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“I hate math,” the girl said to her friends. They repeated it. Then they grew up.
“I hate math,” they all said to their kids. “I hate math,” the kids repeated to their teachers.
“NO YOU DON’T,” the teachers scolded. “It’s fun and you’ll do it!”
“No! We hate math.”
Learn more about 50 Word Friday here.
I had this grand idea when we got married and were hoping for kids – I would teach our children to count starting at 0.
When Daughter was 15 months old, I decided we should start teaching to count with negatives.
But I was wrong on both.
And so is everyone else.
We practice counting 1-10 with our kids. We know (somehow) that before they’re official school age, they should know how to count to 10. And how proud we are as parents if they can count to 20!
But these are just words.
I can teach Daughter to memorize the Fibonacci sequence, but she’d no more know what that means than what counting to 10 means.
In fact, I know this first hand because I used to count to 10 in Spanish. And I’d leave out ocho everytime!
I saw a guy made fun of in Germany because he told a waitress he had fünf people in his party and held up four fingers. (She did it behind his back to another waitress – she wasn’t so rude to say it to his face. (Thank goodness; I would’ve had to go Texan on her.))
We teach toddlers to count for the same reason that we teach them to say please, thank you, yes ma’am and no ma’am – because someday they’ll understand what it means. And in the meantime they can establish good habits.
Regardless if we teach a toddler to start counting with -5, 0 or 1, they start with 2.
-5 to a toddler makes no sense. Teaching -5 to a toddler can only be dreamed up by a math teacher with no kids (i.e. me three years ago).
0 is useless. Why would you even mention that you have zero? Maybe saying that there are zero cookies after she ate them all might work. But generally zero things can’t be seen and by the time you’re down to 0 cookies, there’s probably a meltdown in the works. And we all know there’s no learning during a meltdown.
1 is just as useless. Why count things that are only one? They started with one mom, one dad, one dog, one couch, one bed, one bear,… Almost everything in their world is a single. The number “one” is just as useless to them as the words “the” or “a.”
Daughter was so amazed at the discovery that she had two SnackTraps. Not just the ordinary situation of a bowl of snacks but “TWO BOWLS!”
As soon as multiple copies of things are in her world, she takes note. If you’re an identical twin, the first time your child sees you with your twin might be traumatic. My best friend is the daughter of a twin and she tells horrors stories of this discovery.
This is an extreme, but consider all the pairs of things that kids can notice – two shoes (vs. only one that you can find when you’re freaking out and you’re late), two forks (when you’re begging for yours back from her because you’ve not eaten since breakfast), two cars (when you need to get in one and she insists on going in the other).
And, toddlers really don’t start counting at 2. They don’t start their mathematical careers with counting at all! They start by recognizing multiples. And 2 is the first and fastest multiple.
Keep teaching your kids to count – they still need this skill, just like they need to memorize math facts. But also teach them to subitize (recognize amounts without counting them out). Hold up two of the same items and exclaim “TWO ORANGES!” Then go to another two items and exclaim, “TWO RAISINS!” Stick with one number at a time.
Daughter is on “two,” so we’ll stick with that for a few months. We’ve got plenty of time.
It’s been mentioned more than a few times that ‘math’ is indeed a four letter word. Quickly after that’s said comes a look of, “So why are you claiming otherwise in your website title?!”
I just got an email from someone asking the same question.
When we type “math” we use four keystrokes. But we actually mean “mathematics” which has 11 letters. In some parts of not-Texas, outside the U.S., they abbreviate mathematics as maths. So really, math and maths are both 11-letter words!
The real statement of this site is that math isn’t a four-letter word. It isn’t a bad word. It isn’t something that should be feared, loathed or avoided. In fact, since everyone does it, it doesn’t make sense to fear, loathe or attempt to avoid it.
It’s like saying, “I’m afraid to eat,” “I hate eating,” or “I avoid eating at all costs!”
It’s something you do. You were born with it.
