HP Forum Archive 19

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70% of US students don't understand the equal sign Message #1 Posted by DavidShenk on 11 Aug 2010, 11:42 a.m. Perhaps RPN is more popular than we thought? ;) Story here.

I wonder why. Message #2 Posted by Frank Boehm (Germany) on 11 Aug 2010, 11:49 a.m., in response to message #1 by DavidShenk They probably never heard of "All animals are created equal." Maybe there is some connection. Who knows.

Re: 70% of US students don't understand the equal sign Message #3 Posted by Thomas Radtke on 11 Aug 2010, 12:14 p.m., in response to message #1 by DavidShenk That's probably due to scholarships mostly going to football aces in the US. Or is that just a myth like germans always waering leather trousers? On a second thought, the example given in that article might indicate that these students have an approach to that equation like they would have when ordered to type something into their calculators. But then, why only in the US? No, it's clearly football.

Wonder how many statisticians don't understand the percent (%) sign? Message #4 Posted by Allen on 11 Aug 2010, 1:04 p.m., in response to message #1 by DavidShenk :)

Re: 70% of US students don't understand the equal sign Message #5 Posted by Tim Wessman on 11 Aug 2010, 3:09 p.m., in response to message #1 by DavidShenk Equal sign? You mean this thing? --> = That my friend, is an assignment operator. TW

Re: 70% of US students don't understand the equal sign Message #6 Posted by Mike on 11 Aug 2010, 4:28 p.m., in response to message #5 by Tim Wessman This is an equal sign: = This is an assigment operator: = These are railroad tracks: = Edited: 11 Aug 2010, 4:28 p.m.

Re: 70% of US students don't understand the equal sign Message #7 Posted by David Hayden on 11 Aug 2010, 4:56 p.m., in response to message #5 by Tim Wessman Quote: Equal sign? You mean this thing? --> = That my friend, is an assignment operator. TW Right on baby!!

Re: 70% of US students don't understand the equal sign Message #8 Posted by Crawl on 11 Aug 2010, 9:03 p.m., in response to message #1 by DavidShenk Yeah, the question is ambiguous, so who is to blame? The students, obviously.

Re: 70% of US students don't understand the equal sign Message #9 Posted by DaveJ on 12 Aug 2010, 5:41 a.m., in response to message #1 by DavidShenk Considering that around 1/3rd of US college graduates believe in creationism, stats like that no longer surprise me! Dave.

Re: 70% of US students don't understand the equal sign Message #10 Posted by Mark Storkamp on 12 Aug 2010, 10:15 a.m., in response to message #9 by DaveJ Even scarier is that 2/3 don't. But then politics and religion are two topics that don't belong here anyway.

Re: 70% of US students don't understand the equal sign Message #11 Posted by Martin Pinckney on 12 Aug 2010, 1:55 p.m., in response to message #10 by Mark Storkamp Good science and good religion coexist well. Bad science and bad religion are an anathema to each other. Politics doesn't go with either. IMO.

Re: 70% of US students don't understand the equal sign Message #12 Posted by Juan J on 12 Aug 2010, 9:40 p.m., in response to message #11 by Martin Pinckney Quote: Good science and good religion coexist well. Bad science and bad religion are an anathema to each other. IMO. ...and atheism is not an excuse to act like a jerk. Juan

Re: 70% of US students don't understand the equal sign Message #13 Posted by db (martinez, ca.) on 13 Aug 2010, 1:01 a.m., in response to message #11 by Martin Pinckney Martin; You sound like a candidate for conversion to the Baha'i faith. They believe that god expects people to use science to study his universe, education is obligatory, and that work is worship, among many other sensible things. I'd be one, but i just can't get myself to believe in their god either.

Re: 70% of US students don't understand the equal sign Message #14 Posted by Walter B on 13 Aug 2010, 1:47 a.m., in response to message #13 by db (martinez, ca.) AFAIK buddhism originally had a very similar approach. But during 2600 years these clear ideas became heavily decorated ...

Re: 70% of US students don't understand the equal sign Message #15 Posted by DaveJ on 13 Aug 2010, 1:51 a.m., in response to message #13 by db (martinez, ca.) Dave. Edited: 13 Aug 2010, 2:04 a.m.

