.

Say hi to Ray and Dmitri for me :)

EDIT: While I'm here, I'll add a few other great ones for anyone who's interested:

- Miegakure - A four-dimensional puzzle-platformer by Marc Ten Bosch.
- Euclid: The Game - A geometric proof game by Kasper Peulen.
- Engare - A game about the mathematical patterns in Islamic art by Mahdi Bahrami.

Seabass8122 karma

When one of the kids ask how they are going to use this in the future how do you respond?

MariaDroujkova2 karma

I show them stories of awesome grown-ups who use it now, and invite children to do some of the same things, now. For example, NASA mathematicians fold solar sails to fit into spaceships for launch. And you can make origami models at home - some of the same models they use!

So: show inspirational grown-ups, and help children have some of the same adventures right away.

cadomski2 karma

Great topic! Glad you're doing this! I'm an ex-scientist, now software engineer. I had math up through calc 3, diff eq., and linear algebra. But that was 15+ years ago. I have young kids that I hope to teach these topics to. Where do I start?

MariaDroujkova2 karma

Check out our books: http://naturalmath.com/goods/ You can try activities a few minutes at a time, see what the children like - then do more of it!

Millsy12 karma

Is this really useful long term?

Speaking from my experience, Calculous is a level of math that is basically good to have "so you know what goes on behind the scenes". IE,when your engineering program gets you a result, you sort of know how it did that.

But aside from electrical engineers, low level computer hardware/software engineers, most people will never use Calculus in their daily life.

Is understanding the higher level math providing an advantage to understanding more basic math?

MariaDroujkova2 karma

This is a rich question, and it calls for several different answers.

First, advanced math is there for beauty and joy (not just utility) - like music, or poetry. If we understand how infinitely many pieces can fit into a unit square (1/2+1/4+1/8+...=1) it gives us new perspectives and insights about life, just like art does.

Second, as you say, higher math can help to make sense of more technical math. Some people here in comments said they couldn't do algorithms without knowing why they work. When you learn how our numbers work, it may help to see them in the context of powers of the universe: http://www.nikon.com/about/feelnikon/universcale/

Third, imagine the place where you live, a couple of thousand years ago. The general population knew very different math - and it shaped their society. What if we had a higher math literacy? How would our communities change, our life?

ejly2 karma

I learned calculus through brute force attacks and was surprised when my kid was successfully able to learn calculus in a fun and engaging way his kindergarten Montessori program. I have a few questions:

what specific programs do you recommend for Natural Math learning? Is Montessori (as an example) particularly suited to incorporate Natural math or does it work with other pedagogies?

what can I do as a parent to advocate for natural math vs formal math in public schools?

is there counseling available for those of us who acquired calculus through formal math methods, to find the joy in our relationship with mathematics again?

MariaDroujkova2 karma

Any pedagogy that is open to some adventure and children's choices would work. Reggio Emilia, project-based learning, Sudbury, math circles are some examples of suitably open frameworks. Some flavors of Montessori report good results; it depends on how they set things up. If children are encouraged to integrate a robot or a spaceship or a flower out of the color beads, it probably means a good fit; if they are only told to form base 10 numbers, probably not.

As a parent, one of the best thing to do is to help your school start a math circle. These are informal groups of children, teachers, and parents who get together and do math for love. The school provides roof and children, you and other parents help to organize things, children enjoy!

We have courses for grown-ups with children, and many parents report it as a therapeutic experience: http://naturalmath.com/courses/

Muaddibisme2 karma

Hi Maria, thanks for stopping by to talk to us.

Do you have any material (print or online) that discusses your methods and tactics?

I have a 3yr old and a personal love of mathematics. If I can instill in my daughter her own love of math I would be thrilled.

However, I am unsure how I could approach the topics of calculus without her knowing the underlying algebraic concepts that you use to solve calc problems... hence my request for materials.

Thanks.

Edit: after a bit more reading in this thread I see that you have posted several links already. Thank you. I will explore them soon.

Please feel free to reply with any other items you might want to plug here. I would appreciate them.

