procore

[nigel_k]

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.

I would have said "because 5 - 3 = 2" was the correct choice. The actual choice C is simply stating an answer rather than providing a reason to believe it is correct.

[akwoo]

I am a math professor, and I can say that some of the secondary math education majors who graduate from my department cannot follow (much less come up with) any but the simplest mathematical arguments. (They would have to spend a couple minutes to parse that previous sentence.)

 

These graduates get jobs teaching high school. Their deficiencies are obvious from their transcripts and not a secret.

 

All I can say in our defense is that many universities have even lower standards.

 

The Common Core Standards are going to be a failure, because we do not have enough teachers who can teach it. The schools and the politicians won't say so; they'll just gradually dumb down the tests so that they can be taught to by rote and declare success.

 

Until we make teaching a respected career that many of the smartest people in the country want to pursue, we won't be able to teach reasoning to all of our kids.

[PassedOut]

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.

Thanks much for taking the time to explain this! I see now how the multiple choice questions can be implemented in the way you've shown.

 

My favorite HS class (by far) was geometry, which I took as a sophomore. It was my introduction to formal proofs, and I remember thinking, "Now we are getting somewhere!" I don't see how constructing proofs would be compatible with multiple choice questions, nor how one could actually learn math without working through lots of problems and proofs. But perhaps I am not distinguishing carefully enough between the learning process and the testing process.

[barmar]

I would have said "because 5 - 3 = 2" was the correct choice. The actual choice C is simply stating an answer rather than providing a reason to believe it is correct.

But that wasn't one of the choices; the choice with subtraction said "because 7 - 5 = 2" -- it has the right value of x, but the wrong reason, and the test-taker has to recognize this.

 

 

If they gave both this answer and the actual C, I think it would have to be in a question that allowed you to select multiple answers, since both are correct. One just requires you to understand how variables are used in equations, the other also requires you to know the process used to solve algebraic equations (e.g. changing an addition on one side of the equation to a subtraction on the other side).

[Elianna]

Thanks much for taking the time to explain this! I see now how the multiple choice questions can be implemented in the way you've shown.

 

My favorite HS class (by far) was geometry, which I took as a sophomore. It was my introduction to formal proofs, and I remember thinking, "Now we are getting somewhere!" I don't see how constructing proofs would be compatible with multiple choice questions, nor how one could actually learn math without working through lots of problems and proofs. But perhaps I am not distinguishing carefully enough between the learning process and the testing process.

 

One would not be able to have students construct a whole proof in multiple choice questions, but what I see on the state tests goes something like this:

 

Alex is trying to prove that vertical angles 1 and 3 are congruent. (I'm using <1 as meaning "angle 1")

 

Statements Reasons

1. m<1 + m<2= 180 1. <1 and <2 are a linear pair

2. m<3 + m<2= 180 2. <3 and <2 are a linear pair

3. m<1 = 180 - m<2 3.____

m<3 = 180 - m<2

4. ____________ 4. Transitive Rule of equality

 

And students would be given a multiple choice question to fill in the blanks.

[Zelandakh]

1) x + 3 = 5

c) x = 2 because 2 + 3 =5

d) x = 2 because 7-5 =2

It strikes me that d is just as correct as c since x = 5 - 3 = 7 - 5 = 2. In the old days we also had to show our working and sometimes 60% of the marks were on that so it was important. How hard is it for the students to writesomething of the form in the first sentence rather than resorting to multiple choice? Incidentally, the multiple choice maths tests I did as a child all had negative marking for incorrect answers to prevent guessing and the resulting statistical anomalies this creates. Do the American tests also include this?

[PassedOut]

One would not be able to have students construct a whole proof in multiple choice questions, but what I see on the state tests goes something like this:

 

Alex is trying to prove that vertical angles 1 and 3 are congruent. (I'm using <1 as meaning "angle 1")

 

Statements                            Reasons
1. m<1 + m<2= 180             1. <1 and <2 are a linear pair
2. m<3 + m<2= 180             2. <3 and <2 are a linear pair
3. m<1 = 180 - m<2            3.____
m<3 = 180 - m<2
4. ____________               4. Transitive Rule of equality

 

And students would be given a multiple choice question to fill in the blanks.

