Bill and other math geniuses-what do you think about this?

[smilesonly]

http://systemath.com/2008010521/How-It-Works/The-Decline-of-Math.html

 

i do mean genius in a nice way, btw. :D

Edited November 8, 2011 by smilesonly

[zenjenn]

Is there really a decline? I have educated parents and grandparents and my mathematical knowledge is superior to both generations, and am someone who entered an art & design field.

 

Maybe that's just anecdotal, but I get the feeling we perhaps romanticize the past a bit in this regard. When I encounter someone - even an educated someone - in their 60s, 70s, or 80s, I admit I often anticipate greater wisdom, patience, clarity, and greater literary and historical knowledge - but I rarely anticipate greater mathematical fluency unless the person had a career in science or technology.

[go_go_gadget]

I wonder about this as well. Was the average person in the 19th century really better educated than the average person now, or is it just that more people go to school now and are bringing the average knowledge of school-goers down? Is the 8th grade final exam they cite representative of schools at the time, or is more comparable to an exam from an elite private school today? I know, for instance, that no one in my mother's family got past the modern equivalent of middle school until my parents' generation, because they had to drop out to work the family farm or the like. I think that was fairly typical for a large portion of the population until the second third or so of the last century. I don't think it's necessarily surprising that the children of people who could afford to send their kids to school/not make use of them during the day would be better educated than the average today, when nearly everyone gets through at least the first two years of high school, and most actually finish.

[Crimson Wife]

I wonder about this as well. Was the average person in the 19th century really better educated than the average person now, or is it just that more people go to school now and are bringing the average knowledge of school-goers down? Is the 8th grade final exam they cite representative of schools at the time, or is more comparable to an exam from an elite private school today?

 

:iagree::iagree::iagree:

My grandfather attended a one-room rural schoolhouse growing up and it only ran through 8th grade. The very brightest boys would get scholarships to attend private high school and the rest would either go work on their family's farm, get an apprenticeship in one of the trades, or join the military. The girls tended to either marry straight away or work as a domestic for a few years before marrying. Fortunately, my grandfather won a scholarship to prep school and went on to get his bachelor's and eventually a PhD. One of his brothers got his GED during WWII and then used the GI Bill after the war to get his teacher's credential. His other brother never did get a high school diploma and stayed on the family farm his whole life (his widow last I heard still lives on the few remaining acres at 102).

[Guest]

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[Spy Car]

http://systemath.com/2008010521/How-It-Works/The-Decline-of-Math.html

 

i do mean genius in a nice way, btw. :D

 

Brushing aside the "math genius" appelation, I think this web article is all wet.

 

The School Mathematics Study Group (SMSG) materials were very interesting. I am pleased (thanks to board member Wapiti) to have downloaded many of thir original textbooks in electronic form. Very good stuff! As are the materials from the like-minded Comprehensive School Mathematics Program.

 

When this website says:

 

"This approach stresses the continual introduction of various concepts in mathematics without developing an understanding of the laws behind them. It is also based on memorization and guessing. In math, guessing is called estimating. Like reading, math is systematic and is best taught in a systematic manner"

 

They reveal a total ignorance of the SMSG. These materials, which were developed by top-flight mathematicians, are all about developing an understanding mathematical laws in operation. I'm incredulous that they could make such a profoundly wrong statement.

 

Just as wrong is the assertion that the SMSG favored "memorizing and guessing."

 

I'm inclined to think the Systematic Math folks must be nit-wits :D

 

Bill

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[Spy Car]

Not a math genius but I was always strong in math and science. I agree, but then again I am not conventional in my way of thinking :tongue_smilie:. I have seen what the "spiral" approach to math did to my "mastery" oriented kid by using Horizons math and so I am sticking with math programs that focus on a problem solving way of thinking, rather than memory oriented approach. Memorization may get you there faster, but does not benefit the mathematical way of thinking in the long run. I am replying based on my personal and limited experience ;)! Not as an expert :D of any kind!

 

I agree with you Marie. However, the SMSG has been completely misrepresented by this Systematic Math description. The characterization is 180 degrees opposite of the truth. The SMSG was aiming at teaching very high levels of of mathematical understanding.

 

There were issues of "practicality" when you put materials developed by top-flight mathematicians into the hands of teachers with minimal backgrounds in real mathematics, and some elements (like teaching in alternative bases to "base-10") didn't always work out in the classroom. But the SMSG was developing very deep mathematics materials, ones which emphasized mathematical reasoning (including a deep understanding of the mathematical laws at play) and creative problem solving.

