US Students Struggle With Understanding of the 'Equal' Sign 1268
bickerd--- writes with news of research out of Texas A&M which found that roughly 70% of middle grades students in the US don't fully understand what the 'equal' sign means. Quoting:
"'The equal sign is pervasive and fundamentally linked to mathematics from kindergarten through upper-level calculus,' Robert M. Capraro says. 'The idea of symbols that convey relative meaning, such as the equal sign and "less than" and "greater than" signs, is complex and they serve as a precursor to ideas of variables, which also require the same level of abstract thinking.' The problem is students memorize procedures without fully understanding the mathematics, he notes. 'Students who have learned to memorize symbols and who have a limited understanding of the equal sign will tend to solve problems such as 4+3+2=( )+2 by adding the numbers on the left, and placing it in the parentheses, then add those terms and create another equal sign with the new answer,' he explains. 'So the work would look like 4+3+2=(9)+2=11.'"
Re:Well, that explains things. (Score:4, Informative)
Re:How bad is it? (Score:3, Informative)
Its because you're shoving them into one equation. The scope of your 'working out' is to solve whatever is in that equation so the correct answer to
1+2 = 3
is
3 = 3
true.
Re:Confusing symbols (Score:5, Informative)
No, we use 'x' over here, too.
Re:I guess I'm stupid, too. (Score:5, Informative)
All part of their plan. (Score:0, Informative)
Middle grade levels are most certainly NOT learning programming (in the US anyway).
Warning... political rant follows
Making dumb, helpless, dependent-on-government, indoctrinated drones is exactly what the left-leaning portion of US society who have been running the education systems for decades have wanted, planned for, and are now reaping their harvest.
And we let them do it.
As a bonus, all this is next coming to a health care system near you.
Re:I guess I'm stupid, too. (Score:5, Informative)
I had to read it twice to get what they wanted done. An empty set of parens in proper mathematical expressions is valid and equivalent to (0).
"4+3+2=x+2 solve for x" is the correct way to state that problem.
Re:How bad is it? (Score:3, Informative)
Ya, I've never understood why 1 + 2 = 3 = 4 - 1 isn't okay.
Technically it is as okay as it gets, both sides of each equality operator is equal which is exactly how the symbol works. TFA is about how people don't actually "get" that, if you look at the example in the summary it essentially says 9 = 11 which of course is plain wrong.
The reason "double equalities" might be wrong is if you're solving an equation while showing each step.
Re:Confusing symbols (Score:5, Informative)
we used BOTH an x and a 'box' (as per my other post).
starting out, they taught us to fill in the missing value in the 'box' (square symbol). then, over time, when it was the right time to introduce letters as 'box symbols' they put an 'x' there.
made sense to me. a progression to get the kid up to that level of thinking. a box is empty and can be filled. makes good concrete sense. then later, we 'upgrade' the box to an x. same concept but more steps to get the kid there.
Re:Don't know what () means (Score:2, Informative)
That's because the US educational system is so bad at explaining variables.
I struggled with Math for the entire time I was in the educational system. At the end of each struggle, when the light bulb went on, I almost always thought 'This is so incredibly simple, why didn't they just explain it this way?" I've never seen a math textbook from another country as the article compares to, but I can certainly state that I think the ones written in the US are about at the level of a man page - if you already know the content and just want a quick refresher on some infrequently used bit - they're fine. If you're trying to learn something new for the first time - forget about it!
Re:Confusing symbols (Score:5, Informative)
If you watch the video, they have pictures of the math questions, which makes things a lot clearer. The parentheses are TFA's way of trying to draw a blank space. In the original questions, it's an underlined blank space (so ___ would have been a better choice) -- the same sort of underlined blank space provided in grade school where they want you to fill in the answer. In mathematics classes before algebra, when they're trying to introduce you to algebraic concepts, it's common to use blank spaces for "figure out what goes in this space and write it", rather than writing an "x" and saying "solve for x", which would use a concept the students haven't yet been taught.
Comment removed (Score:3, Informative)
Re:I guess I'm stupid, too. (Score:5, Informative)
The actual notation used in math questions and textbooks is a blank space (e.g., an underlined blank space). The parenthesis are a poor attempt and rendering that in text.
Re:Is it really plural, though? (Score:5, Informative)
Re:I guess I'm stupid, too. (Score:2, Informative)
It's not a trick question. I've seen it before, and I understood it the first time I saw it.
The problem isn't the fact they used odd notions that others assume incorrectly. Instead of ( ), I've also seen the use of an empty box, underline as someone else noted, ?, circles, etc.
