Homeschool Math Blog

(Note: I'm "resurrecting" an old post, with added information and a video. The topic is still very much valid.) Manipulatives are IN, in math education. But do they TRULY facilitate learning to such an extent as people promoting them claim? The entry at Text Savvy, Hands-on, Brains-Off , explains some of the pitfalls in manipulative use. Very enlightening! Quoting from an article at Education Week ( Studies Find That Use of Learning Toys Can Backfire ): In a similar series of experiments at the elementary-school level, the researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10. Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that t...

I just found this news article at EducationNews.org and some of you might find it interesting: U.S. Teachers Not Well Prepared to Teach Mathematics, Study Finds It explains the findings of a new international study of future math teachers in 16 countries.

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...

I wanted to pass on this book link to those of you who are interested in the pedagogy of mathematics: Adding It Up: Helping Children Learn Mathematics (2001) by Center for Education ( CFE ) A big plus is that you can read the whole book online for free. This book talks a lot about how the conceptual understanding of basic arithmetic develops in children during grades 1-8. It explains common misconceptions. The book cites a lot of research about how children learn addition, subtraction, multiplication, division, fractions, and proportions. It doesn't deal with geometry or measuring, because the authors decided to concentrate on "number" concepts. All in all I think it's good basic reading for all of us who teach math. You can read it online by chapters. Here's a direct link to page 187 which starts dealing with single-digit addition . Here's a table comparing various addition and subtraction problem types . And here's a link to the discussion about sin...

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...

Recently there's been a lot of talk about an essay written by mathematician/teacher John Lockhart, called Lockhart's Lament . Some people praise it, some are more skeptical. Lockhart's Lament makes for good reading and he raises some really interesting points, so I can heartily recommend reading it. Personally I don't fully agree with every statement he makes there. But his MAIN point concerns mathematics as an art, and how we should teach it. I went ahead and copied a part of the essay below. This is direct quote from the essay, presenting a VERY GOOD example with the triangle problem. So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G.H. Hardy's excellent description: A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. So mathematicians sit around making patterns of ideas. What sort of patte...

First of all, I've updated my post about the percent problem — just scroll down to it. Then, I thought maybe I have some math teachers in my readership that might be interested in a new Master's degree program offered by the university of Drexel, in collaboration with the Math Forum! Knowing how much expertise the folks at Math Forum have this might be a unique opportunity for those math teachers who want to extend their education. Online Master's in Mathematics Learning and Teaching - "Preparing teachers to incorporate creative, problem-based, student-centered instruction in their classroom." The rest of you... can just continue reading my blog : ) I will post some more about the concept of percent soon.

I am leading a training next week on how to introduce grade level concepts/standards when the students are performing well below grade level, and I am sure that math is going to be an issue. Any suggestions? These are my 2 cents on teaching under-performing students. Let's imagine we have an 8th grader performing on 3rd grade perhaps. I would dismiss for starters geometry and measuring topics, and concentrate on this train of topics, in THIS ORDER: addition subtraction multiplication division fractions decimals. ... the goal being to cover the basic arithmetic up to pre-algebra. Think of mathematics as a building. You need to have the foundation building blocks before you can go forward. Maybe the child stopped understanding the math on 2nd grade or 3rd. We need to find the exact point after which he has not understood everything. You can gauge this by the way by asking the child simple questions such as, I give you an addition 8 + 2 = 10, you give me a subtraction sentence (1st...

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...

I received recently the book Arithmetic for Parents. A book for Grownups about Children's Mathematics by Ron Aharoni. I think this book is exceptionally good, and very worthwhile to read if you're a teacher OR a parent. Ron Aharoni was like many: he thought he could easily teach elementary school mathematics because he knew college level math and beyond (he was teaching math in a university). But... he was in for a big surprise when he entered the fourth, fifth, and first grade classes in a backward town in northern Israel, in 2000. One surprise he had was that... ...he did NOT know how to teach elementary mathematics, in spite of his knowledge of university level math. For example, during his first lessons, he took children outside to measure shadows of trees and buildings, and then also to measure the children and their own shadows. The idea was to use the ratio of a child's shadow to his height in order to find the height of a tree or a building He a...

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...

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...

Math teacher's tools are quite numerous nowadays. First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser. And the book you're using. Then we also have computer software, animations and activities online, animated lessons and such. There are workbooks, fun books, worktexts, online texts. Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on. And then there are games, games, games. The choices are so numerous it's daunting. What's a teacher to do? Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity. There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activi...