 

You (I'm pretty certain) would like these.

 

Bill

[Spy Car]

Looks like I am missing a lot lately by staying off the forums! Would you mind giving me a link, at least to the thread Bill? I would love to check this out and get the downloads :). This has piqued my curiosity now!

 

Try this thread Marie. As I say, it think you will like these materials. And will find they are nothing at all like the false-characterization made about them on the website linked in the OP.

 

http://www.welltrainedmind.com/forums/showthread.php?t=288749&highlight=SMSG

 

Bill

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[Spy Car]

My mistake was in my approach while reading this I think Bill. I made the mistake in taking that part about the SMSG as a given and focusing on the overall message coming from the article, instead of also checking their statements based on the sources they mentioned :tongue_smilie:. I agreed with their overall message but should have spent more time checking their statements/ claims in relation to the SMSG, instead of taking those at face value. Thank you for always being on the ball :) and thank you for the link! I will definitely be taking a closer look!

 

Not your fault. I might have reacted exactly the same way, I just happen to be familiar with the SMSG (at least to some degree) and saw the characterization was false. No worries :001_smile:

 

I also see your point about putting top quality materials in the hands of teachers that do not have the backing of a real mathematical background. I have seen it in my own personal and limited experience while growing up in Greece. The other concern I find also is that sometimes, while a teacher may have a strong mathematical background, they may not have the gift of transferring this information to someone else. This was one thing that fascinated me about Gattegno. He seemed to have the gift of directing the kids to a mathematical way of thinking, without pushing a mechanical way of thinking that does not promote understanding.

 

I'm still working my way through Gattegno, but yes it is great when pure math can be made practical.

 

Part of the problem with "New Math" wasn't just the SMSG materials themselves, but that textbook publishers often jumped on the bandwagon with half-baked approximations of the real thing. The same thing happens now. The math program my child uses in school (while not horrible) is a half-way measure towards Singapore math, but one that comes up short. The same happened with New Math.

 

Bill

[wapiti]

I agree with you Marie. However, the SMSG has been completely misrepresented by this Systematic Math description. The characterization is 180 degrees opposite of the truth. The SMSG was aiming at teaching very high levels of of mathematical understanding.

 

There were issues of "practicality" when you put materials developed by top-flight mathematicians into the hands of teachers with minimal backgrounds in real mathematics, and some elements (like teaching in alternative bases to "base-10") didn't always work out in the classroom. But the SMSG was developing very deep mathematics materials, ones which emphasized mathematical reasoning (including a deep understanding of the mathematical laws at play) and creative problem solving.

 

You (I'm pretty certain) would like these.

 

Bill

 

:iagree:

 

Taking a quick look at the link in the OP, it states:

There was a big change in the approach to the teaching of mathematics at that time. There was a group called the "School Mathematics Study Group" or SMSG, which was formed to counter-act the lead that the Soviet Union had taken in the space race. This group came up with something called "modern math", which was supposed to solve our problems. In fact, it produced a generation of mathematical illiterates. The notion of a "spiral curriculum" was also part of the changes in the approach to the teaching of mathematics.

 

This approach stresses the continual introduction of various concepts in mathematics without developing an understanding of the laws behind them. It is also based on memorization and guessing. In math, guessing is called estimating. Like reading, math is systematic and is best taught in a systematic manner.

 

I think it's odd that he blames the generation of mathematical illiterates on the SMSG of fifty years ago, when IMO, the generation of mathematical illiterates is the result of the fuzzy math of the past twenty years that was an over-reaction to a traditional, memorization-approach, which, in my understanding, did not include the SMSG. Way to gloss over the Math Wars! Google the Math Wars to get a bigger picture, though be aware that many sites are on one side or the other, with some mischaracterization going on (e.g., I don't think anyone seriously advocates teaching memorization without understanding, though obviously the degree to which curricula do this varies quite a bit, and my sense is that there may be not enough choices in the middle, that both seek to teach a solid, conceptual understanding as well as mastery of the algorithms). Dolciani, beloved by many on these boards for conceptual instruction, was part of the SMSG. Moreover, I don't really know what "spiral" has to do with the discussion - curriculum organization is another issue altogether. I don't disagree that instruction should be systematic, and I happen to prefer a conceptual, mastery approach. My sense is that, while fuzzy math had good intentions, to bring conceptual understanding into instruction but perhaps going too far with trying to bring in "real world" interest, a lot of those curricula are reputed to not do enough teaching of the traditional algorithm once they think they've laid the foundation (e.g., Everyday Math).

