Mathematics is the umbrella term and arithmetic is just one type of math. Algebra, geometry and statistics are others.
Arithmetic is addition, subtraction, multiplication and division and is typically covered in elementary school. However, fractions and percentages are also arithmetic and these stump plenty of adults. Hmm, how much should I tip on this $25 check at 20%? If you know the algorithm, you can figure it out on the back of the check. So called "kill and drill" worksheets will get that result. However, mental math is quicker and it comes from understanding the concepts. Most easily, we could move the decimal in $25 to the left a space and know that $2.50 would be 10%. So, we can then double it to get $5 as your 20% tip. Or, you could know that 20%represents the fraction 1/5 and to find 1/5 of $25 you can just divide by 5. You still get $5. You can fugure these things out quickly if you have a good conceptual understanding and you are not a rigid thinker.
Rigid thinkers know the algorithms and they will set up and solve the problem 25 x .2 and get the answer 5 after correctly placing the decimal in the answer. Worksheets and drills promote this type of learning. If this is the way that our students are able to do math, then we need to teach them that way. However, the goal is to get flexible thinkers who can understand and relate concepts from one math discipline to another. To teach to these students, a problem solving approach works best. Teachers need to help them make connections, but then they need to work to solve these problems.
Instead of setting up a lesson in which we find a percentage of a number, I want to set up a problem in which we need to discover the amount of a tip for a restaurant bill. Would a tip be larger or smaller than the bill? They will say smaller. With a bill of $43 what do we know about a percent, say 20%? A percent is a part of a whole. Then, what is the whole? 43. Good, now when I say I want to know 20% of 43, take note of the word of because of means to multiply. So what are we multiplyi g? 43 x .20. Yes! A percent is a decimal. We only have to move the decimal point to the left two places. Now, usually we get a larger number as a product when we multiply. Do you think that will happen this time? No, when you multiply by a fraction or decimal we are only finding a part of the original number, which was 43 in this instance. So go ahead and multiply 43 x .20. You got 86? Don't forget to place your decimal point in the right place because we are looking for a smaller number and 86 is bigger. Yes, move the decimal point to the left and you get 8.6. Yes, now you remember that rule? So, what is our 20% tip on $43? Yes, $8.60. Can you think of another way to do that? What do you notice about the numbers 43 and 86? They're doubled. Good. Why might they be doubled? Because we multiplied by 2 in 20%. Good. What if we multipled by 10? Yes, we could just move the decimal to the left and get 4.30. Easy! That's 10 percent, so we can just double that to get $8.60
Can you see the different ways of teaching. We want to teach like the latter example.This makes sure that kids get a good conceptual understanding. But, not all kids will and then we have to move to direct instruction of the algorithms and use worksheets to reinforce the raw memorization necessary for this approach. Often special ed students need this approach. And that's howcwe give students a passing understanding of math even though we really want to teach to deep conceptual understanding, but it doesn't always happen with every student. The difficulty is having both types of students in the same classes in heterogeneous grouping. Teachers complain about this problem a lot because it really is an inefficiency that would be easily solved by homogenous grouping.