I think the people who proclaim themselves being bad at math comprise more than 10% of the population, for sure.
This is likely true because so few people report on their math skills. There may actually be 10% of people who are bad at math, meaning they can’t add a couple of 2 digit numbers, and maybe some smaller percentage of people who brag about how good they are at math, and the rest don’t care because the average level of math skills is more than sufficient in the lives of the vast majority of people.
ETA: And I have no idea why I’m bad at math.
By the way, I do not find the “reach for a calculator to compute 0.1/0.2” student so alarming. After the student has been taught 2 or 3 or 4 ways to think about this exercise and has given it a try, why not use the calculator to check the answer? We don’t want the student to feel ashamed about it or to grab the calculator out of their hands. Instead, we can turn it into something positive, and maybe next time they will not be so quick to panic and reach for the calculator before trying anything else.
Of course I am not so naive as to think there is one sure-fire right way to teach arithmetic, geometry, statistics, et cetera in schools, though. If there were, people would be doing it already.
People likely have varying aptitude for math, but the inability to do basic practical problems mostly comes down to “Lack of practice”.
Now you may be thinking to yourself “But math class is full of this kind of problem”, but learning how to solve problems in math class is not in and of itself sufficient practice for applying that knowledge and the students who fall further and further behind will have had years and years of math class to solidify the abstractness of math problems, possibly especially the problems that claim to be practical.
Those for whom the style of instruction and pace of learning of standardized basic education worked don’t generally think “Oh, we did this problem in 8th grade!” They think “I know the dimensions, so I can find the volume, I can look up the density and that will give me the weight.”
Remembering concepts and how they interact is much more lasting knowledge than remembering each step of an algorithm you don’t have time to figure out how work. Both kinds of knowledge needs at least some occasional practice to recall 10-20 years later, but practice for the first kind of knowledge can be less frequent and can happen separately for the different concepts.
In other words, although there are many potential gaps in knowledge that may lead someone to ask a question such as described in the OP, the overarching problem is most likely that in math class they learned to solve math problems, and not how to use math to solve problems.
I would be interested to read studies on why people “hit the wall” on math at certain points. I realize there are a bajillion reasons, and I’m wondering what some of them are. For me there was probably a correlation between the number of steps involved and the amount of trouble/frustration I encountered.
I think Senegoid’s point there was about simplifying fractions, not checking the answer. Those of us who are “good at math” look at 0.1/0.2 and instantly see that it simplifies to 1/2, in the same way that 6/9 reduces to 2/3. I can totally understand a student getting thrown off by the decimal points in the fraction, though, and not seeing the simplification.
When I introduce complex numbers in an algebra class, I use this as an icebreaker:
And when I introduce the quadratic formula:
https://cs.appstate.edu/~sjg/class/1010/clicker/CalvinHobbes.jpg
There is a classic fight between Math professors and engineers & technicians.
I took electronics at Southwest Tech in Arkansas. It had just become part of Southern Arkansas University. They were in the process of getting accreditation and hired Math, Chemistry and English lectures.
Our Math teacher was teaching Algebra the traditional way. Lot of explanation and theory.
My electronics teachers were getting frustrated because most of the students couldn’t solve formulas. We hadn’t gotten that far yet in the Algebra class.
Lot of pressure on the Math teacher.
Finally our exasperated electronics teacher spent 2 days of class teaching us himself.
Ok Kirchoff’'s Circuit Laws
You have three resistors. What’s the current in the junction?
Then he showed us how to manipulate the formulas for the unknown value.
Ohms law,. Same thing.
I=E/R
12V, 1/2 amp, what’s the resistance?
IR=E/R * R/1
IR=E
R=E/I
24 ohms
I learned more practical algebra from that electronics teacher’s 2 day class than I did in the entire semester of algebra.
I’m guessing the wall for most people starts at algebra/geometry. Because at that point, we’re no longer dealing with “kitchen table” math, where the problems involve slicing up pizza pies and counting apples. It’s also at this point where most kids can’t rely on their parents for homework help, since their parents have likely forgotten how to solve for x. Without parental reinforcement and example, maybe it becomes easier for a struggling student to not push through the wall.
I don’t have a dog in this fight either way, but what the mathematics professors and others in this thread are saying, is that the more the students are taught to mechanically “solve formulas”, the more it risks becoming an exercise in (easily forgettable) memorization rather than practicing the skills that will allow them to derive and solve formulas on their own. Plus the difficulties involved trying to, e.g., solve the heat equation without at least taking classes in calculus and algebra first.
But there has to be a reasonable middle ground, involving practice working a lot of practical engineering/physics/chemistry/… problems in addition to merely memorizing theorems from the algebra textbook and solving contrived/toy examples.
I agree there are times when math theory is very important. A math major for example needs that type of training.
That’s why colleges offer applied Math classes. Business math and Applied Science math are two examples. Math majors can’t apply these classes towards their degree requirements.