 

What I like about Singapore, MM and AoPS is that they attempt to teach the concepts behind the traditional algorithm, in a very clear, non-fuzzy way. As for the website in the OP, I took a quick look and didn't see any samples, so I can't offer any personal (non-expert) opinion.

 

Part of the problem with "New Math" wasn't just the SMSG materials themselves, but that textbook publishers often jumped on the bandwagon with half-baked approximations of the real thing. The same thing happens now.

 

This is my understanding as well.

 

(ETA, as an aside on the math wars, lately I've been wondering whether fuzzy math took a sharp wrong turn where it could have become something like AoPS)

Edited November 5, 2011 by wapiti

[smilesonly]

Brushing aside the "math genius" appelation, I think this web article is all wet.

 

The School Mathematics Study Group (SMSG) materials were very interesting. I am pleased (thanks to board member Wapiti) to have downloaded many of thir original textbooks in electronic form. Very good stuff! As are the materials from the like-minded Comprehensive School Mathematics Program.

 

When this website says:

 

"This approach stresses the continual introduction of various concepts in mathematics without developing an understanding of the laws behind them. It is also based on memorization and guessing. In math, guessing is called estimating. Like reading, math is systematic and is best taught in a systematic manner"

 

They reveal a total ignorance of the SMSG. These materials, which were developed by top-flight mathematicians, are all about developing an understanding mathematical laws in operation. I'm incredulous that they could make such a profoundly wrong statement.

 

Just as wrong is the assertion that the SMSG favored "memorizing and guessing."

 

I'm inclined to think the Systematic Math folks must be nit-wits :D

 

Bill

 

Bill, you never let me down!;) your answer is exactly what i was hoping for.

 

another question for you. when mr. systematics puts a finger(his;)) on why the US is failing in the area of math- aside from his pointing out the smsg-do you agree with his theory?

 

no more name calling- i promise.:D

 

thx

[Spy Car]

Bill, you never let me down!;) your answer is exactly what i was hoping for.

 

another question for you. when mr. systematics puts a finger(his;)) on why the US is failing in the area of math- aside from his pointing out the smsg-do you agree with his theory?

 

no more name calling- i promise.:D

 

thx

 

Yes and no. I think we have fallen behind because we don't teach for understanding while also developing strong procedural competence. And we end up with pendulum swings that result in programs being chosen that don't do both.

 

The so-called "fuzzy math" programs are not the only ones that are inadequate. "Plug and chug" is not an attractive alternative either. Way too shallow.

 

We need more "Third Way" math instruction that develops strong procedural competence and also teaches the underlying mathematical reasoning, develops logical thinking and creative problem skills. We have a long way to go as a nation...and we better get to it, because the rest of the world is eating our lunch.

 

Bill

[morosophe]

I'll out myself as a complete math ignoramus by saying that the only thing that registered in this post at all is the phrase "New Math," as a result of which I now have Tom Lehrer in my head. Thanks a bunch.

 

Not your fault. I might have reacted exactly the same way, I just happen to be familiar with the SMSG (at least to some degree) and saw the characterization was false. No worries :001_smile:

 

 

 

I'm still working my way through Gattegno, but yes it is great when pure math can be made practical.

 

Part of the problem with "New Math" wasn't just the SMSG materials themselves, but that textbook publishers often jumped on the bandwagon with half-baked approximations of the real thing. The same thing happens now. The math program my child uses in school (while not horrible) is a half-way measure towards Singapore math, but one that comes up short. The same happened with New Math.

 

Bill

 

But, to chime in as a complete math ignoramus: I think at some point you just have to do the boring old fact memorization. It's not fun, and it's not everything, but it is essential.

[Spy Car]

I'll out myself as a complete math ignoramus by saying that the only thing that registered in this post at all is the phrase "New Math," as a result of which I now have Tom Lehrer in my head. Thanks a bunch.

 

But, to chime in as a complete math ignoramus: I think at some point you just have to do the boring old fact memorization. It's not fun, and it's not everything, but it is essential.

