Last night was taco night and my job was to grate the cheese.
I didn’t get too far in before I noticed some math.
For some reason I always start grating on a corner. Then I rotate the block so I’m grating on another corner. After doing this a few times I noticed the angles I was creating:
And what exactly are the shapes, anyway? What is the shape of the grating holes of the grater? And what is the resulting shape of the cheese sliver?
How much cheese is in the pile after you “fluff” it by grating it? What’s the volume of fluffed cheese compared with stuck-together-in-a-block cheese?
More importantly, is there enough for two grownups and a toddler? (When one of the grownups loves cheese!)
I’ve been itching to get into some basic abstract algebra goodies. With the help of the Cuisenaire Rods, Simply Fun Sumology number tiles and the Discovery Toys Busy Bugs, I’m able to do that.
Start with wrap around addition.
This type of math is officially called “modular arithmetic.” We are only going to use the numbers 0, 1 and 2.
It begins as regular addition. And since we are only using those three numbers, all our answers have to be either 0, 1 or 2. So when we add 1+2, we wrap around.
If we were to count in our system, we’d say: “0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, …”
The addition table looks like this:
(Notice you could do this with numbers from 1- 12 and it would be clock addition!)
Now things get buggy.
Switch out all the number tiles with some pretty color Cuisenaire Rods. They don’t have to be the “right” rods. We’re only looking at the colors. Here’s the progression I did:
The end result is a very abstract chart!
You can “bug” two things together.
Like this:
(I know – a spider isn’t a bug. But run with me on this, okay?)
Notice that each of these are directly from the “spider table” above.
You can read this as, “Purple spider green equals green,” just like you would say, “Zero plus one equals one.”
Over the past ten years or so I’ve been hearing this word “creationism.” It seems that it’s the opposite of “evolutionism.”
No problem – until I read about people trying to “prove” creationism. And articles trying to refute it.
I’m not sure what the big hubbub is about. 20 years ago I heard a guy make a simple statement about it all. He proved creationism in 30 seconds.
And it was a mathematical argument.
Before I give you that 30 second super-statement, let’s chat a little about what a real mathematical proof looks like.
Math starts with definitions.
We say, “Okay, here’s the deal. Let’s define a nebino as a number that’s greater than all prime numbers,” or something of that nature. (And yes, you get to make up your own words if you want.)
No math – none – ever starts out with confusing terms. If it does, someone jumps in and says, “You’ve got stuff that isn’t defined clearly.”
Everything shuts down until that gets resolved.
Math assumes… well… assumptions.
Once you have your definitions clear, you get to set up what you assume. This actually might come before the defining part. And often it isn’t said out loud at all.
Which is one reason that scientists sometimes think that they can do math. They’re always assuming the world (i.e. reality). Mathematicians don’t cotton to such vast and willy-nilly assumptions.
Then you get your hands dirty.
You’ve got definitions and you know what you’re assuming. You’re foundation is down. Now you build.
In other words, you create some math.
But things don’t always work out like you planned. So…
If it doesn’t work, you change the definitions or assumptions.
Yep – sometimes we really want something to work, so we just go back and tweak some of the starter points. Which means we change a definition or add (or delete) an assumption.
(Which means if you’re using someone’s math, you have to make sure you’re working with the same definitions and set of assumptions.)
And that’s the best argument for Creationism.
The statement I heard from this Creationist was, in essence, this:
We don’t have to use any evidence of science to prove God created the world in 7 days, 6,000 years ago. God planted the fossils and created all sorts of nifty things like DNA that would contradict the Bible. It was all meant to test our faith.
Voila! Creationism proven.
Brilliant! Change the assumptions, and you’re there.
Beliefs are just that: beliefs.
Which means there’s just no proving them. Kinda like my thoughts on the real line. I don’t believe in it – to the chagrin of my Twitter friend Colin.
So if you want to prove something, change the rules. Or ignore them. Mathematicians do it all the time.
*If you’re really really interested in my beliefs, I’ve shared them here.
Now that school’s officially in session, I’m really thinking about all the homeschooling moms who are taking on teaching math. I can’t help but see the similarities to something they’ve already done – given birth.
For each, I noticed that…
It’s darn scary.
Giving birth is pretty freaky. Especially if you’ve never seen or done it before. If you have the privilege of seeing another arrive in this world, it makes it easier. (I watched both my sisters and my best friend give birth!)
Teaching math might be a whole new world – and pretty strange if you’ve never done it. Dig back to some good math experiences you had as a child and draw on those.
It’s gonna happen no matter what.
Kids get born. They must. And they do. So you might as well make the best of it and enjoy the process the best you can – with a happy and positive pregnancy!
