Far be it from me to reignite the Math Wars of the early 1990s, but I found this article–and the underlying debate–so interesting that I decided to share it.
The question is, when should children use calculators for solving math problems?
Thoughtful people are on opposing sides. On one side are those who say that students learn to do the calculations themselves, without the aid of a device, or the device will do the work for them, and the students won’t understand the mathematical principles. On the other are those who say that people have created and used devices like the abacus to make the use and learning of mathematics more efficient.
I have no opinion since this is not my field. I am glad I learned the times table many decades ago, and it sits securely in my head. But my anecdote is just an anecdote.
Math teachers, what do you think?
Diane Ravitch's blog 
Why learn to print the letter A, when it is easier to point at it on an iPad like a monkey.
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As a regular ol’ tax-paying John Q Public here – and with a high school senior, and high school freshman, and third grader, I definitely see the value of learning times-tables and being able to do basic calculations or accurate generalizations/approximations (what’s my gas mileage, or which is the best deal in the soap aisle at the store).
And indeed, an adult using a calculator is still dependent upon knowing what numbers to work with, and what to do with them.
So – I’d NOT allow calculators up to middle school, and then allow them in the upper grades – with extra-credit non-calculator problems sprinkled here and there on quizzes or tests.
I mean, good heavens! – school districts are busily buying laptops and i-pads and all the rest; I surely wouldn’t get worked up about calculators!
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BAD IDEA : HEAVY RELIANCE ON COOL CALCULATOR TRICKS ON THE SAT
Some of the more expensive calculators out there can solve algebraic equations for x. This is, admittedly, a pretty cool trick, but I’ve found that students with calculators like this tend to think it gives them a bigger advantage than it really does. And sometimes, that turns the calculator into a disadvantage.
If your calculator is on the College Board’s acceptable calculator list, that means the SAT folks don’t think it’s got too much firepower. This should tell you something.
The “solve” command is cool, but really, the SAT doesn’t ask you to simply solve algebraic equations for one variable all that often. Rather, it’ll ask you to solve for one variable in terms of another, or figure out which two algebraic expressions are equivalent to each other using some simple set of rules, like exponent rules, or factoring the difference of two squares.
Students with these high-octane calculators spend an inordinate amount of time trying to wrestle SAT algebra into a form that they can feed into their “solve” functions. If you find yourself doing that, then you might be using your calculator to your detriment.
SAT algebra is not generally time-consuming—do it by hand. Limit your calculator use to graphing the occasional function, and speeding up your arithmetic.
Thanks to: http://blog.pwnthesat.com/2013/01/5-bad-sat-prep-ideas.html#.UmLPyHDUkhw
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I have taught mathematics to elementary students, middle school students, high school students, and college level students over a period of 44 years and I have changed my opinions several times as technology has evolved and what we expect from our students and work force. I am currently a Co-Chairperson of an IEBC mathematics professional learning community comprised of mathematics teachers at all levels from K through university level professors, including Deans and County level mathematicians. We discuss issues such as the proper use of technology at each grade level and opinions are across the spectrum, but I see a trend toward doing more with technology as a tool for solving complex problems starting at the middle school level and many are now arguing that if a student has not learned their basic math facts by the end of middle school, then why not allow them use of the same technology they will have access to in their real world occupation. They will have computer systems at places like McDonalds or Macy’s if they work as a checkout person and certainly they will use computer systems and other technology if they are engineers. For years, my two brothers who were electrical and chemical engineers asked me why we do not allow high school students the use graphing calculators. I believe we are finally coming to the conclusion that if students have not mastered the basics by a certain age, then it is not a good use of time and resources to continue trying to force them. The hope is that they can learn to accomplish any necessary future tasks required of them with the support of new technology and thereby, have a career where they can reach their potential as a contributing member of society.
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Thanks for bringing up Mathmatically Correct. When California was seriously, stupidly, considering Ebonics and Stupid Math and there were State Board of Education hearings at LACOE in L.A. County every educrat, principal, teacher there said Stupid Math was the best thing ever. There might have been 10-12 of us there on the other side being led by Mathmatically Correct. Thank you Mathmatically Correct, because, if they had not been there in force with the rest of us the State Board might have gone with the educrats who were as crazy as you can get. It was stopped.
