procore

[kenberg]

Fairly early on a student learns that (1/2)X2+2 means ((1/2)X2)+2 rather than (1/2)X(2+2). But really that is a conventional agreement, meaning that we could just as well have decided otherwise. . i see no reason to assume that the written phrase "half of 2+2" means (half if 2) plus 2. A person who want to avoid ambiguity could say 2plus half of 2 if that is what he intended. The meaning of "half of 2+2" might be clear by intonation if spoken. When written, I would call it ambiguous.

[Zelandakh]

When you say "What is half of <some calculation>" you are implicitly drawing a bracket around the calculation unless you add an obvious pause.

 

As for the cube problem I think it would be interesting to hear how others visualised this. I mentally split the cube into a 5 x 5 x 3 and a 5 x 5 x 2 and then used the volumes to work out the number of 1 x 1 x 1 cubes to add in at the end. An engineer would probably visualise the entire cube as a single entity. A topologist might come up with some useful transformation. An architect would probably just draw it. Or there is the apple approach of removing the accounted-for sections until only the single areas are leftover. What is great is that a bunch of 6 year olds with building blocks will probably come up with the answer just as fast as some mathematicians!

[gwnn]

The written version "half of 2+2" points very much to "0.5*(2+2)" since there is a clear barrier between the part with natural language and the part of math stuff. On the other hand, no one would ever write or say "half of two plus two" in any real-life scenario other than as a "tricky" math trap (the numbers are too low), so there.

[gwnn]

As for the cube problem I think it would be interesting to hear how others visualised this. I mentally split the cube into a 5 x 5 x 3 and a 5 x 5 x 2 and then used the volumes to work out the number of 1 x 1 x 1 cubes to add in at the end. An engineer would probably visualise the entire cube as a single entity. A topologist might come up with some useful transformation. An architect would probably just draw it. Or there is the apple approach of removing the accounted-for sections until only the single areas are leftover. What is great is that a bunch of 6 year olds with building blocks will probably come up with the answer just as fast as some mathematicians!

I have a transparent 5x5x5 cube in my head, which gets filled up by smaller, opaque cubes. 4x4x4 is clearly too big, so you start by putting a 3x3x3 in one of the corners and then putting as many 2x2x2 as possible, then add up the volumes and see how many little guys we need.

[kenberg]

The written version "half of 2+2" points very much to "0.5*(2+2)" since there is a clear barrier between the part with natural language and the part of math stuff. On the other hand, no one would ever write or say "half of two plus two" in any real-life scenario other than as a "tricky" math trap (the numbers are too low), so there.

 

 

I agree with this. Although I do not have an example handy, I have seen advertisements, warranties, etc where the exact meaning simply is not clear. Sometimes I think this is intentional, sometimes probably just bad wording.

[Cyberyeti]

I agree with tis. Although I do not have an example handy, I have seen advertisements, warranties, etc where the exact meaning simply is not clear. Sometimes I think this is intentional, sometimes probably just bad wording.

 

I was just putting it out there to show how easy it was to put up an ambiguous question. Where I came across it the answer wanted was 3 and it was written down (so no intonation). This was in a MMO and there was no particular consequence for getting it wrong.

[kenberg]

In some ways this all still is addressing the original subject. Gotcha questions may have their place, but in a sixth grade assessment they should not happen. They definitely should not happen intentionally, and care should be taken that they do not happen inadvertently. In the real world we do in fact have to do our best when we encounter badly written stuff. But a sixth grade test, one with substantial consequences, should not test whether the kid can make sense out of carelessly worded questions. There will be planty of opportunity for error when everything is played straight up.

[Cyberyeti]

In some ways this all still is addressing the original subject. Gotcha questions may have their place, but in a sixth grade assessment they should not happen. They definitely should not happen intentionally, and care should be taken that they do not happen inadvertently. In the real world we do in fact have to do our best when we encounter badly written stuff. But a sixth grade test, one with substantial consequences, should not test whether the kid can make sense out of carelessly worded questions. There will be planty of opportunity for error when everything is played straight up.

