October 2021
Welcome to EFM's October Newsletter!
News
Languages: Complete versions of EFM are now available in Arabic, Chinese, Dutch, English, Filipino, French, German, Italian, Korean, Portuguese, Russian, and Spanish!
In order to make EFM available in more languages quickly, we are now using some Google Translated versions of the material (which are clearly marked). If you find yourself groaning or laughing at a bad translation, please consider helping us improve them. The Google Translated files are on a convenient, sharable Google drive and you can make improvements as you like a few minutes at a time in your spare time.
New Activities to Enjoy!
As I prepared my presentation for the mEducation Alliance Symposium at the end of September, I reflected on how essential it was for children to go beyond basic drilling and mechanics; that they should have authentic experiences of being mathematicians and thereby enjoy and benefit from all those experiences have to offer. Two months ago, I wrote in this newsletter about Activities that could be played at multiple levels by making small changes. This month, I want to discuss two EFM Activities that are examples of Activities that can be enjoyed simply as they are, but also invite deeper explorations and play as a child develops their mathematical tools and curiosity. Dan Finkel has a series of talks about rich learning in the classroom, and those classroom ideas carry over nicely to family Activities.
During the Symposium panel discussion, I enjoyed the "Six Bricks" Activities that Care for Education presented, so I thought I would end this newsletter with a short discussion of those. Care's idea is to find a great many ways of using a few simple bricks to further a child's math education.
Chapters 2 — 5 — Nim with One and Two
The simplest version of this game is to put some number of pebbles, say 10, on a surface. One player is given the choice of going first or second. During a turn, a player removes either one or two pebbles. The player forced to take the last pebble loses.
This version can be enjoyed by a child who can barely count, and it is quite satisfying at that level. For such a child, you might start with fewer than ten pebbles.
There are many variations of this game, such as having the person who takes the last pebble the winner, or starting at 0 and using addition to a target total instead of subtraction, or allowing the removal of one, two or three pebbles. However, I would like to look at what happens when a child gets interested in understanding how to win. This is a basic mathematical question - How do I look at a situation that involves lots of experiences, find patterns in those experiences, and then understand those patterns?
There are many important problem solving techniques that you will want your children to learn over the years. Two techniques I would like to highlight here are: 1) Learn from simpler examples and 2) Do lots of examples and look for patterns in those examples. I am often amused when I ask my advanced students in school for a simpler version of a problem. For this problem, they might say "Let's start with five pebbles." Usually you should start as simply as possible, even if it seems absurdly simple — the simplest examples can be done very quickly, and it provides you with information.
Let's start, and we'll assume it's your turn. If there is just one pebble, you will lose. Okay, that was quick and easy. If there are two pebbles, you will win — take one pebble and leave your opponent with a losing position. After just a few minutes the pattern becomes obvious. If there are 1, 4, 7, or 10 pebbles, you will lose, and if there are 2, 3, 5, 6, 8, or 9 pebbles you will win. It's a pattern that progresses in groups of three. Great, we have discovered the pattern. Notice how much easier it was to learn from a few simple examples than trying to start with 10 pebbles and figure it out. This is a habit your child should develop over time.
The next step is to understand the pattern. After some thought you will realize that it comes from always being able to make two successive turns add up to three.
That's all there is to it. Lots of good mathematical experiences found in a simple game! When the time is right for your child to look more deeply at this game, offer the occasional guiding thought ("see what happens in a really simple version"), and then have the patience to let them find their way with all the stumbles, mistakes, and wonderful aha moments along the way.
Chapter 2 — 5 — Connect the Dots — String Art on a Circle
As first given to your child, this is a simple art activity involving basic counting that makes pretty designs. It is wonderful at this level, and as your children learns more mathematics, it invites further exploration.
Draw a circle and place some number of points evenly spaced around the circle. I'll use 8 points in this first example. Number the points starting at 0, so that would be 0 to 7 in our case. Next, choose a number to skip by, say two. Start at 0 and draw a line to the number two away (two), then draw a line from that number to the number two away from it (four), and keep going until you reach where you started. If all the numbers were not reached, start again with a new color at an untouched number, say 1, and proceed as before by advancing by two's. Here are some diagrams that you would get for the first few skip amounts with 8 points on the top row and 9 points on the bottom row:
That's the entire activity. It can also be done with string going around pushpins (or tacks), if you have those available. It produces very pleasing results and it is easy to do.
Looking more deeply, the situation invites some mathematical questions "Why are all the numbers connected sometimes?" and "How can I predict how long a circuit will be?" and "How can I predict how many different circuits there will be?" When the time is right (be patient!), try to let your child come up with these and possibly other questions.
Consider the different skip amounts for 8, the results are: For 1, 3, 5, and 7 there is a single circuit of size 8; for 2 and 6 there are two circuits of size 4; and for 4 there are four circuits of size 2.
Looking at the skip amounts for 9, the results are: For 1, 2, 4, 5, 7, and 8 there is a single circuit of size 9; for 3 and 6 there are 3 circuits of size 3.
Let's do one more. For 10 we get: For 1, 3, 7, and 9 there is a single circuit of size 10, for 2, 4, 6, and 8 there are 2 circuits of size 5; and for 5 there are 5 circuits of size 2.
Lots of good data there. An important problem solving technique when you have lots of data is to organize it so it is easy to look at — make a table of it with your child. For children that understand division, they may notice that if there is no shared factor above 1 between the skip amount and the total, then there will be a single circuit. From there they can start looking at the shared factors and see the relationship between those factors and the number of circuits and the circuit lengths. This activity provides a lovely connection between greatest common factors and art design patterns on a circle!
Chapters 1 — 5 — Care for Education — Six Bricks
This is a rare exception to the EFM policy of not using a commercial toy. These Activities use six Duplo bricks of different colors. You can check out their site at Care for Education. As the cynical among you have already guessed, Lego sponsors the development of this program. To their credit, Lego has donated a huge collection of materials to Care for Education so that they do not need to buy their bricks.
If you own some Duplo bricks, they are an incredibly useful tool for teaching beginning math concepts. Here are just a few quick ideas: count them as you stack them, talk about their different sizes, make towers with them and see which one is highest, and make patterns using sequences of colors. You can snap together groups of 1 through 10 of them and then show how adding works by combining two of those groups. You can also put them in groups of ten and use those groups to illustrate place value ideas for doing two-digit adding. You can place them next to each other and use the array of bumps to illustrate multiplication.
The example Activity that Brent Hutcheson of Care for Education gave was to select any number from 10 to 48, and then challenge your child to find a way to stack the six Duplo bricks so that, when looked at from above, exactly that many bumps are visible (there are 8 bumps on each brick). This is a simple activity that invites experimentation and discussion. Why are some numbers easier to make than others? Are some numbers impossible, and if so, why? What are some different ways to get the same number? This sense of experimentation, play, and discussion is at the heart of doing mathematics!
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
October 18, 2021
chris@kitchentablemath.com