### Friday, July 23, 2010

## Calculators

My scholarly background is in a social science discipline, not math. I have no particular pet theory on the right and proper way to teach math. Frankly, if someone convinced me that counting sheep were the most effective way to do it, I’d gladly requisition a flock or two and tell the soccer team to practice someplace else.

That said, it’s pretty clear at my college -- and at many, many others -- that lower-level math classes (especially developmental) are the most difficult academic obstacles many of our students face. The drop/fail rate in developmental math is embarrassingly and stubbornly high, and the national literature suggests that students who drop out because they feel overmatched in math are among the least likely ever to return. (The same does not hold true of developmental English, interestingly enough.)

In a discussion this week with someone who spends most of her time working with students who are struggling mightily in developmental math, I heard an argument I hadn’t given much thought previously: students who have passed algebra and even pre-calc in high school frequently crash and burn when they hit our developmental math, because the high schools let them use calculators and we don’t.

Among math people, the calculator/no calculator divide seems pretty strong. I’ll admit an uninformed sympathy with the ‘no calculator’ camp, just because I’ve had several experiences in which the ability to guesstimate the ballpark of a correct answer helped me recognize a ludicrous answer when I saw one. Calculators offer precision, but they’re just and only as precise as the numbers you put in. If you hit a number twice, or leave out a digit, or place the decimal point wrong, you’ll get a precisely wrong answer. If you can do the basic math in your head, you’ll have a better shot at recognizing when something is wildly off.

That said, part of me wonders if we’re sacrificing too much on the altar of pencil and paper. It’s great to be able to do addition in your head and long division on paper -- yes, I know, I’m old -- but is it worth flunking out huge cohorts of students because their high schools let them use calculators and we don’t?

At my job, I use statistics all the time. Most of the statistics I use are computer generated. Excel and its progeny (I’m an OpenOffice fan, myself) can crunch huge sets of numbers much faster than I ever could, leaving me free to do other things. Although I like knowing that, in a pinch, I could do a whole bunch of arithmetic myself, I typically don’t. And in most jobs, most people don’t. I agree that it would be better to have the ability than not to have it, but if the cost of holding the line against calculators is turning half a generation away from college, is it worth it?

At this point, the local high schools seem largely to have moved into the calculator camp. Wise and worldly readers, should we follow?

(Program note: next week the gang will be tromping through woods in another state. I’ll resume posting on Monday, August 2.)

That said, it’s pretty clear at my college -- and at many, many others -- that lower-level math classes (especially developmental) are the most difficult academic obstacles many of our students face. The drop/fail rate in developmental math is embarrassingly and stubbornly high, and the national literature suggests that students who drop out because they feel overmatched in math are among the least likely ever to return. (The same does not hold true of developmental English, interestingly enough.)

In a discussion this week with someone who spends most of her time working with students who are struggling mightily in developmental math, I heard an argument I hadn’t given much thought previously: students who have passed algebra and even pre-calc in high school frequently crash and burn when they hit our developmental math, because the high schools let them use calculators and we don’t.

Among math people, the calculator/no calculator divide seems pretty strong. I’ll admit an uninformed sympathy with the ‘no calculator’ camp, just because I’ve had several experiences in which the ability to guesstimate the ballpark of a correct answer helped me recognize a ludicrous answer when I saw one. Calculators offer precision, but they’re just and only as precise as the numbers you put in. If you hit a number twice, or leave out a digit, or place the decimal point wrong, you’ll get a precisely wrong answer. If you can do the basic math in your head, you’ll have a better shot at recognizing when something is wildly off.

That said, part of me wonders if we’re sacrificing too much on the altar of pencil and paper. It’s great to be able to do addition in your head and long division on paper -- yes, I know, I’m old -- but is it worth flunking out huge cohorts of students because their high schools let them use calculators and we don’t?

At my job, I use statistics all the time. Most of the statistics I use are computer generated. Excel and its progeny (I’m an OpenOffice fan, myself) can crunch huge sets of numbers much faster than I ever could, leaving me free to do other things. Although I like knowing that, in a pinch, I could do a whole bunch of arithmetic myself, I typically don’t. And in most jobs, most people don’t. I agree that it would be better to have the ability than not to have it, but if the cost of holding the line against calculators is turning half a generation away from college, is it worth it?

At this point, the local high schools seem largely to have moved into the calculator camp. Wise and worldly readers, should we follow?

(Program note: next week the gang will be tromping through woods in another state. I’ll resume posting on Monday, August 2.)

Comments:

I have that problem in physics. A kid that is getting good marks in algebra screws up their physics equations. i check with their math teacher, and they don't make those mistakes in math class. So I test them myself, and they can manage algebra just fine when x, y, and z are variables and a, b, c, and d are constants. Anything else and they're lost.

Math teachers, at least up here, are consistent, and their students have learned to recognize and apply rules they they didn't mean to teach, such as "only x, y, and z can be variables".

K-12 math does prepare students for calculus and higher math.

At last count, I believe it was all of 8% of students that passed Calc 1.

Good job.

George DeMarse

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I'm in the no calculator camp.

I think there's a distinction between outcomes - the number - and the process.

The aim of Maths is to teach thinking skills. It uses numbers to do this.

Calculators are only as smart as their operators, so if you can't understand the fundamentals you can get bewitched by meaningless statistics.

I think there's a distinction between outcomes - the number - and the process.

The aim of Maths is to teach thinking skills. It uses numbers to do this.

Calculators are only as smart as their operators, so if you can't understand the fundamentals you can get bewitched by meaningless statistics.

DD, Trust your math department in their area of expertise. If they don’t want to let the students use calculators they’re probably right. Computational aids have their place and their use should be taught but if they’re a crutch that obscures the learning of the process they’re no good.

As far as the failure rate for developmental math goes: The reason a college degree is now a minimum qualification for any number of simple clerical jobs is because a high school diploma certifies next to nothing.

As far as the failure rate for developmental math goes: The reason a college degree is now a minimum qualification for any number of simple clerical jobs is because a high school diploma certifies next to nothing.

I'm strictly in the no calculator camp also.

Before I give my answer, here's some anecdotal info. I adjunct at a state university where students either need to pass a basic algebra class (for no credit at full tuition cost) or pass a computerized test at a cost of $20. Most students choose the class because we don't allow calculators on the computerized test and they can't pass it.

In K-12 mathematics, the job is not only to teach thinking skills but also to give students the skills to be prepared for college mathematics. (Calculus and above). These upper level courses require not only theoretical knowledge but the ability to quickly use the processes we teach in lower level classes. If you can't add fractions by hand, you can't add rational functions. If you can't do long division by hand, you can't divide polynomials.

What I see in my remedial math classes is that the older students were the ones who were allowed to skip mathematics in high school. I can't tell you how many students of a certain age never took algebra or never tried anything above algebra. The younger students were given better advice, but have weak computational skills and have been taught with horrible, horrible math curricula.

So these are the students I see in my classes. The college semester calendar does not lend itself to teaching these students. I can't slow down in my class if some of my students are not mastering the material. My cc syllabus requires me to cover all of elementary school and middle school math in one 15 week semester. Maybe some other cc's have a better way.

Before I give my answer, here's some anecdotal info. I adjunct at a state university where students either need to pass a basic algebra class (for no credit at full tuition cost) or pass a computerized test at a cost of $20. Most students choose the class because we don't allow calculators on the computerized test and they can't pass it.

In K-12 mathematics, the job is not only to teach thinking skills but also to give students the skills to be prepared for college mathematics. (Calculus and above). These upper level courses require not only theoretical knowledge but the ability to quickly use the processes we teach in lower level classes. If you can't add fractions by hand, you can't add rational functions. If you can't do long division by hand, you can't divide polynomials.

What I see in my remedial math classes is that the older students were the ones who were allowed to skip mathematics in high school. I can't tell you how many students of a certain age never took algebra or never tried anything above algebra. The younger students were given better advice, but have weak computational skills and have been taught with horrible, horrible math curricula.

So these are the students I see in my classes. The college semester calendar does not lend itself to teaching these students. I can't slow down in my class if some of my students are not mastering the material. My cc syllabus requires me to cover all of elementary school and middle school math in one 15 week semester. Maybe some other cc's have a better way.

