# Posts tagged math education

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A few years ago I viewed “The Hundred Languages of Children,” a travelling exhibit about the Reggio Emilia approach to early childhood education.  If you’re not familiar with this approach it, among other things, considers the environment (of the classroom and other spaces) as a “third teacher”.

Of course, I was drawn to the part of the exhibit that focused on movement and dance as one of the “hundred languages” with which children express themselves.  There was a video that showed the children’s first experiences with an old factory space – a huge room empty except for two rows of large, white columns.  The children were running around and between the columns, peeking around them, and interacting with their friends, all movements and ideas that eventually turned into a formal piece of choreography.

At the time I was just starting to think about creating a math/dance program for preschoolers and my biggest question was how could I encourage that kind of exploration?   It seemed unlikely I would be able to find an empty factory or other interesting environment and get a bunch of preschoolers there on a weekly basis.  And then it hit me – I could create an environment out of tape.  I could define three-dimensional space using two-dimensional lines and colors.

After experimenting with my daughter’s preschool class, I came up with some starting points for parents and teachers who are interested in employing tape in the interest of math and kinesthetic exploration of space.

A simple straight line taped down a hallway becomes a pathway.  It also divides the space in two, and provides a chance to walk on it or jump over it.  Best of all, one can march (or walk, or skip, or slide, etc.) rhythmically down it singing “As I was marching down the street, down the street, down the street…”  Or, tape two or more parallel lines down a space and see what happens when you sing “Down by the banks of the hankey pankey, when the bullfrogs jump from bank to bankey…”

A simple alteration of a child’s environment can deepen their experience and exploration of the space around them.  When my daughter was three her teachers put down a straight line of tape to help the class ‘line up’ before leaving the classroom.  It was a simple, visual learning strategy that appeared to work as envisioned by the teachers.  Later in the year though, I saw pictures of what else the kids had done with the line.  They had used their large blocks to build a wall the length of the tape and then lined up their animals and cars alongside it.

A simple taped perimeter can highlight empty space, as in “Find an empty spot inside the tape and make a shape.”  Floor tape can define and redefine the space it’s in.  Large open spaces encourage a lot of endless running.  The minute you create a large rectangular box on the floor, with corners, you now have enough visual cues to focus a preschooler’s attention to IN (the box), OUT (of the box), AROUND (the sides of the box), CORNERS, and ACROSS, all age-appropriate math terminology.

Ultimately, I would love if every parent or preschool teacher would put down taped lines in their living and learning spaces then stand back to observe how the children interact with them.

Start with one straight line and go from there but don’t bring attention to it.  Let your kids find it and interact with it on their own volition and let us know what you observe!

p.s. FYI, when I talk about ‘floor tape’ I am referring to two different products, both of them sticky.  First, there’s painters tape which is blue and low tack so it can come up easily off both hard surfaces and carpet. There is also the floor tape that P.E. teachers use, which comes in lots of fabulous colors, the better to design with, my dear.

## Math Goggles #2 – Home Patternomics

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It’s time to put on the Math Goggles (not sure what these are? Head over here to find out). Last week I invited you to search for math at your local library. But this week I haven’t had much free time for impromptu field trips. So I’m donning my goggles and hunting for math around the house. To make things a bit more interesting, I decided to only look for one thing – patterns.

I found my first pattern walking down the stairs. Rise-run-spindle-spindle-rise-run-spindle-spindle… I only have one cat, so the pattern was broken there.

Next I had to put some books away and noticed this weaving pattern on a basket on my bookshelf.

And what about a bookshelf itself? Large shelf, two small shelves, large shelf, two small shelves. Moving to an even larger scale, the furniture arrangement pattern in the room (as noticed by my husband) is “furniture to sit on, furniture to pile books on, furniture to sit on, furniture to pile books on…

But of all the patterns I found around the house today, this one had to be my favorite. I had to replace batteries in one of the toys and here it was, positive-negative-positive-negative terminals.

What patterns can you find in your house?

P.S. Once you start noticing patterns, it gets very addictive. So here’s a cool and very relaxing idea for a collaborative pattern hunt video I found on YouTube. Next time you have a few minutes to spare, doodle the patterns you notice. Here’re my doodles, including the furniture arrangement pattern, the stairs pattern (reflected and rotated), the batteries pattern and a few more.

