Homeschool Math Blog

Someone recently sent me a VERY interesting link: The Story of an Experiment Photo by foundphotoslj This experiment in math teaching was done in the 1930s by by L. P. Benezet, and the main gist of it was that formal arithmetic studies were delayed until the latter half of 6th grade. Instead, the instruction concentrated on "teaching the children to read, to reason, and to recite - my new Three R's." They also were taught about numbers they encountered in their reading materials, about time, about measuring units, estimation, and coins. Finally in 5th and 6th grade they also learn skip-counting. Formal arithmetic, meaning paper-and pencil work with addition, subtraction, multiplication, and division using a textbook began in latter half of 6th grade and continued till 8th grade. The experiment was a huge success. Mr. Benezet compared the children's abilities in the experimental classroom to those of the traditional classrooms, and every time the "experimental...

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy. I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue. BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leav...

I just found out an interesting column by Keith Devlin ... he tells elementary teachers to stop telling the students that multiplication is repeated addition. Why? His point is, this idea does not carry through. As soon as the student encounters multiplication of fractions (or of decimals), it won't work. You can't think of 3/4 x 6/11 as repeated addition. He feels it's better to portray multiplication as a scaling process: say 5 x 9 means 9 is scaled by a factor of 5. Then, students can have a true "aha" moment as they discover for themselves that you CAN use addition to find the answer to 5 x 9. But, Devlin says, they should be taught and shown the multiplication idea as a scaling process. Now, I feel that Devlin has a point here... so since I'm constantly in the process of writing math materials for my Math Mammoth series of books, and right now I'm writing lessons on multiplying decimals for 5th grade, I took this idea just yesterday and tried to go wi...

I just received a nice article from Jim Stone, a math teacher at Global Institute of Mathematics . I enjoyed it; I thought you might too. I've seen this issue raised and talked about elsewhere as well. Namely, that it seems to be socially acceptable to admit how bad you're at math... while no one would comfortably admit that they can't read. So here goes: I'm bad at math and I’m okay with it! By Jim Stone For the past eighteen years I’ve been reading articles and editorials lamenting the mathematical performance of America’s school children. It has become an annual ritual for politicians and educators alike to bemoan the results of the latest tests showing American kids falling behind their international counterparts. What is the problem? More importantly, what is the solution? The answer to the first question is almost always laid at the doorstep of our system of education. Blaming the system is safe. No one person represents the system. No one person is held accountab...

What are the incentives needed in order to develop a positive attitude in children and other students towards mathematics? I don't think special incentives is the main factor in developing a positive attitude towards mathematics. I feel it is sufficient to get a few of the basics right, and then that alone will take care of most of it, and then students will like math just fine. Disliking math is not something that is inherent in us or in our kids. Little kids don't dislike math or numbers. They're just fine with them! This "I hate math" or "I don't like math" attitude seems to develop during school years. Now, I also don't think that children are disliking reasoning, because they're happy to do puzzles and play games where you have to think. And, students' negative attitude towards math also is NOT due to (school) math being difficult. The math we learn in school is not difficult. You don't have to be a math whiz to understa...

Recently Brown Center published a Report on American Education with a special sector about the happiness factor in learning. The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data. The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it. American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America! Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts. The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable. I can almost hear my homeschooling readers' anger raising...! But wait a ...

X (however many) Habits of Highly Effective Math Teaching: Part 4: Living and Loving Math You are the teacher. You show the way - also with your attitudes, your way of life. Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day? And then: do you like math? Love it? Are you happy to teach it? Enthusiastic? Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not. Math is not a drudgery, nor something just confined to math lessons. Some ideas: Let it make sense . This alone can usually make math quite a difference and kids will stay interested. Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - se...

Some math curricula are labeled as 'spiraling', and some are said to employ the 'mastery principle'. What does that mean, and which is better? Spiraling mathematics curriculum introduces many topics in one grade level. It does not aim to teach those topics completely in one go, but revisits those topics the next year, the next year, and so on. Mastery approach simply aims to teach to mastery any single topic, before going on to the next. There is a lot of talk against spiraling math curricula, especially in regards to some reformist mathematics curricula. So is the mastery principle then better? Well, I think that dividing this matter into two opposite positions is a mistake. You CAN have good mathematics education employing parts of both principles. For example, a student can learn to add 2-3 digit numbers on grade 2. She can revisit the topic on 3rd grade to learn to add 4-6 digit numbers. She can revisit the topic on 4th grade to learn to add even larger numbers. ...

