Changing Student Misconceptions.

  If you have taught math any length of time, you know students have misconceptions on certain topics.  Right now my pre-algebra class is struggling to get past ignoring minus signs  and  adding everything. 

I got desperate and created a flow chart to follow so they had to stop, look, and think about the signs before completing the operation. . 

Forming misconceptions is a normal part of learning but they can impede the learning process because many times students do not know they have incorrectly learned the material.  Secondly, any new learning filters through the established misconceptions.  Finally, misconceptions become so entrenched that its hard for them to be changed.

One of the best ways to identify student misconceptions is to cut back on lectures and increase student activities.  It is when students are working, their misunderstandings become apparent.  In addition, the best way to replace misconceptions is by changing from a teacher centered to student centered classroom.

Another highly recommended method for eliminating misconceptions is to regularly show problems containing the misconceptions so students can examine each problem, identify the misconception, and discuss why these are misconceptions.  Another name is error analysis so students learn to identify the error.

Unfortunately, many of the misconceptions students have when they hit high school have been with them since elementary school.  One of the standard ones has to do with the idea that if you multiply by 10, you add a zero at the end but that only works with whole numbers.  It does not work when you multiply a decimal by a 10.

I love using the error analysis method in class.  Often students have trouble identifying what is wrong due to their understanding of the material.  It takes several days of showing the same type of misconception for students to begin recognizing it.  It is important to work on eliminating their misconceptions so they do better overall in math.

 Let me know what you think.  Have a great day.

Misconceptions About Fractions

  I teach high school students who still add fractions straight across without bothering with common denominators. I honestly don't know who they get so far without having the basics down.

Several years ago, I worked with a young lady who did not realize when dividing a pizza into pieces, each piece had to be the exact same size.  She didn't know that.

One common misconception is when adding or subtracting fractions you see students pull the 1/2 + 2/5 = 3/7.  As stated earlier, I have quite a few high school students who do this and have no idea why they are marked as incorrect.

Another misconception I've seen is students not understanding when creating fractions with common denominator, they are trying to multiply by a fraction equal to one such as 1/2 x 2/2 = 2/4 so they can have 2/4 + 1/4 = 3/4 instead of 1/2 + 1/4.  Many of my students forget to multiply the numerator by the same number as they multiplied the denominator. Students have been taught they multiply both numerator and denominator by the same number rather than understanding they are multiplying the fraction by an equivalent form of 1.

This misconception comes from teach a process in elementary rather than teaching the concept behind it.  When students do not have this understanding it is difficult for students to find common denominators when working with algebraic fractions.  Although the process is the same, they do not see the connection.

Another thing I've observed is students try to find common denominators when they multiply or divide fractions.  They apply the common denominator rule to all fractions regardless of operation.  I suspect its because they have not figured out the differences in what each operation represents.  In addition, when dividing they often flip the incorrect term because they are told to flip one rather than understanding they are multiplying by a reciprocal.

Furthermore, students see fractions as always being part of a single object such as 1/6 th of a pizza being one piece out of six but 1/6th could represent one red ball out of 6 balls.   They see the whole as being a single object rather than possibly representing a total number of objects.

It seems some students see dividing a whole number by a half the same as dividing in half.  An example would be 4 /(1/2) is the same as 4/2 rather than 4 x 2.  Its why its difficult to create pictorial representations showing 4 divided by 1/2.  Even I struggle with that one because I never learned the concept, only the process.

I'd love to hear what you think.  Have a good day.

9 Common misconceptions.

While researching yesterday's topic, I stumbled across a list of mathematical misconceptions some of which I've had students happily share.

I'm sure you'll recognize some or all of the misconceptions listed below.  I'm also sure some will make you smile at the memory of a teacher telling you that exact thing in elementary school.

I know, I heard them myself.  So here is the list.

1. Three digit numbers are always bigger than two digit numbers.  This rule comes about because when they first learn numbers, they are only exposed to whole numbers.  In that case, this rule is correct but once decimals are thrown into the learning, it no longer applies.  3.24 is not bigger than 6.2.

2. When you multiply two numbers together, the result is always larger than either of the original but that is only true with whole numbers.  Once students begin using fractions or decimals, this may not be true.  one example is 1/2 times 1/6.  The result, 1/12, is smaller than either one.

3. Often students think the fraction with the larger number in the denominator means its larger such as in 1/4 and 1/8.  They sometimes think 1/8 is larger than 1/4 because 8 is larger than 4.  I think this has to do with 8 is larger than 4 normally with what they've been taught so when the context changes their understanding does not.

4. Most students see two dimensional shapes in only one orientation such as a triangle with the base always at the bottom part of the shape rather than placing it at the top with the vertex pointing downward or off to the side.  Teachers need to change the orientation so students do not get in the habit of seeing it one way.

5. In squares the diagonal appears to be almost the same length as the sides and students may assume they are the same.

6. When multiplying by 10, simply add a zero.  This works for a whole number but not for a decimal number.  You could add a zero but it does not help you to remember to change the position of the decimal. 

7. Ratios where students get used to comparing one object to another such as two carrots to three peppers rather than looking at two carrots to five vegetables.  When the situation comes up where they need to set up a part to a whole, they often have trouble.

8. Students often confuse perimeter to area because they count squares for both of them without understanding the whole square inside the shape is counted for area while they are only counting one side of the square for the perimeter.

9. Students often have difficulty determining the scale used by the measuring item. Not all scares are divided into 10's. Many students do not count the markings to figure that out, they assume its always going to be 10.

I understand why students are taught many of these rules when they are in elementary school but it does a disservice teaching these are "rules".  Students need to to quit learning "rules" which only apply to a narrow population of numbers.  Hopefully, teachers will quit doing these so students are more open to learning new situations.

Let me know what you think. I'd love to hear.