## Hidden Math: Book Edition

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My daughter and I have learned so much math by finding it wherever we are and in whatever we’re doing.  For the last year we have been paying attention to the physical world around us and finding as many different examples of math in our lives as we can.  It’s quite stunning how beautiful and full of math even a city sidewalk can be if you have your math glasses on.

Back in May, for example, I wanted to start looking for spirals but only found two examples, one in a garden and one in our local playground.  Long story short, at some point my daughter picked up on the spiral thing and started pointing them out, only to have me say, “No, those are actually concentric circles,” which then lead to a few days of clarification about what a spiral is and isn’t.  Now she sees them everywhere!

We’re a team, her and I.  It’s really fun that things we have taken for granted all our lives suddenly have a new dimension. This is why, I think, that a recent return to reading familiar picture books from our home library made me notice math in books that are not obviously math readers.

My very favorite almost-hidden math story book is Five Creatures, by Emily Jenkins.  It’s about the similarities and differences (attributes!) between the members of a lovely little family.

“Five creatures live in our house,” it begins, “Three humans and two cats.  Three short, and two tall….Three with orange hair, and two with gray.”  We read this book when my daughter was in preschool and it was fun for both of us to look at the pictures to see who matched each description.  The categories of family attributes are not always straightforward, which makes this a wonderfully interactive read.

In Ezra Jack Keats’ The Snowy Day, cut paper illustrations show math from the very first pages.  In addition to great spatial vocabulary (up and down the hills, tracks in the snow, on top of, snowballs flying over the boy’s head) patterns abound. Check out this wallpaper — I love how the pattern units are so different from each other, and yet the overall pattern is so regular:

Parallel lines made by sticks and feet and gates:

The foot prints alternate, making a kind of frieze pattern:

I love this grid pattern in the mother’s dress, and it’s not just a color pattern.  If you look closely there’s another attribute of shading (solid and striped):

This background is a great example of ‘scattered’ like in a scatter plot.  Which section has more dots, and which has less?  How do you know?

In nature, every snowflake has the same structure yet each one is different from every other snowflake.  That’s not exactly the case here.  How many different kinds of snowflakes can you find?  How are they different and how are they the same?

So, now I’m curious what other books are out there that have this kind of ‘hidden’ math?  I just thought of one more book: My daughter listened to the novel Half Magic on CD back in the fall.  In the story, the kids find a charm that gives them half their wish and they quickly learn to wish for twice as much as what they really want.  It’s fabulous.

What other kinds of books have you found that have this kind of hidden math?  I’d love to hear your ideas!

## Different Perspective

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Yesterday I chatted with a friend whom I haven’t seen in a while. Her child, a bright and energetic 8-year old, participates in quite a few extracurricular activities – ballet, gymnastics, tae kwan do, and art. Next year, music lessons might be added to the mix.

I asked my friend how they choose the activities for their daughter. Well, ballet was Mom’s choice since it was something Mom always wanted to do herself, gymnastics – Dad’s, since he was in gymnastics as a boy. Martial arts was a joint decision because it is known to improve child’s discipline. And art was the girl’s own choice.

Then I asked this question: if there was a math club or a math circle near you, would you consider signing up your daughter?

Her answer was “Absolutely! We are actually considering some extra math drills for her since she has some problems in school.”

This reminded me of another conversation, months ago, when another friend said that she wasn’t interested in a math club for her daughter because “she was doing well enough without it”. Yet another friend, this one with a preschooler, said that it was simply too early for her child to learn math.

I find it very interesting why there’s such a difference between attitudes towards, say, a dance class or an art class and a math club. Why the reactive attitude? I mean, why wait until a child falls behind in math class? Why math clubs are thought of as places for remedial math? Why sending a 3-year old to ballet classes is perfectly “normal” while playing advanced math games with her is not (“are you trying to raise a genius?”, “why are you torturing a child?”, “why don’t you let her enjoy childhood for now?”, and similar questions).

What if we looked at art, music, dance, gymnastics, LEGO and other children’s activities from a different perspective. Each and every one of these offers so many opportunities for mathematical discoveries! We just need to help our kids recognize and explore these opportunities. How can we do it? Where do we start?