I attend a small satellite campus of a major university: the experience is very much as you describe on your blog. I, as as student, am in the "no calculator" camp. Whether it be precalc, trig, or calculus my math classes have been designed so that calculators are only necessary for accurate results in real-world models. Ultimately math is about precision...not accuracy; by not using a calculator I can easily estimate to an order of magnitude.

Slightly off the topic of calculators, but on the topic of basic math courses in higher ed-- Kris Gutiérrez's work in this area is really interesting. I imagine that implementing it would be a nightmare from an administrative point of view, but the success she/they had with it makes one wonder..

What are you trying to teach the students?

If it's high-school- or college-level math and their preparatory work has left them calculator-dependent, just go with it. Teaching elementary-school skills is not the point, and here in the twenty-first century, calculator dependency is not the worst of problems.

If you are teaching more basic remedial math going back before high-school then, yes, do require them to do arithmetic by hand. The are supposed to be able to do it, will benefit from the knowledge, and in this case these skills are on topic.

If it's high-school- or college-level math and their preparatory work has left them calculator-dependent, just go with it. Teaching elementary-school skills is not the point, and here in the twenty-first century, calculator dependency is not the worst of problems.

If you are teaching more basic remedial math going back before high-school then, yes, do require them to do arithmetic by hand. The are supposed to be able to do it, will benefit from the knowledge, and in this case these skills are on topic.

with the advent of the TI-model calculators, calculators have become programmable. that means that major functions or sequences of functions can be duplicated with a program on a nice calculator. it's not about pluses and minuses anymore. i could write a program that would let you type in a word problem, and i could rip apart the problem, decide what needed to be solved, and solve it. calculators are not just about arithmetic anymore.

and i would argue that K-12's purpose regarding math isn't to get kids ready for college math. it's to make sure kids can balance their checkbook, understand how percents (and maybe mortgages work), and things like that. K-12 needs to get rid of the calculators. K-12 is simple math. if kids can't do it, theyre screwed. maybe it's a good reflection of the abismal comprehension of mathematics that is represented in every year's graduating classes...

college classes are theoretical. they purposefully design problems so that the numbers don't get overly huge (10+ digits) so that easy arithmetic mistakes aren't made. that means the processes are being tested, not the arithmetic. keep the calculators out of it.

i took calculus in high school, and made a C in calc 1, a B in calc 2, and A's the rest of the way (to my math minor). i attribute my slow start to an adjustment of college culture and college professors (99% of college math professors are awful teachers).

and i would argue that K-12's purpose regarding math isn't to get kids ready for college math. it's to make sure kids can balance their checkbook, understand how percents (and maybe mortgages work), and things like that. K-12 needs to get rid of the calculators. K-12 is simple math. if kids can't do it, theyre screwed. maybe it's a good reflection of the abismal comprehension of mathematics that is represented in every year's graduating classes...

college classes are theoretical. they purposefully design problems so that the numbers don't get overly huge (10+ digits) so that easy arithmetic mistakes aren't made. that means the processes are being tested, not the arithmetic. keep the calculators out of it.

i took calculus in high school, and made a C in calc 1, a B in calc 2, and A's the rest of the way (to my math minor). i attribute my slow start to an adjustment of college culture and college professors (99% of college math professors are awful teachers).

I think this is one example of why getting rid of tenure is a problem: you need people who are experts in their areas, because you aren't.

You're asking a bunch of random folks a question when you have campus experts, people whose qualifications as college teachers your campus has verified, who have a strong opinion. But you don't like that opinion, so you're looking for an alternative.

If there were no tenure on your campus, you could just mandate that everyone allow calculators. Anyone who didn't would lose their job.

This is a bad educational strategy and a bad management strategy.

Talk to the experts on your campus and trust them.

You're asking a bunch of random folks a question when you have campus experts, people whose qualifications as college teachers your campus has verified, who have a strong opinion. But you don't like that opinion, so you're looking for an alternative.

If there were no tenure on your campus, you could just mandate that everyone allow calculators. Anyone who didn't would lose their job.

This is a bad educational strategy and a bad management strategy.

Talk to the experts on your campus and trust them.

Do what your math department thinks is best, but please keep this in mind.

I have a PhD in the Humanities. I can read, write and speak, Russian, Hungarian and German with varying degrees of proficiency. I've taken college level statistics and I use excel on a daily basis. But I am mathematically illiterate (innumerate?).

My wife quizzes me on the multiplication tables but I can't answer the questions from memory like she can. When adding or subtracting two digit numbers I have to work in out on a piece of paper. I cannot do division in my head at all.

My greatest humiliation was elementary and high school math where I was berated by my teachers for being lazy or slow. My parents and I tried everything, flash cards, tutors, and various games but nothing worked and eventually I gave up. I feel like the adults in my life, especially the teachers gave up on me too.

I've accomplished a lot. So don't give up on your students who struggle with basic math. They are not stupid or lazy.

I have a PhD in the Humanities. I can read, write and speak, Russian, Hungarian and German with varying degrees of proficiency. I've taken college level statistics and I use excel on a daily basis. But I am mathematically illiterate (innumerate?).

My wife quizzes me on the multiplication tables but I can't answer the questions from memory like she can. When adding or subtracting two digit numbers I have to work in out on a piece of paper. I cannot do division in my head at all.

My greatest humiliation was elementary and high school math where I was berated by my teachers for being lazy or slow. My parents and I tried everything, flash cards, tutors, and various games but nothing worked and eventually I gave up. I feel like the adults in my life, especially the teachers gave up on me too.

I've accomplished a lot. So don't give up on your students who struggle with basic math. They are not stupid or lazy.

No calculators for me...I see them as a crutch...one time in my calculus class, at the beginning of the semester, before the students became acquainted with how seriously I took the "no caulcator" policy, one of them tried puttin 1/2 - 1 in their calculator. Needless to say, I am not pleased.

Also, I'm happy to hear that you've developed a sense of when an answer is ridiculous...that's one of the lessons I try to teach my students, although when I get negative ages or someone insists that the universe is 36 years old (the best response to which is to tell them to talk to their grandparents), I wonder if I get through to most of them.

Also, I'm happy to hear that you've developed a sense of when an answer is ridiculous...that's one of the lessons I try to teach my students, although when I get negative ages or someone insists that the universe is 36 years old (the best response to which is to tell them to talk to their grandparents), I wonder if I get through to most of them.

Too many math courses teach "non-calculator" using a calculator-dependent textbook, which means students spend all of their time plugging and chugging through arithmetic rather than learning the math they're supposed to be learning. I struggled my ass off through trig in high school for this reason: the textbook believed in calculators and the teacher didn't.

When I took calc, however, it was a non-calculator textbook with a non-calculator teacher, and I learned and more-or-less remember the elegance of the actual MATH, since I wasn't spending hours every night nearly in tears trying to solve things to six decimal places by hand.

When I took calc, however, it was a non-calculator textbook with a non-calculator teacher, and I learned and more-or-less remember the elegance of the actual MATH, since I wasn't spending hours every night nearly in tears trying to solve things to six decimal places by hand.

Yes, let your math faculty decide this one.

But I do have a question. I'm a silly non-math sort of person, but I don't entirely understand why this is an either/or. When my students write a paper for comp outside of class, they have the aid of spell-check and grammar-check - those things aren't banned, and learning to use those tools responsibly is a good thing. When students do in-class writing, they don't have the benefit of those tools, and that's an important skill, too, to be able to write without anything other than a paper and pen and your imagination. Why couldn't the same be true in math - that some assignments are no-calculator and some are yes-calculator?

But I do have a question. I'm a silly non-math sort of person, but I don't entirely understand why this is an either/or. When my students write a paper for comp outside of class, they have the aid of spell-check and grammar-check - those things aren't banned, and learning to use those tools responsibly is a good thing. When students do in-class writing, they don't have the benefit of those tools, and that's an important skill, too, to be able to write without anything other than a paper and pen and your imagination. Why couldn't the same be true in math - that some assignments are no-calculator and some are yes-calculator?

