Hello, I am Maria Droujkova and I organize this collaborative study of learning practices among unschoolers. Please share your story through this form. It only takes a few minutes. It adds to everybody's understanding of how math can be.

Study of Child-Led Multiplication Version 2.0

When educators and students work together consensually, without restrictions of mandatory curricula and testing, we can observe natural patterns of learning. Yet few realize just how great the variety of best practices can be, especially in mathematics! Take multiplication, the topic I've been studying and writing about since 1996. How do kids approach learning "their times tables" if the family has a choice? For example, look at age: do some kids learn the tables as toddlers, in their late teens, or not at all? Let us find out and publicize the findings, to help parents and other educators to be brave and to do well by their kids!

Please share your times tables stories, whatever they are. Thank you!

2/2012 Lindsey S: First understanding conceptually and then memorization. My daughter began to do multiplication and division at four years of age, toward the end of her pre-K year. She had done some of the Montessori manipulatives at the beginning of age four, but she really understood the concept by four and a half years of age. Memorizing the facts happened perhaps in the last few weeks :) She is now seven and in the third grade. She is starting a prealgebra course this week, using AoPS books, so as you can see her ability to conceptualize is way faster than her ability to accept that memorization and mental math will make progression easier. :)

1/2012 anon: We haven't worked on them much at all. They don't know many of these facts.

1/2012 Julie N: My oldest daughter learned them when she was six. I invented a card game to teach her. My son learned them when he was eight using the same game. I actually recently had the game manufactured so others can play it with their children. You can get more information about the game including a video of it being played on my website http://highhilles.com/speed.html

1/2012 rtsallie: My kid knows most of them to the 12s from playing games on her Diji and from playing math attack.

Brenda F: We did other math stuff that did not require multiplication, waiting for her brain to maybe wake up to multiplication. This went on for 3 years. One day I picked a used copy of multiplication songs at the Gathering Place. Within 2 weeks she knew the tables by heart.

Lisa B: She mastered the concept within a few moments and later taught herself double and triple digit multiplication in a way I have yet to understand completely!

Loren R: He wanted to study more complex problems and realized that table recall would help.

Lena H: ...he still hasn't chosen any math, although he sometimes figures things out in his head when needed....he does well with concepts picked up on his own, uses a calculator for other things he needs

Caroline C: Initially, she liked Times Tables the fun way, but the workbooks frustrated her and her interest completely reversed itself. This all happened in a period of 2 months. We have done no math instruction since then.

Sue V: Even though there's really no coercion in his education, I think he likes to keep on the traditional schedule.

Pat S: I have three grown unschooled kids - I have no idea if or when they came to "know" multiplication facts (have them memorized). I know they've gone to college and completed all the required college math with A grades - including calculus and statistics

Janice C: As an unschooler, I don't check whether my child has learned her multiplication tables.

Rebecca S: He has learned most of them this past school year through a combination of Right Start math games and some time on the Times Attack computer game. He is 6 years old.

"capeaches45": When she was in the 1st grade with flash cards,also with computer games.

Linda K: We have always been very relaxed in our schooling, and I had pretty much written off long ago her memorizing the tables in a traditional fashion, but I didn't worry about it. Now, she has decided that, "it is time" for her to learn them. Much to her surprise, she has already learned much of the tables from exposure and use, in spite of herself.

Sandra B: So, with my first child I did everything to help him learn them. Hands-on activities, repetition, everything. Well, I gave up and gave him a break over the summer. When we restarted in September, voila, he knew his tables without review! The same thing happened with my second son in the same way!

Mr. Steve: None of the kids know their multiplication tables, not even the 12 year old. Yet, we had my 12 year old take the Stanford Aptitude Test and he scored beyond 12th grade in the math areas.

Kat S: He still does not have the multiplication table memorized (at age 13), but rather makes the calculation quickly by grouping and adding in his head. As an aside, I realized much later that by playing YuGiOh he had absorbed easily the concept of place value. As another aside, my son has recently chosen to attend math lectures (that are designed for high achieving 15-17 yr olds) at a local university, and he enjoys them.

