March 2023
Welcome to EFM's March Newsletter!
EFM: Supporting families to play, explore, and love math
Early Family Math Turns 2!
Like the young people we work with, we are very pleased with what we have accomplished during our second year! Here are some highlights.
Our website has had 14,700 additional visitors (after 6000 last year) from 100 countries who have made about 11,000 downloads.
Our 160 activities for families are available in 14 languages. We have English and Spanish complete human written versions. We also have French, Chinese, Korean, and Japanese human translations nearly done.
We have 56 annotated storybooks in English and Spanish. These are also being translated into French, Chinese, Korean and Japanese.
We have a mobile app for Android phones bringing all our Activities and Storybooks to families in English and Spanish. Check it out!
We have a program for educators that has 150 Puzzles of the Week, and dozens of Math Games for Classrooms, Classroom Breaks, and to send home to families.
We have Family Math Fest kits that schools, libraries, or community groups can use to introduce families to the advantages of playful math education.
We have a bank balance of approximately $8000 from donations.
We have 160 subscribers to our newsletter (thank you!).
We have a wonderful team of people bringing enjoyable math education to families around the world. More are welcome!
As I wrote last year: I want to thank all who have helped EFM in one form or another during this year. We have a great deal yet to accomplish, and we would love to have your help. Please join us!
The Real World in Math Education
I have known teachers who believed in teaching “real world” applications to reach their math students. I have known adults who didn’t enjoy math until they were taught how math could be used to solve practical problems. You may even be nodding your head in agreement as you read this.
Paul Lockhart in A Mathematician’s Lament writes “… just because a subject happens to have some mundane practical use does not mean that we have to make that use the focus of our teaching and learning. It may be true that you have to be able to read in order to fill out forms at the DMV, but that’s not why we teach children to read. We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas.”
Most people have not experienced the beauty, wonder, and joy of doing math. They have not explored a math puzzle for fun, enjoying their progress in fits and starts, finally realizing a beautiful result only to be lured to continue exploring related ideas. Math may be the most misunderstood of the arts because it is the least experienced and yet people are still confident they know what it is. Consequently, many math education programs are run by people who have a very limited view of math. When they look for meaning and value in math, they look to things they can relate to, such as numeracy skills and balancing checkbooks, not realizing they are missing the core of what math really is and what it has to offer. Math does not need applications to be made more interesting, it is already amazingly engaging and interesting if you just understand what it is!
The Danger to Early Math Education
Most early math education programs are aimed at teaching basic skills and vocabulary. I have talked to nonprofits that explicitly say they teach early math entirely through "real world" applications to make math feel relevant to families. This is teaching a very impoverished version of math. It is throwing away the art and beauty of math for the sake of producing some limited results.
Am I against teaching applications? Absolutely not! Here is the real test: Is an application being done in service to the math, or is the math being done in service to the application? To a two-year old, all math is applied, which is as it should be at that age, and that is the only way to get started in math – counting everything, matching laundry items, describing and comparing all there is to see. In later years, if you want to teach engineering, finance, or science, that’s excellent, just don’t think you are teaching math.
Note that most things are enhanced by dressing them up in a story. As we’ll see further on, the Bridges of Königsberg story of a parade route over some bridges is a more engaging way of presenting a puzzle than talking directly about traversing edges in a graph. That’s okay! The application or story in this case is in service to the math. The math is still there in all its glory and is not simplified, dumbed down, or diminished.
Another Lockhart quote: “Then what should we do with young children in math class? Play games! Teach them chess and Go, Hex and backgammon, Sprouts and nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary.”
Play, Explore, and Sometimes Gently Apply
Let’s look at five math activities that don’t suffer by being associated with a story in the real world.
Fibonacci Numbers
Whole books are written about these numbers, so I’ll just touch on some ideas here. The sequence 1 1 2 3 5 8 13 …, defined by each number being the sum of the two preceding numbers, shows up in many ways. One of the more entertaining is to count how many of each kind of spiral there are on a pinecone or pineapple – they are almost always consecutive Fibonacci numbers.
As another example, male bees have one parent (a female bee), whereas female bees have two parents (a male and a female). If you count the number of ancestors for a male bee at each ancestry level, you’ll see the Fibonacci Numbers once again.
This is a topic rooted in the “real world.” However, there is also fun math to do with linking it with certain puzzles (as well as a lot of advanced mathematics). For example, count the number of different ways of going up steps where you can take the steps with any mix of one or two at a time. 1 step – 1 way, 2 steps – 2 ways, 3 steps – 3 ways, 4 steps – 5 ways, 5 steps – 8 ways, and so on. Why does that happen?
Similarly, count the number of ways of putting stones in a fixed number of empty places so that no stone has a stone next to it (including not having any stones at all). 1 place – 2 ways, 2 places – 3 ways, 3 places – 5 ways, 4 places – 8 ways, and so on. The Fibonacci numbers again – why does that happen?
String Art Around a Circle
I covered this activity in the October, 2021 newsletter, so I’ll be brief. Put some pins evenly spaced around a circle. Then, pick a number, say 3, and put string between every third pin. Look at the beautiful real world art you’ve created.
Experiment with different numbers of pins and gap sizes. You’ll find that some interesting patterns emerge concerning common factors (and at a higher level, modular arithmetic).
Levers and Mobiles
EFM has activities with both of these real world devices. Both of them concern balancing things, so the idea of equality is easy to bring into play. The lever principle also is a fun way to involve multiplication. The lever principle is: the torque caused by one side of a lever is equal to the force applied times the distance of the force from the fulcrum.
For example, suppose you wanted to balance a 4-pound weight that is 6-inches away from the fulcrum on one side of a lever. Where should you put a 3-pound or 2-pound weight? What if you had two 1-pound weights to use – where would you put them? Lots of natural ways to ask about equations involving addition and multiplication. The application facilitates the math and gives it an interesting story. You spend very little time learning the physics of the lever principle.
Bouncing Billiard Ball
Suppose you have a 3 by 4 foot billiard table. If you shoot a ball at a 45-degree angle from one corner, which corner will it hit first and how many bounces will it make to get there?
How does this change for tables of different sizes? Can you find patterns that help you predict what will happen? This involves a real world table and ball bouncing around, but once you get started it’s all math, and it involves similarity of geometric shapes, common factors, and some interesting geometric insights.
Bridges of Königsberg
The story goes that the people of Königsberg created seven bridges over the river in their town. They were so proud of what they’d done that they wanted to have a parade that went over each bridge exactly once.
Can they do it? If so, show how. If they can’t do it, describe why not.
The best way to solve this puzzle is to play with many simpler graphs and learn from those examples. There is lots of math and problem solving involved.
There’s Nothing Wrong with a Good Story
There was a great deal of good math in these activities, and that math was not limited or watered down by being associated with their stories. A good story can pique our interest; however, the best story, the one that keeps us wanting to see what's in the next chapter, is math.
If you have any questions or comments, please send them my way. I would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, I would love to talk with you about ways we can work together!
Chris Wright
March 18, 2023
Chris@EarlyFamilyMath.org
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Early Family Math is a California 501(c)(3) nonprofit corporation, #87-4441486.