February 2023
Welcome to EFM's February Newsletter!
Activities for Educators News
Puzzles of the Week There are now 150 EFM Puzzles of the Week scaffolded into six skill levels. These are perfect for posting on classroom walls, or the walls of libraries and other community spaces! These give you an effortless way of introducing fun and challenging mathematics to children.
Family Math Fests EFM has added 5 more tables of activities. We now have 11 tables that each have three or four games and puzzles for families to play with together. Use FMF’s to introduce families to the fun of playing with math together and to entice them to go to the EFM website to find more fun, free materials.
Math Play and Struggle
Early Family Math believes in playful math the whole family enjoys together. We also believe in challenging math that at times creates an effortful struggle. Some say these goals are at odds with each other.
Challenge, effort, and struggle (CES) should go hand in hand with play, fun, and education!
This was brought home to me when I went back into teaching 17 years ago. I thought my job was to take the math, purée it, and spoon it to my Prealgebra students so that they could effortlessly absorb and master it. Not only was this an impossible goal, it was completely misguided!
Things of true value are achieved through effort, and if that effort is carefully scaffolded, it can be fun, engaging, and rewarding. I learned that deeper understandings and engagement occurred as a result of students being challenged and productively struggling. That personal growth occurred as students persisted and developed grit in dealing with things that were appropriately difficult. My students also learned more of the beauty of mathematics that mathematicians experience, and they learned that taking on hard problems in a supportive environment can be a playful and joyful experience.
CES in Education Literature
The education literature, both mathematical and general, is filled with discussions of the value of CES. Here are just a few samples of what can be found.
Brown, Roedinger, and McDaniel in “Make it Stick” discuss B. F. Skinner’s mistake in advocating for “errorless learning” in education, and they write that accepting and valuing failures is an essential part of learning (research presented by Boaler and others backs that up).
Whitman and Kelleher in “Neuro Teach – Brain Science and the Future of Education” say “Research shows that when students are challenged, especially by something that interests them, their intrinsic motivation increases and the knowledge and skills are more likely to be imbedded into their long-term memory.” “For learning to be fun, it needs to be challenging. Easy quickly becomes boring.” They talk about the Zone of Proximal Discomfort that hits a sweet spot of arousal and performance in the learner.
This last point dovetails well with the ideas of Csikszentmihalyi about when challenge is appropriately matched to skill level to achieve what he calls “Flow.”
Mitchel Resnick in “Lifelong Kindergarten” talks about the educator Seymour Papert’s idea of Hard Fun, and that “most children are willing to work hard – eager to work hard – so long as they’re excited about the things they’re working on.”
Francis Su in “Mathematics for Human Flourishing” devotes an entire chapter to the value of struggle and what he describes as the “struggle to grow.” He puts it well:
… the term productive struggle describes the state of actively wrestling with a problem, persistently trying out various strategies, being willing to take risks, being unafraid of mistakes, and progressing incrementally in understanding the underlying ideas. This wrestling produces a certain kind of endurance, which enables us to be comfortable with the struggle. This endurance produces an unflappable characterthat benefits us in addressing life problems – calming us with the knowledge that it’s okay if we don’t solve a problem right away. We appreciate that not solving a problem can be just as important as solving it – that, as Simone Weil suggested, the effort to grasp truth is itself worthwhile, for increasing our aptitude, even if it produces no visible fruit. In the struggle we acquire competence to solve new problems, and fortify an expectant hope that we will one day solve them. And when we struggle and at last succeed, we build self-confidence. Over time, through incremental and hard-won victories, this leads to mastery.
Su also quotes Fields Medal (the highest honor in mathematics) winner Timothy Gowers:
"It is terribly important to have the experience of finding things difficult but then managing to get past that stage. If the habit of thinking for yourself and solving problems, even though they are difficult, could be instilled at a very early stage, it would make a huge difference."
How do you Mix Play with Struggle?
After all that, I hope you are convinced to include CES in how you do math with your child. However, you may be wondering how to do it in a way that is playful and enjoyable.
This question puts the emphasis in the right place. Don’t avoid CES. Instead, work on creating a playful learning environment that welcomes, nurtures, and celebrates CES. This is intimidating to a lot of caregivers and teachers, and there is art in doing it well. I can’t give you a recipe for what will work best with a specific child, but I can offer general advice.