Before we can answer this question, we need to answer the larger question of "why do we require students to take math courses at all?" Many the student will argue that they'll never use the algebra or calculus or statistics that we're forcing them to take, so why should they be required to take the course. This is a frequent question that I encounter as an Academic Advisor, and one that has direct bearing on your specific question regarding calculators.

I should preface my comments with the standard caveat that these are my opinions only, that I'm not a math faculty member, though I have taught a preparatory math course in the past at the local community college, and that I'm sure there are much better experts on the subject matter. With that said, however, I'm happy to add my 2 cents.

Ultimately I'm firmly of the belief that our requirement for students to take math courses is more about teaching them, as RJ stated, thinking skills. Specifically it's about teaching abstract thinking and how to look at situations as more than just the surface-level, concrete facts that we are presented with. These are skills that are invaluable to us in life - regardless of the specific academic discipline or career area that we pursue.

With that said, let's get back to DD's original question about the use of calculators in math courses. My personal opinion is "it depends." I've been involved in math courses where students are able to plug data into "fancy" calculators and arrive at answers all the while oblivious to the problem they're trying to solve and without having to truly engage their brain or their "thinking skills" at all. This is the reason that I'm fundamentally against the use of calculators in lower-level math courses - especially preparatory or remedial courses.

On the other hand, when students get into some of the upper-level coursework, especially the more applied mathematical courses (e.g., possibly some statistical or numerical methods courses), then the use of calculators (or computer software packages) may be very reasonable and even important - as in day-to-day life, in their actual career setting, these are the tools that would be used to solve such mathematical questions.

When I was pursuing a master's degree in very applied area of psychology, my stats instructor had us learn to do the work by hand first, then via Excel spreadsheet, and then finally via the SPSS statistical package. Ultimately he know that in the "real world", we'd be using SPSS, but he fundamentally believed that we wouldn't really understand the statistics, how to use them, or why to use them, if we jumped right to SPSS and didn't have to more fully engage our brains.

I should preface my comments with the standard caveat that these are my opinions only, that I'm not a math faculty member, though I have taught a preparatory math course in the past at the local community college, and that I'm sure there are much better experts on the subject matter. With that said, however, I'm happy to add my 2 cents.

Ultimately I'm firmly of the belief that our requirement for students to take math courses is more about teaching them, as RJ stated, thinking skills. Specifically it's about teaching abstract thinking and how to look at situations as more than just the surface-level, concrete facts that we are presented with. These are skills that are invaluable to us in life - regardless of the specific academic discipline or career area that we pursue.

With that said, let's get back to DD's original question about the use of calculators in math courses. My personal opinion is "it depends." I've been involved in math courses where students are able to plug data into "fancy" calculators and arrive at answers all the while oblivious to the problem they're trying to solve and without having to truly engage their brain or their "thinking skills" at all. This is the reason that I'm fundamentally against the use of calculators in lower-level math courses - especially preparatory or remedial courses.

On the other hand, when students get into some of the upper-level coursework, especially the more applied mathematical courses (e.g., possibly some statistical or numerical methods courses), then the use of calculators (or computer software packages) may be very reasonable and even important - as in day-to-day life, in their actual career setting, these are the tools that would be used to solve such mathematical questions.

When I was pursuing a master's degree in very applied area of psychology, my stats instructor had us learn to do the work by hand first, then via Excel spreadsheet, and then finally via the SPSS statistical package. Ultimately he know that in the "real world", we'd be using SPSS, but he fundamentally believed that we wouldn't really understand the statistics, how to use them, or why to use them, if we jumped right to SPSS and didn't have to more fully engage our brains.

You might want to consider that the problem is going to become exponentially worse as the Everyday Math curriculum currently in use in many areas of the U.S. doesn't actually EVER TEACH long division and just introduces calculators at that point.

http://www.math.nyu.edu/~braams/links/em-arith.html

http://www.math.nyu.edu/~braams/links/em-arith.html

For a lot of students, the thing that is keeping them from learning math is anxiety, bordering on panic.

For a lot of those students, the calculator is a security blanket.

Any mathematician who puts some thought into it can create questions that require real thought without forbidding calculators (questions that highlight the thought process rather than the arithmetic at the end, questions that incorporate extra variables that make the calculator less useful, all the way to questions that play on the foibles of graphing calculators, and require that the students catch the problem).

Would I like my students to be more comfortable without calculators? Of course. And I absolutely cringe when students grab calculators to multiply by 10. But given the choice between that and students who can think mathematically, I'm always going to choose learning to think.

I think one of the problems with math education in colleges is that people assume that students are coming into classrooms neutral to positive about math, and in remedial classes, that's not the case at all. A large part of teaching math at that level is a psychological game, and I think calculators are a worthwhile tool in that setting.

[Do I occasionally forbid calculators for a particular quiz, or something like that? Sure, with warning, and a great deal of emphasis on them not needing the calculator for this thing X that they're learning. But overall, I'm in the calculator camp.]

For a lot of those students, the calculator is a security blanket.

Any mathematician who puts some thought into it can create questions that require real thought without forbidding calculators (questions that highlight the thought process rather than the arithmetic at the end, questions that incorporate extra variables that make the calculator less useful, all the way to questions that play on the foibles of graphing calculators, and require that the students catch the problem).

Would I like my students to be more comfortable without calculators? Of course. And I absolutely cringe when students grab calculators to multiply by 10. But given the choice between that and students who can think mathematically, I'm always going to choose learning to think.

I think one of the problems with math education in colleges is that people assume that students are coming into classrooms neutral to positive about math, and in remedial classes, that's not the case at all. A large part of teaching math at that level is a psychological game, and I think calculators are a worthwhile tool in that setting.

[Do I occasionally forbid calculators for a particular quiz, or something like that? Sure, with warning, and a great deal of emphasis on them not needing the calculator for this thing X that they're learning. But overall, I'm in the calculator camp.]

Having taught high school math, the problem starts earlier. Most of our students arrive from elementary school accustomed to calculators and unable to tell that 2 + 5 =/= 57 (ie. can't tell they hit the wrong key). Their elementary math teachers mostly weren't math teachers and quite a few had taught their students that it was OK to suck at arithmetic.

Bring back the slide-rule.

On a more serious note, I'm of the opinion that it entirely depends on the class/textbook/professor, like one of the comments above me.

I'll regale you with a story of one of my own past experiences, in Pre-calculus. For 3 weeks leading up to the semester, the class would get bi-weekly emails from the professor, reminding us that we were required to purchase one of the Texas Instruments calculators, at a wonderful price of 120 something dollars. As soon as class began, we were informed that we would only be permitted to use them when necessary, we had to learn to understand the subject without them first. Sounded ok, but halfway into the semester, I hadn't even turned it on once. To add to that, I hadn't even been allowed to reference my textbook if I had difficulty doing in class assignments(not examinations).

To me, this does not make sense. Now I've no problem with learning how to do something by hand, but once I've mastered the skill, I see no reason why I cannot use all the references at my disposal, be they calculators, textbooks, notes, etc. That's how it works outside of college, isn't it?

On a more serious note, I'm of the opinion that it entirely depends on the class/textbook/professor, like one of the comments above me.

I'll regale you with a story of one of my own past experiences, in Pre-calculus. For 3 weeks leading up to the semester, the class would get bi-weekly emails from the professor, reminding us that we were required to purchase one of the Texas Instruments calculators, at a wonderful price of 120 something dollars. As soon as class began, we were informed that we would only be permitted to use them when necessary, we had to learn to understand the subject without them first. Sounded ok, but halfway into the semester, I hadn't even turned it on once. To add to that, I hadn't even been allowed to reference my textbook if I had difficulty doing in class assignments(not examinations).

To me, this does not make sense. Now I've no problem with learning how to do something by hand, but once I've mastered the skill, I see no reason why I cannot use all the references at my disposal, be they calculators, textbooks, notes, etc. That's how it works outside of college, isn't it?

While I personally dislike the speed at which students turn to their calculators when stuck, I believe that ultimately allowing/not allowing calculators doesn't really make that much difference.

The real culprit is the problems in most texts (and hence the problems most teachers put on their exams) were not carefully designed and can be circumvented using technology.

If one is willing to stray beyond the textbook problems, it is not too hard to formulate problems for which the calculators are of no help, or only help after the important critical thinking has taken place. After a few weeks of such assignments, I find that my students stop using their calculators on their own.

