The Order of Operations (OoO for short) is used everywhere in mathematics because it encompasses many of the foundational rules that we’ve agreed to follow.
Alas, students have been given the cheap and dirty version of it for years. “Here, memorize this thing about your Dear Aunt Sally!” What the heck?!
There are subtleties in the Order of Operations that every person over the age of seven should know.
The series begins today.
The order of operations is a set of rules – like the drivers’ handbook for math. If everyone follows the rules, we’ll all be safe. But if someone makes a bad turn, we could be looking at a crash.
But the Order of Operations is only a set of rules for arithmetic! It isn’t even the best practice when it comes to expressions involving a variable like x. I’ll cover what I mean in this weekly series.
Here are the proposed articles:
Intro and mnemonics
Parenthesis
Exponents
Exponents, Multiplication and Addition
Multiplication and Division
Addition, Subtraction and Conclusions
Exponents of Negative Numbers
Another Reason to Ban PEMDAS (aka parenthesis aren’t an operation)
Mnemonics for PEMDAS
Well, there’s one: PEMDAS (pronounced just like it looks). That’s what the cool kids in high school always said. It was the same kids who said “soh-cah-toa” – which I thought sounded really goofy.
And then there’s “Please Excuse My Dear Aunt Sally.” And of course “Piranhas Eat Mostly Decayed Antelope Skin.”
Counting is a big deal in our household these days. Daughter’s 21 months old so teaching advanced multivariable calculus takes way too long for her attention span.
Walking Steps and Counting Them
But we count everything. Things in books, steps, bobby pins, people… you name it.
We count in negatives, too.
If you’re following my tweets, you might know that I count to Daughter in the bathtub – from negative 30 to positive 30.
My neighbor suggested that early introduction to negatives might have helped her friend’s 14 year old son. He now has all sorts of difficulties with math. Not the least of which is arithmetic among positives and negatives.
This makes me even more excited about counting in negatives to Daughter.
Counting in negatives shows order.
You may not need 61 seconds of anything. But you can count from negative 5 to positive 5. The point in the counting of negatives is to introduce the order of the numbers – since negatives seem to go “backwards” when you list them in order.
And you can introduce distances with counting. You can show how distance is different than the number of points. Counting from -5 to +5 is actually 11 numbers. In this video you can see how I “rediscovered” this and then explained it:
Parents often focus on the alphabet and reading in the very early years. There’s some neglect of math things outside of counting to 10 (starting at 1). You can do so much with counting – counting negatives, counting distances, skip counting (2, 4, 6, 8, etc.), finger counting, counting backwards.
Counting is the foundation of all of math. And there’s tons of fun ways to use it in the early years.
Share your ideas about counting in the comments below!
Thanks to Will Summer who inspired this post with a retweet to me.
Seems researchers have previously studied the differences in math skills among children vs. adolescents and grownups. Dr. Vinod Menon with has done new research on early math training that focused on kiddos who where merely one year apart.
Turns out that in one year of math, the brain changes quite significantly!
You can integrate early math learning into just about anything.
Daughter is heavy into Brown Bear, Brown Bear by Bill Martin, Jr. and Eric Carle. This is not a math book. Nor does it pretend to be. But I’ve found a way to use it to develop math skills.
When we arrive at the page with the children, we practice counting the kids. First in one direction, then in the other – as direction matters to kids. It isn’t inherent that if you count one way you’ll get the same number as if you count the other way.
Sometimes we’ll count the top row of children and add it to the number of the bottom row. We can practice the commutative property by adding 5 + 4 = 9 and the adding 4 + 5 = 9.
As an alternative twist, we’ll count and add the kids on the left page to the kids on the right page. This one gets fun, because that little blond girl is half and half!
So we can now add fractions and practice the commutative property at the same time!
What else?
With Dr. Menon’s research, we now know how important it is to get started early with math learning. What other things can you do to start teaching math skills early?
I was playing cribbage with Husband when I found my hand to be 8, 4, 2, 2 and the cut card was also a 2. I focused hard to see if I could come up with 15. (If any group of cards add to 15, you get two points.)