Re: 70% of US students don't understand the equal sign Message #16 Posted by Paul Dale on 13 Aug 2010, 3:38 a.m., in response to message #15 by DaveJ I want that t-shirt. I've educated my daughters on the noodly goodness of the FSM.... Much to their enjoyment. - Pauli

Re: 70% of US students don't understand the equal sign Message #17 Posted by DaveJ on 13 Aug 2010, 4:28 a.m., in response to message #16 by Paul Dale Quote: I want that t-shirt. You're in luck, it's FOR SALE. Dave.

Re: 70% of US students don't understand the equal sign Message #18 Posted by Paul Dale on 13 Aug 2010, 4:39 a.m., in response to message #17 by DaveJ You've made my weekend. And probably my daughters' week :-) - Pauli

Re: 70% of US students don't understand the equal sign Message #19 Posted by DaveJ on 13 Aug 2010, 6:31 a.m., in response to message #18 by Paul Dale May you be touched by his noodly appendage! Dave.

Re: 70% of US students don't understand the equal sign Message #20 Posted by Paul Dale on 15 Aug 2010, 3:48 a.m., in response to message #18 by Paul Dale Ordered one for me and some for my children :-) - Pauli

Re: 70% of US students don't understand the equal sign Message #21 Posted by DaveJ on 15 Aug 2010, 4:31 a.m., in response to message #20 by Paul Dale Quote: Ordered one for me and some for my children :-) More muffin money for me :-D The Zazzle shirts are all top quality, no made in China stuff. I might draw some more FSM designs, I just can't get enough of His Noodleness! Enjoy. Dave.

Re: 70% of US students don't understand the equal sign Message #22 Posted by DaveJ on 13 Aug 2010, 2:07 a.m., in response to message #13 by db (martinez, ca.) Two gods, pictured at the place of creation: Dave.

Re: 70% of US students don't understand the equal sign Message #23 Posted by Martin Pinckney on 13 Aug 2010, 1:38 p.m., in response to message #13 by db (martinez, ca.) Quote: ...Baha'i faith. They believe that god expects people to use science to study his universe, education is obligatory, and that work is worship, among many other sensible things. db, These are all principles that can be gleaned from the Judeo-Christian scriptures, as well, although not what you usually hear emphasized, or perhaps not stated exactly so.

Re: 70% of US students don't understand the equal sign Message #24 Posted by Bill Zimmerly on 14 Aug 2010, 3:27 p.m., in response to message #9 by DaveJ Quote: Considering that around 1/3rd of US college graduates believe in creationism, stats like that no longer surprise me! Dave. Maybe they're just being good engineers and wondering how many generations it took to evolve fully functional reproductive organs?

Re: 70% of US students don't understand the equal sign Message #25 Posted by Palmer O. Hanson, Jr. on 13 Aug 2010, 3:32 a.m., in response to message #1 by DavidShenk Does anyone know how I can obtain a printout of the original article in the Psychological Journal?

Re: 70% of US students don't understand the equal sign Message #26 Posted by Sancerre (Phoenix) on 13 Aug 2010, 11:46 a.m., in response to message #25 by Palmer O. Hanson, Jr. This link was available from Slashdot: Link to Story --Sancerre

Re: 70% of US students don't understand the equal sign Message #27 Posted by Palmer O. Hanson, Jr. on 13 Aug 2010, 10:35 p.m., in response to message #26 by Sancerre (Phoenix) Sancerre: Thank you for your help. Here are couple of interesting quotes from the reference: Quote: Students who exhibit the correct understanding of the equal sign show the greatest achievement in mathematics and persist in fields that require mathematics proficiency like engineering, according to their research. As a fan of algebraic I loved that quote. But I read on and here is another quote that troubles me: Quote: “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. “This response has been called a running equal sign—similar to how a calculator might work when the numbers and equal sign are entered as they appear in the sentence,” he explains. “However, this understanding is incorrect. The correct solution makes both sides equal. So the understanding should be 4+3+2=(7)+2. Now both sides of the equal sign equal 9.” So, here are some questions that may display my lack of familiarity with the way algebra is taught these days: Do modern students really solve problems in algebra that way? Is the use of adjacent opening and closing parentheses without anything enclosed a standard procedure? It isn't acceptable when using the solver in the hp-35s and will result in a syntax error. The syntax error can be avoided by enclosing a variable inside the parentheses, or by simply replacing the () with a variable. I have stumbled on some even stranger examples of students using the running equal sign technique which suggest to me that the reason students in India, China, Turkey and elsewhere who do better do so only because they haven't been allowed to use the running equal sign technique. Palmer