If we, the math faculty, are willing to adapt our curriculum, the calculator issue can become a non-issue.

The real culprit is the problems in most texts (and hence the problems most teachers put on their exams) were not carefully designed and can be circumvented using technology.

If one is willing to stray beyond the textbook problems, it is not too hard to formulate problems for which the calculators are of no help, or only help after the important critical thinking has taken place. After a few weeks of such assignments, I find that my students stop using their calculators on their own.

If we, the math faculty, are willing to adapt our curriculum, the calculator issue can become a non-issue.

I don't teach math; I teach economics. There are some occasions in economics in which students need to be able to do simple algebra (using actual nnumbers or using symbols). For example, there's a very common concept (elasticity of demand) where the elasticity of demand is calculated as

e = (% Change in Quantity) divided by (% Change in something else), or

e = a/b.

First, many of my students cannot calculate the percentage changes from the underlying data (and they are required to demonstrate math proficiency at the level of introductory HS algebra to register for the course).

Second, given the two percentage changes, many of them cannot (for example) consistently divide 10% by 5% to get the answer...And, give them e and a in the simplified equation above, and they calculate b as euqal to a/e...drives me crazy every semester...

And then they go into accounting...

e = (% Change in Quantity) divided by (% Change in something else), or

e = a/b.

First, many of my students cannot calculate the percentage changes from the underlying data (and they are required to demonstrate math proficiency at the level of introductory HS algebra to register for the course).

Second, given the two percentage changes, many of them cannot (for example) consistently divide 10% by 5% to get the answer...And, give them e and a in the simplified equation above, and they calculate b as euqal to a/e...drives me crazy every semester...

And then they go into accounting...

Enjoy the romp, but watch out for Poison Ivy and ticks!

Good timing on this topic, since I am planning how to riff off of a great XKCD cartoon to write about calculators. The comments here give me even more to think about.

In my not so humble opinion, I think calculators in high school are an example of the correlation-causation problem. (I need to remind myself of the - is it four? - conditions you need to infer causation from a statistical correlation.) My opinion is (correct that, there are well designed studies that prove) that many high school "pre calculus" classes are nowhere near the standard set by ours, and even passing ours does not prepare them very well for calculus.

I could not disagree more with part of what Anne Dwyer wrote, although I know what she means. The purpose of K-12 SHOULD be to prepare students for college math (by which I mean a college algebra course that was a HS class for me) but it actually is directed at passing a high-stakes graduation exam with a passing score somewhere in the vicinity of what I call 6th to 8th grade math. Everything else in her comment is spot on with my observations advising, teaching physics, and working closely with the people who teach calculus.

The two big problems these students have is that they can't work with fractions and they can't work with symbols. I blame the former on some truly awful curricula that are used in many schools, curricula that exclude basic algorithms, but calculators may play a role. (I have met students who can simplify fractions symbolically but can't do the same problem with numbers present, suggesting they don't see numbers as symbols.) The latter might be blamed on calculators if (as I suspect from our college classes) weeks or months are spent on learning how to "solve" a linear equation by graphing it to the exclusion of skills that used to be taught back in the day. If they spend a lot of time on numerical problems (although lack of fluency there makes me suspect they don't), that is time taken away from symbolic ones.

BTW, for those of you who can't make the connection between HS and college (DD might be in that camp), our highest developmental course is roughly what I call 7th grade math (basic algebra, the first class where you work with letters) and there is one more course (roughly 9th grade math, where you first meet the quadratic formula) between it and college algebra.

Good timing on this topic, since I am planning how to riff off of a great XKCD cartoon to write about calculators. The comments here give me even more to think about.

In my not so humble opinion, I think calculators in high school are an example of the correlation-causation problem. (I need to remind myself of the - is it four? - conditions you need to infer causation from a statistical correlation.) My opinion is (correct that, there are well designed studies that prove) that many high school "pre calculus" classes are nowhere near the standard set by ours, and even passing ours does not prepare them very well for calculus.

I could not disagree more with part of what Anne Dwyer wrote, although I know what she means. The purpose of K-12 SHOULD be to prepare students for college math (by which I mean a college algebra course that was a HS class for me) but it actually is directed at passing a high-stakes graduation exam with a passing score somewhere in the vicinity of what I call 6th to 8th grade math. Everything else in her comment is spot on with my observations advising, teaching physics, and working closely with the people who teach calculus.

**I also think you should look at the curricula used in your feeder schools, particularly K-5.**The two big problems these students have is that they can't work with fractions and they can't work with symbols. I blame the former on some truly awful curricula that are used in many schools, curricula that exclude basic algorithms, but calculators may play a role. (I have met students who can simplify fractions symbolically but can't do the same problem with numbers present, suggesting they don't see numbers as symbols.) The latter might be blamed on calculators if (as I suspect from our college classes) weeks or months are spent on learning how to "solve" a linear equation by graphing it to the exclusion of skills that used to be taught back in the day. If they spend a lot of time on numerical problems (although lack of fluency there makes me suspect they don't), that is time taken away from symbolic ones.

BTW, for those of you who can't make the connection between HS and college (DD might be in that camp), our highest developmental course is roughly what I call 7th grade math (basic algebra, the first class where you work with letters) and there is one more course (roughly 9th grade math, where you first meet the quadratic formula) between it and college algebra.

I teach math. Both transfer level and developmental. Our college allows calculators in some of the developmental classes. The success rates are still abysmal, even for students who did ok in high school math classes. I question whether the calculator/no calculator issue is the reason students who did ok in hs math "crash and burn" in college math.

Don't know the actual reason. Just don't think calculators are it. More likely it is a combination of factors, rather than one silver bullet reason.

Don't know the actual reason. Just don't think calculators are it. More likely it is a combination of factors, rather than one silver bullet reason.

When I was in high school, the teachers subscribed to the use of calculators for math because they were allowed on the state proficiency test. In fact, Texas Instruments designed a calculator specifically for the state's math proficiency test. While you didn't need a calculator to complete the test, the teachers received grants and funding to make sure you knew how to use the calculator because it was believed that you stood a better chance of passing. While faulty logic and certainly a disservice to the students, it seemed to be the driving reason when I was in high school.

I tend to agree with km's post. I teach a variety of math courses from developmental to graduate level. Noticed abysmal performance in dev math with or without the calculator. The calculator is not THE culprit.

Here is my list of a set of complex factors conspiring together for the dismal performance of students in developmental math.

1) middle school math is more focused on algebra as early as 7th grade. So students don't have enough mastery of fractions, percents etc . As a parent of a middle schooler, I'm teaching my son these concepts on the side.

2) So (1) leads to a poor algebra course for all but the most skilled in math.

3) More students attending college- so the lack of good high school prep is more evident.

4) Content of dev math courses in college are aimed at preparing students for a precalc/calculus track. But those going into sociology or psychology need to know basic math, but in more detail. They don't need a bunch of algebra tricks for calculus.

My solution would be a beginning dev math course should go into a small number of basic topics, but in detail. Should have plenty of word problems in non-physical science contexts. Those not wanting to be in a STEM major can then take a for-credit general education survey of math course or a basic stats course, and be done.

The ones aiming for precalc/ calculus can take more dev math algebra courses.

Here is my list of a set of complex factors conspiring together for the dismal performance of students in developmental math.

1) middle school math is more focused on algebra as early as 7th grade. So students don't have enough mastery of fractions, percents etc . As a parent of a middle schooler, I'm teaching my son these concepts on the side.

2) So (1) leads to a poor algebra course for all but the most skilled in math.

3) More students attending college- so the lack of good high school prep is more evident.

4) Content of dev math courses in college are aimed at preparing students for a precalc/calculus track. But those going into sociology or psychology need to know basic math, but in more detail. They don't need a bunch of algebra tricks for calculus.

My solution would be a beginning dev math course should go into a small number of basic topics, but in detail. Should have plenty of word problems in non-physical science contexts. Those not wanting to be in a STEM major can then take a for-credit general education survey of math course or a basic stats course, and be done.

The ones aiming for precalc/ calculus can take more dev math algebra courses.