 

There are multiple ways to learn "math facts," rote-memorization is not necessarily the best way to get to the desired end as it can hide lack of basic conceptual understanding. You want to build on a solid foundation from the beginning, and not be fooled by a false illusion of competence.

 

Bill

 

Bill

[Guest]

There are multiple ways to learn "math facts," rote-memorization is not necessarily the best way to get to the desired end as it can hide lack of basic conceptual understanding. You want to build on a solid foundation from the beginning, and not be fooled by a false illusion of competence.

 

Bill

 

Bill

 

:iagree:

[smilesonly]

Yes and no. I think we have fallen behind because we don't teach for understanding while also developing strong procedural competence. And we end up with pendulum swings that result in programs being chosen that don't do both.

 

The so-called "fuzzy math" programs are not the only ones that are inadequate. "Plug and chug" is not an attractive alternative either. Way too shallow.

 

We need more "Third Way" math instruction that develops strong procedural competence and also teaches the underlying mathematical reasoning, develops logical thinking and creative problem skills. We have a long way to go as a nation...and we better get to it, because the rest of the world is eating our lunch.

 

Bill

 

ah, yes. well i was afraid you would respond with such reason and insight. me thinks me must not ignore my gut, hold my little guy's hand and explore areas he rejected in the past.

 

.

 

as a very social person, i welcome the company.:D

[Laura in STL]

Yes and no. I think we have fallen behind because we don't teach for understanding while also developing strong procedural competence. And we end up with pendulum swings that result in programs being chosen that don't do both.

 

The so-called "fuzzy math" programs are not the only ones that are inadequate. "Plug and chug" is not an attractive alternative either. Way too shallow.

 

We need more "Third Way" math instruction that develops strong procedural competence and also teaches the underlying mathematical reasoning, develops logical thinking and creative problem skills. We have a long way to go as a nation...and we better get to it, because the rest of the world is eating our lunch.

 

Bill

:iagree: I have never understood why the argument always seems to be for a focus on one or the other. It seems so obvious that both procedures and concepts as well as problem solving strategies are necessary in mathematics instruction.

 

As a graduate of an elementary education program, I can certainly see why "New Math" programs could fail in the classroom. I received more mathematics education in high school than I did in my "teacher's math" classes in college. I was lucky to have gotten all the way through calculus before college, but I can't imagine some of those students teaching kids math past 3rd grade.

[wapiti]

:iagree: I have never understood why the argument always seems to be for a focus on one or the other. It seems so obvious that both procedures and concepts as well as problem solving strategies are necessary in mathematics instruction.

:iagree: Yes. It makes me want to say, "duh"

[Beth in SW WA]

There are multiple ways to learn "math facts," rote-memorization is not necessarily the best way to get to the desired end as it can hide lack of basic conceptual understanding. You want to build on a solid foundation from the beginning, and not be fooled by a false illusion of competence.

 

Bill

 

Bill

 

:iagree:

After experimenting with my oblivious guinea pigs I have discovered that simply 'doing' math daily with constant application is much more effective than daily timed practice and review. My dds have mastered their times tables by using them daily.

 

I was told on this board a year ago that dd should not begin fractions/long div/etc until she mastered her facts. My gut told me otherwise. She is now 'undoing equations with negatives' today in her TT math lesson (which I model with our HoE cubes/pawns). I never slowed her math progression due to lack of mult fact mastery.

 

Over the summer we drilled the fact cards for a few days and they were thrilled to buzz through the stack in record time. I can't imagine doing that exercise daily.

 

ETA: To the OP, not a math genius here. Just a mom. :)

Edited November 8, 2011 by Beth in SW WA

[FairProspects]

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[smilesonly]

:iagree:

ETA: To the OP, not a math genius here. Just a mom. :)

 

i edited my thread title.;) hey-we're all in this together.:grouphug:

[morosophe]

There are multiple ways to learn "math facts," rote-memorization is not necessarily the best way to get to the desired end as it can hide lack of basic conceptual understanding. You want to build on a solid foundation from the beginning, and not be fooled by a false illusion of competence.

 

Bill

 

I think this post, which was in response to a post of mine, kind of missed my point...

