Inspired by Calculus online course: Newsletter February 1, 2016

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Join Inspired by Calculus: Online course for parents, teachers, and their children ages 5-12

  • What: An intensive three-week online course on calculus for young children
  • Why: Learn how to help your children play with rich and beautiful math.
  • Who: 25 parents, teachers, math circle leaders, and their children, with Kalid Azad and Shelley Nash as organizers
  • When: Live meetings February 5, 6, and 27 at noon EST (New York)
  • Where: Online video-talk software Zoom (similar to Skype)
  • Price: Registration is $40. Work-trade stipends are available upon request.
  • Supplies: Paper, markers, scissors, glue, tape, building blocks

Sorry, the course is full. Sign up  for the waiting list, or check out the self-paced Multiplication Explorers!

Inspired By Calculus February 2016

We are running a three-week online workshop for 25 parents, math circle leaders, and teachers with their children ages 5 to 12.The goals of Inspired by Calculus are to help adults develop their own math intuition, to see the joy and beauty of math (especially Calculus), and to help adults develop the confidence to try out calculus activities with their own circle, family, or class.

Let’s Explore!

Calculus is usually presented as a tool for engineers and physicists. It misses the bigger picture: Calculus is a point of view that helps you see everyday objects, from a humble banana to a flipbook, in a new light. Imagine having “X-Ray” vision, where you can look at any shape and see how it was put together. Or, practice your “Time-lapse” vision and learn to predict how shapes will change in the future, like peeking ahead in a flipbook. These analogies provide more insight than years of dry textbooks, and can be enjoyed by children and adults alike. Join us to play and explore a new point of view!

Here’s how it works:

Adults meet twice in a webinar format at the beginning of the workshop (Feb 5, 6th). During the first workshop, our goals are to become familiar with Natural Math methods and activities and to explore in the context of Calculus. You’ll be introduced to Math Sparks and you’ll try them out live!

Between the first and second workshop, you will try out a math activity with your group of children or students. We want you to bring your experience back to the second workshop. Bring the successes, the challenges, the struggles, whatever happens we want to know!

The goal of our second workshop is to give you the support, feedback and confidence you need to try out even more math sparks on your own. We will discuss your experiences and share feedback and ideas with one another.

After the initial two workshops, you get to explore with your family, students, or math circle! For three weeks try to dive into our support materials and discuss your experiences in an online forum. You can try as much or as little as you like. Kalid and Shelley will be actively answering questions in the forum and you are encouraged to share your experiences and help others, too!

The last workshop (Feb 27th), we’ll wrap up the experiences, and our goal is to be sure we have all gained in our confidence and abilities and love of math, too! The support materials and forum will always be available to you even after the close of the live workshop session.

Want to make a difference for your children and see mathematics in a new light? Join this new Natural Math adventure!

Here is a sample of our Math Sparks. We’ll have Sparks like that to go with every activity. The goal is to start thinking about ideas – to spark curiosity. Click to see the full-size PDF and ponder the questions in it.



Kalid Azad is an extremely curious person who loves sharing insights with others. While studying Computer Science at Princeton University Kalid started a site to explain concepts as he would have liked to learn them. BetterExplained is a continuation of that idea. The book Math, Better Explained is a well-received Amazon bestseller.

Shelley Nash is the homeschooling mom of seven amazing and challenging kids ages 13 to 1. For over 20 years, she has been driven by an insatiable desire to understand learning, education, education’s role in our society and personal development, and the role of technology in learning. She is passionate about letting children own their education and learning, and about discovering the elements that inspire learning in all of us.

Natural Math Principles

Bridges is one of the seven Natural Math main principles, and the focus of this class. Bridges connect and unite. In the class, you will learn to link math ideas to one another, math to other human endeavors, and people to math-rich communities. Connections help us to make sense of math, and to use math to make sense of life.

A video message from Maria Droujkova: 5 year olds can learn calculus

This is a presentation the founder of Natural Math gave at a SparkCON festival, building on 5 Year Olds Can Learn Calculus article in The Atlantic. It’s a short and fast-paced intro to what Natural Math is all about, and to calculus with young children.

A video message from Kalid Azad: calculus in 1 minute

Calculus is a special way of understanding the world. In this video, I give a 1-minute intuitive overview of what calculus is all about.

A letter from Shelley Nash

Is math a struggle at your house? Do you want to do it differently? Do you want it to be fun? Profound? Joyful? Do you want to know why this subject is so important beyond balancing your checkbook?