Like others, I'm in the "it depends" camp. (As background, I was a stand-out math student in high school, went on to pursue a masters degree in applied math and now work in a field known as operations research. Basically, all I do for a living is math.)

One thing it hugely depends on is the curriculum. If the goal is to teach computational skills, then by all means, no calculator allowed. But if the goal is to teach processes, and the computation is secondary, then either allow the calculator or write the curriculum such that students aren't lost in computational issues. But don't leave me having to calculate 123x456 (or something similar) by hand for every problem. That's a waste of time, and disastrous as an earlier commenter pointed out.

The only thing a student should have to do with 123x456 is estimate the answer so he/she knows if they fat fingered an input.

And quite frankly, so what if a student forgot how to do basic arithmetic by hand. I can think of several things that I use computers for these days where I've forgotten how to do the underlying math by hand. Calculus is one of them.

One thing it hugely depends on is the curriculum. If the goal is to teach computational skills, then by all means, no calculator allowed. But if the goal is to teach processes, and the computation is secondary, then either allow the calculator or write the curriculum such that students aren't lost in computational issues. But don't leave me having to calculate 123x456 (or something similar) by hand for every problem. That's a waste of time, and disastrous as an earlier commenter pointed out.

The only thing a student should have to do with 123x456 is estimate the answer so he/she knows if they fat fingered an input.

And quite frankly, so what if a student forgot how to do basic arithmetic by hand. I can think of several things that I use computers for these days where I've forgotten how to do the underlying math by hand. Calculus is one of them.

Here is an interesting note that I bet is NOT unique to my CC. Many students fail in remedial math. I will NOT say that all of them would pass with the aid of a calculator, but some significant fraction of them would pass. And the reward for passing all their remedial math without a calculator is that we AGAIN grant them the use of a calculator in the remainder of their credit bearing courses.

That's right. We think it's so important they do so much without a calculator that we hand it right back to them in the credit bearing classes. This is EXACTLY what got them to be so innumerate in the first place. They were handed a calculator in high school and promptly forgot any arithmetic skills they developed before that. And guess what? Our students promptly forget the arithmetic skills we teach in the remedial classes as soon as they get to the credit bearing courses.

Many of these students, with the aid of a calculator, do not do well in the credit bearing course either. This just suggests that the calculator is not the entire problem, and it is also not the entire solution. But it is interesting how we treat the calculator in our own curriculum.

If you see the notation that I deleted a previous comment, this is the same comment with a typo corrected. More typos likely still exist! :-)

That's right. We think it's so important they do so much without a calculator that we hand it right back to them in the credit bearing classes. This is EXACTLY what got them to be so innumerate in the first place. They were handed a calculator in high school and promptly forgot any arithmetic skills they developed before that. And guess what? Our students promptly forget the arithmetic skills we teach in the remedial classes as soon as they get to the credit bearing courses.

Many of these students, with the aid of a calculator, do not do well in the credit bearing course either. This just suggests that the calculator is not the entire problem, and it is also not the entire solution. But it is interesting how we treat the calculator in our own curriculum.

If you see the notation that I deleted a previous comment, this is the same comment with a typo corrected. More typos likely still exist! :-)

DD, I'm pretty much in the same (non) math camp as you. All I know is this: my son attends a college prep private middle/high school where probably 98% of the graduates go on to college, and probably a fair percentage of those go on to various ivies and prestigious sorts of schools. They've won tons of awards in terms of science programs and merit scholars and all of the rest. My son, who is entering the eighth grade, is pretty good at math and is in the most advance classes.

At this school, all the math classes (as far as I can tell) use calculators, and not just your dime store add/subtract/divide calculators, either. I'm talking scientific calculators that a) can do all sorts of things I do not understand, and b) cost me too much money.

Like I said, I don't really know. But if it's good enough for these kids....

At this school, all the math classes (as far as I can tell) use calculators, and not just your dime store add/subtract/divide calculators, either. I'm talking scientific calculators that a) can do all sorts of things I do not understand, and b) cost me too much money.

Like I said, I don't really know. But if it's good enough for these kids....

Solution: divide math classes randomly in terms of both teachers and students. Assign one group to teach with calculators and one group to teach without. Repeat for a couple of semesters and see what happens.

If calculators improve passage rates, use calculators; if not, don't. This is something measurable, so why not measure, rather than tell stories to one another?

If calculators improve passage rates, use calculators; if not, don't. This is something measurable, so why not measure, rather than tell stories to one another?

The problem with calculators is that they help with arithmetic, not math. Students who use them often fail to learn algebra as symbol manipulation, algorithms such as finding the common denominator, etc. Then they hit calculus, and all the numbers are gone -- it's all symbols. And then then calculator is no help, because it doesn't know what the goal is.

I fully agree with Michael. I worked with 9th grade algebra students who did exactly as Michael describes. Given an all numbers problem, they can use a calculator to figure it out. Given a problem with a variable they can't solve it. Even a problem such as x-8=2. After a few months of working on problems like that, they finally seemed to be getting it. But when they got to a worksheet that used all different letters as variables, they got completely lost all over again. Obviously they did not grasp the concept of using a symbol. Too much reliance on calculators in early math? I'm not sure.

"This just suggests that the calculator is not the entire problem, and it is also not the entire solution."

So true!

I've been teaching developmental math for the past two years, and I'm very much in the no calculator camp, at least for this level of math. We're trying to teach our students (1) a basic understanding of numbers, (2) how to do basic arithmetic, (3) how to perform some very basic symbolic manipulation, and (4) how to apply all of these to basic, real-life situations (among other things, of course). From what I've seen, most students who are allowed to use calculators and rely heavily on them don't learn (1) or (2). This doesn't bode well for them being successful in (3) and (4), so they still might fail.

This relates to another problem we have with math. Students aren't taught to understand math; they're just taught to do it. Math builds upon itself, and the most successful math students are the ones who understand in the beginning what's going on, so they can follow along and understand in the end what's going on. If students don't have a basic understanding of how fractions work, for example, they're never going to understand more complicated operations involving them. If a student doesn't understand why you can cross-multiply to solve a proportion, they're going to struggle with more advanced equations.

I feel that allowing calculators removes students from the math a bit, which makes it easier for them to pay less attention and understand less of the mathematics behind the arithmetic, which in turn leads to less learning in the longrun.

So true!

I've been teaching developmental math for the past two years, and I'm very much in the no calculator camp, at least for this level of math. We're trying to teach our students (1) a basic understanding of numbers, (2) how to do basic arithmetic, (3) how to perform some very basic symbolic manipulation, and (4) how to apply all of these to basic, real-life situations (among other things, of course). From what I've seen, most students who are allowed to use calculators and rely heavily on them don't learn (1) or (2). This doesn't bode well for them being successful in (3) and (4), so they still might fail.

This relates to another problem we have with math. Students aren't taught to understand math; they're just taught to do it. Math builds upon itself, and the most successful math students are the ones who understand in the beginning what's going on, so they can follow along and understand in the end what's going on. If students don't have a basic understanding of how fractions work, for example, they're never going to understand more complicated operations involving them. If a student doesn't understand why you can cross-multiply to solve a proportion, they're going to struggle with more advanced equations.

I feel that allowing calculators removes students from the math a bit, which makes it easier for them to pay less attention and understand less of the mathematics behind the arithmetic, which in turn leads to less learning in the longrun.

This question leads to a larger one, which is whether or not the "class" model is the correct way to approach remedial math.

I still contend that it isn't, and the astronomical fail rates are support for this. Of course, you do at least help the folks who are easy to help, and that's not useless.

I still contend that it isn't, and the astronomical fail rates are support for this. Of course, you do at least help the folks who are easy to help, and that's not useless.

"The ability to guesstimate the ballpark of a correct answer," as you put it, seems like a very reasonable requirement for a college degree. People who can't do this are likely to put their family's finances in danger -- or make bad publicity for their employers, etc. Consider the problems with Verizon's ".002 dollars vs .002 cents":

http://consumerist.com/2006/12/transcript-verizon-doesnt-know-how-to-count.html

So even if you do allow calculators for most of the developmental-math sequence, I don't see anything wrong about holding students back until they can AT LEAST pass a test of ballpark-guesstimation-proficiency WITHOUT a calculator.

http://consumerist.com/2006/12/transcript-verizon-doesnt-know-how-to-count.html

So even if you do allow calculators for most of the developmental-math sequence, I don't see anything wrong about holding students back until they can AT LEAST pass a test of ballpark-guesstimation-proficiency WITHOUT a calculator.

