by Josh Rappaport
Quick! For homeschooling parents, what is the “M”-word?
In other words, what word or phrase starting with “M” represents the most unpleasant (or feared) concept in the homeschooling world?
Could it be the dreaded Mitosis/Meiosis process?
What about the tricky English concept of a Misplaced Modifier?
Or perhaps the oft-forgotten Social Studies notion of the Monroe Doctrine?
Or, might it be an ENTIRE subject area, the one called … “MATH?”
If you folks are like the homeschoolers I’ve known, I’d venture that for you or others, MATH holds a palpable “yuck” factor, a “fear factor” or both.
If so, I’ve got my work cut out for me. My job, as I understand it, is to write an article (about MATH) every month, for 12 straight months. I am supposed to capture your attention (about MATH), provide interesting information (about MATH), and make your thoughts (about MATH) more pleasant than they ever have been.
Hey wait! Are you about to click off the page? Come on now … give me a fighting chance!
Before anything else, let me at least answer one question you may have: Who am I? Put another way: What kind of nut would take up this challenge?
Truth be told, my name is Josh Rappaport. I’m best known in homeschooling circles as “Who?” — or, if it’s my lucky day — as the author of something called the Algebra Survival Guide. But I’m not supposed to talk about that here. So I won’t. Still I can tell you my background, inasmuch as it relates to writing this column. Here goes:
— Majored in math in college.
— Along with my lovely wife (who did a lot more of it than I did), homeschooled our two children (myself teaching MATH) from the get-go through 4th and 7th grade.
— Taught for four years in the local public school system (Santa Fe, NM) — not MATH, if you can believe that. I taught communications, creative writing and history.
— Worked — and still work — as a professional tutor for the last 21 years, tutoring primarily MATH, though I also tutor language arts, study skills, etc.
— Author of several books on guess what … MATH (shh! Note at end of article if you’re curious).
Through my tutoring, homeschooling and writing, I’ve “cut my teeth” in math education. And just recently, I accepted the honor of writing this column because those experiences have led me to insights that will, I hope, be useful to your math teaching endeavors.
In this first article, I’d like to present an overview of the topics I’ll be delving into over the year. Then in the space remaining, I’ll share detailed info on a subject that’s critical for solid beginnings in math education: a concept called “number sense.”
Important side note: Since I’d like to get to know my audience, I’m interested in dialoguing. What kind of math topics would you like me to address? What issues in math education are important to you? Feel free to share your thoughts, by writing: josh@SingingTurtle.com
Topics I Intend To Address in this Column:
Healing Math Angst by Developing an APPRECIATIVE Attitude toward Mistakes
Math with Wings: Teach Kids to FLY with Mental Math
Using Color to Clarify Math Concepts
StoryTime: Using Extended Analogies to Make Math Ideas Come to LIFE!
Walk on the “Fun Side”: Teach Math through Games
Success Strategies for Word Problems
Evaluating and Choosing Math Curricula and Programs
Exploring Non-Traditional Math Topics (the good stuff they don’t make time for in school):
Math in Art
Math in Nature
Math in Games
Math in Crafts, Cooking, Quilting … the list goes on!
Making Peace (and Having Fun) with Calculators
Taking “Math Walks” Around your Neighborhood
Making it Human: Learning about Key People in the History of Math, and Discovering that Math is still a Developing Field
Reading Kids’ Books and Stories in the new field of “Math Literature,” to Enrich your Children’s Math Education Experience
Teaching Kids to Keep Creative “Math Journals”
Helping Children Plan and Carry out In-Depth Math Projects
As you can probably tell from this list, I’m one of those unusual people who believes that teaching and learning math can actually be a blast. And I’d like to convert you to my point of view.
So much for the overview, though. On to “NUMBER SENSE.”
First of all, what is “Number Sense”? What does this term really mean?
As originally defined by educational theorist Tobias Danzig in the 1950s, Number Sense had a narrow meaning. It referred to the human ability to glance at a collection of objects (like six buttons) and to be able to tell if that number of objects was changed while the person was not looking. In other words, the ability to notice a difference in number when glancing at a collection of objects twice.
Later the concept of Number Sense expanded to include such things as:
1) determining at a glance exactly how many objects are in view (subitizing), 2) comparing two sets of objects to tell which one is greater, and 3) being able to count any number of objects with accuracy.
In the last 10 – 20 years, though, the term Number Sense has been significantly broadened by professional educators, so that the term now includes a wide range of cognitive steps, including but not limited to:
* Understanding and using our base-10 number system.