 

I didn't say rote memorization, although I think that can be a valuable tool. In fact, in an attempt to make everything less painful, I got my son the Flashmaster, which allows him to review his math facts without the endless chanting I remember from my school days. (And it seemed to work--he enjoys "playing" with it occasionally, which is all I ask for.) But I think math facts still have to be learned to the level of automaticity somehow--if you don't want to call that process "memorization," then use whatever term you feel fits, but I stand by the fact that it is an essential but not complete part of math.

 

(I do review my son's phonogram sounds with flashcards for his spelling program, whereas I don't remember ever, ever doing any kind of useful phonogram review myself as a kid. I already knew my sounds to a level of automaticity by the time we got to them on the stupid phonogram chart. So am I perpetuating boring, stupid rote memorization? Eh, maybe.)

 

By the way, note that my son uses Math-U-See, which does try really, really hard to make sure that children are learning the concepts behind the math. I have nothing against concepts being learned--in fact, I thought I was agreeing with the majority opinion on this. Oh, well.

[Spy Car]

I think this post, which was in response to a post of mine, kind of missed my point...

 

I didn't say rote memorization, although I think that can be a valuable tool. In fact, in an attempt to make everything less painful, I got my son the Flashmaster, which allows him to review his math facts without the endless chanting I remember from my school days. (And it seemed to work--he enjoys "playing" with it occasionally, which is all I ask for.) But I think math facts still have to be learned to the level of automaticity somehow--if you don't want to call that process "memorization," then use whatever term you feel fits, but I stand by the fact that it is an essential but not complete part of math.

 

 

I definately take the point that children need to have facility with "math facts."

 

In our case we spent a lot of time working math strategies to develop skills with addition and subtraction, and then added games and other reinforcements to build quick recall. Had we skipped the concrete stages of learning with C Rods, then following up with conceptual work, then working strategies, and then (lastly) sharpening the speed-skills, I think we would have lost a good deal vs approaching these "math facts" as something to be learned via flash-card memorization alone. And we got to "automaticity" without ever using a flash-card.

 

With multiplication and division I will admit that in addition to the precursor conceptual work we have also done memory work. I don't disagree that very fast recall is beneficial for a child. It certainly helps the work get done faster and more accurately.

 

I'm sure we have common ground :001_smile:

 

By the way, note that my son uses Math-U-See, which does try really, really hard to make sure that children are learning the concepts behind the math. I have nothing against concepts being learned--in fact, I thought I was agreeing with the majority opinion on this. Oh, well.

 

I'm sorry if you felt misinterpreted. As I've said, I think we need to strive for both procedural competence (which includes a solid-recall of "math facts") and the pursuit of mathematical reasoning, conceptual understanding, and creative problem solving. Seems you are like minded.

 

Bill

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[Guest]

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[Roadrunner]

This is an interesting discussion. Can somebody help me understand what you mean by math fact memorization? Am I understanding correctly that a child should be able to say 15 in an instant when asked about 8+7? So, if a kid takes a second to think through (8+7 is 8+2+5) that's not good enough? I am asking for the sake of my kid because I know he takes that second to think. So does that mean we need to just drill instant recall? Am I misunderstanding something?

[morosophe]

This is an interesting discussion. Can somebody help me understand what you mean by math fact memorization? Am I understanding correctly that a child should be able to say 15 in an instant when asked about 8+7? So, if a kid takes a second to think through (8+7 is 8+2+5) that's not good enough? I am asking for the sake of my kid because I know he takes that second to think. So does that mean we need to just drill instant recall? Am I misunderstanding something?

 

If your child has to add on his fingers by the time he gets to multiple-digit multiplication, he'll probably get very frustrated doing his math problems.

 

But if he just takes that second when he's learning single-digit addition, he'll probably have it down all the way by the time he gets through carrying. He won't even have to break it down into 8+2+5 by then: he'll just know it.

 

So I would think your son is fine, depending on where he is. (Then again, I don't pretend to know everything, here.)

[Roadrunner]

If your child has to add on his fingers by the time he gets to multiple-digit multiplication, he'll probably get very frustrated doing his math problems.

 

But if he just takes that second when he's learning single-digit addition, he'll probably have it down all the way by the time he gets through carrying. He won't even have to break it down into 8+2+5 by then: he'll just know it.

 

So I would think your son is fine, depending on where he is. (Then again, I don't pretend to know everything, here.)