Big Thanks to Peter Gray, Sandra Dodd and Pam Sorooshian

Peter Gray was one of the big inspirations behind this study, namely his blog posts at Psychology Today talk about "Less is more" in mathematics, and kids easily learning math when they control learning. I interviewed Peter for the Math 2.0 series in 2010. You can see the recording of his online event "Freedom to Learn." Peter keeps collecting information about open learning environment. Here is a quote from his recent email:

Peter: Maria, great collection of responses. Thank you for sharing it with me. I'll add this to my collection of "math stories."

I'll be speaking at the New England Unschoolers' conference this weekend. I plan to ask for volunteers for a survey, which I'll email to people later. I may include some questions about math in that. I like your way of focusing the math question on just one rather clearly defined accomplishment.

When I started the study in July 2010, I asked Sandra Dodd to help me by sending it out. She helped tremendously - by refusing, and explaining in detail why she thinks it's a bad idea. She also forwarded my letter to Pam Sorooshian who had a similar reaction, and added further details. Here are some quotes from what they told me.

Sandra: I think it harms unschooling to have it be studied, for the parents to have an overlay between them and their children.

If you rephrase so that you make clear that you know some people never do memorize it, then some who see your writings will flip out. The idea that "the times tables" must be memorized has to do with training clerks and bookkeepers 120 years ago. Same with "a nice round hand," so that the handwriting would be legible to anyone who came along to read the books. Those needs are long, long gone. Measuring 21st century kids by that is the problem, not the details of how they might learn 19th century skills.

I'm short of time, but there seems to be an underlying assumption that multiplication facts will be memorized - all at once, at some age.

I have three grown unschooled kids - I have no idea if or when they came to "know" multiplication facts (have them memorized). I know they've gone to college and completed all the required college math with A grades - including calculus and statistics, so I would guess that they're pretty quick with multiplication by this time (ages 18, 22, and 25).

Your questions aren't going to elicit sensible answers from unschoolers. The premise, that at some time a person will memorize multiplication facts, is problematic.

I thought their responses were brilliant, and also that I should have known better. I spent the rest of the summer thinking of how to fix the idea of the study. Here is what I think is a much better version. Thank you, Sandra and Pam!

## What About Times Tables?! Learning and free choices

## Table of Contents

## Study of Child-Led Multiplication Version 2.0

When educators and students work together consensually, without restrictions of mandatory curricula and testing, we can observe natural patterns of learning. Yet few realize just how great the variety of best practices can be, especially in mathematics! Take multiplication, the topic I've been studying and writing about since 1996. How do kids approach learning "their times tables" if the family has a choice? For example, look at age: do some kids learn the tables as toddlers, in their late teens, or not at all? Let us find out and publicize the findings, to help parents and other educators to be brave and to do well by their kids!

Please share your times tables stories, whatever they are. Thank you!