But indeed, yes. Math, the English language construct, has four characters:
So I can’t really argue.
Or I can argue all day.
And THAT’s the real fun of being a mathematician!
Originally posted on MathsInsider.com as a guest post.
Are you thinking about tutoring your own children?
Are you hesitant?
The toughest thing about giving instruction to a family member is how close you are to the situation.
Hire a math tutor and you can watch him peacefully manage your sweet offspring.
Try it yourself and they turn into maniacs.
And why?
Because you’re still frustrated that they haven’t learned to pick their clothes up off the floor. And you’re annoyed that they don’t clean up the milk they spill. They want to go to the movies and you won’t let them. And they feel like you aren’t letting them be themselves. And…
But you can pull away from all that and still tutor!
It’s time to be creative, parents. If you’ve tried to tutor as “the mom” and failed, well, try to tutor as “Mrs. Fibonacci.” Get a smock and a hat or pair of fake glasses. Put on some earrings that you would never wear. Don’t get too goofy, but have fun with it.
Let your sweet dear know that you’ll be tutoring as Mrs. Fibonacci and set aside a specific amount of time. Depending on the needs of your child, you might try one hour on Tuesdays and Thursdays or a half hour each day. Go to a set place and close the door. Get the rest of the family on board with this, too, so they won’t interrupt.
Also pay for the math tutoring. Instead of paying an outside tutor, set up a reward system for both you and your child. Every hour that you successfully tutor, you get an hour of fun on the weekend doing something as a treat. Or you earn points toward a trip. Or you can use monetary rewards – depending on how effective they are with your family.
Many math tutors will ask what the subject and topic is a few days ahead of time. This gives them the opportunity to brush up.
Find out what the next few lessons are and get help before the next session. That way when you’re Mrs. Fibonacci and in session, you’ll have some of the answers on hand.
And allow yourself to be wrong – mathematicians do it all the time. If he asks a question that you don’t know the answer to, let Mrs. Fibonacci offer to do research and get back to him in the next session.
Let the tutoring session be professional.
As Mrs. Fibonacci, speak differently to your child. Forget the clothes he left all over the floor this morning and the milk he spilled that he didn’t clean up. Those are all out. He’s a new, fresh kiddo. Beautiful, smart and ready to learn.
And have him call you Mrs. Fibonacci. Keep that role alive through the whole session. You’ll feel the difference and be able to talk math in a professional way to him.
Are you ready? You might start getting your “kit” together now – in the summer before the school year starts. Create your outfit and choose your name. “Mrs. Fibonacci” is okay, but you might want to find another mathematician’s name that speaks to you more.
Start getting the whole family on board. Let them choose the location and times. Get ready. Get set. So when the school year starts, you can GO!
I discussed empty sets in a previous article. I hadn’t yet listened to the recent podcast of The Math Dude, Jason Marshall, in which he introduced sets and subsets. Such a curious coincidence; I thought I would continue down our united path.
This week, the sweet Math Dude talked about union and intersection of sets. So I’m heading there too – and incorporating the empty set.
These symbols represent “operations” (like addition or multiplication) that you do to a pair of sets. They are called the cup and cap and look like, well, a cup and a cap. Or a U and an upside down U. The Union is the U and the intersection is the other one – that’s how I remember it and teach it.
For the sake of illustration, I’ll use Daughter’s handy SnackTraps like in the picture above.
The sets I’m using are
L ={green cube, blue half circle thingie}
R = {green cube, red triangular prism, orange rectangular prism}
I’m using the letters L and R for left and right in the picture.
The union of the two sets results in all the pieces from each set, all crammed together. (And if there is a repeated item, you only take it once.)
Let’s take the set of all states that border Texas and the set of all states that border Oklahoma, like this:
T = {Louisiana, Arkansas, Oklahoma, New Mexico}
O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}
Then the union of the two would be “The set of all states that border both Texas or Oklahoma or both” and would be T U O = {Louisiana, Arkansas, Oklahoma, New Mexico, Colorado, Kansas, Missouri, Texas}
Notice we don’t keep duplicates.