My resolve was strong. Then I remembered that for any group of numbers to add up to an odd number, one of them must be odd.
While shuffling for the next hand, I started thinking about adding evens and odds:
Even + Even = Even
Odd + Odd = Even
Odd + Even = Odd
This looked familiar. Then I remembered the rules of multiplying positives and negatives:
Pos x Pos = Pos
Neg x Neg = Pos
Neg x Pos = Neg
I was so excited to notice this. Alas, I’m pretty sure I’ve noticed it before. I love rediscovering patterns as much as discovering them!
Look for patterns everywhere.
Always keep your eyes peeled for similarities. When you see them, if your children are old enough, use leading questions to get them to observe the same thing. If they aren’t old enough, keep a journal to share things with them later. Encourage your kids to look for patterns in everything. Use toys and manipulatives to improve this, but also use nature and everyday activities. What patterns or nifty things in math have you noticed recently?
Do you struggle explaining to your students what “simplify” means? Do you wonder why they won’t reduce fractions to their lowest terms?
Here’s how to explain and teach some of the top confusing terms in mathematics – without throwing you or your students into a tizzy!
Where
Really! “Where.”
As in “Where do the lines intersect?” or “Where is the highest point of the graph?”
“Where” in math almost always means “at what x-value?” Every now and then “where” can mean “at what point?” In the first case, “Where do the lines intersect?” can be just as easily satisfied by a point (x, y) as the x-value only.
When you get into calculus, the word “where” is used everywhere (pardon the pun). “Where is the slope of the tangent line equal to zero?” and “Where are there discontinuities?” among others.
Keep your eyes peeled in the textbooks for this tricky term.
Reduce
Typically this means to reduce to lowest terms. Which isn’t really helpful, since you still need the definition of “terms.”
Numerators and denominators (tops and bottoms of fractions) that have one or more common factors are not officially “reduced.” Once they have no common factors at all (except 1), they are said to be reduced or reduced to lowest terms.
A reduced fraction helps in many ways. You can “feel” the value more easily: 2/5 is a lot easier to picture than 14/35. Also, when you add fractions, starting out with all the numbers in their reduced form will often help to find common denominators.
Zeros, Real Zeros and Roots
Zeros, real zeros and roots all mean the same thing. It is where a graph passes through the x-axis. We can write “where are the zeros” meaning at what x-value (“where” from above) does the graph pass through the x-axis? We also will write “what are the zeros” and mean the same thing.
To find the zeros or roots, we set a function (or any equation solved for y that involves x) equal to zero. Then we solve for x. This will give the result that we want – the x-values where the graph will go through the x-axis.
We call them zeros because they are where y = 0. We call them real zeros because there can be complex or imaginary zeros that don’t exist on the graph.
Finding the zeros or roots of a function help when graphing it. If you know all the places it goes through the x-axis, you can then determine where it’s above or below the x-axis. Then you can roughly sketch the graph.
Simplify
Of all the terms in mathematics, this one is the most slippery. This meaning of this term is always in context to the book or situation. What it means is “manipulate the expression.” The end result may or may not be “simpler” than the starting expression.
Better textbooks use phrases that are clearer, like “multiply the polynomials” and “combine like terms.”
If “simplify” is the instruction to a problem set in a textbook, consult the examples to see what type of manipulation to do.
In later courses, students have to recognize when a situation requires a different form of the expression. Practicing various forms of manipulation (or “simplifying”) gives them tools to manipulate the expression to satisfy their needs.
Do you run into other terms in mathematics that are hard to explain to students? Hard to understand yourself? Share them in the comments!
I was so moved by this letter I received from a homeschooling mom in Australia. She has generously allowed me to reproduce it.
Of course ALL children ARE gifted! ALL people ARE gifted!
What people seem to refer to by the use of the word ‘gifted’, is actually referencing ‘intellect’ or ‘intelligence’ or just plain good ‘ole ‘smarts’! or is it speed? ie, the child can do the work faster than other children and have more time to spare. Or is it their ability to concentrate better in such a busy environment as a classroom? aren’t easily side-tracked? extremely focused?
