Alternatives to Acceleration in Math

For awhile now, I've been getting to know many parents of gifted children, a population where there seems to be a disproportionate interest in math.  I identify a lot with many of these kids.  I had a propensity for math, and it was a subject in which I took pride because I could get the high scores that made me stand out.  At the same time, I didn't like the math that I learned, not until calculus.  When I look at how the math-gifted kids are dealt with, I am somewhat befuddled.  Here we are, gifted and many of us homelearning:  We acknowledge that there is no one size that fits all, not in pacing, not in interests, and not in learning styles.  Yet, it seems to me that nearly all the math-gifted kids are offered the same accommodation, differing only in pacing.

In the schools, both brick and mortar and online, advancement in math seems easy to deal with.  The student takes some tests and gets accelerated to the appropriate grade.  Now, for many kids, that is a good way to go.  Acceleration means more difficult material delivered in a neat package.  The package is important because many parents are unsure of their own knowledge in math.  Many kids thrive on the challenge.

Sometimes, acceleration isn't possible or it isn't enough.  Then students might be offered contest problems or lateral thinking puzzles.  Very occasionally, they are shown math from other cultures or given some math history.  These topics and exercises are fun and interesting, but they're also piecemeal and lacking in the area of actually educating in math.

Additionally, what about those whose thinking is divergent, who learn through problem solving, or whose language or development makes them unready to tackle coursework at their mathematical level?

How about depth instead of pacing?  Making connections?  Cultivating intuition?

BatBoy is 7 now.  His placement tests put him somewhere in the middle of grade 5 for math.  He doesn't know all of his multiplication tables.  He can do short division but not long.  Multiple digit multiplication is messy for him.  He has a need to know numbers in depth.  He loves prime numbers, families of exponents, roots, and factor trees.  He senses there is more to know about fractions and wants to dig in.  He is not ready for the procedural task of traditional algebra.  Contest problems freak him out.  So what do we do?  What have we done?

I write this because we have delved into deeper math with BatBoy, and I am convinced that other children would also love to explore math in this way.  BatBoy's past year has looked like this:  Along with regular skills, such as listing combinations, and surface explorations, such as of fractals, he's also continued to explore numbers in depth.  The topic of prime numbers led us to explore 0 and 1.  0 and 1 are special; in what other ways are they special?  What are the roles of 0 and 1 in multiplication, fractions, exponents, negative numbers?  What is interesting about other numbers, such as 6, 7, 8, 9, 36, 49?  What happens when we try to take 0 to the power of 0?  What does it mean to get one answer when you look at it one way (anything to the power of 0 is 1) and another answer when you look at it another way (0 to any power is 0)?  It means there's no solution, because in math, a solution is valid if you can reach it no matter how you get there!  How do we divide by fractions?  The math books will say that you multiply by the reciprocal.  This is a huge pet peeve of mine, because it is around this time that many students seem to throw up their hands and decide that math is meaningless and arbitrary.  Who can blame them really?  Nobody can explain why we multiply by the reciprocal; nobody even attempts an explanation.  BatBoy would not accept (nor remember) such a procedure.  What do we do?  We make a foray into the idea of mathematical proof.  Turn a concrete example into a representation that could be any numbers m and n, and we prove that dividing by 1/2 is the same as multiplying by 2.  (For older students, I have begun with the idea of the fraction as a division.  Once they see that dividing by 2 is the same as multiplying by 1/2, the reverse translates easily.  BatBoy did not accept the reverse as obvious.)

BatBoy may not extend these mathematical habits of mind into other problems right away.  However, the seeds are planted.  By filling out the world of mathematics beyond definitions and skills, we create context.  This is the kind of context a "mathy" kid needs.  This world of math is the context in which connections between numbers and concepts are made.  And if we can be patient with the slow and unsystematic way a child makes these connections, the foundation is laid for encouraging the development of mathematical intuition.

Creating a Culture of Math: Books

Children love books.  Books are a window to the world greater than their own.  Within books, children can play with personas, meet friends, explore ideas, discover people different from themselves, and have great adventures.  Books can be either informative or fantastical.  Either type can be a good way to ensure math is included in your family culture.

For the youngest children, we have counting books and shape books readily available.  One of our favourites is The Very Hungry Caterpillar, by Eric Carle.  I also recommend Math Fables and Math Fables Too, by Greg Tang .  They are everywhere you can find books.  Pick one or two that your child enjoys and put it on your bookshelves, or borrow them from your local library.  Read them with your child; enjoy the story and count the pictures together. 