Kids learn math. They must. And they do. So you might as well make the best of it and encourage them in the best way – with a happy and positive math environment!
You get to choose how you do it!
You can give birth at home, in a hospital, with a doctor, with nobody, with drugs, without drugs… pretty much any way you want.
We’ve got more freedom now that we ever had. In Texas, where Betsy and I live, a homeschool is an independent private school. We get to teach in an unschooling way with lovely things like Math on the Level.
It may not go the way you want,
You’ve got a great plan. You know what to do and how to do it. You’ve taken the classes and my goodness! You’re in hard labor for 32 hours and you’re still at labor station -4.
You might think you have the best curriculum in the world. You’re all prepared. You start teaching an holy cow! Your sweet student couldn’t be less interested in everything you’re presenting.
…so be flexible.
Yes you’re frustrated. Remember the goal – a healthy child. Put away the frustrations and go for the C. Make it up by being a breastfeeding champion.
Likewise, you’re wanting a happy, healthy child. Choose a different curriculum. Let your child choose a different one. Experiment. Make it up by taking the crew to a great museum.
There are professionals for this.
Get a good ob/gyn, doula or midwife. You don’t have to go-it alone.
Get a tutor, enroll in a program like Kumon, join a homeschool co-op. You don’t have to go-it alone.
There is lots of information online.
With books like Betsy Dewey’s Birthright and sites like this one, there’s no reason to go into either blind.
It’s hard,
Giving birth is no piece of cake. However you choose to do it, you’re gonna have pain – in some way. Not to mention the nine months of puberty-like hormone swings!
Teaching math is a challenge. It’s not about conveying information, it’s about following a child’s natural curiosity and pointing out the math that happens along the way. Which means having to recognize math and the learning/curiosity style of another human. Not an easy task.
…and it’s worth it.
AH, the prize! When you are presented with a slimy little mini-human, you can’t help but think how cool it is!
And when a munchkin comes to you and says, “Did you know…” your heart fills with the glee of knowing that she extrapolated to get that – not just memorized it.
Are there more?
Any other similarities that I missed? Share them in the comments!
Betsy Dewey, my cousin, was a huge help and resource when I was pregnant with Daughter. She was pregnant with her second at the same time – that made it even cooler!
Betsy is an advocate of natural home birth and homeschooling. This article is written in honor of her and everyone else that listens to the beat of their family’s drum – and marches to it!
I’m fascinated by how the simple is really the most complex. Just proving that the two assumptions
3 + 0 = 3
and
3 + x = 3
will force x to also be 0 is a serious and important proof. And harder to do than you think.
When we get even more basic and talk about equality and comparison, things get impossibly complex.
And that’s the fun in the challenge inspired by 2nd grade math teacher, Suzanne Weider. She wrote:
Another idea I have been looking into has to do with the equal sign: teaching it as a symbol that means “the same as” as opposed to the sign that comes before the answer.
Equality and comparison are everywhere.
Some people are obsessed with fairness. Almost all people have some sense of fairness. This means that things need to be roughly equal, if not the same, for all situations in life.
We go into every situation assessing what’s the same and what’s different. Which means a judgment call on equality.
The beauty of this is that you can tap into how our natural comparison connects with the comparison tools of mathematics. These are tools like the greater than, less than and equals signs. This can set your children up for some hard-core success in algebra, geometry, calculus, and Radon-Schure-Greenlee Mathematics. (I made that last one up, but I mean crazy everything math.)
Equality and comparison are so complex that a single article won’t hold all there is to write about it. So this is the first in a series.
Children need a strong foundation in beginning math skills, like counting and basic addition and subtraction, in order to succeed with higher level math operations.
You probably already know this, but it’s easy to feel pressure to move your child on to the next level in math even when these basic foundational skills are stabilized. It’s easy to assume she’ll just “get it” with enough exposure.
Except that isn’t true.
Children that don’t pick up basic math skills with a decent amount of exposure and practice likely need a different kind of math stimulation.
Take basic addition and subtraction. Learning facts like 3+4=7 requires both a strong understanding of the concept of addition as well as a strong picture for the fact itself. In many ways, math facts are like sight words. Even when we understand the meaning of 3+4=7, we still need automatic fluency with recognizing and remembering it.
If you treat basic facts like sight words, it’s possible to help your child develop a strong mental picture for the fact – just like you can close your eyes and picture the letters in the word ‘teach.” (There are five symbols – letters or numbers/signs – in each!)
You can develop imagery for basic facts using sensory processing by using a “see, say, feel” multi-sensory approach.