Using the same L and R sets from above, I can take the intersection.
Let’s go back to our set of all states that border Texas and Oklahoma:
T = {Louisiana, Arkansas, Oklahoma, New Mexico}
O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}
Then the intersection of the two would be “The set of all states that border both Texas and Oklahoma, at the same time” and would be T ∩ O = {Arkansas, New Mexico}.
When you start throwing the empty set into the mix, you follow the same rules.
For the union, you throw everything in both sets into one bag. Well, since there’s nothing in the empty set, “everything” is the original other set!
For the intersection, you take only the stuff that’s in common between the two. Since the empty set has nothing – there’s nothing in common.
Does this remind you of anything? What similarities do you see between this and addition or multiplication (or subtraction or division)? Ask your children these questions too.
Get out some snack traps and blocks (or a map) and go for it! Let me know how it goes in the comments.
Most parents aren’t professional mathematicians. But there are a few. This is the first in a series of interviews with mathematician parents with the goal of helping parents integrate math teaching into parenting. See the list of interviews here.
I had the privilege of interviewing David Chandler, a physics and math teacher. He runs Math Without Borders, an ongoing project to supplement high school math textbooks with Home Study Companions, to make them useful for homeschoolers and adults for self-teaching.
Last week I published an article about mazes that was inspired by some of his work with homeschoolers. He does some really cool things!
MathFour: Thanks for taking the time to answer some questions, David. First, what’s your degree and career? And how long have you been in math?
David: I have a BS in physics (1970), an MA in education (1975), and an MS in mathematics (applied math) (1997). I have taught physics and math (high school, Jr. college, some K-12) since 1972 and done other math-related projects on the side (inventions, design projects, publishing, curriculum innovations).
MathFour: Tell me about your family – how many kids do you have and how old are they? Are any of them more or less interested in math than the others in the family?
David: Two daughters born in 1974 and 1979, making them currently 37 and 32. B1, my older daughter, went through Alg II. She had some excellent teachers and some rather poor teachers. She would freely come to me for help and discussion. My younger daughter, B2 didn’t want the discussion. She tended not to come to me because she just wanted to know how to do the immediate problem at hand and got irritated when I said any more. As an adult she went back for more math as a prerequisite to other course work (involving logs, trig, etc.) and got tutoring from me over the phone. She values the explanation thing more now. My wives have not been mathematicians.
MathFour: Did you have any worries about your daughters academically? In particular, did you think they would do better in math than in other subjects because of your influence?
David: I think they both got the point that math is about understanding and problem solving and creativity. Both are academically well rounded.
MathFour: How did you play with your girls? How do you play with your grandchildren? Did/do you view playtime as different in any way than other “non-mathematician” parents?
David: I engage in a lot of banter that is probably math-influenced. When they say to me “Happy Birthday” I say back to them, “Happy Birthday to you too,” in the sense that she can be happy on my birthday. This year B1 sent a Facebook message, “Happy Birthday to us all.” That’s not explicitly mathematical, but it is the kind of thing Sheldon might say. I came up with a song that eliminates the asymmetry of l-m-n-o-p so you could sing the alphabet both forward and backward and taught it to my grandkids. Every rainbow we see is a physics excursion into dozens of phenomena. Theme park rides are, of couses, a learning experience. We have lots of “shared learning experiences.” Whenever something triggers a new insight in me, they are definitely exposed to the excitement it generates. They know what it is like to experience the world through eyes informed by physics/math awareness.
MathFour: Do you think you speak with your daughters or behave differently than other parents because you have a math background?
David: I don’t talk down to them, but I don’t overwhelm them in jargon either. I wonder about a lot of things out loud. I critique the world with a quantitatively informed (and order of magnitude informed) crap detector in their presence. I model inquisitiveness and observation and show appreciation when it is a shared endeavor. I think that attitude has rubbed off on both of them, possibly genetically as well as environmentally.