We have/had the same issue in Computer Science. It originally taught people how to program and get jobs. I saw it change when I was in college. They created a separate Data Systems Applications degree for people who wanted practical hands on programming training. Computer Science became more theoretical. My shop hired some of the Computer Science graduates later in my career. It took a lot of training to get these youngsters up to speed in real life business programming. My colleagues and I found it extremely frustrating that these CS graduates weren’t taught the basics in college.
I understand that some teachers take formulas too far.
Some electronics teachers for example teach Ohms Law by memorizing three formulas. The students plug in the numbers.
I wasn’t taught that way. We learned one formula. I=E/R and used Algebra to solve for unknown value. I’m very thankful now that he insisted that we understand the Order of Operations to solve equations.
We didn’t just plug numbers into an equation.
My mom understood what I was trying to do, and my older brother did as well. I hit the wall at algebra, but didn’t have as much of a problem with geometry. Our oldest never hit the wall, and our youngest didn’t hit it until he took calculus his senior year (fortunately he was able to pass with help from his brother).
I’m not bad at math.
I have tutored kids who claimed to be bad at math. From an outside-looking-in perspective, it seemed like they didn’t understand how anything connected to anything else. For example, they’ll color in 3/4ths of a circle and 6/8ths of a circle and 9/12ths of a circle because the instructions say to, but it never really hits them that it’s all the same thing. And then later on when the lesson is on reducing fractions, they learn to divide the top and bottom by 2 (because that’s what was taught in class), but not quite get how that relates to the circles they colored in or how the circles related to each other.
This also seems true - often times, it doesn’t feel like they’re working the problems, it feels like they’re taking random math-centric guesses.
My experience with anyone who is “bad at math” is that, at some point, they didn’t understand, and then that part was part of what they needed to know for the next step, and so on. Since math is (as taught in school, at least) cumulative, it only takes not having enough time to have gotten one thing to get stuck forever.
And then, after that, where they’ve tried but failed so often, they give up.
As for memorization: I’ve repeatedly said it’s not a useful way to learn, because you will most likely forget. It is exactly why calculus is my least remembered subject–all those differential formulas. Yes, we may “prove” a few, but I had no intuition of why they worked. So I don’t remember any rules save the power rule, which I could sorta see by looking at how acceleration, velocity, and displacement work.
Hell, I remember being awed when someone finally showed me that a sine curve is just an unrolled circle. Why wasn’t that in school? Why did we never cover what dx meant? Ugh.
How is figuring out you have a learning disorder when you are in your late 30s significantly better than in my late 50s?
Yes. I’m pretty sure that what I’m seeing here is a student who may be able to plug the right numbers in, but doesn’t understand what’s actually going on, what those numbers actually mean.
If you understand the meaning, not just how to plug into the formula, then it’s obvious that 0.1 ÷ 0.2 is going to produce the same answer as 1 ÷ 2, or 10 ÷ 20, or 100 ÷ 200, and so on; or for that matter 50 ÷ 100. If all you know is the formula, those are five different problems, one of them with decimal points thrown in to make it look harder.
Which is why I figured it was going to kick me in the head eventually that I didn’t understand what was going on, even if I could plug things into the formula.
I dunno. I found it liberating to realize that it wasn’t that I was specially stupid but in fact had a real named disorder. I went about telling everybody I knew, much to their patient boredom. I wish I had gotten that information earlier in my life so I could be relieved that many years sooner.
Not merely formulas, but in this case we also want the student or statistician to understand that “obviously” 0.2 = 1/5 so dividing by 0.2 is the same as multiplying by 5. Also 0.1 is 1/10, 99% means 99 out of 100 and 95% means 1 out of 20 miss the mark, and so on. Nobody cares, I think, that the student be particularly good at mental arithmetic per se.
I am reasonably competent at understanding the principles of mathematics. Basic arithmetic is a problem.
Some of this is due to my early education. I was reading fluently before I started Kindergarten. I honestly don’t remember NOT knowing how to read. Back then (when dinosaurs ruled the earth) first grade was entirely about learning to read, and the teachers didn’t know what to do with me. My mom was chastised for teaching me, but all she did was read me stories and I just picked it up. Not knowing what else to do, in January they advanced me to second grade with no warning. Second graders were already doing arithmetic and I had NO idea what was going on.
I still have difficulty doing simple arithmetic without writing it down.
Some of it is also, I think just the way one’s brain is wired. Both my daughters also read well before starting school. My younger daughter also did arithmetic in her head before the age of four. She used to ask me for the discarded number tickets at the deli counter and would add them up just for fun. During one shopping trip we were passing the eggs, and she asked me what the word “dozen” meant. “Twelve of anything,” I answered. “So,” she mused, “each of those boxes has 12 eggs?” “Right.” “So two boxes would have 24.” “Uh, yeah.” “And three boxes would have 36.” “Yes.” She kept on until she got into three-digit numbers. In her head. While sitting in the baby seat in the shopping cart. Nobody taught her that; she just figured it out.