Thanks again! I do see that this testing approach allows for useful and measurable results. With uniform measurements over large populations, the effectiveness of different teaching approaches can be compared and tweaked. Looks reasonable and promising to me.

[PassedOut]

It strikes me that d is just as correct as c since x = 5 - 3 = 7 - 5 = 2. In the old days we also had to show our working and sometimes 60% of the marks were on that so it was important. How hard is it for the students to writesomething of the form in the first sentence rather than resorting to multiple choice? Incidentally, the multiple choice maths tests I did as a child all had negative marking for incorrect answers to prevent guessing and the resulting statistical anomalies this creates. Do the American tests also include this?

To me, the word "because" rules out d.

 

In my (long ago) school days, professionally developed multiple choice tests did penalize incorrect answers. Perhaps Elianna will update us on this too.

[kenberg]

Actually, the word because is at least a little tricky. If you believe that the equation x+3=5 leads to x=2 because 2+3=5 then you might also reason that x^2+3=7 leads to x=2 because 2^2+3=7. This would be wrong, all we can conclude from x^2+3=7 is that either x is 2 or x is negative 2. Really, in the problem as stated, we need to say that x+3=5 leads to x=2 because 2+3=5 and, if x is taken to be any number other than 2, the sum of x and 3 is not 5.

 

I was reluctant to note this (yes of course it is picky) until your post because I absolutely do not think this issue would cause trouble for any student, and anyway Elianna was just illustraing an idea. But if we get down to saying that we really are going to teach reasoning, then the fact is that 2+3=5 is only part of the reason that we can conclude from x+3=5 that x=2.

 

I find most "because" problems to be iffy when push comes to shove. I very much favor teaching reasoning, and although testing on reasoning is, I think, very difficult I favor that also. I also favor listening to teachers such as Elianna. But I think that it is tough.

 

I know I have told this before, but it bears on the issue of reasoning and shows just how widespread the problem is. I was in a Burger King or its equivalent when, next line over, a woman paid for her order and got the change back for a 20. But she had given the cashier a 100, not a 20. The cashier looked in the drawer and agreed. The cash register announce the change for a 20, no one in the BK had any idea how to cope with this. No one knew how to reset the cash register. I barged in. Thankfully everyone agreed that 100 was 80 more than 20, and they then accepted my solution that the woman should receive an extra 80 in change. Whew!

 

This very basic reasoning is useful, and you would think anyone with an 8th grade education could do it. My father in fact had an 8th grade education and I am confident that he could have done it. But many can't, so I heartily endorse teaching this basic reasoning., My preference would be to place such reasoning, most of the time at least, in a real world setting. Part of the problem with the x+3=5 setting is that the student has learned that you subtract 3 from both sides of the equation and so, from his view, the answer "because 5-3=2" would make sense to him, while the answer "because 2+3=5" may not seem right. True, he is possibly confusing technique with logic, but if we look at a real world situation often we get more clarity.

 

I want to repeat that I do not mean to dis Elianna's example. Rather I think that as soon as words such as "because" get into the act, there almost always can be issues. Often, as with this example, the "problem" is really a non-problem But things can, and sometimes do, go wrong.

 

 

Final comment: Three cheers for Bill Gates fro taking this on. I hope he and everyone involved can make it work.

[Elianna]

In my (long ago) school days, professionally developed multiple choice tests did penalize incorrect answers. Perhaps Elianna will update us on this too.

 

Historically, the ACT did not penalize incorrect answers, and the SAT did on the multiple choice sections. (Both of these are used as college entrance exams.)

 

I know that the old California Standardized Tests did not penalize, but I am not sure about the new tests. What I do know is that as they are adaptive, my guess is that students cannot skip questions, and so they do not penalize per se for missed questions, but they adjust future questions based on results of the previous questions, so if you miss the middle-of-the road questions you will start getting easier questions, but have less of an opportunity of getting a higher score.

