Mfomsoɔ kyerɛ nnipa nyansa, ɛno nti deɛ ɔse ɔnyɛɛ mfomsoɔ da no, ahwere adeɛ.
Akan Proverb
Mistakes teach people wisdom, that’s why the person who says he has not made mistakes before has indeed lost a thing (an opportunity to learn).
With over a decade of experience as a classroom educator and over two decades as experience as a tutor I have come to notice several trends among most of my students. Here are the top four mistakes students make in mathematics and math-based science classes and how to correct those mistakes today.
1. Arithmetic errors
We’ll be working on a complex calculus or physics problem and my student will be executing the procedure beautifully, and then…
7 x 4 = 11 …Doh!
Simple arithmetic errors are very common mistakes that can single-handedly render pages of good problem-solving technique incorrect. Mastery of arithmetic comes from practice. Building efficiency and consistency with respect to the recalling rules for multiplying negative numbers, or dividing fractions, or adding decimals is like building muscle memory. The mind is a muscle. The more you train a skill, the faster and more efficiently your brain is able to execute that skill.
I recommend to my students that they download an arithmetic app on their phone and dedicate a few minutes each day to practicing their arithmetic. Choose an app that includes the appropriate level of arithmetic, i.e. adding, subtracting, multiplying, and dividing with integers (positive and negative whole numbers), fractions, and decimals. Be sure to set a daily alert that prompts the student to complete his daily practice. Apps that feature rewards and incentives are best; I find that tokens or other recognition for consecutive days of practice and efficiency of practice are very motivating.
2. Not showing all steps
If I’m at a restaurant I’ll definitely calculate the tip for the waiter in my head. If I’m in the field and want to determine whether or not a design idea is feasible I’ll most certainly do a quick back-of-the-envelope calculation; and after decades of experience I might not bother to write each and every step down. But for homework, classwork, tests, or any assignment that I am submitting to another human being for review then I’ll definitely show each and every step.
The two solutions above are both correct answers to the same calculus problem. The solution on the left shows all steps, while the solution on the right does not. Again, both solutions are correct. So, what is the benefit from including the detail as shown on the left? Consider that this is just one of 20 other homework problems that this student is working on. In addition, this student has been learning about the chain rule for a week so he is very confident in how to solve this problem. The issue is that weeks or months after solving this problem the student may forget the chain rule. If the student refers back to his homework and the only thing that he finds is the solution on the right, will this solution be very helpful to him?
We show all of our steps because not only does it reduce the likelihood that we’ll make a mistake, but it also gives us a good study tool to refer to in the future when the topic is not so fresh on our minds. Notice how the solution on the left not only mentions the method required to solve the problem (the chain rule), but it also lists the formula for the chain rule and continues to list step by step each procedure required to solve the problem.
Finally, as an added bonus, listing all steps is a great test-taking strategy if the problems for an assignment or a test are graded using partial credit. Listing your steps shows the teacher that you know what you are doing; therefore, if you make a small error but follow the correct procedure you can still get credit for your work. But, if you do not show your steps and you make an error then there is less of an opportunity for your teacher to give you partial credit. Our solution above on the left explicitly shows knowledge of the chain rule, the power rule, and how to differentiate natural logs while the solution above on the right jumps directly to the answer without any indication of how we reached it.
3. Not reality-checking answers
A student is knee-deep in crunching numbers to solve a word problem until, finally, an answer: –51.6. Okay, circle the answer and quickly move on to the next problem… Whoa, not so fast slick! The problem asks how long (in minutes) will it take the boat to arrive and the answer that you got was –51.6. Negative minutes? Don’t think so.
Students know that they can often check their answers for correctness by substituting their answer back into the original problem or equation. Most will choose not to do this, however, since it is time consuming. But reality-checking answers is a much faster process. It simply takes a moment to look at the answer that you’ve gotten and ask if it makes sense given the context of the problem. There are no negative times or lengths, the rocket is not traveling at a speed of 4.5 m, and it’s unlikely that each of your friends would receive 2.42771 apples.
4. Not reviewing solutions after the test
Too many students see the test as the end of the learning on a topic. Whatever questions they got correct represent the totality of their knowledge on the subject from that point onward. Whatever questions they got incorrect will forever remain shrouded in mystery. This, of course, is missing out on the best opportunity for true learning. After the test, when you have the solutions available to you gives you a great opportunity for self-assessment and correction. Comparing the solutions with your (incorrect) thought process will give you a clear indication of exactly where there are flaws in your thinking as well as any other skills that you need to tighten up (such as those mentioned above).
The test is just one assessment of your capacity at a single point in time. It does not, and should not, represent the totality of your knowledge on a particular topic. Strive for mastery regardless of whether or not your grades officially recognize it.
That’s it!