 

True but it happens. In my university final maths exams, one of the questions (one I didn't answer thankfully, it was an "answer 5 of 7" type paper) was either unanswerable, or took 20 minutes instead of 3 (I forget which) because a plus sign was printed as a minus or vice versa. It was the chaos theory paper so I suppose it was somehow appropriate.

[blackshoe]

…nowhere to be found in our galaxy…

Assumption. Where's your evidence?

[blackshoe]

The main thing for young people to learn about credit cards is that they are the invention of the devil.

No. Credit cards are the invention of bankers. Bankers are the invention of the devil - with apologies to my Uncle Rep, who at one time owned two banks.

[blackshoe]

The written version "half of 2+2" points very much to "0.5*(2+2)" since there is a clear barrier between the part with natural language and the part of math stuff. On the other hand, no one would ever write or say "half of two plus two" in any real-life scenario other than as a "tricky" math trap (the numbers are too low), so there.

A comma might be useful here (as opposed to the many times I see them where they don't belong).

[chasetb]

No. Credit cards are the invention of bankers. Bankers are the invention of the devil - with apologies to my Uncle Rep, who at one time owned two banks.

No, you have that wrong. LAWYERS are the invention of the devil, bankers just lie down with them.

[gwnn]

Assumption. Where's your evidence?

Do you refer to extraterrestrial or terrestrial logic? In any case, it was not an assumption, it was an assertion exaggerated on purpose for comedic purposes. And my evidence to back it up is only my own experience and that of others that I have come across and discussed with. You are free to disagree with my opinion, I did not put a QED at the end of it (except the Carlin quote at the bottom, I guess). I do not think pure logic is the overriding concern in the lives of most people. I do use it from time to time but much too often only as a post mortem measuring stick to assess how badly I failed to use it.

[blackshoe]

Do you refer to extraterrestrial or terrestrial logic? In any case, it was not an assumption, it was an assertion exaggerated on purpose for comedic purposes. And my evidence to back it up is only my own experience and that of others that I have come across and discussed with. You are free to disagree with my opinion, I did not put a QED at the end of it (except the Carlin quote at the bottom, I guess). I do not think pure logic is the overriding concern in the lives of most people. I do use it from time to time but much too often only as a post mortem measuring stick to assess how badly I failed to use it.

Neither. I asked for evidence, not logic — although logic in these discussions is always welcome.

 

Hyperbole: exaggerated statements or claims not meant to be taken literally.

I suppose that's fair enough — and I did recognize it as such, but it seemed an unnecessary exaggeration, so I wondered if you were serious.

[gwnn]

Neither. I asked for evidence, not logic — although logic in these discussions is always welcome.

 

I can read. I know that you asked for evidence, I was asking only what you asked the evidence for... OK, for you, here is my question in a correct, logical wording:

 

I find your question slightly unclear. You are asking for evidence to support my assertion, but which assertion exactly?

 

a) People on Earth rarely follow rigid logical rules in their day-to-day communication and thought processes. (this was the point I was trying to make)

b) No form of life in this galaxy could ever conceivably follow rigid logical rules. (this was the over-the-top restatement)

 

Anyway, moving on, I was very much serious in that in my opinion, linguists are better equipped to interpret day-to-day communication than logicians. (I am unable to provide evidence for this and your opinion may well differ. I don't think it's a kind of statement that is easy to (dis)prove with concrete evidence, anyway) For example, there is no logical difference between saying "more than 50 people" and "more than 500 people" if the actual number was 10,000 (say, people at a concert), but a linguist will immediately recognise how both of these statements are misleading in a likely malicious way, if the speaker knows the correct figure. At the same time, I admit that a lot of illogical expressions irk the logician in me. For example, "Do you mind helping me?" "Sure."

[mycroft]

It's terribly easy to get this wrong, what's half of 2 + 2 ?

2 ⫞ ?

 

sorry, did you want the top half?

 

[barmar]

The only test I remember taking in grade school that had a "trick" to it was the one that begain with "Read the entire test before answering any questions." The trick was that at the end of of the test there was an instruction saying to answer only question 1. So if you just started in answering all the questions, it meant you didn't properly follow the first instruction.