After counting, math books seem to vanish from the bookstore shelves.  We need to look a litte harder to find them.  After counting, children like to see numbers in use.  Some they might like include

Creating a Culture of Math: Conversations, Part II

... continued from Part I

3)  Introduce Vocabulary -  When things or ideas are important in a culture, we name them so that we can talk about them.  Children intuitively understand that when we have a name for something (particularly if there are names for nuances of an idea) it is important to us.  Using math vocabulary communicates the value of math.

Often in a setting of structured learning, children are introduced to vocabulary at the beginning of a unit so that they understand concepts and details taught during the unit.  In an unstructured setting, I have found it works much better to introduce vocabulary after a concept is broached.  Just as when we first learn to speak, first we see the object, action, person, etc., then we want the name for it.  The name now has meaning.  For example, I observe (to myself) that my child is frequently adding 2+2+2+2.  I can then find an opportunity to observe aloud that yes, 4 twos is 8, and inject, 2 times 4 equals 8.

Creating a Culture of Math: Toys and Games

Children are naturally interested in the things that surround them, especially when those things are being used by Mom and Dad.  We'll address tools that adults use to go about daily life in subsequent posts, but in this post, let's consider things that children have free access to.  Children love toys.  While it may be true that they don't need many or complicated toys, what they have access to, they will play with and learn about.  Math may or may not come naturally to a particular child, but toys made available give them opportunity to explore and speak to what their family culture values.

Creating a Culture of Math: Conversations, Part I

As our children enter the world, their first experience of culture comes from their household, the people who spend the most time with them, usually their parents.  We know that modelling is extremely important in showing our children how the world operates.  We don't talk much though, in my experience, about showing children what is valued through our engagement and conversations with them.  Conversation is a wonderful way for a parent to be truly present with a child, create pleasant memories, play with ideas, and learn about one another.  It is through pleasant memories and engagement with important adults that children learn to value particular types of experiences.

When I first started reading novels with my daughter, I struggled to talk with her about the parts of the story.  She could tell me that she enjoyed the story, and I could say that I liked the story, but the conversation kind of halted there.  I would try to think back to my school days and I would remember things like foreshadowing and point of view, but she wasn't really there yet.  What I lacked was a knowledge about how learning about stories develops so that I could meet her where she was.

Conversations about math are the same as any other conversations in that we meet one another in a place of commonality and each add something in turn.  We add our observations, our reactions, our wonderings, and our suggestions for action.  We talk and we also really listen, and we always try to keep some common ground. 

So, what is there to talk about?  Here are a few key things we can try:

1) Make observations and ask for a response - At some point, each of my children enjoyed making shape patterns with counters.  For example, my son would bring 12 counters into the kitchen and arrange them into a rectangle.  "I see a rectangle," I observed.  A smile.  He knows I am paying attention and willing to engage. 

"How many are there?" he asks.  I know he is already proficient at counting by ones but hasn't gone on to skip counting yet, so I insert a new idea into the exchange. "1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12," I count.  

Later on, we would count, "3, 6, 9, 12."  Or when he was exploring multiplication, I would count 12 and then conclude, "3 x 4 = 12.  (pause)  I wonder if we could arrange the counters in a different way." 

Sometimes, he would bring more counters to continue the game with a new number.  If we had 7 counters, he would spend some time trying to make it into a rectangle and find that he could only put them all in a long line.  "I can only arrange them this way," he'd say.  "Yep," I agree, "7 only makes a long, skinny rectangle.  7 is prime."

Creating a Culture of Math: A Series

During one of my forays into internet groups, a question came up about "speaking math" to one's child when math is not one's forte.  I addressed it briefly and invited more specific questions with the idea that I might begin a series of blog posts about connecting with children on the topic of math in a variety of situations.  The invitation was largely ignored, but that's fine, I'm doing it anyway.  AND, I could use any input from you:  What's useful or not useful?  What would you like to see?  This has been an interest of mine and I would love to know the ideas that are helpful and why.  So, please and thank you, post any constructive feedback in the 'comments' section.


There is the idea that, while not new -- there is documentation of it from a journal of the National Council of Teachers of Mathematics from 1938 -- has come to my attention in the past year or so.  The idea circulated by professionals and scholars originally was that of postponing formal mathematics education until age 10 or later.  In homeschooling circles, however, I have heard the idea expressed as postponing mathematics until age 10-12.  But there is a huge difference between postponing formal math education and postponing math education at all!

It is true that once a child realizes the value of money, he or she begins to explore the ideas behind adding, subtracting, and grouping.  But we use math every day much more than for just counting our coins.  The thing is, most of us use math in our heads where it is invisible to the people around us.  Just as we surround children with books to support literacy and a love of reading, we need to surround children with math to support numeracy and a love of mathematical concepts and processes.  More than surrounding them, we need to live it, to share it, to engage with it, to play with it, with our children and in front of our children.  This series will talk about specifics of intentionally creating a culture that embraces math.