I wanted to briefly touch some more on this topic of "grind" or daily grind that learning math might sometimes become that I mentioned briefly last time. I didn't mean to imply that learning to add, subtract, multiply, and divide whole numbers, decimals, fractions, percents, numbers with exponents, and integers has to be such a boring task. You can avoid that type of feeling. One way: Make connections between the concepts . Don't make math appear as fairly separate compartments of "fractions" and "decimals" and "percents" and "geometry". I sometimes wonder how students feel when every year (on 4th, 5th, 6th, 7th, and 8th grade) they have a chapter on fractions, a chapter on decimals, a chapter on whole numbers, a chapter on geometry... Almost the same thing over and over. We need to make sure they can see how these things connect. An example: A jogging track is 3.5 km long. If you jog 2/5 of it, how far was that? How many percen...

Recently Spunky wrote about reform math, and I feel I want to say something about it, too. She was critizing the movement heavily. Some people call reform math 'new new math' or 'fuzzy math' (in a degrading way) - and like you can read in Spunky's post, sometimes when new ideas are implemented, it can cause math to become 'fuzzy' to the children. But that is not necessarily a fault of the reform on a whole - maybe the teacher in question didn't understand how to implement the methods, or for some reason (unwisely) totally abandoned 'traditional' math fact drills. I would say not all of the IDEAS that have come from this mathematics education reform are bad. For example, encouraging kids to explore and investigate mathematical concepts, or discovering their own rules, can be a good thing. Group work CAN be a good thing. The reformists promote understanding of concepts, problem solving, critical thinking. They try to distance themselves from the ...

I just read a very inspirational article Teenager or Tyke, Students Learn Best by Tackling Challenging Math (PDF) ( html ). It tells about two teachers who frequently employ open-ended problem-solving sessions in their teaching - and the students (almost all) like it well and are very motivated. In math education, OPEN-ENDED problem usually means it doesn't have a specific step-by-step solution. You can solve it in many different ways. Or, it may have more than one solution. The problems these teachers use are often from real life, and not quick to solve. Instead it can take some time and struggling to get anywhere. (Hey, that's how problems in real life often are, too!) But, struggling can be valuable. One of the teachers featured in the article, Heidi Ewer, says: "Struggling helps them see this as an investment of their own time and energy. It makes them more willing to learn," Ewer says. "Struggling to solve problems requires students to use their intuiti...

For the purpose of this post, we could divide word problems to three different categories: 1) routine word problems 2) non-routine word problems 3) algebra word problems Actually you could divide algebra word problems to routine and non-routine as well, but I want to now talk about word problems kids encounter in school before algebra - in grades 1-8 usually. J.D. Fisher suggested in the comments section of my previous post on word problems that kids are encouraged to think linearly, step-by-step. Then, when the word problems they encounter don't anymore follow any step-by-step recipe, they are lost. You might want to go back and read that. Don't typical math book lessons kind of follow this recipe: LESSON X --------------------- Explanation and examples. Numerical exercises. A few word problems. In other words, the word problems are usually in the end of the lesson. (That might make solving them a rush.) Then, have you ever noticed... If the lesson is abo...

You might think, "Proof? You need it before high school geometry?" Sure! But, we are talking about a different form of 'proving' here. What IS proof, first of all? It is something that the person hearing or reading the 'proof' will become CONVINCED that whatever you're proving is, indeed, true. You will want to convince your youngsters or students that what YOU ARE telling them, is indeed true. That is 'proving', in a general sense. But it doesn't have to be on the same level as later. For example, when you are showing them how the multi-digit multiplication works and WHY it works (2 &times 371 is the same as 2 × 300 + 2 &times 70 + 2 &times 1), you are proving - or maybe we should say "justifying". When you take fraction manipulatives and demonstrate why 5 × 2/3 is 3 1/3, you are 'proving' - or demonstrating. (You take 5 times 2/3. You combine the thirds until you get 3 wholes and one third.) Oftentimes diagrams ...