 

While the wording of some questions may not be perfect, if you just interpret them according to normal common sense you'll usually come to the right conclusion. Like the question about credit cards. Deciding that b is the correct answer because they say "accrued interest" rather than "accrual of interest" means you're expecting it to be a trick question based on grammar, not a normal question about how credit cards work. Basically, unless the question is something like "Which of the following is has correct/incorrect grammar?", I wouldn't expect a detailed grammatical analysis to be necessary.

Regarding the reindeer question. The right way for reading comprehension tests to be worded is to start with something like "Based on the text you've just read, answer the following questions." It's been a while, but I think that's how the SAT English Comprehension tests were written. So if the text says that reindeer are the largest deer, you can't argue that "moose" is the correct answer -- it may be true, but this isn't a biology or zoology exam, it's a reading exam, and you're supposed to show that you understood what you read.

[blackshoe]

Anyway, moving on, I was very much serious in that in my opinion, linguists are better equipped to interpret day-to-day communication than logicians. (I am unable to provide evidence for this and your opinion may well differ. I don't think it's a kind of statement that is easy to (dis)prove with concrete evidence, anyway)

What you said was "logicians describe how people ought to think and speak in an ideal world that is nowhere to be found in our galaxy". First, I don't know that logicians actually do that - sure they describe what logic is and point out the illogic in some speech or writing, but I don't think I've ever seen one try to define an "ideal world" where people think and speak in the way the logician thinks they should. IAC, the thrust of my objection was that you cannot have evidence that your statement about "nowhere to be found in our galaxy" is true - unless you've been everywhere in our galaxy or know someone trustworthy who has. I will grant that the assumption is probably pretty safe B-) but it's still an assumption.

 

I do agree that linguists are generally better equipped to interpret day-to-day communication.

[PassedOut]

I thought of this thread when I read this article today: How Bill Gates pulled off the swift Common Core revolution

 

Bill Gates was de facto organizer, providing the money and structure for states to work together on common standards in a way that avoided the usual collision between states’ rights and national interests that had undercut every previous effort, dating from the Eisenhower administration.

 

The Gates Foundation spread money across the political spectrum, to entities including the big teachers unions, the American Federation of Teachers and the National Education Association, and business organizations such as the U.S. Chamber of Commerce — groups that have clashed in the past but became vocal backers of the standards.

 

Money flowed to policy groups on the right and left, funding research by scholars of varying political persuasions who promoted the idea of common standards. Liberals at the Center for American Progress and conservatives affiliated with the American Legislative Exchange Council who routinely disagree on nearly every issue accepted Gates money and found common ground on the Common Core.

 

One 2009 study, conducted by the conservative Thomas B. Fordham Institute with a $959,116 Gates grant, described the proposed standards as being “very, very strong” and “clearly superior” to many existing state standards.

 

Gates money went to state and local groups, as well, to help influence policymakers and civic leaders. And the idea found a major booster in President Obama, whose new administration was populated by former Gates Foundation staffers and associates. The administration designed a special contest using economic stimulus funds to reward states that accepted the standards.

 

The result was astounding: Within just two years of the 2008 Seattle meeting, 45 states and the District of Columbia had fully adopted the Common Core State Standards.

 

The math standards require students to learn multiple ways to solve problems and explain how they got their answers, while the English standards emphasize nonfiction and expect students to use evidence to back up oral and written arguments. The standards are not a curriculum but skills that students should acquire at each grade. How they are taught and materials used are decisions left to states and school districts.

With my sons out of school, I hadn't focused on how the Common Core had been developed and implemented, so lots of the information in the article was new to me. Surely something needs to be done to improve US education, particularly in the area of drawing conclusions based on evidence and reason. I hope this approach helps.

[kenberg]

I thought of this thread when I read this article today: How Bill Gates pulled off the swift Common Core revolution

 

 

With my sons out of school, I hadn't focused on how the Common Core had been developed and implemented, so lots of the information in the article was new to me. Surely something needs to be done to improve US education, particularly in the area of drawing conclusions based on evidence and reason. I hope this approach helps.

 

 

I also heve not been following it closely, I also hope for the best. As with so many things, a lot depends on interpretation and implementation. Based on experience, I am concerned about "explain how they got their answers".