 

There are two companies that are making tests for common core, and states have signed up for them in roughly equal proportion. California signed up for Smarter Balanced (SBAC), and I forget the other one.

 

I know that the example I came up with was not great, but on my behalf, I thought of it in 5 minutes. If someone wants to see actual sample questions, they can go to the smarter balanced webpage, and try out the sample test: http://sbac.portal.airast.org/practice-test/

[kenberg]

Your example was fine, I didn't think it was intended as the absolute last word on anything. My point was solely that it actually is tougher than it looks to get these exactly right.

[gwnn]

How about "x=2 because 3+2=5 and because any first order polynomial can have at most one root, as per the fundamental theorem of algebra"?

[PassedOut]

Your example was fine, I didn't think it was intended as the absolute last word on anything. My point was solely that it actually is tougher than it looks to get these exactly right.

No doubt it is difficult to get the questions exactly right. But now I have an idea of how the Common Core works, whereas I had none before. It seems to me that this, or something like it, is the only way to get a handle on how to improve educational performance across the US. I have a granddaughter now :) so I might have an opportunity to see some of the results...

[kenberg]

No doubt it is difficult to get the questions exactly right. But now I have an idea of how the Common Core works, whereas I had none before. It seems to me that this, or something like it, is the only way to get a handle on how to improve educational performance across the US. I have a granddaughter now :) so I might have an opportunity to see some of the results...

 

Congratulations!

[blackshoe]

I know I have told this before, but it bears on the issue of reasoning and shows just how widespread the problem is. I was in a Burger King or its equivalent when, next line over, a woman paid for her order and got the change back for a 20. But she had given the cashier a 100, not a 20. The cashier looked in the drawer and agreed. The cash register announce the change for a 20, no one in the BK had any idea how to cope with this. No one knew how to reset the cash register. I barged in. Thankfully everyone agreed that 100 was 80 more than 20, and they then accepted my solution that the woman should receive an extra 80 in change. Whew!

I wonder what happened when they closed out the cash register at the end of the shift or end of the day or whenever they do that. I can imagine it reporting that the till was 80 bucks light. Well, perhaps somebody knew how to deal with that problem. :blink:

[jeffford76]

I wonder what happened when they closed out the cash register at the end of the shift or end of the day or whenever they do that. I can imagine it reporting that the till was 80 bucks light. Well, perhaps somebody knew how to deal with that problem. :blink:

 

Why would the till be light? $80 it didn't know about got put in (the $100 instead of the $20), and $80 it didn't know about got taken out (the extra change).

[kenberg]

I wonder what happened when they closed out the cash register at the end of the shift or end of the day or whenever they do that. I can imagine it reporting that the till was 80 bucks light. Well, perhaps somebody knew how to deal with that problem. :blink:

 

Light why? I'm not seeing the problem here.

 

Since I am typing, let me bounce back to the problem about credit cards:

 

Which of the following does NOT occur when using a credit card instead of currency?

 

a. possible service fees

b. accrued interest on unpaid baalnce

c. increased spending limit if overspending occurs

d. less cash carried and an expedited checout process

 

 

Zel observed that the point was "The children are taught that overspending with a credit card has consequences and they cannot rely on the card company raising their credit limit if they get into debt. You can therefore tell this is the right answer even if you do not agree with the others."

 

Yes, probably he is right about this. Basically it is a propaganda question. I believe we had many such questions in high school. I remember learning that we always were to wear a tie and coat when applying for a job. The book said so, and this was therefore the correct answer. Mostly these questions taught me (or confirmed what I had already concluded) that you should ignore a lot of what you are told by adults. However, if we really wanted to propagandize, I would suggest that students learn that if they ever have spent so much on their credit card that the spending limit is an issue then they need to take a very serious look at their spending habits. Financial emergencies happen, but usually not to fifteen year olds and even with adults the emergency is far more often bad planning.

 

Really I have no idea from experience what the correct answer is, and I would be skeptical of anything someone told me, especially if they had a view that they were pushing. I would be seriously worried about any child who did, from experience, know what the correct answer is. .

