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.

Baby Steps

On the topic of supporting the development of mathematical communication, today I want to address patience.  One of the greatest obstacles to learning is the fear of making mistakes.  In other areas of study, we know that mistakes are the path to learning.  In Math class, we get points taken away for every mistake we make, and if the teacher doesn't understand the steps you made to get to the wrong answer, we get no points at all.  But in fact, we don't want children to know what steps to show; we want them to understand how to show enough so that their audience can follow them. We, as guiding adults, need to exhibit patience with our children as they learn to communicate their mathematical ideas.

When children are learning to write in the early years, we encourage them to get their ideas on paper without fussing about perfect spelling and punctuation.  When they are learning to express themselves as toddlers, we give them the words we think they are trying to use.  We need to do the same with "showing steps."  Before we ask children to show all of their work on paper, we need to talk out their reasoning with them.  We have the advantage in that we can make educated guesses about how they came up with a solution to a problem or what steps they might take.  The back and forth of a conversation allows the child to make baby steps in learning to communicate by both watching and hearing the adult model, and making small attempts themselves and getting immediate feedback.  As we converse, we might write the parts on paper that we think would go into "showing work."  Eventually, the child will make their own written attempts.  

The process of learning what steps to show takes time and is unevenly paced.  A child not only needs to learn the language and vocabulary of math, but also takes years to understand that different audiences require different steps to be shown or different ways to show them.  We need to embrace that what goes on inside the head of a child is usually more than they can express well.  Rather than the patience of waiting -- waiting for the child to mature when one day they will magically "get it" --  what is required is the patience of building.  Celebrate each small success.  Laugh over misunderstandings.  Learn from mistakes.  It's all valuable.

Supporting Communication Through Understanding A Child's Process

Earlier I wrote about the importance of supporting the development of mathematical communication in a child.  I thought break my thoughts up into chunks.  Today, I'd like to talk about listening to understand a visual thinker.

One of the difficulties in communicating with a visual-spatial person, especially a child, about how a solution is reached is that they often understand things in pictures.  For one thing, there isn't really a sequence of steps they took to arrive at "an answer."  Rather, everything is there all at once in the picture.  For another, since we converse using language, ask children to explain themselves using language, and often model using language to explain, children expect to explain themselves using language.  As children get on in Math, this might evolve into a habit of trying to explain steps in a process using numbers or equations.  We need to break out of these two boxes. 

Communication II

Continued from an earlier post...

Now where was I?  (That's the trouble with taking quiet time in the morning, rather than at night:  there's no way to steal from your sleep to get a blog post finished.)  Right, communication.

What brought on these thoughts about helping children communicate their math?  It was a combination of things that grabbed my attention this year.  One contribution was the oft-repeated assertion that visual-spatial thinkers "just know" solutions to problems.  The other was the similarity in approaches used by teachers in a variety of subject areas:  In The Writer's Jungle, Julie Bogart emphasizes the importance of helping children express themselves by first listening to them and scribing for them, then offering your own words and structures to clarify or help them to better communicate their ideas.  Grammar, spelling, punctuation are all there to support communication; they aren't ends in themselves.  When my daughter brought her music composition to show her teacher, her teacher didn't critique her very unusual timing (In fact, she commented that she found it interesting.) but rather helped her by showing her how to add the bar lines and time signatures that would allow readers to understand the sounds she wanted to create.  Visual art, too, has been described as sharing with the world what you see the way you see it.  Learning to communicate in math needs the same type of support as learning to communicate in visual, linguistic, and musical mediums.  

In North American society today, we tend to think of Math and Sciences as going hand-in-hand, perhaps because the Sciences so regularly use Math to communicate and understand their own ideas.  However, Math is not itself a Science.  There are parts of Math that are irrefutable and reproducible; those are the parts that seek truth in the way Sciences do.  Math is also about beauty and creation; those parts are like the Arts.  To validate only the mathematical ideas of a child that agree with a textbook is akin to accepting only the drawings of a child that are exactly like the given sample.  To ignore or correct a child's own mathematical ideas is to dampen the mathematical spirit in him.

How then do we support the development of mathematical communication?

To be continued in... 

• Understanding a Child's Process
• Baby Steps

Communication

Lately, I've been hearing a lot about kids who "just know" an answer to a problem and don't know how they got it.  I must say, at first I felt a bit perplexed.  In all my years of teaching and tutoring, I have never come across a single student who "just knew" an answer consistently and couldn't show work.  Even if they began by saying that they didn't know how they knew, we could always tease out the line of thinking that led to their answer.  What I believe, then, is that it's not okay to become complacent with these very intuitive kids.  Showing work for marks is not necessarily a goal for everyone, but being able to communicate thoughts is an important step in learning.  Yes, communication is necessary for the sharing of ideas that leads to piggybacking and synergy.  More importantly, though, successful communication ensures that a child will understand that their mathematical ideas are valuable, that their logic is valid, that they are capable of "doing math."


Continued...

Supporting a Global Learning Style

A few months ago, I wrote about exploring the "right-brained" style and sequence of learning, the details of which are written about extensively at "The Right Side of Normal" website.  In my previous post, I speculated that perhaps we are a family of "right-brained learners."  We are not.  (My son and I are "whole-brained" when it comes to math.  This plausibility of this claim is supported by a recent article citing studies that investigate brain activity of youth with a predilection for math and/or music.)  However, I do have one child who learns math in a way that constantly surprises me.  She does, indeed, follow the "right-brained" way of doing things. 

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.

How, Not What, to Think

BatBoy has for the past year or so been very keen on matchstick puzzles.  I picked up a book of them at Chapters and he often likes to work on them while SpiderGirl has her classes.  One day, we were with a good friend as BatBoy worked on the puzzles, and our friend wanted to engage with him a bit -- which is great!  And she did what people often do when they help someone with a puzzle:  She figured out the answer in her head, and then hinted at what the solution was.  Of course, he then quickly figured it out and went on to start another puzzle.