One important facet of math is that students need to have fluency of math facts. I realize with all the calculators around, there is debate on if students still need to know their math facts.
Math fact fluency is defined as the ability to quickly recall basic facts for all four operations, addition, subtraction, multiplication and division. As they become more fluent, there are changes in the brain and this ability goes into long term memory so they can focus on other parts of solving equations.
In addition, this fluency allows students to focus on learning problem solving, new concepts and skills which are needed for higher level mathematics.
These math facts are the building blocks for higher level concepts so if a student is fluent, they will find these concepts easier to learn. In addition, it helps build their confidence in their ability to do math.
According to one article I read, the use of calculators can hinder students who do not have math fact fluency will it can help those who are fluent. It has been found that having students use a calculator to check their work is much more effective because it provides immediate feedback.
One of the most compelling arguments I've read for having students learn their basic number facts is if they make a mistake entering the numbers in, they know the result is off. The same applies to entering more complex problems in the calculators, if its not put in correctly, the answer will be wrong.
In addition, knowing math facts is good when working with fractions because you have to multiply or divide fractions with unlike denominators to get a common denominator. Very few calculators are easy to work with fractions. Sometimes, they give a decimal answer and other times, it is a pain just to enter the fraction.
Not knowing their multiplication and division facts makes it more difficult to identify fractions as equivalent because they don't know multiples. In addition, this carries over into decimals and percents so overall these students struggle in algebra and higher level math classes.
Furthermore, having your multiplication facts down cold makes it so much easier to list possible factors when factoring trinomials into binomials. If you don't know them, you won't be able to factor properly.
Most people who use calculators use them for more complex mathematics. I recently saw Hidden Figures which illustrates this point. The IBM produced two different results for John Glenn's landing spot and they called in the woman to double check the figures. She did the simple calculations on a fancy adding machine but she knew her math facts so incorrect answers did not slow her down. She knew.
Let me know your opinion on knowing math facts fluently. I'd love to hear. I'm still struggling with the use of calculators because I know my math so well, its often faster to do it by hand than with a calculator.