After reading Mathematicians Lament about how we kill the joy in math, I’ve wanted to teach math differently and use more discovery and interaction. But since I didn’t learn math this way, it has been really challenging to figure out how to do it this way. I have felt silly, or like I’m groping around in the dark. I’ve realized that I have never really “done Math” in the sense a mathematician uses the word, and reading about how to “do math” isn’t enough. So I never did much differently. I bought lots of books and read about doing different, but I still really didn’t know how it should “look!” Then I found mentors at Natural Math.

When you first start this mathematical journey, it can feel like embarking to a foreign land. A mentor helps you see the way, but a group of peers encourages you to keep going and to gain confidence in your abilities. Sharing with others will comfort you that your struggles and feelings are a normal part of the process. With others you can find more diverse ways of thinking about math and you can share ideas with others about how this new way of doing math is looking in your home. When you are connected to other families or groups you can make more lasting changes to your learning practices.

What do you get from Inspired by Calculus?

  • A highly interactive experience where you make models, talk, and collaborate with other parents and teachers.
  • Leading your children or students on playful math adventures.
  • The first meeting in a math circle format (you as a student!) to see how to run these activities, and to get inspired.
  • A day to try activities with children and friends
  • The second meeting to answer your questions, overview other activities, and prepare to do them with children.
  • Three weeks of mentoring and peer discussions on the forums.
  • The third meeting to recap, ask more questions, and stay inspired by calculus!
  • A dozen reference cards (math sparks) to help you try out the activities with your children.
  • Access to the Natural Math online forum to ask questions and share resources, even after the class is over.
  • Most importantly, you’ll get the confidence in your own ability to do math differently in your family!

Questions? Email or ask in comments to this page.

  • Connection and devices: you will need fast internet to watch and listen. It’s better to have a microphone and a webcam so you can show and tell as well, but you can also use text chat. We recommend larger screens rather than phones.
  • Software: Zoom is a tool for talking, like Skype or Google Hangouts. Please download and try it, here: The same page has the link to a tech support center, in case you need it.
  • Recording: The meetings will be available as YouTube videos.
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Posted in Make & Grow, Newsletter

Poetry and Graphing Lines

I recently struggled to understand limits in Calculus. One math problem in particular made me really unsure of what I was doing.


I ended up asking Maria to help me understand the limit as x approached infinity. As she and I played around with similar graphs and functions on the graphing site, I began to see  patterns and the graphs were so beautiful!  I was filled with a joy that math can be so playful and inspiring, and I had these bits and pieces of thoughts running through my mind.

So, I took those bits and made a poem. I hope you enjoy it!



graphing lines

coming together

to show the tension and grace

Of their powers.

the exponents pulling, pulling

forming an entirely new shape.

The struggle of powers

Births a new creation

of beauty, grace, and wonder

all it’s own.

graceful, fluid, moving

now toward infinity, now away

now toward zero, now away

but it can never arrive.

there are limits to its

powers, curves and grace—

even empty spaces at times.

what a beautiful pattern

of numbers, space, and time.

what lovely art and joy.

Graphing Lines.





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Posted in A Math Circle Journey

Easy Complexity and Special Snowflakes: Newsletter January 21, 2016

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Easy Complexity Course

If you are a parent or a teacher reading this, you probably want to offer math activities that are full of joy and beauty. It may sound simple when you read blogs or books – a child’s play. Yes, give children blocks or patterns to play and have fun, use a few math words along the way, ask a question or two, and voila! – you have an amazing math adventure for your students or family!

The reality is often different. Our dark math past may hold us back from playing. Even if we manage to organize an open, joyful activity, we have this nagging feeling that it’s not advanced enough, not deep enough, not hard enough. If it’s not difficult and a bit painful, is it even real math?

This two-day online math circle will help you learn how to deal with both dangers. First, we will offer activities that can get you playing, give you the much-needed sense of ease. Next, you will learn to ask great questions and find bridges to more complex explorations within these now-familiar activities.

Learn more and sign up at the Natural Math site.


Talks and blogs

Since we announced the 1001 Circles Facebook group in our previous newsletter, it’s been lively and energetic. There are 530 members as of this writing. If you use Facebook and are interested in family math and math circles, join the group. If you prefer longer posts, here are two thoughtful stories sparked by the conversations in the group.

Sue VanHattum at Math Mama Writes blog has a post titled, Does your kid hate math? Try a new angle

Long before I became a parent, in my teaching (of community college students), a number of them told me how bad they were at math even though their mom or dad taught it. I figured the parents pushed too much or something. (Blame the parents much, do we?) I ‘knew’ I wouldn’t do that.
Well, I don’t think I pushed. But my son hates math, and is consequently way behind his peers. (He unschooled for years and there was no ‘behind’. But he chose to go to a regular middle school this year, where the other kids have mostly had the standard schooling.) So when two people I respect got into a meaty conversation about this, my antennae popped up.