MathFour: Have you ever had either of the girls express negative thoughts about math?
David: Not really, except for B2’s impatience for my explaining in too much detail, and a tendency to go on too long when her attention has already shifted.
MathFour: Have you ever disagreed with one of your daughter’s math teachers?
David: Yes. B1 took Alg II in India where I was teaching at an international school. Her teacher was a young, very traditional Indian woman who spent a week or so teaching them how to read log tables, including interpolation, funky work-arounds for logs of numbers less than 1, etc. It was a total turnoff to the class, all of whom had calculators by then, and seeing this drove me up the wall.
MathFour: What happened; how did you handle it?
David: I discussed with B1 how all of this was now outdated and how using logs to solve exponential equations, using log paper, modeling perceptual phenomena with log scales (dB’s, astronomical magnitude scale, octaves, etc.) was really where it was at. I approached the teacher about this issue, as tactfully as I knew how. She listened politely, and didn’t react in an overtly negative way, but she may have been overwhelmed by me. She didn’t change what went on in class. I taught B1 a lot of things not in the book, and better ways to do things, but this didn’t generally lead to direct interactions with the teachers.
MathFour: Now to change direction a little to a more worldview of math. What do you see as the biggest challenge in math education today?
David: Getting kids to quit memorizing things and start digging for understanding. Also seeing math not so much as a body of knowledge as a mindset.
MathFour: What do you see great happening in the world of math education?
David: Internet communication for discussions like these. New insight-generating computational tools (Geogebra, Geometer’s Sketchpad, Tracker, spreadsheets, etc.) and their use in the classroom. What goes on in my (and presumably a lot of other) classrooms despite regressive influences like high stakes testing, minimum standards which become maximum standards, etc.
MathFour: What advice can you give to non-mathematician parents that might help them raise their kids to like and appreciate math.
David: Learn some math, engage in problem solving, problem posing, observation of the world, cultivating a sense of wonder, taking curiosity a step further and digging a little, model these for your children, etc.
MathFour: Thank you for taking the time to answer our questions, David!
How about you – do you have any questions for a mathematician parent? Share them in the comments – we’ll see if we can get David in here to answer them!
This is the 5th in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.
Last week’s article on the order of operations included a quick mention that division is the same as multiplication – but different. Now’s the time to explain that a bit.
The only thing mathematicians like to do more than create, is destroy. That’s how we get imaginary numbers, dividing by zero and raising things to infinity.
Once we create with multiplication, we want to know what happens when we destroy with the opposite of multiplication.
Multiplication is created from the need to quickly add a bunch of numbers that are all the same. They each must be equal to make this work:
6 + 6 + 6 + 6 + 6 is shortcutted to 5 X 6.
Technically we can break 30 up into these 5 pieces: 4, 8, 7, 6 and 5. But “division” requires (or implies) that we are dividing equally. So 30 would have to be broken up into 5 equal pieces of 6 each.
Of course this is in a purely mathematical world. When you get into a toddler world, things will be different.
We represents multiplication with little x‘s or dots or stars. Like this:
3 X 5 or 3 • 5 or 3 * 5
We represent division with this cute little symbol:
But we can write it in many more ways. Notice the cute symbol looks like a fraction with dots on the top and bottom. That’s not a coincidence. A fraction means division.
Remember the poem:
When dividing fractions
Don’t bat an eye
Just flip the last
And multiply!
Well, you can change this to:
When dividing numbers
Don’t bat an eye
Just flip the last
And multiply!
And “flipping” the last just means taking the “assumed 1” that’s underneath it and putting it on the top.
So when you’re teaching the MD in PEMDAS, the order of operations, remember that D is the same as M. And if things get a little confusing, demand some parenthesis before doing the problem.
Share your thoughts in the comments!
