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.
Tuesday, December 24, 2013
Monday, October 28, 2013
Zooming the Number Line
Presented with a problem of rounding in Museum of Mysteries, by David Glover, BatBoy makes a very common mistake: He wants to round 7651 to 7600. At least I know he is not just memorizing the answers (or at least it is very clear when he has just memorized an answer) because he makes this same mistake every time he comes to this point in the story. 7651 seems to be closer to 7600 than to 7700 because there appear to be more similarities between 7651 and 7600.
Yet, when presented with a number line, BatBoy can find the midway point between 7600 and 7700, and then deduce that 7651 is, in fact, closer to 7700. But we need to draw this diagram over and over again.
Yet, when presented with a number line, BatBoy can find the midway point between 7600 and 7700, and then deduce that 7651 is, in fact, closer to 7700. But we need to draw this diagram over and over again.
Sunday, October 13, 2013
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
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
Saturday, October 12, 2013
Operations in Fractions
BatBoy understands fractions in the concrete sense, as equal parts of a whole. We haven't really touched on concepts like finding common denominators or multiplying fractions. Instead, he has spent time and energy really getting to know a few basic fractions and how they "work together." As he has learned to add and subtract, and then multiply and divide, he has been able to apply these operations to these basic fractions (halves, thirds, quarters, eighths, and occassionally, sixths). Using his visual understanding, he has solved problems involving adding or subtracting fractions of the same "family" (halves, quarters, and eighths; or thirds and sixths) including their mixed numeral relatives. He is able to find answers with negative values, such as the solution to 1/8 - 1/4. Today, he was playing with the sequence of dividing numbers by two. Beginning with 8, he divided by 2, again and again, until he got to 1/16 and couldn't (his words) "multiply 16 by 2" without paper.
SpiderGirl plays with concepts less and so, seems less drawn to mathematics than BatBoy. But given problems, she solves them quickly. She gets mental blocks when she is anxious. If she thinks that there is a "right" way to solve a problem (a way that she is less than confident with), or that I expect a particular answer, solved in a particular way, she freezes and claims she has no idea. But, if the problem is presented in a low pressure environment, she can shine brightly indeed. SpiderGirl understands fractions visually. Even when fractions are not of the same "family," she can puzzle it out using manipulatives. She used manipulatives to figure out to find common denominators in order to add or subtract. When she gets more comfortable with multiplication, factors, and multiples, I have every confidence she will gain a more methodical way of finding common denominators. When manipulatives are not available, SpiderGirl is also to present reasonable guesses to problems involving fractions, decimals (to hundredths), and percents. And she is learning that there is value in her ability to estimate.
SpiderGirl plays with concepts less and so, seems less drawn to mathematics than BatBoy. But given problems, she solves them quickly. She gets mental blocks when she is anxious. If she thinks that there is a "right" way to solve a problem (a way that she is less than confident with), or that I expect a particular answer, solved in a particular way, she freezes and claims she has no idea. But, if the problem is presented in a low pressure environment, she can shine brightly indeed. SpiderGirl understands fractions visually. Even when fractions are not of the same "family," she can puzzle it out using manipulatives. She used manipulatives to figure out to find common denominators in order to add or subtract. When she gets more comfortable with multiplication, factors, and multiples, I have every confidence she will gain a more methodical way of finding common denominators. When manipulatives are not available, SpiderGirl is also to present reasonable guesses to problems involving fractions, decimals (to hundredths), and percents. And she is learning that there is value in her ability to estimate.
Monday, September 23, 2013
When It Rains, It Pours.
Have I mentioned that BatBoy likes math? Yep, I think he does. He likes math games, he likes manipulatives, and he loves worksheets. What? Who loves worksheets?? Seriously, this boy drags them around like a security blanket. He brings them everywhere we go. Whenever he can spare a minute, whether he's waiting for his sister to finish her class or riding in the car or hanging out at home, he works on his worksheets. He even drags the worksheets to bed and refuses to sleep until he is done what he is working on. Between his obsession and his willingness to brush off and learn from errors, BatBoy is gaining arithmetic skills at a rapid rate.
Saturday, August 24, 2013
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.
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.
Monday, August 12, 2013
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.
Friday, August 9, 2013
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."
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."
Wednesday, July 31, 2013
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.
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.
Sunday, June 23, 2013
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.
Thursday, June 13, 2013
What Have We Learned This Year?
As I complete weekly reports throughout the year, I ocassionally purposely slip in a little math, just for the record. But really, this year, I had been feeling that we hadn't done a ton of math. SpiderGirl never mastered the multiplication tables, worked much with area and volume, or advanced in fractions, all things that had been on her learning plan. BatBoy has been exploring many mathematical concepts, but still, not much to go into a report. So I thought. When I began working on their final reports for our Distributed Learning school, I realized just how much they have learned this year.
Wednesday, March 27, 2013
Isn't That Odd?
BatBoy has been exploring odd and even numbers for awhile, just categorizing them in his head and finding patterns. He came to a conclusion a month or two ago that odd numbers had 'middles' and even numbers didn't. I've tossed in the idea lately that the pieces of even numbers can be paired off; in essence, they are divisible by two. But, that didn't (yet) disagree with the 'middles' theory and BatBoy seemed to disregard it. In the past week, however, BatBoy has encountered two troubling numbers: zero and one.
Sunday, March 17, 2013
Puzzle Books
Having embraced the idea that my kids are pretty right-brained when it comes to learning math, I feel more comfortable introducing problems that would normally be several "steps" ahead of where they are. Enter into our lives: the Maths Quest series by David Glover.
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| The Museum of Mysteries focuses on number manipulation. |
Thursday, January 24, 2013
Many Paths to an Endpoint
When I was in university, I had a professor for a geometry course that all math teachers were required to take. I found it interesting at the time that he was a professor that students seems to really like or really dislike, in terms of teaching style. At the time, I chalked it up to his habit of bringing in ideas and skills from related courses, and the lack of good mathematical understanding by some students, which, I thought, should allow them to integrate the different areas of mathematics.
Friday, January 18, 2013
Squares
Sometimes I'm not crazy about having so many tiles in our house. But they are rather handy for math.
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