It is easy for a educator to fall into the trap of just becoming a "teller" and not a "teacher". As a developmental college math teacher I use a lot of basic number theory to help students understand fundamental mathematics concepts. It is my hope that they can then build on these concepts and be successful in a college level math course. It is in this endeavour that I often find myself undoing damage done by some other math instructor from a previous course somewhere in the student's near or distant past. Much of this damage has often been done by an instructor who "told" the student how to do something instead of teaching them why and how a fundamental process is happening.

"Telling" makes teaching a lot easier! I prefer to consider myself more of the classroom facilitator than just the provider of information. When students are "told" why a mathematical process works, then there is no discovery, and the logical processes that make the mathematical learning sequence transferable to other disciplines is lost. Most developmental mathematics students will NOT be pure mathematicians, engineers, or scientists. Most students have mathematics in their degree plan for the logical procedures and reasoning skills that are present in mathematical processes.

Once the process of "telling" begins, holes in the logical sequence of mathematical phenomena start. If the acquisition of a new math skill set requires building on "told" mathematical processes, there is no concrete understanding for the student, and they most likely will need to be "told" the new skill set as well, and no learning is taking place.

This list will in no way be comprehensive, but I want to highlight some basic math skill sets that are often "told" and not discovered. When these topics are introduced and discovered in my developmental college math courses I always ask the students if they were just "told" that this is how it works, and almost every hand goes up.

• Dividing Decimal Numbers - Most students have no idea for why when dividing decimal numbers by hand we force the divisor to be a whole number and not a decimal number. Most were "told" that you just move the decimal point on the divisor as many places to the right as it takes to make it a whole number, and then do the same thing to the dividend. Why is this necessary, and what gives the authority in mathematics to just move decimal points around? When students are simply "told" that this is how it is done, fractions become a mystery and math becomes a kind of hocus-pocus with no logic or reasoning.
• The Product of Two Negative Numbers is a Positive Number - This is a classic concept that is often "told" and not discovered. To teach this concept properly simply involves basic number theory and the introduction of a new real number identity. Many instructors get so engrossed in trying to teach students about numbers that they lose focus of how the mathematical sequence is supposed to be benefiting the student in their overall cognitive development. Math is not magic. Mathematics is about known relationships that have been discovered through repeated processes over time. As a facilitator of mathematics I feel it is my duty to expose and develop an understanding of these repeated processes, not just "tell" them.
• Multiplying and Dividing Fractions - Students are often "told" that to multiply two fractions together you simply multiply the numerators together and the denominators together. Simple! Except for the fact that the student has no concept that when a value is multiplied by a proper fraction that original value gets smaller, and if you multiply some value by an improper fraction it makes that original value larger. They also lose the concept that when multiplying by a fraction the denominator is dividing the original value into that many equal parts, and the numerator is how many of those parts you now have. Dividing fractions is even worse! The student is usually just taught pattern recognition.

I am grinding some what of a personal axe with this Hub. I too was a victim of being told a math concept, and I was greatly embarrassed when I was corrected by an 8th grade student who informed me that I had been "told" a concept incorrectly. It was my senior year in college and I was in my second semester of student teaching. I was in a class with students who were working on independent lessons. One of the students was working on a problem involving the use of the "order of operations". It had been quite a while since I had used this concept in an elementary-type math problem, so I tried to recall something I remembered being "told" in my formative schooling about the order of operations.

My memory recalled the acronym PEMDAS because that had been "told" to me sometime in my childhood and that was the recall memory associated with the phrase "order of operations". In my recall I remembered the teacher "telling" us that the acronym had to go in order. Of course when I reminded the student that "M" came before "D" so that probably meant to do multiplication before division, the student looked at me with a puzzled look and told me that they had been instructed different. When I asked if they had learned the acronym PEMDAS they informed me that they had not.

If I had learned or discovered the order of operations when I was supposed to have in my formative schooling instead of being "told", this embarrassing situation could have been avoided. When instructing students on topics that others have developed an acronym or mnemonic device of some sort to learn the concept better, I never tell the students about them. If they have heard of some mnemonic device or acronym that is supposed to aid in student understanding, then they may discover these devices on their own. I try my best not to lead the student to them.

Of course there are several other math learning concepts that are "told" and not discovered that I could add to the list, but these listed are some of the most blatant and common. It also was not the intention of this article to bore the reader into submission but to inform learners both young and old that it is not acceptable to just be told things. This is not thinking, and this also not my understanding of what it means to "learn" something. Without discovery, there is no solid foundation on which to develop future concepts. I believe this to be one of the reasons many students can not advance through college level mathematics courses and why they struggle with the developmental math sequence in college. Somewhere in the student's development of the mathematical learning sequence, too many of the concepts have been "told" and not "taught".

Feel free to add to the mathematics "told" list by commenting on this Hub.