IQ tests often use "something is missing, please complete", problems. Cognitive theory has it that the brain does not process information straight forward, it takes pieces of what it sees and reconstructs it. That's why we can understand misspelled words, geometric shapes with broken lines, and why visual tricks work on our brain. Understand broken English sentences.
The problem is students are growing up lacking in critical thinking skills. When left with a blank or a void, their minds fail to fill it in properly.
We're heading into a generation that can't understand things unless it's spelled out for them and that is a shame. And if you really have a BS in CS and genuinely could not understand the problem listed above, rather than stubbornly proving a point, then that is very sad indeed.
Re:Wrong (Score:3, Informative)
Re:Home School (Score:3, Informative)
Re:I guess I'm stupid, too. (Score:2, Informative)
Re:Wrong (Score:1, Informative)
Yes. Which leads to stupid problems like this one: :=. C (et al) have ==. BASIC is just crap and needs to die.
x = y = 1
In a sane language, y would be assigned the value 1, then x would be assigned the value in y.
In BASIC, x is assigned the value of the expression y = 1 which is a test of whether the value of y is equal to 1.
BASIC is anything but. Side-effects are side-effects, and too many of them make code unreadable and difficult to learn. Equality and assignment aren't the same operation, and they shouldn't use the same operator. Pascal has
Re:Wrong (Score:5, Informative)
In most of the world we study Mathematics. I didn't realise that there was only one Mathematic studied in the US.
This is a dialectical thing about American English. We use singular verb inflections with collective nouns.
Queen's English: "Aerosmith are playing Wembley Stadium."
U.S. English: "Aerosmith is playing the Verizon Center."
This is why you hear "the data is" over here.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:2, Informative)
It is "fill in the blank" - but filling the blank with 9 doesn't make it a correct equation. Filling it with 7 does.
Re:I guess I'm stupid, too. (Score:1, Informative)
An empty set of parens in proper mathematical expressions is valid and equivalent to (0).
Oh, really? Can you site an authority for this?
I've had a fair amount of mathematics in my 6 years of college, and I have never once encountered this.
Re:Wrong (Score:4, Informative)
"x = 1" is only a legal BASIC statement in the first place because interpreters have been relaxed for programmers too lazy to use "Let".
It's not really laziness. Mostly it's for historical reasons. My first computer (a TRS-80 model 1, "Level 1") had 4K of RAM. That's right, 4096 bytes. When you've got that little space for your BASIC programs, removing the requirement for "let" in your BASIC code freed up valuable bytes. (other similar shortcuts existed back in the day, such as "?" available as a replacement for "print".) ...and it wasn't just TRS-80s. Most computers of that era had similar restrictions. Today, space is virtually unlimited, but these legacy shortcuts remain.
Now get off my lawn.
Re:Wrong (Score:3, Informative)
there are just as many overweight people there as in the US
Did you actually count them? As a percentage of population? Some people did [wikipedia.org] and there are a lot, but not as many. The figures are around 33% for the US, 22% for the UK being classified as obese. For men, obesity in the EU ranges from 10-27% of the population, depending on the country - all of them are lower than the 33% figure for men in the USA. It's slightly different for women, where the figure for the USA is 35%, while the EU ranges from 10-38%.
So, there are a few countries in the EU where a higher percentage of he female population is obese than in the US, but no country where more men are obese.
Re:Wrong (Score:3, Informative)
It would be a lot harder to type by accident when you really meant to type just one.
Re:Wrong (Score:1, Informative)
You might do something like this in C:
[code]
BOOL Status;
if (Status = some_true_false_function())
x++;
return Status;
[/code]
Re:Wrong (Score:4, Informative)
Re:Is it really plural, though? (Score:5, Informative)
Sice everybody seems to be too lazy to check, here's an excerpt from Wikipedia.
Etymology
The word "mathematics" comes from the Greek (máthma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.[9] Its adjective is (mathmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, (mathmatik tékhn), Latin: ars mathematica, meant the mathematical art.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural (ta mathmatiká), used by Aristotle, and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[10] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.
Re:Wrong (Score:4, Informative)
Re:I guess I'm stupid, too. (Score:2, Informative)
Re:Wrong (Score:4, Informative)
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:5, Informative)
I saw that in textbooks right on the cusp of learning algebra, to ease you into it. Only, it's not really "( )", it's, well...Slashdot doesn't support unicode so I can't show you...but, it's supposed to be a circle. TFA didn't use one either. IDK if it's because they're not using the same textbooks I've seen, or just because they don't know how to type unicode, either. At any rate, students would, from their earliest years, be used to seeing "2 + 2 = ( )" or "2 + 2 = [ ]" where those are supposed to be circles or boxes for them to put the number in. Or, perhaps "2 + 2 = ___" a blank line for them to put the answer in. The point was that, with no explanation of the equal sign, they come to the wrong conclusion about that circle. They see "4 + 3 + 2 = circle + 2" and they do what they've always done, by rote, and put 9 into the circle, then proceed on to the next little bit, which is +2, there, 9 + 2 is 11, they wonder why there isn't another circle, and make one.