Re: 70% of US students don't understand the equal sign Message #28 Posted by Don Shepherd on 14 Aug 2010, 12:04 a.m., in response to message #27 by Palmer O. Hanson, Jr. Quote: Do modern students really solve problems in algebra that way? No. At least my students don't. The example quoted might be used to teach the student that the equality must be "balanced." That is, the value to the left of the equal sign must be the same as the value to the right of the equal sign. So they would fill in the parentheses with the value that would make that true. That's fine for teaching how things must be balanced, but it is not "solving" an algebraic equation. An "algebraic equation" must contain a variable, something to solve for. The example shown is not an algebraic equation. I wouldn't read too much into this example. Kids are learning algebra today the same way we learned it years ago, at least in most schools.

Re: 70% of US students don't understand the equal sign Message #29 Posted by bbeamer on 14 Aug 2010, 10:35 a.m., in response to message #28 by Don Shepherd I'll second what Don said. I also teach algebra the old fashioned way, with x's and y's. I think the problem may illustrate the idea of equations needing to be "balanced", but I don't think it's a good problem. As far as the running equal, that's something I've not really seen at all. Brian

Re: 70% of US students don't understand the equal sign Message #30 Posted by Crawl on 15 Aug 2010, 12:18 a.m., in response to message #29 by bbeamer Again, the question is ambiguous. There is no established meaning for (), so it's up to the student to guess what the question is asking. Really, the correct interpretation of the results is not "Students don't understand the equals sign" but rather "Students don't understand what the meaningless "()" symbol means ... but why should they?" I recently saw another study -- was it linked here? -- that ALSO said that students don't understand the equals sign. But the test-makers in that case meant it to mean something completely different! There example was a=5 b=2 a=b So now what? Of course, the answer is supposed to be that now a=b=2, but if you think "equals" always means "you must balance", then those statements made no sense. In my opinion, teachers often do a very poor job of asking unambiguous questions. One example from my education I remember was when I was asked in an oral example about things traveling backwards in time. I had no idea what was being asked. I ventured, "Are you talking about tachyons?" That seems to me, then as now, to be a reasonable interpretation of the question. Then they rephrased the question, "No, I mean does time have an arrow?" Well, phrased that way, I immediately knew they were talking about entropy. Although in that case, it wasn't so much a question testing my understanding of entropy so much as a test of me having read A Brief History of Time, and knowing that specific phraseology from it.

Re: 70% of US students don't understand the equal sign Message #31 Posted by bbeamer on 15 Aug 2010, 4:46 p.m., in response to message #30 by Crawl Yes, context is important. In this example, the context seems to be more related to programming than to algebra and the = is interpreted as "is assigned the value" . Brian

Re: 70% of US students don't understand the equal sign Message #32 Posted by Don Shepherd on 15 Aug 2010, 6:02 p.m., in response to message #31 by bbeamer Quote: 4+3+2=( )+2 I think I would agree with crawl. Just that statement with no other directions would be ambiguous as best, meaningless at worst. If this was a problem in a math textbook or workbook and the directions for the problem stated: "In the following problems, enter a value within the parentheses that makes both sides of the equation equal, such as 3+5+1=(4)+4+1", then that would be reasonable. In the absence of a clear instruction like that, the problem is meaningless and it would be impossible to draw any valid conclusions from the answers provided by the students. That's my opinion. Edited: 15 Aug 2010, 6:11 p.m.