@~Me: The problem is that most people are not mentally capable of making even the smallest abstract leaps. It's one of the nastier aspects of modern living, that a talent which was nigh-useless (and drove people crazy) until five hundred years ago is now more or less required to be a functional citizen.

Also a social scientist, but with extensive math training.

I say no calculators until students know whether the answer they got is off by a factor of 100. A sense of when something can't possibly be right is SO important for so many things. Think about the nurses administering your future medication.

I say no calculators until students know whether the answer they got is off by a factor of 100. A sense of when something can't possibly be right is SO important for so many things. Think about the nurses administering your future medication.

*Given a problem with a variable they can't solve it. Even a problem such as x-8=2. After a few months of working on problems like that, they finally seemed to be getting it. But when they got to a worksheet that used all different letters as variables, they got completely lost all over again. Obviously they did not grasp the concept of using a symbol.*

I have that problem in physics. A kid that is getting good marks in algebra screws up their physics equations. i check with their math teacher, and they don't make those mistakes in math class. So I test them myself, and they can manage algebra just fine when x, y, and z are variables and a, b, c, and d are constants. Anything else and they're lost.

Math teachers, at least up here, are consistent, and their students have learned to recognize and apply rules they they didn't mean to teach, such as "only x, y, and z can be variables".

Other than to say I'm in the no calculator camp, I wanted to add another voice to the chorus pointing out how much trouble students have when we change the letters around. My stats students usually remember the equation y=mx+b, but when I tell them that in my class we use y=a+bx, too many of them get lost. They learned the symbols they were taught, but never really understood them.

Perhaps what needs to be considered is appropriate transition from calculator to non-calculator. If you have students who are used to calculators, have them do the math both ways. This will help them gain confidence in their own calculations, and yet provide them the solace of the calculator in understanding what is being taught.

I think that anyone who blindly accepts the results of a spreadsheet is in no position to criticise students for just accepting the number a calculator gives. For many problems there is a better solution than a spreadsheet, especially excel. Spreadsheets take the garbage-in/garbage-out factor of a calculator and multiply it by a million.

All,

It seems that one nice local approach would be to follow this advice:

jseliger.com said...

Solution: divide math classes randomly in terms of both teachers and students. Assign one group to teach with calculators and one group to teach without. Repeat for a couple of semesters and see what happens.

If you'd prefer to not experiment on your own students perhaps a search of mathematics education research could offer some guidance. The evidence is mixed. See the next comment

Big picture point: Yes, your local people (and folks here) have 'on-the-ground' knowledge...

But, it's worth asking for additional support for either position. In this case there there is significant research done on this topic, much more than I've cited. Maybe you disagree with the results of this research? Do your own and show your claims are correct. This thread was problematic for me to read because everyone was arguing from their experience but even based on that experience most statements were for the form: I think they're bad/good/it depends.

We're in academia because we enjoy thinking in a rigorous way about a problem. But, when it comes to this conversation there was no reference to rigor or research. Especially at teaching schools I would argue that the scholarship of teaching and learning is exactly what people in our profession should be doing. That means more than collecting our experience and talking from that...

Heck, there's a whole movement about it:

http://en.wikipedia.org/wiki/Scholarship_of_Teaching_and_Learning

It seems that one nice local approach would be to follow this advice:

jseliger.com said...

Solution: divide math classes randomly in terms of both teachers and students. Assign one group to teach with calculators and one group to teach without. Repeat for a couple of semesters and see what happens.

If you'd prefer to not experiment on your own students perhaps a search of mathematics education research could offer some guidance. The evidence is mixed. See the next comment

Big picture point: Yes, your local people (and folks here) have 'on-the-ground' knowledge...

But, it's worth asking for additional support for either position. In this case there there is significant research done on this topic, much more than I've cited. Maybe you disagree with the results of this research? Do your own and show your claims are correct. This thread was problematic for me to read because everyone was arguing from their experience but even based on that experience most statements were for the form: I think they're bad/good/it depends.

We're in academia because we enjoy thinking in a rigorous way about a problem. But, when it comes to this conversation there was no reference to rigor or research. Especially at teaching schools I would argue that the scholarship of teaching and learning is exactly what people in our profession should be doing. That means more than collecting our experience and talking from that...

Heck, there's a whole movement about it:

http://en.wikipedia.org/wiki/Scholarship_of_Teaching_and_Learning

Some papers to read:

1) The Arithmetic Gap

Educational Leadership, v61 n5 p55 Feb 2004

Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

2) The Effects of Non-CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta-Analysis

School Science and Mathematics

Volume 106 Issue 1, Pages 16 - 26

Summary: 42-studies included in a meta-analysis.

The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.

3) Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis

REVIEW OF EDUCATIONAL RESEARCH September 2008 vol. 78 no. 3 427-515

Link: http://rer.sagepub.com/content/78/3/427.abstract

Effects of Computer Assisted Instruction were moderate.

4) Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis

Review of Educational Research June 1, 2009 79: 839-911

Summary:

Effect sizes were very small for mathematics curricula and for computer-assisted instruction. Positive effects were found for two cooperative learning programs. Outcomes were similar for disadvantaged and nondisadvantaged students and for students of different ethnicities. Consistent with an earlier review of elementary programs, this article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

There is, in fact, an entire organization of mathematics education researchers devoted to studying undergraduate education: www.rume.org

There are, I'm sure, resources to be found there.

5) This isn't actually a study but rather a report. The Mathematical Association of America, an association of mathematicians and mathematics educators with a focus on undergraduate education, has a committee called:

Curriculum Renewal Across the First Two Years

They have issued a comprehensive report on College Algebra.

http://www.contemporarycollegealgebra.org/goals/CRAFTY_2year.html

They say:

Use of technology - every student is expected to have daily access to a graphing calculator and/or computer. The ability to use technology for plotting and computation is a very important skill.

1) The Arithmetic Gap

Educational Leadership, v61 n5 p55 Feb 2004

Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

2) The Effects of Non-CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta-Analysis

School Science and Mathematics

Volume 106 Issue 1, Pages 16 - 26

Summary: 42-studies included in a meta-analysis.

The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.

3) Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis

REVIEW OF EDUCATIONAL RESEARCH September 2008 vol. 78 no. 3 427-515

Link: http://rer.sagepub.com/content/78/3/427.abstract

Effects of Computer Assisted Instruction were moderate.

4) Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis

Review of Educational Research June 1, 2009 79: 839-911

Summary:

Effect sizes were very small for mathematics curricula and for computer-assisted instruction. Positive effects were found for two cooperative learning programs. Outcomes were similar for disadvantaged and nondisadvantaged students and for students of different ethnicities. Consistent with an earlier review of elementary programs, this article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

There is, in fact, an entire organization of mathematics education researchers devoted to studying undergraduate education: www.rume.org

There are, I'm sure, resources to be found there.

5) This isn't actually a study but rather a report. The Mathematical Association of America, an association of mathematicians and mathematics educators with a focus on undergraduate education, has a committee called:

Curriculum Renewal Across the First Two Years

They have issued a comprehensive report on College Algebra.

http://www.contemporarycollegealgebra.org/goals/CRAFTY_2year.html

They say:

Use of technology - every student is expected to have daily access to a graphing calculator and/or computer. The ability to use technology for plotting and computation is a very important skill.

Some papers to read:

1) The Arithmetic Gap

Educational Leadership, v61 n5 p55 Feb 2004

Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

2) The Effects of Non-CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta-Analysis

School Science and Mathematics

Volume 106 Issue 1, Pages 16 - 26

Summary: 42-studies included in a meta-analysis.

The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.

3) Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis

REVIEW OF EDUCATIONAL RESEARCH September 2008 vol. 78 no. 3 427-515

Link: http://rer.sagepub.com/content/78/3/427.abstract

Effects of Computer Assisted Instruction were moderate.

4) Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis

Review of Educational Research June 1, 2009 79: 839-911

Summary:

Effect sizes were very small for mathematics curricula and for computer-assisted instruction. Positive effects were found for two cooperative learning programs. Outcomes were similar for disadvantaged and nondisadvantaged students and for students of different ethnicities. Consistent with an earlier review of elementary programs, this article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

There is, in fact, an entire organization of mathematics education researchers devoted to studying undergraduate education: www.rume.org

There are, I'm sure, resources to be found there.

5) This isn't actually a study but rather a report. The Mathematical Association of America, an association of mathematicians and mathematics educators with a focus on undergraduate education, has a committee called:

Curriculum Renewal Across the First Two Years

They have issued a comprehensive report on College Algebra.

http://www.contemporarycollegealgebra.org/goals/CRAFTY_2year.html

They say:

Use of technology - every student is expected to have daily access to a graphing calculator and/or computer. The ability to use technology for plotting and computation is a very important skill.

1) The Arithmetic Gap

Educational Leadership, v61 n5 p55 Feb 2004

Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

2) The Effects of Non-CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta-Analysis

School Science and Mathematics

Volume 106 Issue 1, Pages 16 - 26

Summary: 42-studies included in a meta-analysis.

The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.

3) Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis

REVIEW OF EDUCATIONAL RESEARCH September 2008 vol. 78 no. 3 427-515

Link: http://rer.sagepub.com/content/78/3/427.abstract

Effects of Computer Assisted Instruction were moderate.

4) Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis

Review of Educational Research June 1, 2009 79: 839-911

Summary:

Effect sizes were very small for mathematics curricula and for computer-assisted instruction. Positive effects were found for two cooperative learning programs. Outcomes were similar for disadvantaged and nondisadvantaged students and for students of different ethnicities. Consistent with an earlier review of elementary programs, this article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

There is, in fact, an entire organization of mathematics education researchers devoted to studying undergraduate education: www.rume.org

There are, I'm sure, resources to be found there.

5) This isn't actually a study but rather a report. The Mathematical Association of America, an association of mathematicians and mathematics educators with a focus on undergraduate education, has a committee called:

Curriculum Renewal Across the First Two Years

They have issued a comprehensive report on College Algebra.

http://www.contemporarycollegealgebra.org/goals/CRAFTY_2year.html

They say:

Use of technology - every student is expected to have daily access to a graphing calculator and/or computer. The ability to use technology for plotting and computation is a very important skill.

Some papers to read:

1) The Arithmetic Gap

Educational Leadership, v61 n5 p55 Feb 2004

Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

2) The Effects of Non-CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta-Analysis

School Science and Mathematics

Volume 106 Issue 1, Pages 16 - 26

Summary: 42-studies included in a meta-analysis.

When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.

3) Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis

REVIEW OF EDUCATIONAL RESEARCH September 2008 vol. 78 no. 3 427-515

Link: http://rer.sagepub.com/content/78/3/427.abstract

Effects of Computer Assisted Instruction were moderate.

4) Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis

Review of Educational Research June 1, 2009 79: 839-911

Summary:

Effect sizes were very small for mathematics curricula and for computer-assisted instruction.

This article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

5) This isn't actually a study but rather a report. The Mathematical Association of America, an association of mathematicians and mathematics educators with a focus on undergraduate education, has a committee called:

Curriculum Renewal Across the First Two Years

They have issued a comprehensive report on College Algebra.

http://www.contemporarycollegealgebra.org/goals/CRAFTY_2year.html

They say:

Use of technology - every student is expected to have daily access to a graphing calculator and/or computer. The ability to use technology for plotting and computation is a very important skill.

1) The Arithmetic Gap

Educational Leadership, v61 n5 p55 Feb 2004

Summary: The students using calculators in school classrooms result in lower math scores than students who never use them.

2) The Effects of Non-CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta-Analysis

School Science and Mathematics

Volume 106 Issue 1, Pages 16 - 26

Summary: 42-studies included in a meta-analysis.

When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.

3) Effective Programs in Elementary Mathematics: A Best-Evidence Synthesis

REVIEW OF EDUCATIONAL RESEARCH September 2008 vol. 78 no. 3 427-515

Link: http://rer.sagepub.com/content/78/3/427.abstract

Effects of Computer Assisted Instruction were moderate.

4) Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis

Review of Educational Research June 1, 2009 79: 839-911

Summary:

Effect sizes were very small for mathematics curricula and for computer-assisted instruction.

This article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

5) This isn't actually a study but rather a report. The Mathematical Association of America, an association of mathematicians and mathematics educators with a focus on undergraduate education, has a committee called:

Curriculum Renewal Across the First Two Years

They have issued a comprehensive report on College Algebra.

http://www.contemporarycollegealgebra.org/goals/CRAFTY_2year.html

They say:

Use of technology - every student is expected to have daily access to a graphing calculator and/or computer. The ability to use technology for plotting and computation is a very important skill.

Anecdotally speaking, I can attest that some people simply cannot memorize basic math facts--addition, subtraction, multiplication, and division tables. I know, because I'm one of them.

My parents and teachers did all they could humanly do to force-feed arithmetic to me, and to this day I could not tell you what 9 x 6 is without a calculator. I don't

However, my high-school math and science teachers thought I was stupid. When a university recruited me at the end of my junior year, the school refused to give me a GED, despite the fact that I had more than enough credits to qualify for graduation, because the principal felt I could not succeed on the university level. My chemistry teacher, puzzled by this odd opportunity for his C-minus student, decided to give his class two exams: one with no formulae entailing valences, and one that was all formulae. I aced the first and flunked the second, for no other reason than that I can. not. remember. math. facts.

As a girl, I wanted a career in astrophysics. By the time I finished trig, however, it was clear that even if women had been welcome in the hard sciences at that time (we were not), I didn't stand a chance, because I couldn't do basic arithmetic, because I couldn't remember what 7 x 8 is. That's how I ended up with a low-paying career in the liberal arts instead of going into the much better paid field I really desired.

If I'd been allowed to use a calculator in that chemistry class (if we'd had them -- I used a slide rule), you can be sure that today I wouldn't be retiring on a pittance and wondering whether I'll be living under the Seventh Avenue Overpass in my old age.

My experience (equally anecdotal) in all those years of teaching English is that there are lots of students out there just like me.

My parents and teachers did all they could humanly do to force-feed arithmetic to me, and to this day I could not tell you what 9 x 6 is without a calculator. I don't

*think*I'm stupid...last time I looked, I had an IQ of 140, a B.A. in French, and a Ph.D. in English.However, my high-school math and science teachers thought I was stupid. When a university recruited me at the end of my junior year, the school refused to give me a GED, despite the fact that I had more than enough credits to qualify for graduation, because the principal felt I could not succeed on the university level. My chemistry teacher, puzzled by this odd opportunity for his C-minus student, decided to give his class two exams: one with no formulae entailing valences, and one that was all formulae. I aced the first and flunked the second, for no other reason than that I can. not. remember. math. facts.

As a girl, I wanted a career in astrophysics. By the time I finished trig, however, it was clear that even if women had been welcome in the hard sciences at that time (we were not), I didn't stand a chance, because I couldn't do basic arithmetic, because I couldn't remember what 7 x 8 is. That's how I ended up with a low-paying career in the liberal arts instead of going into the much better paid field I really desired.

If I'd been allowed to use a calculator in that chemistry class (if we'd had them -- I used a slide rule), you can be sure that today I wouldn't be retiring on a pittance and wondering whether I'll be living under the Seventh Avenue Overpass in my old age.

My experience (equally anecdotal) in all those years of teaching English is that there are lots of students out there just like me.

I am a mathematician, and I can say definitively...

No calculators. Just like you said, the ability to spot a wildly incorrect answer is valuable. I would venture to say that the entire point of taking, say, a low-level trig or college algebra course is to learn to spot preposterous answers.

After all, the folks who aren't going into STEM will be using Excel or calculators if they do any math at all. So either teaching them mathematics is unnecessary in the first place (which is a universally unpopular idea) or the only point is to teach them to do it with pencil and paper. Otherwise, the class would be called "Excel 101" not "College Algebra." They can take basic computing if they need to learn how to use a calculator or Excel.