After looking at a fact flash card, have your child write the fact horizontally in the air with her dominant pointer finger. Encourage her to really watch her finger and focus on picturing the equation as she writes it
As she writes the numbers and signs, have her say what she is writing out loud. “Three plus four equals seven.” This should happen simultaneously as she writes.
After she writes the equation, ask imagery questions like: “What number do you see in the middle?” or “What sign do you picture after the three?”
This simple process stimulates your child’s sensory processing in three fundamental ways all at the same time.
It stimulates her visual processing as she focuses on creating a mental image for the fact.
It stimulates her auditory processing as she says and hears the equation out loud.
It stimulates her kinesthetic processing as she literally feels her finger drawing the equation and develops muscle memory for writing it.
Using a multi-sensory approach to teaching basic math facts benefits all math students.
For children who really need to strengthen their visual, auditory, or kinesthetic processing for learning, this technique fills a need that additional worksheets, timed tests, and more flash cards games just can’t. For kids who don’t necessarily need the multi-sensory stimulation, it serves to accelerate learning. They are more engaged in each problem.
Now if only they made scratch-and-sniff flash cards – that would probably really get their attention!
How did it work for you? Please share your experiences in the comments.
Beth McKeon, of Bright Brain Studio, is a brain-based educator on a mission to demonstrate that every child has the capacity to learn. She has spent the past ten years customizing instruction for individual students and teaching teachers and parents how to engage the whole brain in the learning process. Her workshops and coaching provide practical techniques parents can use to reduce the frustration and resistance around homework assignments.
Daughter is into pickles. Like way into pickles. If I were to start a blog just for her, it would be called www.PicklesAndPretzels.com. (She’s also into pretzels.)
So when Husband grabbed two instead of one jar yesterday, it seemed natural.
Unloading the groceries, I saw the two jars a little more closely.
“Holy cow,” thought I. They’ve made ellipses (pickle ovals) out of segmenting cylinders (the whole pickles)! And they’re marketing them!
I had the pleasure of assisting Sarah Shah in her appearance on Great Day Houston yesterday.
While preparing for the show, I observed Sarah and the host, Deborah Duncan, in the makeup room having a conversation about math.
When I said to Sarah later, “that was an interesting math conversation,” she looked at me with anticipation, encouraging me to share what I heard. She had no idea I was referring to her conversation!
The math conversation was fully on-topic.
It was national thrift store day, and Sarah was going to share with GDH viewers some tips on shopping at resale shops. The topic of the show inspired their kibitzing behind-the-scenes about buying gold jewelry.
Deborah was talking about how there’s a difference (sometimes big) between the cost of the gold in a piece of jewelry, and the sale price.
The cost of craftsmanship should be close to its value.
Deborah was making the point that there’s value on the design of an object based on the workmanship that went into it. And this goes only so far.
Right now gold prices are around $1700 per ounce. Since an ounce is around 28 grams, gold is valued at about $60 per gram.
The QVC bracelet in the picture is 9 grams. It’s selling for $530 – pretty much exactly the value of the gold contained within.
If the value of gold for a 9 gram bracelet is around $530, charging $3000 for it means you’re paying about $2500 for the craftsmanship!
Unconscious math is all around.
Aqua And Gold Fractal by Sharon Apted
It was a wonderful experience to see two intelligent, educated women having a lively and entertaining conversation about math. It was quite disheartening, though, that Sarah didn’t even recognize it. In a previous life she was a physicist.
How many other conversations about math are ignored? How many people who claim they aren’t good at math have these conversations every day?
Look around at your conversations this week. How many of them are about math? Share your conversations in the comments. And with your kids!
Our discussion on fractions and pizza on #mathchat yesterday reminded me of a story.
Image via Wikipedia
My best friend lost on this deal back in 1978. But she gained a valuable lesson.
My friend, let’s call her Linda…
She was a terribly terribly slow eater. Like annoyingly so.
We would sit down to a Totino’s pizza on a summer afternoon. I would finish my half while she was still working on her first piece!
One day I suggested a different method.
“Let’s NOT divide the pizza in half – equally,” said I in my sweetest voice. “We’re best friends, and dividing food up is so primal. Let’s just eat like normal people.” (Okay, I’m sure I didn’t say, “primal,” but you get the point.)
She agreed. Even though she could manipulate the heck out of me, I certainly had my share of the talent.
She got two pieces.
A whopping 1/4 of the pizza.
She still tells this story.
And guess what? She totally knows the difference between her 1/4, our 1/2 and my 3/4.
What about you? Do you have any lessons you learned from friends in math? Share them in the comments.
Please note that my bottom still reflects this bad pizza eating habit. Perhaps writing it down for the world to see will help me knock of the over-pizza-eating habit. That and the LoseIt! iPhone app.