I want to discuss this topic because of a possible confusion. Recently in a blogpost I linked to the article Math anxiety , which mentions this math-related myth: MYTH #4: IN MATH, WHAT'S IMPORTANT IS GETTING THE RIGHT ANSWER. Then, in another recent post I linked to an article Things not to learn in school where the author strongly convinces us that getting 70% in a test is not enough since in real life you need to get it 100% right. So is it important to get the right answer when doing a mathematics problem? Well, yes but it depends. This is not a simple cut-and-dried question. It IS important if you're drilling multiplication facts. But many times it's not your focus. For example, when a student is learning to do long division or multiplying 4-digit numbers, the emphasis should be in learning the procedure and why it works. Calculation mistakes might yield a wrong answer, but if the way the division was done is right, then some credit should be given. And obviously all...

Have you ever met anyone who feels, "I hate math", or "I don't understand math at all", or "I'm not a math person" ? Wonder when and where those feelings got started? I recently read an article called Formula to bring back thrill to math . The author argued that calculus should be reserved to college studies, not high school. His point was, while the nation is suffering from a shortage of math teachers, the good math teachers are needed, not at advanced-placement calculus classes, but instructing 'the masses' so that they would learn to love math. He also said that besides a drastic shortage of math teachers, there is also a severe shortage of math majors. In other words, fewer and fewer kids are studying math as their major in college, and there are fewer and fewer (qualified) math teachers... Of course the result is that schools then hire just about anyone to teach math. I just wonder, is there a connection between the lack of good math t...

Someone emailed me just recently with this question: "How do we know that pi is indeed non-repeating? Do we have proof? What is that proof?" (The person who emailed me this question is from Japan, based on the email address.) I think it's an excellent question! And I hope every eight-grader is thinking about that when they are first told about pi. You know, kids learn about pi when they are studying circles, and so they take several circles and measure the circumference and the diameter of them all, and they calculate the ratio and they get 3.3 or 3.1 or 3.45 or 3.274658 or whatever. I tell you a truth: In all your measuring you will never stumble upon the fact that pi is irrational. Proof: Because your measuring results are all rational numbers, and so you're diving a rational number by a rational number. It's not something that would be found with observation or exploration of that sort. Students are just plain announced the fact that this ratio is called Pi an...

With my daughter, we've been concentrating on simple addition problems and the topic of tens and ones. Right now we are working on these goals: Understand subtraction concept - she partially does Tens and ones - she partially understands. (I realize mastering place value well takes often years.) Understand something + 1 type problems - she partially does Memorize answers to double 2, double 3, etc doubling facts And, we're heading towards these goals that are further away, yes, but I am keeping them constantly in mind: Memorizing all basic addition facts Learn to use those to do subraction problems Having definite goals is important. Good math teachers know where students are supposed to be heading, and also how to get there. Make sure you know (and understand!) what the final and intermediate goals are. If you are simply following the math book page by page, it's like walking on a road while looking down at the road - not knowing exactly where the road is leading. When...

In elementary school, I was under the impression that math was all about NUMBERS. Before 7th grade, I was thinking that there couldn't be much more math since we seemed to be doing the same things over again (spiraling curriculum). After that, the math I was learning seemed to be all LETTERS: x's, y's, a's, b's, f's etc. And soon it was Greek letters too! But math is actually neither. Math is all about PATTERNS. The whole structure of our base-10 number system is a certain pattern. Multiplication, division, etc. are full of patterns to notice. Functions and relations are all about making general rules for patterns. When first learning about sine in right triangle , it may seem to students like a disjointed piece of information - but if they could get past the initial stage and see the sine curve ! Patterns! But I especially wanted to mention trees, ferns, coastlines, snowflakes, and such. Because they have a pattern too, and mathematicians have unveiled th...

An interesting article discussing how yearly standardized testing has affected school mathematics: teachers are required to teach a curriculum that is 'mile wide and inch deep' and don't have time to teach any topic in depth so students could truly understand it. They don't get to master much anything. In homeschool this is much easier to avoid. U.S. children left behind: When tests drive curriculum, no one masters anything .