The environment you create is crucial, and the relationship each child has with their environment is equally important. What works for one child may be the wrong speed, attitude, or level of challenge for another. Watch your child closely, and continually adjust the level of challenge appropriately to avoid damaging levels of anxiety and stress when over challenged, and levels of boredom and listlessness when under challenged.
Here are some more broad guidelines:
Encourage playful exploration that often follows your child’s lead. Value the efforts and strides a child makes more than the successes they achieve.
Support risk taking, effort, and struggle, and avoid any stigmatization when successes do not occur.
Use roadblocks as opportunities to look for fresh approaches and hone techniques – both for the student and for the guide.
Embrace mistakes and being stuck as learning opportunities. Explicitly identify them as such and examine them together with interest and curiosity.
Celebrate the things your child discovers, even if it is not what you had in mind.
It’s also important not to restrict the label of play to unstructured silliness. Any activity your child enjoys enough to get engrossed in and want to do repeatedly should be considered play. Doing a math game together can be very playful. A math puzzle your child feels compelled to think about is play. Finding patterns in shaped blocks or in the behavior of a set of numbers can also be play.
Let’s Play and Struggle with Some Math!
Recently I was part of a group that put on an EFM Family Math Fest one afternoon at a local library. Here are some math activities the children enjoyed and steps the adults took to keep them playful.
Sum Difference
This simple activity can be played anywhere at any time. One person thinks of two numbers. They then tell the other person the sum and difference of those numbers. The challenge is to figure out the original two numbers.
I introduced the game to a parent and child and then, by way of illustration, asked the child to find two numbers whose sum was 9 and difference was 1. The child immediately said 4 and 5.
Not knowing the child, but hearing how easily the child answered the first question, I gave another puzzle: I have two numbers whose sum is 12 and whose difference is 6. The child paused for a while and struggled. The child said 10 and 2. I asked what the difference of 10 and 2 was and the child said 8. More time passed.
The child was over challenged and also was put in the position of performing for a stranger. This was causing anxiety and the child was shutting down. The parent wisely stepped in and suggested the child make a table of all the ways two numbers could add up to 12. The child perked up, wrote down the series of additions, and fairly quickly spotted the one that had a difference of 6.
By resetting the environment and giving the child an appropriate level of challenge, the child was having fun again and was eager to continue with more of these puzzles. Note that the challenge wasn’t removed and the child still had to exert effort to find the answer.
Soma Cube
These are a set of seven blocks that are made up of three or four 1x1x1 cubes. These seven blocks can be put together to form a 3x3x3 cube in 240 distinct ways. At my old school, I have seen students enjoy playing with these for hours while they chatted with friends.
At the library event, we set out many sets of these blocks, colored in various ways. While the children were aware of the challenge of putting them together to form a larger cube, most chose instead to make flat patterns or tall structures of their own devising. Rather than force them to follow our goal, we let them have fun creating their own challenges. They learned how to balance some pieces on top of others and how to create symmetries and patterns.
Lately, some people have been experimenting with replacing the standard Soma block set with sets that have simpler pieces. They say that the standard set is too challenging, even for adults. Most adults have grown up seeing math as something you either succeed or fail at, and that if you can’t do a problem quickly you must not be very smart. They have never learned to enjoy exploring in a mathematical space. They have grown to hate CES and avoid it at all costs. They want to protect children from that awful experience. This attitude, and its transference to children, is an important way that CES can fail.
Shapes with Symmetry and Patterns using Pattern Blocks
We put out a set of pattern blocks and offered up two activities involving them. One was to put the blocks together to form symmetric shapes, and we had pieces of paper available with mirror lines that would help with that. Another activity was to create pattern sequences together that the child and adult could identify and extend.
Instead of following our activity, most of the children chose to make designs of their own. They had a great time fitting these blocks together in ways that appealed to them.
They came up with their own goals, challenged themselves, and learned a lot about how the different shapes and colors worked together. And they played very happily and productively for long periods of time completely ignoring the wonderful activities we had in mind for them. It was great!
Play and Struggle Belong Together
I hope you have a wonderful time playing with your children, and that your children grow as people and as mathematicians as they playfully respond to interesting mathematical challenges created by others and by themselves.
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
February 18, 2023
Chris@EarlyFamilyMath.org
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Early Family Math is a California 501(c)(3) nonprofit corporation, #87-4441486.