I just finished Dan Pink’s book A Whole New Mind in which he discusses the difference between a maze and a labyrinth. It made me think of this wonderful video of dissecting a maze into two walls to solve it:
I learned about this method from David Chandler of www.MathWithoutBorders.com in the Math Future Google Group.
From the discovery that David and his class of Math Explorers made, I learned that you can take two colors and create a maze. So off I went…
I’ve been dying to use this elementary school graph paper I got the other day so out it came. I grabbed a purple marker and started:
I immediately began creating the “rules” of what would be good maze design. And almost as quickly I began seeing that I was all wrong.
After a while I got out the orange marker:
More rules… more “…no, that’s not really a requirement” thoughts from me.
After a while I just gave up on making rules and decided to make sure that my purple and orange would come out at some point together.
Then with a little photoshop magic, I made it all black. You can download it here and play it!
This might not be something to teach, but rather something to do at home together. It’ll be a learning experience for the whole family.
Grab anything with lines or a grid on it and two different color pens or markers and give it a shot. Let me know how it goes. You can even post a link to the pictures of them in the comments!
Check out the interview with David Chandler here.
This article was previously part of a We Are That Family “Works for Me Wednesday” post.
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Are you teaching sets? Some of the stuff that you talk about in set theory is pretty clear – you have a basket { } and you put stuff in it {♥, ✂, ☎, ✿}. No problem.
But what’s the deal with the basket with no stuff in it?
That’s the empty set. It’s written like Ø or simply as the “basket” like this: { }.
But kids might be tempted to write {Ø} – and then they get it wrong. BUT WHY?
Suppose you have four empty sets. Do you have nothing? Nope – you have four containers. Just check your cabinets – I’m guessing you have a handful of empty sets in there with the Rubbermaid® or Tupperware® logo on them.
Supposed you put one inside the other, like this:
This is the same as {Ø}. Would you be willing to put leftovers inside the bigger one while the smaller one is there? It would be something like this:
Instead, wouldn’t you be more likely to remove one empty set from the other and have just Ø and Ø (no curly brackets) like this:
What do you think? Does this help you understand the empty set? Will it help you teach it?
This is the 4th in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.
The Order of Operations can be boiled down into three “real” operations.
Parenthesis are merely a way to group things – they aren’t a real operation. So they doesn’t count as a real operation.
Since division is really just multiplication turned upside down, we don’t need to include it separately, either. Likewise, subtraction is addition on its ear. So we throw him out, too.
Now we have only three: exponents, multiplication and addition.
In a previous article about remembering exponent rules, I recorded this video about exponents:
You see that 3 x 3 x 3 x 3 is 34. The exponent is the shortcut for multiplication.
Likewise, when we get overwhelmed with adding up the same numbers over and over, like in the video, 4 + 4 + 4 + 4 + 4, we can use multiplication to shortcut it: 5 x 4.
The shortcut evolution is like this
So when you do arithmetic, we should do the “recent” shortcuts first (exponents) and then the “older” shortcuts (multiplication) and then the “regular” arithmetic (addition). Remember that subtraction is addition and division is multiplication.
And keep in mind that we need to watch the grouping or anything isolated with parenthesis, absolute value bars or in a fraction. The grouping/isolation tools always trump the other operation rules.
Some textbooks actually have multiplication done before division. This means that the problem 9 ÷ 3 x 2 will show a different answer (3/2) in that textbook’s solution pages than what you would get following the OoO I’m describing here (6). These texts are rare, but I’ve seen them. So keep your eyes peeled.
The Order of Operations is a set of rules that we’ve agreed on. Which means that as long as a textbook clearly defines their order that they will follow, they can do things like this.
If you get confused as to which to do first, demand parenthesis. (Yes, you can do that.) Or put parenthesis into your child’s textbook to help them out.
The problem sets should be there to enforce, not confuse.
What do you think? Does this help or hinder the way you’ve always viewed the Order of Operations? Share your thoughts in the comments.
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