 

One of the many stupid things that I have done in life: When I moved from Minnesota to Maryland I had to change my driver's license. No problem right? So I turned in my Minnesota license and asked for a Maryland one. Oops, there is a written test, multiple choice. Example: You are driving 50 mph and put on the brakes. How many feet does it take to stop? When I took my Minnesota exam some twelve years earlier, I had memorized all of this stuff. But now? Of course it is simple. The answer is the largest number that appears. The whole thing is a joke. You do not actually have to know anything except how to take a multiple choice test. Many many tests go that way. I got an A in my very boring undergraduate psychology course. I never came to class except for the multiple choice exams. No doubt psychology is an interesting subject. I accept this on faith, the course would never lead one to that view. One of the questions (really): You feel a tight band around your head and believe people are following you. This is an example of a. Normal behavior ... I forget b c and d, but I managed.

[Fluffy]

How about "x=2 because 3+2=5 and because any first order polynomial can have at most one root, as per the fundamental theorem of algebra"?

 

This is similar to how I'd solve it, this might sound weird but for some reason when I saw the equation I tested to substract numbers from 5 untill they gave me 3 instead of substracting 2 from 5, obviously took me a small fraction of a second anyway, but still a bit weird.. Kenberg explained why this is not right in theory, but I also know about first order polynomials.

 

Linguists* describe how people use language in their day-to-day lives, logicians describe how people ought to think and speak in an ideal world that is nowhere to be found in our galaxy. Why is it unfortunate that lawyers (who work with real people in real life, not robots from outer space) use language as other people do? I agree, though, that sloppy wording is in no one's best interest.

Lawyers, at least on my country, do not use the same language as I do. They might share grammar and some words, but it is far from the same language.

[blackshoe]

Why would the till be light? $80 it didn't know about got put in (the $100 instead of the $20), and $80 it didn't know about got taken out (the extra change).

You're right. Disregard my last. :o :blink:

[kenberg]

If you are up for it, there is a long pdf on the math standards at

 

http://www.corestand...h_Standards.pdf

 

 

Browsing, I saw something that triggered a memory:

 

Use the method of completing the square to transform any

quadratic equation in x into an equation of the form (x – p)2 = q

that has the same solutions. Derive the quadratic formula from

this form.

 

In fact my high school teacher, without previous warning, called on me to come to the board and derive the quadratic formula referred to above. [For the many who don't remember and or don't care, this gives a formula for solving the equation a x^2 + b x +c=0]. I did it. I believe there were, perhaps, two other kids in class that could have done it. My 1956 high school graduating class was 208, give or take a couple. So I ask: Will the typical high school graduate in, say, 2020 be able to derive the quadratic formula? Is this what we are thinking? We might also ask if he should. I walked to school with Fred. Fred took "shop math" as it was then called, and planned to become a plumber. I am not saying that Fred could not have learned the derivation (if you put a gun to his head), although I doubt it would have made much sense to him and I am positive he would never have any use for it.

 

Readers of this thread, those not engaged in mathematics, might ask themselves if they can derive the quadratic formula and, if the answer is no, ask if it has held them back in their career.

[nige1]

Historically, the ACT did not penalize incorrect answers, and the SAT did on the multiple choice sections. (Both of these are used as college entrance exams.)

I know that the old California Standardized Tests did not penalize, but I am not sure about the new tests. What I do know is that as they are adaptive, my guess is that students cannot skip questions, and so they do not penalize per se for missed questions, but they adjust future questions based on results of the previous questions, so if you miss the middle-of-the road questions you will start getting easier questions, but have less of an opportunity of getting a higher score.

There are two companies that are making tests for common core, and states have signed up for them in roughly equal proportion. California signed up for Smarter Balanced (SBAC), and I forget the other one.

I know that the example I came up with was not great, but on my behalf, I thought of it in 5 minutes. If someone wants to see actual sample questions, they can go to the smarter balanced webpage, and try out the sample test: http://sbac.portal.a.../practice-test/

Thanks Elianna. The tests seem like a good idea to provide feedback on student attainment and teacher proficiency.