* Developing an “internal” (mental) number line as a tool for comparing, adding and subtracting numbers. Example: by using a mental number line, students can “see” that 8 – 5 means the distance of 3 between 5 and 8 on the number line.
* Using knowledge of our place value system to invent original strategies for adding and subtracting. For example, realizing that you can add 9 by first adding 10, then taking away 1. [e.g.: 14 + 9 = 24 – 1 = 23]
* Breaking numbers apart, aka: “decomposing” numbers. For example, realizing that the digits of 55 refer to 5 groups of 10, and 5 ones.
* Checking the reasonableness of an answer through estimation. For example, noticing that 45 is NOT a reasonable answer for 9 + 5.
* Getting a grip on time. For example, recognizing that a car trip to a vacation resort will likely take a number of hours, not a number of minutes (much as children might hope it would!).
* Realizing that numbers can serve as tools for measuring real-world objects, and getting experience making measurements. Projects, projects, projects.
* Recognizing that the whole numbers have interesting properties. For example, any positive whole number is either odd or even, either prime or composite.
* Recognizing patterns in number facts. For example, recognizing that the multiples of 5 always end in 5 or 0.
As you can see, the concept of number sense has expanded dramatically so that it now refers to bedrock foundation on which children’s ability to work with numbers is based. That means that if we want our children to succeed at math in the long term, we need to pay attention to number sense in the short term, knowing what it is and taking advantage of daily opportunities that can enrich, expand and build number sense.
How can we do this? Here’s one suggestion for quite young children, say children around 3 – 8.
Have your children make a “Numbers Book.” Here are the guidelines for this fun activity.
Include a cover (to be decorated as a final step), and devote at least five pages for each of the numbers from 0 – 10. By the way how many numbers are there from 0 – 10, inclusive? Turns out there are 11 — that concept itself is part of “number sense.”
Use sturdy paper: construction or scrapbooking paper.
Start with zero. Make a title page that shows a big 0. Let your children decorate it.
Then, on each of the the subsequent pages for 0, let your child explore and address questions such as these:
NUMBER IN NATURE, IN THE WORLD: How does zero show up in the world?
In other words, can you find something in the world that has a quality of zero? If so, have your children draw it, or photograph it, and put that in the book. One example would be an empty bird’s nest, as it contains zero eggs. Or an empty cookie jar. As you and your child start to think about this, looking around for examples, you will see examples.
PROPERTIES: What properties does the number zero have? Is it odd or even? Is it prime or composite? (zero is neither, as it is a special case) What follows zero on the number line? What comes before it?
NUMBERS IN OUR LIVES: How does zero come into play in your life? Is there anything that you’d be happy to have zero of? (zero cavities at the dentist, for example!) Is there anything that you’d be afraid to have zero of? (zero friends, for example).
NUMBERS IN STORIES: Can you think of any stories or folk tales, in which the number zero came into play? For example, in The Little Red Hen, the other animals do zero work, and as a result, get zero to eat.
NUMBERS IN GEOMETRY: What shapes can you make with zero sides? Of course, this question is more relevant with positive numbers, but it still can be explored with zero, if you are willing to “think outside the box.”
And when you feel like you’ve thoroughly thought about zero, move on to 1. And then 2, all the way till you get to 10.
You may find yourself more challenged with some numbers than with others. For example, how does seven show up in the world? A quick Google search reminds us that there are 7 main colors in a rainbow. Who does not like to draw a rainbow?!
I guarantee you that if you and your child make this book and address all of the numbers from 0 – 10, your child will start thinking about numbers as you’ve never seen him or her do before. You will both see that numbers are not all the same. They have different properties, characteristics. It’s almost as if they have different “personalities.”
In any case, this is a great project for getting kids started noticing numbers in their daily lives, and it is fun to make the books, to boot!
For older children, you can extend this project. Focus more on the properties of numbers and on the numbers in geometry. Also, go beyond 10. Go all the way up to 30 or 50 or further.
A good resource for this project is a book called: Number Freak, by Derrick Niederman, published by Penguin. In this book, Mr. Niederman performs an analysis like this for all of the numbers from 1 through 200. His approach is focused more on the mathematical properties of numbers, but he does touch on stories, history and culture. A good read.
Have fun with this project. I look forward to sharing more ideas next month.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling book, Algebra Survival Guide, and the companion Algebra Survival Guide Workbook. Josh is also the author of the Card Game Roundup books, and has also written a playful approach to positive and negative numbers, PreAlgebra Blastoff!
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Check it out @ http://www.mathchat.wordpress.com Josh also leads workshops on math education for school districts large and small, and at homeschooling conferences. Josh can be reached at: josh@SingingTurtle.com
Quick! For homeschooling parents, what is the “M”-word?
