Homeschool Math Blog

Denise wrote something interesting  relating to very LARGE NUMBERS, sort of a misconception that we all seem to have. She asked students to mark 1 million on a number line that went from $20 to $1 billion. Here's the number line. Can you do it? Give it an honest try! Then go read how her students fared and see the right answer ... you might be surprised! She said, " Our intuitive number sense seems to work like a semi-logarithmic scale, with the small numbers evenly spaced and the bigger numbers crowded together at the end. "

An interesting piece of research has just come out on the misconceptions with the equal sign (=). Students' understanding of the equal sign not equal According to the research, US students exhibit this misconception much more often than students in other countries. It has to do with thinking of the = sign as an operator. Kind of like thinking that = means "to do" the operation. For example, a student with that misconception tends to solve the problem 7 + 6 = ____ + 2 by adding 7 + 6, and placing the answer on the empty line. The correct way is of course to think of the equality: 7 + 6 equals 13, so the other side has to equal 13 too. 11 fulfills this little equation: 7 + 6 = 11 + 2 I have known of this problem for years, and have tried to include problems in my Math Mammoth books to help children NOT to develop this wrong idea. For example, children solve 200 + 50 + 6 = ____ + 200 + 50 in the place value section. Or, I use problems where they have to...

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 found this link just today, via Mr. Barton's essential math freebies page . It is a collection of classic math mistakes, presented as posters with silly titles. You can post them on your wall and let students figure them out, because the poster doesn't actually explain the mistake; it just shows an example of it and the funny title. Check it out: Classic Math Mistakes - Gallery of Posters

Today I have a "goodie" for you all: a free download of some pan balance (or scales) problems where children solve for the unknown: Just right click on the link and "save" it to your own computer: Balance Problems (a PDF). This lesson is also included in my book Math Mammoth Multiplication 2 and in Math Mammoth Grade 4 Complete Worktext (A part). The problems look kind of like this: These can help children avoid the common misconception that equality or the equal sign "=' is an an operation. It is not; it is a relationship. You see: many students view "=" as "find the answer operator", so that "3 + 4 = ?" means "Find what 3 + 4 is," and "3 + 4 = 7" means that when you add 3 and 4, you get 7. To students with this operator-view of equality, a sentence like "11 = 4 + 7" or "9 + 5 = 2 × 7;" makes no sense. You might also find these resources useful: Balance word problems from ...

It's very good to know something about the most common misconceptions students might have. The website CountOn.org has 22 examples of them at http://www.counton.org/resources/misconceptions/ . Here are some examples: #2. Multiplication always increases a number... is that really so? Well, to small kids it appears to be so - but only if you just try whole numbers. Take 10 for example. If you multiply it by 2, 3, 4, 5, etc., it does get bigger. But all you have to do is multiply it by a fraction less than 1, or by a negative number, and 10 does not get bigger... 1/2 x 10 is 5. 1/4 x 10 is 2.5. -3 x 10 is -30. We must remember that repeated addition is not the only meaning or definition for multiplication . That's what it is for whole numbers. For fractions, 1/3 x 12 is better understood as 1/3 of 12. #3. As 1 x 1 = 1, then 0.1 x 0.1 = 0.1. This one was new to me. Quite curious. A child might think of 0.1 as some kind of "unit" like 1. But seriously, 0.1 is one tenth....