Author: Bon Crowder

  • Can You Teach All K-12 Math in 8 Weeks?

    Can You Teach All K-12 Math in 8 Weeks?

    Elizabeth, @Ser3nd1pity, tweeted an excerpt from a book by David H. Albert called Have Fun. Learn Stuff. Grow.

    Reading through it, the thought that came to mind over and over was, “Finally.”

    Here was my favorite part:

    …the subject matter itself isn’t all that hard. What’s hard, virtually impossible, is beating it into the heads of youngsters who hate every step.

    Indeed there is a gracious sufficiency of beating – and resistance – when teaching math.

    But everyone already knows math!

    In the book, Albert wrote:

    If you never teach a stitch of math, in a mathematical culture your kids will learn heaps of it anyway. …learning math along the journey is a difficult thing to avoid.

    You can’t not do math and exist. It’s like not breathing. Impossible.

    We survived for a very long time without the written word. But we’ve never existed without math.

    Early humans knew that to divide a chunk of meat between two people would yield too little:

    meat2<what I need\frac{\text{meat}}{2} < \text{what I need}

    He might not have have had the fancy way to write it – but he could learn how to write it in 8 weeks.

    Daughter playing with washers: counting them and learning math in her world.

    Can you teach all of K-12 math in 8 weeks?

    The short answer – it depends on to whom. In Albert’s article, he’s teaching children. You can totally do this with kids. And cavemen, I suspect.

    Grownups are a different story. Innumerate adults already have anxiety, anger or fear associated with math. There’s a whole lifetime of un-doing that would have to take place.

    But once you undo this, then the 8 weeks would work.

    The way I teach college classes is this: we’re going to spend 48 contact hours together – I’m going to wow you in ways you’ve never expected. When we’re done, you’ll be much calmer. You’ll be able to learn math much better, both on your own or in a classroom.

    I don’t force learning. I don’t even require learning. I let it happen. It’s the closest I can get, in the classroom, to what Albert is saying.

    Can you get closer to the 8-week method?

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  • What is Subitizing?

    What is Subitizing?

    Suzanne at Guided Math Study Group emailed me about subitizing the other day.

    I used perceptual subitizing to “know” that there were three green hair curlers here.

    Alas, I had never heard this term. So I did some research and thought I’d share what I’d learned.

    Subitizing is a way of instantly counting. In fancy math terms it would be getting to the cardinal number of a set (how many) without without going through the ordinals (counting each one’s position).

    There are two types, perceptual and conceptual. Perceptual is perceiving the number of objects immediately. Conceptual is putting a little effort into it.

    For instance, I glanced at the hot rollers Daughter had scattered on my side of the bed. I perceived there were three on the nightstand. But I had to do a little conceptual subitizing to arrive at the number of curlers on the floor (see the pictures).

    I had to mentally “chunk” these hair curlers into smaller perceivable bundles to conceptually subitize the number here.

    Here is a list of resources I found about it:

    In the next article, I’ll attempt to answer the two questions I see associated with subitizing:

    • If you do it, how did you learn to do it?
    • How do we teach children to do it?

    What do you think of these questions? Do you do it? Do you teach it?

    (By the way, it is pronounced with ooo – like “Ooo! Subitizing is neat!”)

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  • Counting Isn’t an Inherent Concept

    Counting Isn’t an Inherent Concept

    We spent our vacation a couple of weeks ago in central Texas. We had lots of fun – including lots of counting.

    One to one association of fingers to Discovery Toys’s Giant Pegboard pegs

    I noticed that some of the basic counting principles grownups know, aren’t inherently known to kids.

    I was observing Daughter as I was counting. She continued to look in deep concentration as I counted from one direction and then recounted from the other. Then I would rearrange the items and count them again. She was still enthralled.

    We teach our young children to count to 10, but never realize they don’t know what that means.

    Counting is a way to associate how much with a group of objects.

    If there are three things in a bundle, we associate the word “three” and the symbol 3 with that bundle. This number is called the cardinality of the set/bundle and means the number of things in it.

    My lone Calculus classmate in high school decided to create her own number system when we were freshmen. Everyone else laughed at her, but I got it. She was noticing that there was no rhyme or reason that we called a set of two objects “two.”

    We’ve decided to say out loud “two” and label it two and 2 and that means this many things: X X

    Counting is a way to order things.

    A bunch of things can be lined up and counted – even if they aren’t technically in a line. Each subsequent number is associated with another object. These numbers are the ordinal numbers. The final number that you count ends up being the cardinality of the set (from above). In this way you use ordinals to determine the cardinality.

    Counting is a way to compare one group of objects with another.

    Take two groups of objects. Pair one object from one group to an object from the other group – set them up in a one-to-one fashion. This shows that the two groups have the same number of objects. It doesn’t determine how many there are, but very young children don’t have to know the numbers to grasp the concept of “the same.”

    This eventually leads to the concepts of equality as well as less than and greater than.

    Counting isn’t dependent on which object you start with.

    This was the craziest concept for me. I noticed this when reading Brown Bear, Brown Bear one night.

    To mix it up (to keep my sanity) I would count the children in the book in differnet directions. After 3,000 nights of reading the same book over and over, something occurred to me. There is no reason for a 2 year old to know that counting in one direction will yeild the same number as counting in another direction.

    This is taught – not directly, but through experience. After counting a bazillion times, we eventually figure out that no matter which way you count things, you’ll get the same number.

    Well, unless Little Brother starts eating those things.

    Counting can be stopped and picked up where you left off.

    This is another concept that grownups “just know.” If you can mark your place (and Little Brother isn’t involved), stopping and coming back won’t change the result. This is the forerunner to addition, too.

    Counting is the foundation of all mathematics.

    This is the kicker. Counting is the beginning of it all.

    If you can get your kiddo to count, the rest is cake. And not just saying, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10,” but really getting him or her to understand the totality of the concepts.

    • How much is there?
    • Is there an order?
    • Does one group have more, less or the same as another group?
    • Did the number of objects change when you counted differently?

    Grownups get it, but we aren’t born with it. Imagine that you don’t know these things inherently. How does that change the way you observe the world?

    This article was originally shared on Homeschool Creations.

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  • The Order of Operations Explained: Intro and Mnemonics

    The Order of Operations Explained: Intro and Mnemonics

    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:

    1. Intro and mnemonics
    2. Parenthesis
    3. Exponents
    4. Exponents, Multiplication and Addition
    5. Multiplication and Division
    6. Addition, Subtraction and Conclusions
    7. Exponents of Negative Numbers
    8. 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.”

    What’s your way to remember it?

  • Teaching Math with Counting

    Teaching Math with Counting

    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!

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  • Practicing Math Skills Early in Life Is a Brain Changing Event

    Practicing Math Skills Early in Life Is a Brain Changing Event

    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?

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  • How Adding Evens and Odds is like Multiplying Positives and Negatives

    How Adding Evens and Odds is like Multiplying Positives and Negatives

    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?

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  • 6 Confusing Terms in Mathematics

    6 Confusing Terms in Mathematics

    Thanks to David Ginsburg for inspiring this post.

    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!

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  • What’s the Difference Between Gifted and Disabled?

    What’s the Difference Between Gifted and Disabled?

    There are many definitions of giftedness. And there are many reason to test for giftedness. But the bottom line is how that giftedness is treated.

    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?

    We do.

    But I won’t say it’s always easy to do so 😉

    Thanks for letting me share this!

  • Should You Test Children to See if They’re “Gifted”?

    Should You Test Children to See if They’re “Gifted”?

    In a previous article, I wrote What it Means to Be Gifted in Math. Now the question is, should you test for this?

    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.

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