Many students struggle with mathematics and the comprehension of its principles from elementary school to college. For some, calculations may come naturally, but for many others they require a lot of attention, effort and practice. Sources of initial difficulty may come from a variety of places; for instance, never really learning basic number facts and concepts, harsh or ineffective teaching methods, poor study habits or problems with computation and comprehension. These and many other reasons may be the cause behind the difficulties that people experience with calculations and mathematics in general.
By definition, a calculation is a "mathematical determination of the size or number of something." And we're all very familiar with many forms of calculations. From basic arithmetic such as multiplication and division to upper forms of algebra, trigonometry and calculus. Though, outside of basic calculations many students struggle with high school and college level mathematics. And a lot of common misconceptions that take root often include things that occur outside of the classroom; such as improper study techniques and exam preparation, holding a poor image of math in general, and having trouble thinking mathematically.
So in addressing calculation difficulties, the first obstacle to tackle is learning how to 'think mathematically'. The inability to see the bigger picture or think of math as something other than a strict set of rules and guidelines can undoubtedly hinder one's ability to be successful. This major shortcoming can be addressed by first acknowledging it as a problem and then taking the necessary steps to change one's particular mode of thinking.
How to think mathematically
The idea to 'think mathematically' is a common one and may mean different things to different people. But in general the point is to look at math as more than just numbers and formulas. Limiting it in this way can make it hard to fully comprehend and apply mathematical concepts and ideas. Experts in the field encourage looking at math as a means of investigating patterns, applying specific techniques to find solutions, and connecting abstract concepts to the real-world. So to help yourself see math in a new light, try to implement the following;
- See math as a field that is constantly evolving
- Learn the language of math
- Connect mathematical concepts to the outside world-even if it seems impossible
- Try to understand how a formula works rather than just knowing what it is
A great deal of comprehension in math deals with not just memorizing facts and steps but also knowing and understanding the meaning behind formulas and calculations. For instance instead of simply memorizing the quadratic formula ax^{2} + bx + c = 0 its more important and suitable for a long term benefit to also understand that in fact a, b, and c are constants that cannot change and that x represents an unknown variable. A simple breakdown such as this can help you to better apply calculations consistently over time and really comprehend the material being presented.
And in addition to 'thinking mathematically' it's also important to note some common errors people experience; this may include errors in arithmetic as well as complex and advanced calculations.
Common calculation problems
Misunderstanding signs
In a book by Thomas Carpenter on the subject of math skills in elementary aged children, the author noted the abundance of children in many grades that harbor an improper understandings of the = equal sign. Amazingly enough, the simple equation
8+4= ___ +5
received a variety of wrong answers by students due to the misinterpretation of the = sign-many students interpreted it as a command to add rather than to show the relation between two equal amounts. If this is the case with elementary aged students, one can only imagine the trouble that can come with more complex signs and symbols at higher levels of mathematics. So not really knowing what certain symbols and signage represent can definitely lead to calculation difficulties.
*Other signs along this line that may be misinterpreted are the approximately equal sign ≈ and the not equal to sign ≠).
Not reading the problem correctly
Not only in math, but in many other subjects, people often assume they know what the question is asking without actually reading through the instructions. Not paying attention to this step can cause major calculation difficulties - for instance, things may not add up correctly, or the end result is not what it should be even though all the steps were followed correctly-if these instances occur, this could be the source of the problem.
Poor handwriting and communication
Handwriting is usually the primary means of communication in math classes whether in keeping your own notes or sharing what you know with your instructor. Messy or sloppy writing can lead to you not being able to grasp the meaning of your own notes or cause you to make major mistakes in calculation simply because you thought you wrote something that you didn't! Likewise, you may get an answer wrong because your teacher misinterprets your writing as well.
*Poor communication in general can also be extended to the student not fully understanding either the teacher's explanation of a problem or a textbooks explanation of it. Sometimes math wording is difficult to interpret so students may run across calculation problems because they do not understand what is being asked of them or the steps that they must follow to obtain the correct answer.
The above mistakes are common to many areas of mathematics and may be applied generously. Some other mistakes are more subject specific-algebra in particular is important to highlight because most people encounter it as high school students as well as college students. And as with other types of math there are some typical problems that students usually fall into-two are detailed below.
Common errors with Algebra
Problems with distribution
When using the distribution property students may sometimes trip up by not properly distributing for all numbers. For example;
5(3 + 1) = 5 x 3 + 5 x 1
means that you need to distribute the 5 for all numbers in the parenthesis, meaning the 3 and the 1, not just the 3. Students may fall into the mistake of simply multiplying the 5 with the 3 only and leaving off the 1, or just adding it to the product.
Problems with negative signs
Many students from an early age encounter problems with negative signs. Its important to remember that a negative sign does not always indicate that a number is itself negative. This concept can be a bit confusing to some. For example, -x would be considered to be a positive number if the number itself is positive (4 for example) but if the number is negative, lets say x=-4, then the number would be positive, that is -x=4. This concept may also be coined as 'double negatives' and is a very important one to understand to eliminate any calculation problems in the long run.
In addition to these examples with algebra, students also encounter many other general problems such as fractions, order of operations, the proper use of parenthesis, etc. So a lot of attention and effort needs to be put into understanding basic rules and operations. Also an important point to note is that math is cumulative; so as you develop as a student you continue to rely on previously learned skills and material-not being in tune with a lesson or missing vital information can cause major problems when trying to apply new formulas or concepts. Being alert, actively involved, taking great notes, and asking for help when needed, are all crucial elements to success in mathematics.