## “The Cult of Right Answers” by David Albert

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After we posted “Can you let them fail?” there was a lovely discussion at our Facebook group. As a part of it, David Albert (above) offered the Moebius Noodles community an excerpt from his latest book. David describes himself as father, husband, author, magazine columnist, itinerant storyteller, and speaker. Here is the gem from David:

Ring the bells that still can ring.
There is a crack in everything.
That’s how the light gets in.
- Leonard Cohen,

“Anthem”

I grew up in the cult of right answers. To this day, I’m sure I don’t know quite why my schools thought it all that important to initiate me into the cult. My wife in her antiquarian pursuits balances the family checkbook. She says the bank is virtually always right, so if things are more than a few cents off, she assumes she got something wrong, and adjusts accordingly to make things conform. Meera, my future-accountant daughter, would likely do the same, though she’d bother about the pennies as well.

In my quantitative work in my day job, when my hunch is that things are “off”, I’ve learned over time that it is almost never the result of a math error, but because of a problem at the data entry point, a computer shut-down, or any number of exigencies having virtually nothing to do with arithmetic. I have learned to look for results that are consistent; unless I have alarm bells set off elsewhere, I almost never examine them to see whether they are right.

When I go to the grocery store to buy a jar of spaghetti sauce (\$1.89 on sale) and a package of spaghetti (\$.99), if I want to know whether I’ve got enough money, I always add from left to right (just how I was taught not to do when I was in school), I estimate well enough, and I pay absolutely no attention to the fact that the final digit in the addition is “8”. Since I am likely to pay for it with my debit card, I am not going to count the change either.

I do the same with estimating the time the bread is baking in the oven (have any of you actually checked the accuracy of the electronic timer?), how long it will take to defrost the chicken in the microwave, my gas mileage or the distance to my destination, or the amount of salt to put into the dish when the recipe calls for “a pinch”.

And so, looking back at my school experiences, it is difficult to see what purpose the cult of right answers, which extended far beyond the world of mathematics, served, other than as an odd kind of social sorting mechanism. The successful competitors (including me) sat on the edges of our seats, ready to perform our next trick and obtain a herring from our trainers, the less successful got hungrier in the back until many forgot what food was. But the actual purpose for my initiation was, I believe to this day, to indicate that they thought they “owned” me, and that, deep down, I was one of them. Am I? In more than 30 years of deschooling, this is something that I am still trying to figure out.

There is nothing wrong with right answers of course. When I drive over a bridge, I am depending on the architect and builder having calculated the load capacity correctly. I need to be sure that the loan calculator used to figure out my mortgage payment is accurate, and I want to feel certain when I read that someone other than Ichiro has won the American League batting championship.

That’s all well and good, but I think it can’t be emphasized enough how the cult of right answers can be damaging to one’s capacity to learn, and can stymie both creativity and curiosity. And this is true whether one turns out to be good at it or not.

I was most definitely a science and math nerd in school, and was well-rewarded and advanced in the cult for my right answers. I also had some very concrete experience of what happened when the “right” answer (you’ll see the reason for quotations marks in a minute) wasn’t forthcoming.

This is a tale not even my mother knows. I was nominated for an all-expense-paid, two-week trip to a newly established (soon to be prestigious) math camp, for which we had to take a competitive exam. There were six scholarships awarded. There were 50 questions, each worth two points. Six of the competitors scored 100 points on the exam. I scored a 99. Why? In a geometric proof, I left out A=A. Well, duh! This is more Gertrude Stein (“A rose is a rose is a rose”) than Isaac Newton. I can’t imagine there is a mathematician in the entire universe who would have cared (though I never really met a mathematician until I was in my mid-20s), but I learned my lesson well.

I never took a single math course after high school; now I use quantitative analysis in my daily work. Go figure.

I often recommend the wonderful book The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger to homeschoolers. The book is engaging and very well written, nicely illustrated, and opens up a world of mathematics to children and youth that they might otherwise never know was even there. It was originally written for 10-11 year olds, I think, but I have actually discovered that children as young as seven (some of whom having had it read to them) find great pleasure in it.