Re: 70% of US students don't understand the equal sign Message #33 Posted by Vieira, Luiz C. (Brazil) on 15 Aug 2010, 6:36 p.m., in response to message #1 by DavidShenk I vote for the KR "C"´s implementation: = is attribution, so 'A=2' is not "A is equal to two", instead it is "Attribute 2 to variable A"; == is a relational operator, so 'A==2' is not "Attribute 2 to variable A", instead it is "A is equal to two", and a 'TRUE' or a 'FALSE' flag returns. No doubts about, no double interpretation. But hey, when the equals sign was created there was no such thing as a programmable computer nor programming languages, right? Cheers. Luiz (Brazil)

Re: 70% of US students don't understand the equal sign Message #34 Posted by Gerson W. Barbosa on 16 Aug 2010, 2:15 p.m., in response to message #33 by Vieira, Luiz C. (Brazil) Hello, Luiz! Quote: I vote for the KR "C"´s implementation: = is attribution, so 'A=2' is not "A is equal to two", instead it is "Attribute 2 to variable A"; == is a relational operator, so 'A==2' is not "Attribute 2 to variable A", instead it is "A is equal to two", and a 'TRUE' or a 'FALSE' flag returns. Pascal totally avoids the ambiguity by using := for assignment. So does the LET instruction in BASIC (when it's not omitted). Cheers, Gerson.

Re: 70% of US students don't understand the equal sign Message #35 Posted by Palmer O. Hanson, Jr. on 15 Aug 2010, 10:19 p.m., in response to message #1 by DavidShenk I found the following discussion at this site: http://castingoutnines.wordpress.com/2010/08/11/student-misunderstanding-of-the-equals-sign/" Quote: The biggest problem I seem to encounter with “=” sign use is that students use it to mark a transition between steps in a problem. For example, when solving the equation for x, you might see: 3x - 2 = 10 = 12 = x = 4 The thought process can be teased out of this atrocious syntax, but clearly this is not acceptable math — even though the last bit of that line (x=4) is a correct statement. If the student would just put spaces, tabs, or even a semicolon between the steps, it would be a big improvement. But many students are so trained to believe that the right answer — the ending “4? — is all that matters, they have little experience with crafting a good solution, or even realizing that a mathematical solution is supposed to be a form of communication at all. Do some mathematics teachers really allow stuff like that? We would have discouraged it when I was teaching algebra by grading not only the solution but also the solution process. Palmer

Re: 70% of US students don't understand the equal sign Message #36 Posted by Don Shepherd on 16 Aug 2010, 6:23 a.m., in response to message #35 by Palmer O. Hanson, Jr. Quote: Do some mathematics teachers really allow stuff like that? Good God no!!! As I teach algebra, the equals sign separates the two sides of an equation, and there better be one per each line when you re-write the modified equation, and each line is a step of solving the equation. It better look something like this: 3x-4=8 +4 +4 3x =12 /3 /3 x =4

Re: 70% of US students don't understand the equal sign Message #37 Posted by Walter B on 18 Aug 2010, 1:51 a.m., in response to message #36 by Don Shepherd FYI, we were taught this way some decades ago to solve such problems: 3x-4=8 | +4 3x =12 | :3 x =4 Unambiguous, one line per step, easy to follow, economic. But only possible where you write a real 1. Sorry d8-)

Re: 70% of US students don't understand the equal sign Message #38 Posted by Martin Pinckney on 18 Aug 2010, 12:49 p.m., in response to message #37 by Walter B I was taught exactly like Don showed, 47 years ago. But I agree, Walter, your system is certainly more efficient.

Re: 70% of US students don't understand the equal sign Message #39 Posted by Don Shepherd on 18 Aug 2010, 1:51 p.m., in response to message #37 by Walter B Walter, I kind of like that style, but what do you mean by "only possible when you write a real 1"? Don

Re: 70% of US students don't understand the equal sign Message #40 Posted by Karl Schneider on 18 Aug 2010, 2:47 p.m., in response to message #39 by Don Shepherd Quote: Walter, I kind of like that style, but what do you mean by "only possible when you write a real 1"? I didn't originally make the effort to 'get the reference' at first, but I see it now. Europeans write the numeral "one" with a 'streamer' connecting to the top. Americans tend to write only a vertical line segment, which could be confused with the vertical bar in Walter's notation. -- KS

Re: 70% of US students don't understand the equal sign Message #41 Posted by Don Shepherd on 18 Aug 2010, 3:10 p.m., in response to message #40 by Karl Schneider So he's really referring to the delimiter between the equation and the steps you take to solve it, here. Like I say, I kind of like this approach but it's not taught this way in the US and, as a teacher, I would be reluctant to teach it to my students when the next math teacher they have may not understand it.