 

Ah! No fingers. Just in the head. Thanks. :001_smile:

[Spy Car]

This is an interesting discussion. Can somebody help me understand what you mean by math fact memorization? Am I understanding correctly that a child should be able to say 15 in an instant when asked about 8+7? So, if a kid takes a second to think through (8+7 is 8+2+5) that's not good enough? I am asking for the sake of my kid because I know he takes that second to think. So does that mean we need to just drill instant recall? Am I misunderstanding something?

 

Perfect example.

 

There are (at least) two ways to work learning something like 8+7 (but I'm going to keep it simple :tongue_smilie:)

 

Way number one. You break-out flash-cards (or similar) and you do memory drill.

 

Way number two. You learn/teach re-grouping strategies, such as knowing that 8 needs two to become 10, and if 7 gives up 2 it become 5, so you have 10 and 5, which is 15.

 

The first way may be "faster", but the skills are not scalable.

 

The second way requires the patience of working out the strategy repeatedly, but you end up with re-grouping skills that can be scaled up and used with larger numbers, and it reenforces the place value nature of math at a critical age for developing this sort of understanding. Without developing solid place vale understanding you are building on a foundation of sand.

 

I strongly prefer the second option. Every single addition problem we did starting out included the step of having my son explain the mental process he chose for finding the proper solution. I'm convinced the time put into this sort of learning approach is worth the effort vs leaning via rote.

 

Bill

Edited November 8, 2011 by Spy Car

[Roadrunner]

Perfect example.

 

 

I strongly prefer the second option. Every single addition problem we did starting out included the step of having my son explain the mental process he chose for finding the proper solution. I'm convinced the time put into this sort of learning approach is worth the effort vs leaning via rote.

 

Bill

 

Thanks! I had a moment of panic and confusion:001_smile:.

[Guest]

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[Roadrunner]

This is something to watch out for too, I feel. This is what my son was doing and it was essentially like counting fingers in his head. The C-rods and Singapore's number bonds were the two approaches that helped him cement the concepts and "get it". I am not as eloquent as Bill at explaining what I want to say :tongue_smilie: but I hope you understand what I mean.

 

I don't understand. Singapore is what he is using in his head. I guess the question is are kids allowed to calculate? or is it suppose to be a simple fact memorization at lower numbers and why. If I am given a problem like 250 + 65, I don't have instant recall, I would add 250+60+5 in my head and get you an answer. He can do three digit addition in his head (mental math from SM 2A-B) but he needs time to think a little. So, why shouldn't kids be able to think to calculate lower numbers? What's wrong with that split second when somebody is thinking? or generally what's wrong with thinking while doing mental math? I see a lot of emphasis on timed tests. 25 questions in 90 seconds. What's the harm if a kid takes 2 minutes instead?

[Spy Car]

This is something to watch out for too, I feel. This is what my son was doing and it was essentially like counting fingers in his head. The C-rods and Singapore's number bonds were the two approaches that helped him cement the concepts and "get it". I am not as eloquent as Bill at explaining what I want to say :tongue_smilie: but I hope you understand what I mean.

 

It's because a child could, theoretically, count in their head that I really think it is essential that when they first start doing addition and subtraction problems that they explain their strategies. Every. time.

 

And, you're right C Rods and number bonds are great way to conceptualize the re-grouping strategies.

 

Bill

[Roadrunner]

So I am just not understanding. By counting you mean going 8, 9, 10, 11, 12, 13, 14, 15 in the head? Sorry, a slow day here today. :001_huh:

[Spy Car]

I don't understand. Singapore is what he is using in his head. I guess the question is are kids allowed to calculate? or is it suppose to be a simple fact memorization at lower numbers and why. If I am given a problem like 250 + 65, I don't have instant recall, I would add 250+60+5 in my head and get you an answer. He can do three digit addition in his head (mental math from SM 2A-B) but he needs time to think a little. So, why shouldn't kids be able to think to calculate lower numbers? What's wrong with that split second when somebody is thinking? or generally what's wrong with thinking while doing mental math? I see a lot of emphasis on timed tests. 25 questions in 90 seconds. What's the harm if a kid takes 2 minutes instead?

 

So I am just not understanding. By counting you mean going 8, 9, 10, 11, 12, 13, 14, 15 in the head? Sorry, a slow day here today. :001_huh:

 

Solving 250+65 might involve re-grouping to

 

250+50+15

 

300+15

 

315.

 

That is different that going 250, 251, 252....

 

Bill

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Edited September 28, 2015 by Guest