My children are extremely gifted, and I have never thought of them in any other way except that. Yet they have been diagnosed with several ‘disabilities’ (I refer to them as ‘diffabilities’ – different abilities) and if the doctor’s had their way, they’d be diagnosed with more!
So, why are they considered ‘disabled’? because they meet a set of criteria which isn’t the norm. Why is someone considered ‘gifted’? because they meet a set of criteria which isn’t the norm. Why is one considered to be on one end of the ‘intelligent/ability’ spectrum and the other on the opposite end? semantics? perception? social conditioning?
What is normal anyway?
If I compared my children’s IQ test with most children, my children would win – only saying that from the numbers on a piece of paper, not pride.
Yet, because of many other challenges they have, they can’t even survive in a classroom environment, and would be considered for remediation classes, not the advanced classes. They would be considered intellectually disabled and shunted to a special needs learning unit (separate ‘schooling’ on same campus as ‘normal’ school and the aim is to integrate them into the ‘normal’ school classroom).
Yet, my gifted, brilliant munchkins are struggling with learning higher math concepts, because, although we have spent more than 7 years trying to learn and retain the basic math facts, it just seems out of their ability to grasp. Yet we do university level science and computer studies, etc.
Gifted? Disabled?
All I know is that it’s a daily challenge for our munchkins and many others like them, who may or may not benefit from labelling in our society. I sincerely have no idea whether labelling is a wise course to take or not. Unfortunately we don’t have parallel lives so that we can compare the two with the same people and situations – wouldn’t that be fabulous?
One thing I didn’t understand for many years was this particular comment from parents, therapists and professionals alike “it’s so obvious you love your children” – what the??? doesn’t everybody love their children??? But sadly, I started seeing that although parents LOVED their ‘gifted/disabled’ children deeply and painfully, they just couldn’t tolerate their behaviour, so their interaction with their child looked like intolerance, disgust, despair…
…and a lot of shame and guilt in the mix.
It’s so sad that we don’t feel we have the freedom to show how much we love our children in our society, especially when our children just don’t ‘fit’ anywhere. They are too different.
But isn’t that what we should be celebrating? their difference?
I heard this story once about testing and learning:
A group of people were given a test and then separated into two rooms. One room of people was told that the test showed they had an aptitude for learning welding. And so they were being taught welding. The other set was told that their tests reflected a lack of natural ability to weld. But they were being taught welding, anyway. The group that was told they were gifted at welding, excelled. The group that was told they had no aptitude, did poorly.
Curiously, the tests were never graded and the people were separated arbitrarily!
Perception changes things.
As soon as the people in the “bad at welding” class perceived they couldn’t do the job, they didn’t try as hard. It became part of their internal belief system that they wouldn’t be good at it.
And once the gifted people realized that welding was their “thing,” they believed they would be great, so they tried harder.
If it is part of your and your children’s internal belief system that they’re mathematicians at heart, then they will be. They will excel regardless of the method of teaching you choose. They might still decide to be political scientists or English professors, but they will do well in math.
Do you test your child for Gifted & Talented?
There’s a saying among corporate trainers: “Don’t ask for feedback about something unless you can, and intend to, change it.”
Only test your child if you will act on the results of the test.
Children who are part of a classroom school system will be tested before being allowed into an honors or GT class. If you are a homeschool system, you can teach “GT style” without ever testing.
But you may be interested in “testing just out of curiosity.” Keep the story of the welding students in mind as you make that decision. As soon as you “know” something about your child, you will treat them differently. We’re human; we can’t prevent this.
If you’re curious, and the result of a GT test is, “Nope, your child’s just plain normal,” there’ll be disappointment.
Indeed there are anomalies – prodigies, math intuitives, etc. But unless you’re sure that your child falls into one of these categories, and you intend to act on that knowledge, don’t have them tested.
Treat your child as gifted.