Re: 70% of US students don't understand the equal sign Message #42 Posted by Walter B on 18 Aug 2010, 4:41 p.m., in response to message #41 by Don Shepherd Don, there is a single vertical stroke in each line, no continuous line over the whole calculation. Behind that stroke stands the operation to be applied to both sides of the equation in the respective calculation line, resulting in the next line. The stroke will stay close to the right end of an equation. So particularily in the process of reducing long formulae, the stroke will wander to the left from line to line. Another nice thing is you may use this notation in matrix algebra as well. And Karl is right, at least in good ol' Germany people write 1 like printed here, but without the horizontal bar at the bottom. OTOH, a handwritten 7 often gets an extra horizontal stroke through its middle.

Re: 70% of US students don't understand the equal sign Message #43 Posted by Don Shepherd on 18 Aug 2010, 5:23 p.m., in response to message #42 by Walter B Thanks Walter and Karl. Yeah, my wife puts horizontal bars through her sevens, and she's not even German! Don

Re: 70% of US students don't understand the equal sign Message #44 Posted by Walter B on 19 Aug 2010, 2:51 a.m., in response to message #43 by Don Shepherd The "1" is written like this: It took me some time, however, to find a quotable sample for the "7" in the internet, but here it is: Edited: 19 Aug 2010, 3:22 a.m.

Re: 70% of US students don't understand the equal sign Message #45 Posted by Massimo Gnerucci (Italy) on 18 Aug 2010, 6:07 p.m., in response to message #42 by Walter B Quote: And Karl is right, at least in good ol' Germany people write 1 like printed here, but without the horizontal bar at the bottom. OTOH, a handwritten 7 often gets an extra horizontal stroke through its middle. So I was taught, here in Italy, back in the days... :) Greetings, Massimo

Re: 70% of US students don't understand the equal sign Message #46 Posted by Jeff Kearns on 21 Aug 2010, 7:41 p.m., in response to message #45 by Massimo Gnerucci (Italy) The Continental Europeans and French Canadians among others (I was educated in French in Quebec), and presumably most non-anglophone Americans (North and South), generally do not confuse the 1 and 7 in hand-written form, by adding the horizontal line across the seven. In a high school math book, I recall learning that there is apparently a historical reason why this is so, as well as why the '1' is written with the upward slash almost like an inverted 'v' but without the horizontal line at the base; if you write all numbers from 1-9 'properly', using straight lines, you should have a number of acute + right angles equal to the number itself. A little creativity is required for the 9, as I recall... This explains the horizontal line in the 7. Try it! The zero is no exception, of course; no angles at all in a circle. Jeff Kearns p.s. Not sure about the Brits, Aussies, Kiwis, etc... Dave J.?

Re: 70% of US students don't understand the equal sign Message #47 Posted by Jeff Kearns on 21 Aug 2010, 7:57 p.m., in response to message #46 by Jeff Kearns Why numbers are written that way

Re: 70% of US students don't understand the equal sign Message #48 Posted by Manolo Sobrino on 17 Aug 2010, 8:57 a.m., in response to message #1 by DavidShenk That's because less and less students have heard about Logic these days. I used to explain the = sign with the Leibniz definition: a=b iff every (and all) property that a has, b has it too. Tarski has a nice elementary book called Introdution to Logic (very cheap now), that provides just enough background in Logic for any Maths student. (I'm a physicist and sometimes have a hard time trying to cope with the poor formal reasoning in my mates' "proofs"). Edited: 17 Aug 2010, 8:58 a.m.

Re: 70% of US students don't understand the equal sign Message #49 Posted by John B. Smitherman on 22 Aug 2010, 10:00 p.m., in response to message #48 by Manolo Sobrino Thanks for the tip. I picked up a copy and am making my way through it. Regards, John

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