I hate myself. I can never answer questions like this, and I exhaust myself from trying so hard.
Throughout school, I was absolutely HOPELESS. I couldn't do anything when it came to basic math, except some algebra. These questions (even faced with the formula to solve it) still bugger my head. I tried to read books, get tutors, do everything to better myself (who ever heard of a computer guy who couldn't do math!).
I can easily write algorithms, do algebra, write complex programs, do anything on a computer but faced with a question like this my brain shuts down very quickly.
Holy smokes, thank you, thank you, and thank you so much for posting this! I had no idea that this had a name called "Dyscalculia". I wonder if there is a way to medically verify if I actually have this condition. To this day, I cannot read an analog watch. I cannot do math in my hand, although I do perfectly fine on paper. For even simple calculations, like 23 + 45, I simply cannot keep both numbers in my head. It is truly a curse. I actually physically get dizzy when I'm asked to perform computations. When I was in high school, I still played games like "Number Munchers", asking myself.. "What is wrong with me?", despite the fact that I had successfully made it to calculus and maintained a very high GPA.
It is amazing then, that I somehow have managed to complete both an undergraduate and Master's degree in Electrical and Computer Engineering. I compensated for my lack of numerical ability by heavily relating on calculators throughout my college education. In fact, one of the reasons I pursued computers in the first place is because I recognized that I could survive, and hence, fake it, by offloading such "trivial" computations to a machine.
Many standardized tests, such as the GRE, don't allow calculators unfortunately.
I was diagnosed in college. My adviser suspected something was wrong and suggested I see a specialist. It's been more than 20 years, so I'm a little hazy on the details - but from what I remember, the tests mostly involved reasoning and abstraction.
If the ball cost $0.10, the bat would need to cost $1.10 (a dollar more than $0.10), which would mean that the pair together would cost $1.20, which we know isn't true. To get the right answer, you have to solve the simple algebra equation in Retric's sibling comment.
For the lily pads, the percentage of the lake covered doubles every day, so the lower bound percentages for the last few days look roughly like this: 12.5%, 25%, 50%, 100%. On the 24th day, you'd have to double in size 24 more times in order to fill the pond, rather than the once it'd take on the 47th day.
The first problem is operational and the second problem is change on a slope.
People use what they know to deal with problems, so facing these two they use basic math (subtraction and division) being ignorant of higher-level math concepts such as algebra or calculus. The answers they come up with appears right to their known level of logic.
If you were the type of person that got to learn about high level math concept, and are the studious type to double check answers, then these two problems are condescendingly seen as trivial.
I see it differently. The whole point of the article is that these problems affect smart people.
>For one thing, self-awareness was not particularly useful: as the scientists note, “people who were aware of their own biases were not better able to overcome them.” This finding wouldn’t surprise Kahneman, who admits in “Thinking, Fast and Slow” that his decades of groundbreaking research have failed to significantly improve his own mental performance. “My intuitive thinking is just as prone to overconfidence, extreme predictions, and the planning fallacy”—a tendency to underestimate how long it will take to complete a task—“as it was before I made a study of these issues,” he writes.
It has nothing to do with knowledge of higher level mathematics because these problems are easily solvable with arithmetic. Lacking calculus doesn't kill you on this problem. An intuitive gut feeling that you've already arrived at the right answer and laziness is the source of confusion.
I've even read about that damn bat and ball problem and it STILL tripped me up this time. I could easily have double checked my answer, but I wanted to read the article. Even a child knowing nothing other than addition could get it right with a little bit of trial and error. I hope after admitting that you see that I don't find the problems condescendingly trivial.
Personally, I found the second problem much easier... probably because programmers have a better intuitive grasp of powers of 2. Bringing in slope is stretching it a bit. Working in reverse from the completely covered lake, it should be obvious that going back one day halves the lily pads. However, I could imagine how someone more familiar with linear processes would get the wrong intuitive result.
>The answers they come up with appears right to their known level of logic.
A studious habit sure, but checking your answer isn't a higher level math skill.
It has nothing to do with calculus, or math at all.
My natural reaction to the bat and ball problem was to parse the problem statement verbally and search for a plausible answer among the tokens. The algorithm retrieved "USD$1", and then a second background process took over and said "wait, that sounds a little bit too right". It might have taken me full 5 seconds before I realizes I had to switch to math_mode!!!