At any rate, your solution of "just use algebra" is absurd, they haven't learned it yet. Algebra is what they're trying to teach them with this. And the point is, it doesn't matter. If they show them "x = 2 + 2, so x is 4", they just might get it. But, if they see "4 + 3 + 2 = x + 2" they would do the same as before "x is 9, so x + 2 is 11". They're just assigning too low a priority to equality in the order of operations, really...and also thinking in C I suppose, where (x = 9) + 2 does equal 11 ;)
Re:Go back and read your basic English book! (Score:3, Informative)
Nope. That's not what it means at all. The apostrophe (that's its name) is used to denote the possessive e.g. JeremyP's post. It's also used to show that you've missed out some letters, as in "you've" or "it's" or "couldn't". So technically "math's" is correct, but nobody ever writes it like that.
Re:Wrong (Score:1, Informative)
"Maybe your TRS-80 should have tokenized keywords to conserve memory."
In fact, it did.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:2, Informative)
1 + 1 does not always equal exactly 2.
Yes; yes it does.
1 + 1 always equals 2. You aren't adding 1 + 1 when you're adding .5 and .5, you're adding .5 and .5. There's a 50% difference; your answer will reflect this difference.
To use a first grade example, if I have one apple and put it in a bag with someone else's apple, that other person's bag now has two apples. Now, if I have taken a bite out of my apple, the other person has one bite less than two apples - which, as any person might be able to see, is absolutely less than two apples. Relatively speaking, however ("depending on your point of view") the person with the bag "has two apples".
This might also be called rounding, which, while not incorrect, is not true equation; it's an approximation. This does not change the absolute nature of 1 + 1 = 2.
Re:Wrong (Score:3, Informative)
Re:Wrong (Score:3, Informative)
Most people just call them all British.
Just don't call a Scot "English," or you'll likely start a fight.
Re:Wrong (Score:1, Informative)
That hasn't been true since the civil war. Get with the times.
Re:Wrong (Score:3, Informative)
That's a bit of a strong statement.
I don't want to pretend I'm some godlike C programmer, but competent coding and a simple review process catches this stuff pretty easily.
I wouldn't say I'm a godlike programmer either, but I've been doing it for a living for 22 years. I've learned a thing or two in that time and seen many times what happens when programmers don't follow best practices.
In my experience competent programmers use all the automated tools at their disposal; exceptional ones find more things to automate.
Running a static code checker a 10,000 line program takes less than a second and catches 100% of stupid errors. Reviewing the same program by hand takes hours and is significantly less than 100% effective. Wasting hours of your time doing something the computer could do better in under a second is extremely foolish and unprofessional -- in other words, it's a sign of incompetence. That time could be better spend checking things that automated tools CAN'T easily catch, like critiquing the high-level design, writing automated test cases, etc.
If you don't consistently follow best practices, you're not a competent programmer. Considering that I've made a very lucrative career out of cleaning up other people's messes, I can say there's definitely no shortage of incompetent programmers working in the field.
And sometimes that there if(a=b) is exactly what you mean.
Can't think of any situations off hand, but I'm sure there have been some.
If you had bothered to read the splint output, you'd have seen this:
If an assignment is intended, add an extra parentheses nesting (e.g., if ((a = b)) ...) to suppress this message
Reading is fundamental.
Re:It should be: 4+3+2=x+2 (Solve for x) (Score:3, Informative)
I presumed it to be the text equivalent of an empty box on a printed worksheet. In my hazy memory, a blank like that was the standard way of writing algebraic questions like this in elementary school. It requires less abstraction than the 'x' option, since you don't have to get into the business of "assigning" a value to x and that whole business. You just visually complete the box (or empty line) or whatever so that there's a true statement on the page.
If you had seen this on a sheet of paper, I think it'd be more obvious.
Re:= for assignment not unique to C-style language (Score:3, Informative)
Fortran (at least up to Fortran77) uses "=" just for assignment. For comparisons we have to use (x.eq.y). The system works pretty well, and much simpler than C. However, typing .eq., .ne., .gt., .and., etc at every logical group is a royal pain in the ass.
Re:Wrong (Score:1, Informative)
Most people just call them all British.
Not in Scotland they don't.
Well, not more than once anyhow.