In lieu of testing, just treat them as gifted from the get-go. It’s not about if your child is gifted, it’s about if you believe they are gifted.
Thanks to the great parents at the LivingMathForum for the discussion that inspired this post.
Do you wonder if your children should be using a calculator “at their age”? Are you a fan of calculators, but have friends who aren’t? Are your friends “into” calculators while you oppose them?
I often hear people say that children 50 years ago understood math concepts more quickly. Although our parents weren’t taking classes called algebra in the 7th grade, they were doing algebra in the 7th grade.
Algebra is arithmetic.
There are two fundamental and rarely understood facts about algebra:
Algebra is arithmetic with one or more numbers in disguise.
Algebra has exactly the same rules as arithmetic.
Which means if you can do arithmetic you already know how to do algebra!
Our parents or grandparents, 50 or even 30 years ago, weren’t using calculators. They had to apply all the rules of arithmetic to get the job done. Which means that they had to apply all the rules of algebra.
Teaching them a class called “Algebra” was much easier because of this.
What are the rules?
The basic rules that non-calculator users must apply are the distributive property and the order of operations. The distributive property is the thing that calculator use eliminates.
Children could get practice mentally multiplying things like 3 x 86 and do 3(80+6) = 240+18=268. With this practice, they are ready for 4x(3y+2z) = 12xy+8xz.
If they never have to multiply 3 x 86 in their head, they never get the experience of the distributive property. Which means teaching them 4x(3y+2z) = 12xy+8xz will cause anxiety and frustration. They see it as “magic” or “something you made up just to confuse me.”
Give them the tools they need.
Refuse to let students have the calculator. Let them have the tool of the distributive property for algebra before you teach them “Algebra”. Give them the benefit our parents and grandparents had!
Does your child struggle when you put him down in front of his math book? Are you frustrated in your attempts to get him to do math classwork or homework?
Maybe he’s a top-down learner. If so, you might not know based on his current “regular” work. It will help in his education if you know he needs the big picture before the details – or the big theories before the steps.
I remember learning to do derivatives when watching the foster kids that lived with us. I was eight. I’m a top-down learner. Here’s how to find out if your child is one too:
How the “green beans” con works.
My mom used to leave green beans open in a can on the table. We would walk by and eat them. If she put them on our plate, we would refuse them. So she got us to eat vegetables without asking us to, by just making them available.
You can use the “green beans” con as a test.
For math, put out the harder stuff. Find some books at Half Price Books or someplace cheap in your area. Open up the book. Sit and do some of the math yourself while mumbling aloud. Then walk away.
If he sees his folks (or older sibs) working through those problems, he might be interested. Watch to see if he goes up to the book to check it out (steal a green bean). Be available to answer questions if he asks.
If he can grasp some of that “higher level” stuff, he’s probably be a top-down learner. He won’t want the building blocks until he sees the plans for the whole house. This could be the cause of some of the struggle and frustration – he’s been given the building blocks instead.
Let him have the big stuff – start “allowing” him to do more of the advanced books. He’ll back up on his own to learn the “lesser” stuff so he can understand the big stuff better. You won’t have to force the work on him anymore.
Share your experience with your top-down or bottom-up learning in the comments!
I love finding nifty ways to use tools for teaching math. Especially tools that aren’t supposed to teach math. Or at least the math I’m trying to get it to teach.
I have this very cool balance that I got from Discovery Toys that would normally be a science toy. But, alas, I’m a mathematician, Jim, not a doctor. So I’ve taken the fancy science toy and turned it into a way to teach subtraction.
You can, of course, use it to teach addition and later I’ll do a post on using it to teach multiplication and division.
If you have children who struggle with math concepts, teaching them with hands on bits (manipulatives) sometimes helps. Here’s how to teach subtraction using a balance:
This nifty trick can be done with any balance as long as you have weights appropriately sized. Sometimes that’s not so easy to find. Order a colorful balance that’s similar to the Discovery Toys one in video here.
Did it work? How did your children receive this method of learning arithmetic? Please share your experience with it in the comments!