And, yet, I have a confession to make. I recently retrieved my copy down from the shelf and read it for the third time. Each time, I experience the same response, a falling feeling in the pit of my stomach. I want to put it down, but I fight through the queasiness. The number devil poses conundrums that cannot be solved, provides answers that are clearly correct but cannot be explained, and are intended to excite a general sense of wonderment at the wide, beautiful world of mathematics. But the feeling of wonderment when it comes to mathematics was confiscated from me when I joined the cult’s inner circle. And like my permanent record, it seems to have disappeared into the ether long ago, and it is awfully difficult to cultivate it anew.

The cult of right answers is based at bottom on the fear of wrong answers, fear of failure, fear of error, fear of disappointing those upon whose favor one has come to depend on for one’s self-esteem and self-worth. It may work short-term, but in the longer run, fear is a very poor motivator for learning, and a prime cause of apathy. For while a continuous string of right answers might assuage fear for a while, an equally effective way of coping with fear is to cultivate a feeling that learning really doesn’t matter, or to deliberately (even if sometimes unconsciously) respond with wrong answers so that external expectations are lowered, or to simply attempt to absent oneself from the entire enterprise. After one experiences failure enough times and sees that the short-term consequences are really not so devastating, apathy born of failure becomes an acceptable response. Once having become discouraged, humiliated, baffled, or fearful, an apathetic silence can become a welcome refuge.

This silence can easily be reinforced, of course. Most of the days I passed in school I spent with the ‘know-it-alls’. They expended most of the day talking at me. It might have been the case that they were there to answer our questions, but it quickly became evident, as virtually every school child knows, that the know-it-alls (the teachers, of course) asked 95% of the questions, and 98% of the time already knew the answers. Sometimes there were “discussions”, which never really resembled anything like discussions in the real world; they were just manipulative tools for soliciting right answers. How many times I can remember the leading questions being answered by a classmate in a way already predetermined to being just not quite right, and the teacher calling on someone else to remedy the ‘ignorance’ displayed by the first responder. Shame and learning cannot occupy the same psychological space at the same time, for when they are forced into the same box, the only likely result is loss of self-respect….”

Some of Davids’ books:

## Teaching Number Concepts

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I am trying to teach my son a concept of positive whole numbers being made up of other, smaller, positive whole numbers. This has been a tough going so far, full of unexpected obstacles. There was, for example, the part where I tried to explain and show that although a larger number can be made up of smaller numbers, it doesn’t work in reverse and a smaller number cannot be made up of larger numbers.

An even more formidable obstacle was (and still is) showing that a larger number can be made out of various combinations of smaller numbers. Say, 5=2+3, but also =4+1 and even 1+2+2. And by showing I mean proving. And by proving, I mean having my son test the rule and prove (or disprove) it to himself.

That’s why I was very happy when I got a hold of Oleg Gleizer’s book Modern Math for Elementary School. By the way, the book is free to download and use.  We’ve been building and drawing multi-story buildings (mostly Jedi academies with x number of training rooms) ever since. If this sounds cryptic, I urge you to download the book and go straight to page 12, Addition, Subtraction and Young Diagrams.

And just yesterday I found this very simple activity on Mrs. T’s First Grade Class blog, via Love2Learn2Day‘s Pinterest board. All you need for it is a Ziploc bag, draw a line across the middle with a permanent marker, then add x number of manipulatives. Took me like 2 minutes to put it together, mostly because I had to hunt for my permanent marker.

The way we played with it was I gave the bag to my son and asked him how many items were in the bag. He counted 8. I showed him that the bag was closed tight, so nothing could fall out of it or be added to it. I also put a card with a large 8 on it in front of him as a reminder. At this point all 8 items were on one side of the line. I showed him how to move items across the line and let him play. As he was moving the manipulatives, I would simply provide the narrative:

Ok, so you took 2 of these and moved them across to the other side. Now you have 2 on the left and how many on the right? Yes, six (after him counting). Two here and six here. Two plus six. And how many items do we have in this bag? Good remembering, there are 8. So two plus six is 8. Want to move a few more over?

It went on like this for a few minutes until he got bored with it. Overall, I thought it was a good way of teaching, especially for children who do not like or can’t draw very well yet. Plus upping the complexity is really easy – draw more than one line on the bag and create opportunities for discovering that a number can be made of more than two smaller numbers.

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