Also, having started my career in math education, I completely agree with your comments about the terrible methods in which it is currently taught.

If your interested in a follow up question: how do you think we could go about changing the way math is taught (and really education in general s presented) in our current school environment?

MariaDroujkova3 karma

Hello - I love "Dune" too. It has some good thoughts on education: mentors and hands-on experiences. But I want to add peers to that, because Paul/Muaddib there was a bit lonely... One of the good ways to make a difference for your daughter is to start a little playgroup for her and her friends, where you can delight them with toys, construction sets, pretend-play, and maybe gentle puzzles, all rich in mathematics. Your personal love of math can help her make connections to rich math ideas, to grow her math eyes, and to notice patterns everywhere. Offer her to choose what to do, support any path she takes, and offer new possibilities to explore.

And much of the same can be said for older students in your school! Start a math circle. Invite them to explore math for love, to make connections to what they like. That's what makes a difference: children making choices, with strong support from grown-ups.

stl911 karma

What's the most important advice to give elementary school parents about how to help their children succeed in math?

MariaDroujkova12 karma

Choice.

Whatever you do in math, arrange it so your child has three or more choices of that. Want to offer a problem? Have a selection ready, and help the child select problems. Learning multiplication tables by heart? Look at three or more methods (pattern-based, spaced repetition, number tricks) and help your child try them out, then pick what works. "Do you like algebra, geometry, or calculus more?" Well, expose your self and your child to other sixty large areas of mathematics, such as logic, topology, number theory - and help to choose the topics that speak to your child most.

Help children make informed choices.

j_stin1 karma

As a student going into his first year in high school, how can I make myself WANT to learn geometry and do my work? I worded that very strangely but hopefully you get my question!

MariaDroujkova3 karma

I think I get what you are saying, j_stin. There are two parts of to it. First, you need to acknowledge the realities of your situation. For one, is it *your* work - as in, did you give informed consent to doing it? It's more than a little difficult to love what you don't choose. Past this difficult question, a lot of desire for work comes from good support and good logistics. Here are some things to arrange:

- Always have someone handy who can answer questions and discuss things. A friend, a relative, help center, online Q&A place, etc. It helps not to feel alone.
- Eat before doing math (and snack during) - not on sugar, though, because sugar crash+math=pain. Math burns a lot of calories, like vigorous exercise.
- Give yourself about 5 times more time than you think you need on problems at home. Spend the time looking at several different materials about the topics: videos, sites, books. Math takes more time than we think.
- Look at topics 2 weeks ahead of the time they come up in class. Just look, read a bit, watch a video on YouTube. It helps.
- Find someone who likes geometry (sincerely) and hang out with them a bit.
- Explore GeoGebra. Love the tool!

I hope it helps.

frozenflameinthewind1 karma

Do the majority of parents that seek out your expertise do it in order to be a partner in their children's education, or are they more of the "tiger mom" type of parent that pushes their children to be exceptional?

Edit: Typo

MariaDroujkova5 karma

Natural Math focuses on adventure, open problems, and explorations; such activities don't give direct immediate advantage in very competitive situations. The type of parent parodied in the "Tiger Mom" novel (and it was satire, by the way!) seeks ways for their child to win (over others). I don't see that pattern of behavior in our community - it's very collaborative. Parents and teachers doing Natural Math tend to be awesomely nurturing and supportive, of their own children and others. They may do some "tiger mom" stuff outside of our activities, but if so, I just don't get to see it!

A lot of these parent behaviors are driven by fears. If the community and math activities are built on love and care and helping each person to be brave, people behave gently.

Ovicior1 karma

How're 5 year olds doing calculus? Is it playing with it, or do they actually solve problems like in high school?

MariaDroujkova2 karma

Children are playing with ideas - say, can you imagine day and night repeating again and again and again (infinity)? Playing with things, such as building different slopes (functions) out of LEGO blocks. Solving puzzles, such as making a circle out of squares (integration). Formal problems with a lot of number crunching come later.

communikay456 karma

Why do so many children (and adults) hate advanced math? Is it how it's taught, or what is taught?

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