Call Annapolis and talk to the Dean’s Office as I have and here is what they say “When your computer on your gun goes down in a fire fight you die unless you can do the aiming of the gun by hand.” At the top private school in Silicon Valley computers and caculators are not allowed. Do you think I could do what I do only with a caculator and computer. NO WAY. Long ago there were no such thing. I did not have ball point pens until high school. I remember the day my dad, a top engineer at Lockheed whose boss, and mine later also, was Kelly Johnson, came home and told me about the first engineer with a hand held caculator like the one we buy now at the 99 Cent Store for $.99 each for $180 then $1,000 now. Until then it was slide rules. How did we do it without computers, caculators, airconditioning and not even fans? Who are the wimps now? We are making them wimps physically and mentally. Wrong answer. If you cannot do it by hand you do not get a caculator or computer as they are only tools. I used to have to do it all by hand when there was no internet or PC’s. Serious advantage to me and anyone else who can work by hand and not need those tools. They are only tools and nothing else. Just like I can fix a car with my mind and the simplest of tools without a problem. Maybe, art has something to do with creativity. After all, for 37 years that is how I made my living in my own business.
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I urge every viewer of this thread to zero in on this particular comment by George Buzzetti: “At the top private school in Silicon Valley computers and ca[l]culators are not allowed.”
He makes other good observations so this particular one may go unnoticed. It is critical to understanding what “education reformers” see as appropriate schooling for THEIR OWN CHILDREN and what they consider appropriate [and mandate] for OTHER PEOPLE’S CHILDREN.
Link: http://www.nytimes.com/2011/10/23/technology/at-waldorf-school-in-silicon-valley-technology-can-wait.html?pagewanted=all&_r=1&
For those who feel I have oversimplified, please click on the following selected links:
Link: http://www.ucls.uchicago.edu [re Mayor Rahm Emanuel]
Link: http://www.lakesideschool.org/academics [re Bill Gates]
Link: http://schools.cranbrook.edu/home [re Mitt Romney]
Link: http://www.delbarton.org [re Governor Chris Christie]
Link: http://www.sidwell.edu [re President & Mrs. Obama]
Regardless of particular instructional and technological offerings—and everything else including the kitchen sink—when education rheephormers say “choice” they mean they get to make substantive and real choices for THEIR OWN CHILDREN; for OTHER PEOPLE’S CHILDREN, they want to mandate and limit choice to the rheeal options they think we deserve.
That is, the option to learn low-level skills and docility and obedience at the nearest Centre of EduExcellence.
A “better education for all” with or without calculators and the like? Irrelevant. It’s all about $tudent $ucce$$.
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Thank you Krazy one. That is a very salient point.
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I’ve taught 6th grade math the past 9 years. Students add, subtract, multiply and divide decimals, whole numbers, fractions, mixed numbers and fractions greater than one w/out calculators. They couldn’t use a calculator on the CST, so I didn’t let them use one in class. In the spring, though, when we get to data, statistics, and probability we use calcs. Students need to be able to mentally compute basic math facts w/ automaticity. The debate is what is automaticity for each grade/age/student? I’ve used calculators to help students understand division to relate decimals and fractions.
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In grades 1 through 4, students should learn to add, subtract, multiply and divide. These skills advance through the introduction of fractions and decimals in the later primary school years, but only incrementally.
In secondary school, if mathematics problems are such that they rely on, or are reduced to calculations, then the problems are really primary school math, and are not grade-appropriate. Unfortunately, this is what the typical American classroom usually offers: dumbed down mathematics. The calculator debate only exists because students continue to be asked to do low end calculations.
If mathematics properly advanced into more difficult (and grade-appropriate) problem solving rather than continuing with rote calculations, we wouldn’t be having this discussion.
Here are but two examples of “real” problems where advanced thinking is necessary and having a calculator is irrelevant:
http://fivetriangles.blogspot.com/2013/10/109-lined-up-triangles.html
http://fivetriangles.blogspot.com/2013/10/104-equal-area-proof.html
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In the first question I think the given area should have been 36 instead of 60. The second question should read, line segement EF is parallel to… IMHO.
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In #109, the total area doesn’t matter because the ratio remains constant, so you may choose any number to start. 60 or 36 or something else, the choice is arbitrary.
In #104, you may reword the question as you choose; it does not alter the underlying proof. We worded the question to adhere closely to the wording in the problem’s original language.
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Yes, thanks, I enjoyed those examples and agree with your point on concepts vs. calculators. As for the 36, I like whole numbers so if 36/9 x 1.25 = 5. The second question I had to read several times to figure out what it was saying, so I thought naming the line segement is easier to understand.
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Marilyn Burns has a good story, in Math: Facing an American Phobia, about using the calculators to help explore a topic that would have been beyond the students otherwise. They were exploring decimal answers to a division problem I think, and were not yet ready to deal with the procedures involved. (Maybe in third grade?) She was helping them to make logical connections between multiplication and division, and helping them strengthen their number sense.