 

I have encountered more than few people who have very rigid ideas about what the correct explanation should be. There are a great many students taking the test, and so a large supply of graders is needed. More often than you might think, "correctly" explaining how the answer was obtained means conforming rather rigidly to something that has been put on a list of acceptable explanations. I have not kept close track of the story here in Maryland, but I am pretty sure that there has been a significant reduction in the number of "explain your reasoning" questions. This is not because anyone is opposed to reasoning, rather it comes from the difficulty of fairly and consistently evaluating responses with a large number of test takers and a large number of test graders. Perhaps they can cope with this, but it is tougher than it sounds.

 

Imo, the best way to test and encourage reasoning is to give problems where getting the right answer requires careful reasoning.

[PassedOut]

Imo, the best way to test and encourage reasoning is to give problems where getting the right answer requires careful reasoning.

I certainly agree. My interpretation of:

 

The math standards require students to learn multiple ways to solve problems and explain how they got their answers, while the English standards emphasize nonfiction and expect students to use evidence to back up oral and written arguments.

is "show your work," as my math and geometry teachers required all through school. Harder to grade than multiple-choice questions for sure, but then I think that US teachers should be paid a lot more than they are now...

[barmar]

Harder to grade than multiple-choice questions for sure, but then I think that US teachers should be paid a lot more than they are now...

And we should have more of them so the class sizes will be manageable.

 

Unfortunately, public school teachers are paid out of tax dollars, and no one wants their taxes to go up so we can have more and better paid teachers. Everyone says it's a crime that teachers are paid so little, but few people are willing to put their money where their mouth is.

[PassedOut]

And we should have more of them so the class sizes will be manageable.

 

Unfortunately, public school teachers are paid out of tax dollars, and no one wants their taxes to go up so we can have more and better paid teachers. Everyone says it's a crime that teachers are paid so little, but few people are willing to put their money where their mouth is.

I think that the US anti-tax propaganda is short-sighted to the point of being nuts, and I gather that you agree. Our parents and grandparents paid plenty of taxes to build roads and schools for us to use, and they supported an exciting space program to boot. Today's it's-all-about-me politicians and voters seem happy to let that decay around us instead of improving the country for those who follow.

[kenberg]

We have to take people as they are. I was in high school in the mid 1950s in St. Paul. The schools needed more money and, at the time, this had to be put to the voters. It failed. So some big cheese in the school system announced that there simply was not enough money to do everything, so as of next fall there would be no high school football games. They put the funding up for another vote. It passed. Fast forward fifty years or so. I am at one of these 4th of July lakeside galas where politicians and their flunkies, oops I mean top level assistants, come to press flesh. I am speaking to this guy about funding for the University. He wants to talk about the Terps (Terps=Terrapins, Fear the Turtle is our school slogan). I do not have a solution for this.

 

Some things seem very clear to me, for example it seems clear that education benefits society as well as the individual. For some reason this is not so clear to everyone.

[Elianna]

A bit of background for those who don't know me: I teach HS Math, and have been to many district meetings about common core (at least at the MS/HS level).

 

The "explain your reasoning" types of questions take place in different forms: (I'm using VERY simple Math so everyone could understand)

 

1) x + 3 = 5

 

a) x= 8 because 3 + 5 = 8

b) x = 8 because you have to add 3 to both sides

c) x = 2 because 2 + 3 =5

d) x = 2 because 7-5 =2

 

Basically, students have to not only choose the correct answer, but have to choose the correct reasoning. Note, the choice between c and d could be confusing if d said "because 5 - 3=2", but I haven't seen ones where there is a distractor (incorrect choice) with plausible reasoning.

 

2) Constructed response questions where students have to come up with answer and explain their reasoning. As a teacher, as long as the reasoning is mathematically valid (i.e. I couldn't use the same logic to reach an incorrect answer) I accept their reasoning, but like others have said, I worry about the readers on the actual test.

 

I should note that my students are much more taken aback by the multiple choice questions than by the constructed response. They especially struggle with MC questions that require multiple responses, and are right if they only if they choose EXACTLY the correct responses.