Yelena McManaman at Duct Tape Rocket blog shares her thoughts about a discussion of renewed math wars, started by James Tanton:

James uses an example of subtraction with borrowing. Before I get to that, I’m going to give an example of addition with carry.


Simple enough, right? Applying the standard algorithm, a child would re-write this, placing 56 under 237, then add starting with units, then tens, then hundreds, arriving at the answer = 293.

That’s how I was taught. And that’s how I’ve been doing it for many years except… Ok, I’ll return to the “except” part in a second.

For Rocket Boy, however, “standard algorithm” is the least effective. Because of his neurological condition, writing is hard for him and writing neatly is impossible. And writing one number under another, aligning units, tens, hundreds falls into the impossible category. So he figured out his own way of doing addition, relying on mental arithmetic and working his way left to right.

… except… I also add left to right whenever I don’t have pen and paper in front of me. Which is most of the time.

Read more at Yelena’s blog.


Dr. Maria Droujkova and the Natural Math crew


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One Family’s Math Story

Math has been an interesting journey at our house. We homeschool and my oldest daughter taught herself to read at about 4 ½. Learning came really easily to her and she loved it! She loved all the school things that looked like school.

But then we hit a roadblock – math. I had pretty strong opinions on education and learning (that’s why we homeschool, right?) but we did math the same way it had always been done for me. Basically, do a lesson and some worksheets.

This worked out okay until about 3rd or 4th grade. Somewhere in there, the concepts became too abstract for my daughter. We had used a conceptual program and my daughter refused to move on until she understood a concept, but she couldn’t explain what she didn’t get, and I didn’t know what else to do after explaining it a bunch of different ways. Some days we were both reduced to tears of frustration.

We decided not to worry about math in a formal sense for a while since it became a strain on our relationship. Around 6th grade my daughter was really bothered that she was “behind” in math. I wasn’t about to spend money on anything only to have her refuse after a few pages, so I told her she could try Khan Academy. She loved the progress she could see, but she refused to watch the videos. Instead, when she got stuck she would yell at me to come and explain it.

I would ask her, “Have you watched the help video right there?!” And invariably she would answer, no.

Eventually I learned that the videos just confused her more, but she somehow thought she was supposed to just magically understand each concept as it came along. She felt that by doing each set of problems she would be prepared for the next set of problems without any real effort to learn.

The problem was, most math programs we had, kind of felt this way. “Here is one quick explanation, maybe two, now you go do it. Surely, you must understand this new concept after my 5 minute video explanation!” So I set out on a quest for the perfect curriculum and never really found it!

So my first insight: most math programs don’t really teach math. You really, really need a live person!

My second insight: my daughter doesn’t think about math the same way I do.

My third insight: I have no idea how to explain math in ways other than how I learned.

I did find two great resources though. Ko’s Journey was a game and story based math program for middle school students and my daughter enjoyed it and was proud of herself for finishing it. Jump helps me understand how to teach math concepts without skipping steps I don’t realize I’m skipping.

Fourth insight: My daughter learned better in context and story form.

Ok now we are getting somewhere! So we ditched curricula for living math, and I stumbled on as well. I started to really re-think how we experience math and what it means to be good at math.

One summer I taught Trachtenburg Speed Math online just for the fun of the skill. My students had so much fun learning tricks and patterns and when they could do the multiplication really quickly, they were amazed and they enjoyed looking “good at math.” That was my whole goal for the class!

Then we read, A Beginner’s Guide to Constructing the Universe by Michael Schneider. This online book discussion was so fun! My young students (9-13 years old) and I used our compass and straightedge to make polygons; we built things; we explored the symbolism and sacred side of numbers. We found math in architecture, nature, chemistry, time, everywhere! We were all changed by this experience. My daughter would jokingly accuse me of hiding math everywhere! But I think she began to see that math wasn’t what she had closed it into before. It was bigger, and more amazing and magical.

The next step on my journey was to reach out to Maria Droujkova and ask for help learning to teach math in a Natural Math style. So she and I partnered on Inspired by Calculus. This completely changed me! (I’ve blogged about it here.) My daughter enjoyed some of the activities and was just grateful not to be forced to do formal math. But, ironically, she was plodding along in the Khan Academy 6th Grade math and was working on it daily.

My daughter kept bringing up her frustration that she was ‘behind’ in math. She was worried that she would never get past 8th grade math. She worried that her goals would be hampered by her lack of math knowledge.