I read the book many years ago, and don’t remember all the details, but what stands out for me is that playing around with a calculator, and asking yourself (or your students) interesting questions, can be a great way to learn. Using the calculator because you don’t trust yourself to multiply 20*30 is a big problem. I teach at community college, and often see my students pulling out calculators for questions they should be doing in their heads. I talk often about mental math, and get them to play along.
The problem we face is not whether or not to use calculators, but how to make all math classes a time for real thinking.
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In grades 1 through 4, students should learn to add, subtract, multiply and divide. These skills advance through the introduction of fractions and decimals in the later primary school years, but only incrementally.
In secondary school, if mathematics problems are such that they rely on, or are reduced to calculations, then the problems are really primary school math, and are not grade-appropriate. Unfortunately, this is what the typical American classroom usually offers: dumbed down mathematics. The calculator debate only exists because students continue to be asked to do low end calculations.
If mathematics properly advanced into more difficult (and grade-appropriate) problem solving rather than continuing with rote calculations, we wouldn’t be having this discussion.
Here are but two examples of “real” problems where advanced thinking is necessary and having a calculator is irrelevant:
http://fivetriangles.blogspot.com/2013/10/109-lined-up-triangles.html
http://fivetriangles.blogspot.com/2013/10/104-equal-area-proof.html
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Where’s the answers?
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Is the first answer 20 square cms?
Not sure why but that’s about what it looks like.
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I too am not a math teacher. However, the purpose of education is for people to learn to use their complete brain power, in all its many aspects. Perhaps over reliance on gadgetry may get in the way of that development, especially in the developmental stage of human existence.
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Isn’t going back to the 90s exactly what we need to be doing? Isn’t that when the ideas now prevailing began to gain momentum?
I think we should totally have bold conversations that did not transpire in the 90s (and could have, perhaps should have) now. Yes yes yes. We have to find the mis-steps.
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Using calculators from early age doesn’t allow a good number sense to develop. Children (and adults) need a healthy number sense to be able to quickly and correctly estimate the expected results of simple arithmetic operations, be it while driving, standing in the checkout line, or solving a chemistry problem. Calculators are helpful to deal with unwieldy numbers quickly and accurately, but K-8 math doesn’t need to deal with unwieldy numbers — unless it is trying to *force* the use of calculators for ideological reasons. Unwieldy numbers mostly start showing up in science and labs in high school and that’s when calculators start being needed. Most of HS math doesn’t need calculators either, with only a handful of potential exceptions.
The abacus example is a bit misleading. One needs to have a good number sense to use abacus effectively, somewhat similar to what we used to need for using a slide rule. In contrast, you can punch *any* number in the calculator and it will (mostly) not complain of typos and just produce an erroneous number. Very dangerous unless one knows the approximate result to expect.
Finally, just as a reference, most high achieving countries (South Korea, Singapore, Japan) don’t allow calculators in K-8 math classes, sometimes even in not HS math (but do allow them in HS science).
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Relatively few people have argued the most extreme pro-calculator position: calculators on demand for all kids regardless of age. And I don’t recall seeing too many teachers, if in fact any, doing that in elementary grades. But I simply cannot concur with George Buzzetti’s view of Mathematically Correct. And I find it difficult to have a productive conversation about these issues with someone who, like the main players at MC, insists on using such pejorative language about the ideas and people on the other end of the spectrum. What does it accomplish to refer to “Stupid Math”? And what does “Ebonics” have to do with a discussion of calculator use? Those are inflammatory terms used politically by people with very combative points of view. I have, over the last couple of decades, tried various approaches to engage people from MC, NYC-HOLD, etc.: from polite disagreement and question-posing to fighting fire with fire. In my experience, nothing works. And that’s why we’re still stuck in a mire when it comes to mathematics education, one that the Common Core is not going to get us out of, because the fundamental conflicts that arose in the ’80s and ’90s have not been settled in any way, and I see no provisions in CCSSI to meaningfully bring some sort of closure to the “Math Wars.” (And so, Diane, there’s no danger of ‘reigniting’ something that has never ended).
I think that a fair minded person looking at the issues would see a lot of space for balance, be the issue calculators or lots of other things. In the Common Core math Practice Standards, we find at #5:
“Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.”
There is nothing in that statement that specifically prescribes or proscribes tools by grade level, and I think that’s a good call. Instead, the issue becomes what proficient students do, and here the key words are “Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.”