## Study Results: Family Stories

You can read some longer stories here.Seeing the great variety of viable learning practices is powerful! Thank you very much for contributing!YOUR NAME HERE! Please contribute!Brenda F:We did other math stuff that did not require multiplication, waiting for her brain to maybe wake up to multiplication. This went on for 3 years. One day I picked a used copy of multiplication songs at the Gathering Place. Within 2 weeks she knew the tables by heart.Lisa B:She mastered the concept within a few moments and later taught herself double and triple digit multiplication in a way I have yet to understand completely!Loren R:He wanted to study more complex problems and realized that table recall would help.Lena H:...he still hasn't chosen any math, although he sometimes figures things out in his head when needed....he does well with concepts picked up on his own, uses a calculator for other things he needsCaroline C:Initially, she liked Times Tables the fun way, but the workbooks frustrated her and her interest completely reversed itself. This all happened in a period of 2 months. We have done no math instruction since then.Sue V:Even though there's really no coercion in his education, I think he likes to keep on the traditional schedule.Pat S:I have three grown unschooled kids - I have no idea if or when they came to "know" multiplication facts (have them memorized). I know they've gone to college and completed all the required college math with A grades - including calculus and statisticsJanice C:As an unschooler, I don't check whether my child has learned her multiplication tables.Michelle:We've made Waldorf style charts like this clock http://robinsunne.com/robinsunnes_multiplication_clockRebecca S:He has learned most of them this past school year through a combination of Right Start math games and some time on the Times Attack computer game. He is 6 years old."capeaches45":When she was in the 1st grade with flash cards,also with computer games.Linda K:We have always been very relaxed in our schooling, and I had pretty much written off long ago her memorizing the tables in a traditional fashion, but I didn't worry about it. Now, she has decided that, "it is time" for her to learn them. Much to her surprise, she has already learned much of the tables from exposure and use, in spite of herself.Sandra B:So, with my first child I did everything to help him learn them. Hands-on activities, repetition, everything. Well, I gave up and gave him a break over the summer. When we restarted in September, voila, he knew his tables without review! The same thing happened with my second son in the same way!Mr. Steve:None of the kids know their multiplication tables, not even the 12 year old. Yet, we had my 12 year old take the Stanford Aptitude Test and he scored beyond 12th grade in the math areas.Kat S:He still does not have the multiplication table memorized (at age 13), but rather makes the calculation quickly by grouping and adding in his head. As an aside, I realized much later that by playing YuGiOh he had absorbed easily the concept of place value. As another aside, my son has recently chosen to attend math lectures (that are designed for high achieving 15-17 yr olds) at a local university, and he enjoys them.## Big Thanks to Peter Gray, Sandra Dodd and Pam Sorooshian

Peter Gray was one of the big inspirations behind this study, namely his blog posts at Psychology Today talk about "Less is more" in mathematics, and kids easily learning math when they control learning. I interviewed Peter for the Math 2.0 series in 2010. You can see the recording of his online event "Freedom to Learn." Peter keeps collecting information about open learning environment. Here is a quote from his recent email:

Peter:Maria, great collection of responses. Thank you for sharing it with me. I'll add this to my collection of "math stories."I'll be speaking at the New England Unschoolers' conference this weekend. I plan to ask for volunteers for a survey, which I'll email to people later. I may include some questions about math in that. I like your way of focusing the math question on just one rather clearly defined accomplishment.When I started the study in July 2010, I asked Sandra Dodd to help me by sending it out. She helped tremendously - by refusing, and explaining in detail why she thinks it's a bad idea. She also forwarded my letter to Pam Sorooshian who had a similar reaction, and added further details. Here are some quotes from what they told me.

Sandra:I think it harms unschooling to have it be studied, for the parents to have an overlay between them and their children.If you rephrase so that you make clear that you know some people never do memorize it, then some who see your writings will flip out. The idea that "the times tables" must be memorized has to do with training clerks and bookkeepers 120 years ago. Same with "a nice round hand," so that the handwriting would be legible to anyone who came along to read the books. Those needs are long, long gone. Measuring 21st century kids by that is the problem, not the details of how they might learn 19th century skills.Pam:Sandra sent me your email - I'm also owner of the UnschoolingDiscussion email list and moderator on AlwaysLearning.I'm short of time, but there seems to be an underlying assumption that multiplication facts will be memorized - all at once, at some age.I have three grown unschooled kids - I have no idea if or when they came to "know" multiplication facts (have them memorized). I know they've gone to college and completed all the required college math with A grades - including calculus and statistics, so I would guess that they're pretty quick with multiplication by this time (ages 18, 22, and 25).Your questions aren't going to elicit sensible answers from unschoolers. The premise, that at some time a person will memorize multiplication facts, is problematic.I thought their responses were brilliant, and also that I should have known better. I spent the rest of the summer thinking of how to fix the idea of the study. Here is what I think is a much better version. Thank you, Sandra and Pam!

## Old July 2010 Version