This gets down into what the point of mathematics education is. The vast majority of people who use math on the job learn it on the job, or use tables to look things up and never do their own calculations anyway. We aren't teaching people math to teach them to do their own calculations, but to teach them where all that information comes from, and to teach them how to think logically and precisely. You can't learn that with a calculator.

No calculators. Just like you said, the ability to spot a wildly incorrect answer is valuable. I would venture to say that the entire point of taking, say, a low-level trig or college algebra course is to learn to spot preposterous answers.

After all, the folks who aren't going into STEM will be using Excel or calculators if they do any math at all. So either teaching them mathematics is unnecessary in the first place (which is a universally unpopular idea) or the only point is to teach them to do it with pencil and paper. Otherwise, the class would be called "Excel 101" not "College Algebra." They can take basic computing if they need to learn how to use a calculator or Excel.

This gets down into what the point of mathematics education is. The vast majority of people who use math on the job learn it on the job, or use tables to look things up and never do their own calculations anyway. We aren't teaching people math to teach them to do their own calculations, but to teach them where all that information comes from, and to teach them how to think logically and precisely. You can't learn that with a calculator.

No way, Dean Dad!

I DO teach math, and calculator use in the high schools are a HUGE problem!

One important reason is that many (a LOT, not just a few) students learn to depend on them to the extent that they do not know the multiplication tables -- I mean, they cannot multiply two times four without a calculator. If they do not know their multiplication tables, they will hit many intractable snags in algebra. The square root of eighteen? Factoring quadratic equations? Students are totally incapable of learning to do these and other basic tasks if they have depended on a calculator their whole lives.

Students who can't manage fractions and can't add negative numbers and don't know their multiplication tables? Those are the ones flunking your developmental math classes -- and if they don't learn these basics, they will NEVER make it through algebra.

No! No calculators!

I DO teach math, and calculator use in the high schools are a HUGE problem!

One important reason is that many (a LOT, not just a few) students learn to depend on them to the extent that they do not know the multiplication tables -- I mean, they cannot multiply two times four without a calculator. If they do not know their multiplication tables, they will hit many intractable snags in algebra. The square root of eighteen? Factoring quadratic equations? Students are totally incapable of learning to do these and other basic tasks if they have depended on a calculator their whole lives.

Students who can't manage fractions and can't add negative numbers and don't know their multiplication tables? Those are the ones flunking your developmental math classes -- and if they don't learn these basics, they will NEVER make it through algebra.

No! No calculators!

Funny About Money:

That's a genuinely strange thing. Straight memorization is something that cuts across talent for different disciplines. Hunh.

That's a genuinely strange thing. Straight memorization is something that cuts across talent for different disciplines. Hunh.

I decided to blog, extensively, about this topic. My thoughts on the specific topic of developmental math and calculators is here, while one about higher-level classes will probably appear sometime tomorrow.

Thoughts more specifically about calculators themselves are in two articles from earlier today that you can find from the link to my blog.

Thoughts more specifically about calculators themselves are in two articles from earlier today that you can find from the link to my blog.

I am well aware of who goes into developmental math, and it is most assuredly not students who have ever passed a genuine pre-calculus class, or even a genuine algebra class.

CSU gives you a pass out of remedial math with an SAT/ACT score of 530/23. UC gives you a pass out of placement at 700 on the SAT, and if you are in a humanities program you don't have to take any math if you received higher than a 600.

It defies all logic that community college courses have higher standards than the CSU or UC programs, which means that all of you going on and on about how you have five degrees in language or a law license but by golly, you just couldn't do a bit of math are really, really not whose under discussion here.

The real issue is why we are calling it "college" when students are taking eighth grade level math, and why we are suggesting giving college degrees eventually to students who can't even manage high school level algebra.

CSU gives you a pass out of remedial math with an SAT/ACT score of 530/23. UC gives you a pass out of placement at 700 on the SAT, and if you are in a humanities program you don't have to take any math if you received higher than a 600.

It defies all logic that community college courses have higher standards than the CSU or UC programs, which means that all of you going on and on about how you have five degrees in language or a law license but by golly, you just couldn't do a bit of math are really, really not whose under discussion here.

The real issue is why we are calling it "college" when students are taking eighth grade level math, and why we are suggesting giving college degrees eventually to students who can't even manage high school level algebra.

In considering goals for Mathematics Education at your school see the report "Crossroads in Mathematics: Standards for Introductory Mathematics before Calculus" produced by the American Mathematical Association of Two Year Colleges (www.AMATYC.org). I have been teaching mathematics for 22 years. Based on my experiences, their recommendations seem sensible and responsive to the changes in society and the economy over the last 20 years.

RJ from the first comment here again.

We didn't use calculators until part way through fourth form (O level/School Certificate level), and then only to speed up what we could already do.

What we used in school aged 3 or 4 upward were the wonderful Cuisenaire Rod

http://en.wikipedia.org/wiki/Cuisenaire_rods

http://www.canplay.org.nz/catalog/product_info.php?cPath=33&products_id=1514

We had a set at home and I loved to make patterns with them (although if you stood on the 1cm cube it hurt).

They make learning fractions easy too. Here's a simple way of learning fractions.

http://www2.nzmaths.co.nz/frames/Animations/cuisenaire.html

No coincidence that we four children (who played with cuisenaire rods and wooden blocks and puzzles) all easily coped with maths.

The other thing we did a lot of was baking - the "normal" way - 8 oz flour, 4 oz sugar etc - measuring by weighing or cup measures. Doubling cupcake and pancake recipes are the easiest way to learn maths! (We always cooked form ingrediencts, not pre-packs - maybe the lack of cooking from scratch in the USA is also affecting mathematic skills?)

We didn't use calculators until part way through fourth form (O level/School Certificate level), and then only to speed up what we could already do.

What we used in school aged 3 or 4 upward were the wonderful Cuisenaire Rod

http://en.wikipedia.org/wiki/Cuisenaire_rods

http://www.canplay.org.nz/catalog/product_info.php?cPath=33&products_id=1514

We had a set at home and I loved to make patterns with them (although if you stood on the 1cm cube it hurt).

They make learning fractions easy too. Here's a simple way of learning fractions.

http://www2.nzmaths.co.nz/frames/Animations/cuisenaire.html

No coincidence that we four children (who played with cuisenaire rods and wooden blocks and puzzles) all easily coped with maths.

The other thing we did a lot of was baking - the "normal" way - 8 oz flour, 4 oz sugar etc - measuring by weighing or cup measures. Doubling cupcake and pancake recipes are the easiest way to learn maths! (We always cooked form ingrediencts, not pre-packs - maybe the lack of cooking from scratch in the USA is also affecting mathematic skills?)

For statistics, one important drawback of computing by hand or even with a hand calculator is that you can only look at very small data sets.

Moderately large data sets have qualitatively different behaviour in some important ways

- the law of large numbers and the central limit theorem really work

- graphs can't usefully make every individual point visible

- differences can be convincing without being visually obvious

Whether or not you want people to do some old-fashioned calculation as well (I'm not convinced), you really should also include some larger data sets that require computers.

Moderately large data sets have qualitatively different behaviour in some important ways

- the law of large numbers and the central limit theorem really work

- graphs can't usefully make every individual point visible

- differences can be convincing without being visually obvious

Whether or not you want people to do some old-fashioned calculation as well (I'm not convinced), you really should also include some larger data sets that require computers.

I am presently fighting the high school in my town. This reliance on a calculator isn't acceptable. Has anyone looked at the tests for college admission? Financial Planner, Medical College, Military, Business school,Pharmacy Tech,GED,etc. No calculators allowed.

Is everyone proficient in Math? I doubt it. Look at what happens in the supermarket when a cashier is given the extra 2 cents to round off the total? They don't know what to do. I am researching what jobs won't allow calculators. Does anyone know some? Shame, shame on our schools. Mrs. C.

Is everyone proficient in Math? I doubt it. Look at what happens in the supermarket when a cashier is given the extra 2 cents to round off the total? They don't know what to do. I am researching what jobs won't allow calculators. Does anyone know some? Shame, shame on our schools. Mrs. C.

K-12 math does prepare students for calculus and higher math.

At last count, I believe it was all of 8% of students that passed Calc 1.

Good job.

George DeMarse

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