Smart mom that I am, I didn’t really listen. I only heard the desire to catch up, so I tried to find curriculum again, or to encourage her to just keep going. We had both had such bad experiences with typical curricula and math that I decided to avoid them and try something different.

I formed an online math circle for teens and we tackled James Tanton’s book Arithmetic: Gateway to All. This started out as a very bad experience! We all struggled and my daughter and another student floundered. We felt like we were drowning in math thoughts. I wanted to understand them, and figured eventually I would, but my daughter took this as the final sign of failure in all things math.

I was at my wits end. My daughter wanted to be able to do math (whatever that meant), my relationship with her was strained, and I had tried everything. So I asked Maria Droujkova to rescue me!


It’s an honor to be invited into this story for a bit. The first and main thing I want to say: it takes a village for a child to find meaning in math. The village doesn’t have to be Silicon Valley. The virtual neighbors you call for help don’t have to be specialists; they do need to care about you, your child, and your project.

We talked with Shelley in a face-and-voice online meeting, as two parents of teens, and then I talked with her wonderful girl. I heard the same message Shelley did: that the young lady is frustrated by her math progress, and that she is anxious. But different people make different stories of what they see and hear. That’s why it takes a village: we need other people’s stories to get unstuck.

One story I hear: Shelley’s daughter wants serious power, including math power. She wants to be, and already is, dedicated, diligent, hard-working, and ambitious. So the messages that she can just go slower, or stick to lower-level math, or play around every which way – these messages make her sad and even angry.

The second story (hard to recognize in the everyday minutiae of family life) is of deep respect and deep weight of words. As open and accepting as Shelley is, one of the main difficulties her daughter has with math is the double jeopardy of failing a problem – and also facing the disappointment of one of her favorite, most important people in the world (mom). I remember how surprised and hurt I was when I’ve discovered a similar pattern with my child. My husband and I worked hard toward respect and openness, and yet the anxiety of judgment seeped into our house, probably from the larger society.

I think these stories speak about choices. How do you see power in everyday life? Those who wield power are free to choose, but also responsible for choosing. When are you anxious about judgments? When the judge is the one who chooses the task and decides if your work is good enough.

So, my main suggestion was for our brave young lady to choose some of her own math quests. And if Shelley is not the quest giver, she is free to be a fellow adventurer!

I want to point out, once again, that none of the above required my specialized math knowledge. You absolutely can, and should, help your friends that way no matter where you are in your own math journey. If you feel stuck, talk to someone you know who cares, and encourage your children to do so. By our powers combined, possibilities are infinite :-)



Maria’s advice to me was hard to hear, but I really appreciated knowing how my daughter felt. We were able to discuss her feelings, acknowledge them and let her know how much we loved her and admired her for all her hard work.

Maria’s most important insight for me: Using one math curriculum means there is a perceived authority and you must be able to do it one way or you are the problem. So use more than one authority and give your child permission to find those different authorities.

So, I made a drastic change to my online math circle and we deviated from one authority to many. For example, I asked the students to do anything they wanted with Negative Numbers (our next chapter). They could learn the history of it, google the word, find examples, see if there were games or activities online, try the activities from the chapter in Gateway. Anything! I explained to them the idea that we all can get stuck on one way of doing things and that we were going to challenge that feeling a little bit.

I’d like to say we suddenly had math lovers doing research and problems all week, but we didn’t! What we did have was a gradual increase in confidence; a gradual belief that I really would be fine with whatever their efforts produced as long as they gave effort; and a gradual understanding and acceptance that there really isn’t just one way to “do math.”

Sending my students and my daughter, in particular, to explore the world of mathematics online helped her to see the huge diversity of ideas, methods, problems, information, etc. that exists out there beyond any one curriculum of any kind.

I recently asked her if she wanted to work on a math curriculum that I’ve found and love (Jump Math).

She politely said, “No. I have a list of stuff I need to learn and I prefer to just google the concept and see what comes up. That seems to work better for me.”

I was thrilled, and I trust her to fill in the gaps of her own knowledge just fine. She knows I will help her or Maria will help her or that somewhere online, she’ll find the explanation that makes sense to her.

After being taught a bit about Geogebra a few days ago, I couldn’t get my computer back for an hour because she was working on using sliders to animate her circular design. (She doodles circles on her notebooks, so animating her doodles was intriguing to her.) As she finally walked out of my office, I heard her say to her brother.

“Have you ever heard of Geogebra? Come here, I want to show you. It is really cool!”

Now we are finally back on track!




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