In other words, what word or phrase starting with “M” represents the most unpleasant (or feared) concept in the homeschooling world?
Could it be the dreaded Mitosis/Meiosis process?
What about the tricky English concept of a Misplaced Modifier?
Or perhaps the oft-forgotten Social Studies notion of the Monroe Doctrine?
Or, might it be an ENTIRE subject area, the one called … “MATH?”
If you folks are like the homeschoolers I’ve known, I’d venture that for you or others, MATH holds a palpable “yuck” factor, a “fear factor” or both.
If so, I’ve got my work cut out for me. My job, as I understand it, is to write an article (about MATH) every month, for 12 straight months. I am supposed to capture your attention (about MATH), provide interesting information (about MATH), and make your thoughts (about MATH) more pleasant than they ever have been.
Hey wait! Are you about to click off the page? Come on now … give me a fighting chance!
Before anything else, let me at least answer one question you may have: Who am I? Put another way: What kind of nut would take up this challenge?
Truth be told, my name is Josh Rappaport. I’m best known in homeschooling circles as “Who?” — or, if it’s my lucky day — as the author of something called the Algebra Survival Guide. But I’m not supposed to talk about that here. So I won’t. Still I can tell you my background, inasmuch as it relates to writing this column. Here goes:
— Majored in math in college.
— Along with my lovely wife (who did a lot more of it than I did), homeschooled our two children (myself teaching MATH) from the get-go through 4th and 7th grade.
— Taught for four years in the local public school system (Santa Fe, NM) — not MATH, if you can believe that. I taught communications, creative writing and history.
— Worked — and still work — as a professional tutor for the last 21 years, tutoring primarily MATH, though I also tutor language arts, study skills, etc.
— Author of several books on guess what … MATH (shh! Note at end of article if you’re curious).
Through my tutoring, homeschooling and writing, I’ve “cut my teeth” in math education. And just recently, I accepted the honor of writing this column because those experiences have led me to insights that will, I hope, be useful to your math teaching endeavors.
In this first article, I’d like to present an overview of the topics I’ll be delving into over the year. Then in the space remaining, I’ll share detailed info on a subject that’s critical for solid beginnings in math education: a concept called “number sense.”
Important side note: Since I’d like to get to know my audience, I’m interested in dialoguing. What kind of math topics would you like me to address? What issues in math education are important to you? Feel free to share your thoughts, by writing: josh@SingingTurtle.com
Topics I Intend To Address in this Column:
Healing Math Angst by Developing an APPRECIATIVE Attitude toward Mistakes
Math with Wings: Teach Kids to FLY with Mental Math
Using Color to Clarify Math Concepts
StoryTime: Using Extended Analogies to Make Math Ideas Come to LIFE!
Walk on the “Fun Side”: Teach Math through Games
Success Strategies for Word Problems
Evaluating and Choosing Math Curricula and Programs
Exploring Non-Traditional Math Topics (the good stuff they don’t make time for in school):
Math in Art
Math in Nature
Math in Games
Math in Crafts, Cooking, Quilting … the list goes on!
Making Peace (and Having Fun) with Calculators
Taking “Math Walks” Around your Neighborhood
Making it Human: Learning about Key People in the History of Math, and Discovering that Math is still a Developing Field
Reading Kids’ Books and Stories in the new field of “Math Literature,” to Enrich your Children’s Math Education Experience
Teaching Kids to Keep Creative “Math Journals”
Helping Children Plan and Carry out In-Depth Math Projects
As you can probably tell from this list, I’m one of those unusual people who believes that teaching and learning math can actually be a blast. And I’d like to convert you to my point of view.
So much for the overview, though. On to “NUMBER SENSE.”
First of all, what is “Number Sense”? What does this term really mean?
As originally defined by educational theorist Tobias Danzig in the 1950s, Number Sense had a narrow meaning. It referred to the human ability to glance at a collection of objects (like six buttons) and to be able to tell if that number of objects was changed while the person was not looking. In other words, the ability to notice a difference in number when glancing at a collection of objects twice.
Later the concept of Number Sense expanded to include such things as:
1) determining at a glance exactly how many objects are in view (subitizing), 2) comparing two sets of objects to tell which one is greater, and 3) being able to count any number of objects with accuracy.
In the last 10 – 20 years, though, the term Number Sense has been significantly broadened by professional educators, so that the term now includes a wide range of cognitive steps, including but not limited to:
* Understanding and using our base-10 number system.