 

 

 

 

 

 

[mike777]

1) Note that teacher tenure in public schools ruled to violate the law in Calif of all places.

2) Many of us confuse education and scholarship, they are not the same.

3) the arrow of discovery

4) the belief that university knowledge generates economic wealth, consider changing the direction based on empiricism.

5) In education consider nonlinearity (convexity) The gains are large in relation to potential side effects.

6 Charlatan and the academic..to use education in medicine as just one example. Medicine misleads people because for so long its successes were prominently displayed, and its mistakes are literally buried.

7) encourage autodidact inclined students

8) Many of us often very often accept what experts say as true because they are experts, not based on testable evidence or theory. This is often true in my favorite sport, baseball.

[mike777]

If you are up for it, there is a long pdf on the math standards at

 

http://www.corestand...h_Standards.pdf

 

 

Browsing, I saw something that triggered a memory:

 

 

 

In fact my high school teacher, without previous warning, called on me to come to the board and derive the quadratic formula referred to above. [For the many who don't remember and or don't care, this gives a formula for solving the equation a x^2 + b x +c=0]. I did it. I believe there were, perhaps, two other kids in class that could have done it. My 1956 high school graduating class was 208, give or take a couple. So I ask: Will the typical high school graduate in, say, 2020 be able to derive the quadratic formula? Is this what we are thinking? We might also ask if he should. I walked to school with Fred. Fred took "shop math" as it was then called, and planned to become a plumber. I am not saying that Fred could not have learned the derivation (if you put a gun to his head), although I doubt it would have made much sense to him and I am positive he would never have any use for it.

 

Readers of this thread, those not engaged in mathematics, might ask themselves if they can derive the quadratic formula and, if the answer is no, ask if it has held them back in their career.

 

Ken I remember something, a conversation when I went back to MBA school in the middle of my life.

 

In having a chat with the head of the marketing dept she said that knowing the words, words we never use in normal life and their meaning can help you master a subject.

 

 

For example in your post:

"derive the quadratic formula"

 

If in math class we are taught the words, "derive" and "quadratic formula", I bet many more of us could do it.

 

Funny enough just using the words in a lecture in not teaching us but I bet you of all people knew that which makes you a wonderful math teacher.

 

IN my algebra I class I would often very often get really pissed off because I could see the teacher was moving far to fast over basic beginner concepts and losing most of the class the rest of the week. The story of many of my classes in education

 

As an older student in my MBA classes I knew most of the words, I had heard of and used many of the words taught in my classes. Those who had not were left behind.

 

I remember one girl sitting next to me on the first day in my first class with ten color coded pens to write everything down.

 

I wrote less than nothing which was not best.

 

 

what I learned most important that I needed a business writing class which I took on the side on my own. I strongly recommend this. If anything on these forums I love it is the writing of posters. Second is how posters explain math in clear terms, such as Ken, Helene, Richard and others.

 

the rest was flash,,,lots of flash and nonsense.

[WellSpyder]

Readers of this thread, those not engaged in mathematics, might ask themselves if they can derive the quadratic formula and, if the answer is no, ask if it has held them back in their career.

I use maths to some extent in my career - I'm an economist - and I was one of those to whom maths at school always came fairly easily. But I don't think I could derive the formula - and indeed I don't actually remember its derivation being discussed at school, although I suppose it must have been. In fact, I happened to read a derivation only a few months ago and thought it was pretty neat, but it hasn't stuck.

 

As an aside, I could certainly still recite the formula, and indeed understand and apply it, but I'm not convinced that if I couldn't do this it would have held me back in my career either!

[helene_t]

You mean the roots in a second degree polynomial? It was somehow derived in the 1G class, that is children aged 16. All I remember is that it involved writing the polynomial as A(x-alpha)(x-beta), which by the way was generalized to n'th degree polynomials although we obviously didn't get the root formulas for higher than 3rd degree. I am not sure how stringent it was proved that alpha and beta are the only roots.

 

I have used the formula on a few occasions in my work. The derivation of that particular formula I haven't used but a general ability to derive formulas is essential in my job.