Seven (?) Habits of Highly Effective Math Teaching: Habit 2: Remember the Goals What are the goals of your math teaching? Are they * to finish the book by the end of school year * make sure the kids pass the test or do you have goals such as * My student can add, simplify, and multiply fractions * My student can divide by 10, 100, and 1000. These are all just "subgoals". But what is the ultimate goal of learning school mathematics? Don't we want our students to be able to navigate their lives in this ever-so-complex modern world? This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely. All that requires good understanding of parts, proportions, and percents. Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it ...

I've seen homeschooling blogs lately filled with seven habits of highly effective school year... so I thought of doing something similar, basically just by myself, about math teaching. You know, Seven (?) Habits of Highly Effective Math Teaching . And I'm going to take these one by one so you have better chance to think about these. And I'm not even sure if I'll get to seven! Anyway... Habit 1: Let It Make Sense Let us strive to teach for understanding of mathematical concepts and procedures, the "why" something works, and not only the "how". This understanding, as I'm sure you realize, doesn't always come immediately. It may take even several years to grasp a concept. For example, place value is something kids understand partially at first, and then that deepens over a few years. This is why many math curricula use spiraling: they come back to a concept the next year, and the next. And this can be very good if not done excessively (like for ...

As you're probably aware, each state has learning standards for various school subjects and grades. What many don't know, though, is that the MAJORITY of the math standards are poorly written. This is what Thomas B. Fordham Foundation has found in its research. They have published the findings in their State of State Math Standards 2005 . They gave state math standards grades - and 29 of the states get Ds or Fs! Only three states - California, Indiana, and Massachusetts - received grade A! So why is that? The executive summary (makes for excellent reading) lists nine major, widespread problems within most states' standards: 1. Calculators Typically state standards emphasize the use of calculators. "But for elementary students, the main goal of math education is to get them to think about numbers and to learn arithmetic. Calculators defeat that purpose." 2. Memorization of Basic Number Facts Many states don't require that students memorize basic facts. But th...

What are the strategies for solving simple equations? I got this question in the mail just today. I assume the person means LINEAR equations - those where you only have one variable (usually x), and that x is not raised to second or third or any other power, nor is it in the denominator or under square root sign or anything. Just x's multiplied by numbers and numbers by themselves, such as: 2x - 14 = 9x + 5 OR 1/3x - 3 = 2 - 1/2x OR 2(5x - 4) = 3 + 5(-x + 1) Here are the strategies for solving these: * You get rid of paretheses using distributive property * You may multiply both sides of the equation by the same number * You may divide both sides of the equation by the same number * You may add the same number to both sides of the equation * You may subtract the same number from both sides of the equation You might think, "Which one of those will I use, and in which order?" That depends. There is no clear cut-n-dried answer. Whatever you do, you try to transform your equa...

I still want to continue on the topic of word problems - not because it's my 'soap box' subject but because I feel parents and students can use lots of help with them. If you didn't catch the earlier post on how your typical math books subtily teach kids NOT to think carefully with word problems, read it here . The easy 'routine' word problems in early grades usually require just one operation to solve. I would recommend studying a bunch of such word problems WITHOUT calculating the answers but only thinking and finding which operation is needed to solve each problem. After you do that enough times, the student should start associating the types of situations with the appropriate operations: Total is divided into so many parts/containers, each part having same amount. This is the multiplication/division situation : ( number of parts ) × ( amount in each ) = total If you know how many parts and how much in each , MULTIPLY. If you know the total and the number o...

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...

I was trying to think today why it is that so many kids feel that word problems or story problems are difficult. It didn't make sense to me initially. See, Most kids just love stories. Usually kids love words, too, based on the fact they use them a lot. And problems - I can't imagine that kids don't like word problems just because they need find an answer to something. Most of us even adults get fascinated by puzzles, for example. So what is the problem with word problems? It surely can't start on 1st grade. You know, someone tells you a story problem such as: There are five ducks on the lake and three on the shore. How many ducks are there total? And often the math book has a nice picture there to accompany it. Surely kids don't think that as being difficult. My child has gotten to like "subtraction stories" pretty well - just simple situations where someones or some things go away. She has even made up some herself. Could it be that the...