* Developing an “internal” (mental) number line as a tool for comparing, adding and subtracting numbers. Example: by using a mental number line, students can “see” that 8 – 5 means the distance of 3 between 5 and 8 on the number line.
* Using knowledge of our place value system to invent original strategies for adding and subtracting. For example, realizing that you can add 9 by first adding 10, then taking away 1. [e.g.: 14 + 9 = 24 – 1 = 23]
* Breaking numbers apart, aka: “decomposing” numbers. For example, realizing that the digits of 55 refer to 5 groups of 10, and 5 ones.
* Checking the reasonableness of an answer through estimation. For example, noticing that 45 is NOT a reasonable answer for 9 + 5.
* Getting a grip on time. For example, recognizing that a car trip to a vacation resort will likely take a number of hours, not a number of minutes (much as children might hope it would!).
* Realizing that numbers can serve as tools for measuring real-world objects, and getting experience making measurements. Projects, projects, projects.
* Recognizing that the whole numbers have interesting properties. For example, any positive whole number is either odd or even, either prime or composite.
* Recognizing patterns in number facts. For example, recognizing that the multiples of 5 always end in 5 or 0.
As you can see, the concept of number sense has expanded dramatically so that it now refers to bedrock foundation on which children’s ability to work with numbers is based. That means that if we want our children to succeed at math in the long term, we need to pay attention to number sense in the short term, knowing what it is and taking advantage of daily opportunities that can enrich, expand and build number sense.
How can we do this? Here’s one suggestion for quite young children, say children around 3 – 8.
Have your children make a “Numbers Book.” Here are the guidelines for this fun activity.
Include a cover (to be decorated as a final step), and devote at least five pages for each of the numbers from 0 – 10. By the way how many numbers are there from 0 – 10, inclusive? Turns out there are 11 — that concept itself is part of “number sense.”
Use sturdy paper: construction or scrapbooking paper.
Start with zero. Make a title page that shows a big 0. Let your children decorate it.
Then, on each of the the subsequent pages for 0, let your child explore and address questions such as these:
NUMBER IN NATURE, IN THE WORLD: How does zero show up in the world?
In other words, can you find something in the world that has a quality of zero? If so, have your children draw it, or photograph it, and put that in the book. One example would be an empty bird’s nest, as it contains zero eggs. Or an empty cookie jar. As you and your child start to think about this, looking around for examples, you will see examples.
PROPERTIES: What properties does the number zero have? Is it odd or even? Is it prime or composite? (zero is neither, as it is a special case) What follows zero on the number line? What comes before it?
NUMBERS IN OUR LIVES: How does zero come into play in your life? Is there anything that you’d be happy to have zero of? (zero cavities at the dentist, for example!) Is there anything that you’d be afraid to have zero of? (zero friends, for example).
NUMBERS IN STORIES: Can you think of any stories or folk tales, in which the number zero came into play? For example, in The Little Red Hen, the other animals do zero work, and as a result, get zero to eat.
NUMBERS IN GEOMETRY: What shapes can you make with zero sides? Of course, this question is more relevant with positive numbers, but it still can be explored with zero, if you are willing to “think outside the box.”
And when you feel like you’ve thoroughly thought about zero, move on to 1. And then 2, all the way till you get to 10.
You may find yourself more challenged with some numbers than with others. For example, how does seven show up in the world? A quick Google search reminds us that there are 7 main colors in a rainbow. Who does not like to draw a rainbow?!
I guarantee you that if you and your child make this book and address all of the numbers from 0 – 10, your child will start thinking about numbers as you’ve never seen him or her do before. You will both see that numbers are not all the same. They have different properties, characteristics. It’s almost as if they have different “personalities.”
In any case, this is a great project for getting kids started noticing numbers in their daily lives, and it is fun to make the books, to boot!
For older children, you can extend this project. Focus more on the properties of numbers and on the numbers in geometry. Also, go beyond 10. Go all the way up to 30 or 50 or further.
A good resource for this project is a book called: Number Freak, by Derrick Niederman, published by Penguin. In this book, Mr. Niederman performs an analysis like this for all of the numbers from 1 through 200. His approach is focused more on the mathematical properties of numbers, but he does touch on stories, history and culture. A good read.
Have fun with this project. I look forward to sharing more ideas next month.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling book, Algebra Survival Guide, and the companion Algebra Survival Guide Workbook. Josh is also the author of the Card Game Roundup books, and has also written a playful approach to positive and negative numbers, PreAlgebra Blastoff!
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Check it out @ http://www.mathchat.wordpress.com Josh also leads workshops on math education for school districts large and small, and at homeschooling conferences. Josh can be reached at: josh@SingingTurtle.com