Of course, teachers have to be able to make sound decisions about tools as well. I don’t see how that is possible if these issues are addressed by closed-minded people who reject out of hand CERTAIN technology with epithets. Nor will sound decisions be made by people who simply say, “Anything goes, all the time,” because that is not considering the serious issue of “appropriateness.” We’ve all heard “horror stories” about students who grab for calculators when asked to compute, say 23 x 10. For nearly any numerate person, that would be a clear-cut example of inappropriate use of a tool. Going further, I’d argue that for most kids, grabbing pencil and paper to solve that problem would ALSO be a clear-cut example of inappropriate use of a tool. By some point in elementary school, students should know and understand place value well enough to be able to conclude from mental arithmetic that 23 x 10 = 230, and it’s a distinct advantage to be able to do so.
But as calculations become more intricate, numbers get very large, non-integer numbers come into play, the lines become more blurred as to what tool(s) make sense.
If I’m asked to find an approximate answer for the circumference or area of a circle with radius of 3 units, there’s a good chance that it will suffice to for me to answer 18-20 units for the circumference and 27-30 sq. units for the area, simply by knowing that pi is slightly greater than 3 and knowing the appropriate formula for each (2*pi*r and pi*r^2, respectively). It’s silly to drag out a calculator with a pi key to crunch those numbers.
But where do we draw the line? I’m quite adept at mental math, but there are times when I don’t choose to use it: I may be tired, ill, or just feeling lazy. I think we should educate students to be well-positioned to make informed, reasonable choices like that. But I don’t think you get those sorts of kids by making calculators part of something called “dummy math” and yelling at or humiliating students who make poor choices.
There has long been snobbery and elitism associated with mathematics that I find offensive and counter productive. I believe groups like MC and NYC-HOLD help promote those attitudes, and they have done it through their rhetoric and their uncompromising attitudes. I’m sure they believe they are completely justified in their actions and writing. Maybe they believe they started out being reasonable and accommodating and met inflexible opposition that drove them to becoming so harsh and adamant. But regardless of who cast the first stone in the Math Wars, there is no question that they have thrown quite a large number of them. Calculator use is merely one of many issues in this decades-long conflict. It serves as a microcosm for how these wars have been conducted.
I still believe that one side has been far less flexible, far less accommodating, and far more combative in addressing these important ideas and policies. I think George’s comments help highlight why it’s hard to find a middle ground (and why CCSS will NOT settle anything). I coach teachers to think before they hand out calculators and I tutor students to think before grabbing for electronic (or even pencil-and-paper) tools. I’ve used many examples to try to give both teachers and students (and parents, if they’re interested) good examples of where calculators may be a waste of time, and where they may provide a host of insights that students would have difficulty gaining without technology. I know that I really appreciate having spell-check even though I’m a much better than average speller. it’s an excellent time-saver. But I like knowing that I can spell a lot of words without needing “help.” Why calculators and other electronic technology evoke such harsh reactions may be a much more philosophical and political question than a mathematical one.
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Well said. I would add that it’s not rote learning versus calculators. It’s also making sense of numbers. For example, 5 + 6 is not just 11, it is also one more than 5 + 5.
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@Detroiter: At this point, I’ve given up believing that it’s possible to reconcile with some folks on all these issues. They appear to me to always be spoiling for a fight.
I’ve never been a fan of memorizing through rote repetition IF there are good alternatives. And often there are. Much depends on what it is I’m trying to memorize. Mnemonics are lovely for ‘arbitrary’ things like the order of the cranial nerves, given that there’s no logical reason that I know of that they should come in the order that they do (plus of course, the names change from language to language, and those words are arbitrary, too). I’m hardly alone in thinking that FOIL is a mnemonic that we’d all be better off without, particularly as it leaves a lot of students completely stumped about multiplying everything except a binomial by a binomial. Seems like a pretty stunted mnemonic.
Then there’s the issue of rote (as opposed to memorizing SOME way or other). Drilling the addition and multiplication tables certainly appeals to a lot of folks, but I think there’s reason to suspect it’s not necessarily the best way for a lot of kids. The approach John van de Walle takes for helping students achieve ‘automaticity’ with those tables strikes me as more thoughtful (it builds on simple facts some students already know, such as doubling, one more than, two more than, etc.; plus commutativity, to give students not only ways to play with numbers while gaining recall, but also builds the kind of number sense you mention in your comment. Since I think that is invaluable, I’m in favor of approaches that are likely to help kids gain that sort of familiarity with numbers.
A mathematician I had for a Calculus 2 class in NYC was the first to reveal to me that it’s not necessary to memorize nearly as much as I had imagined. It was liberating to see that a professional felt free to pick and choose which facts he felt were essential for him to know, and which he felt comfortable deriving. In my own experience, things I need to know and use a lot generally “stick” sooner or later.
I understand the fear that giving students calculators “too early” means that they will never learn their basic facts. So then we must ask: what about students keeping fact tables and being allowed to consult them? Is that better than allowing unlimited calculator access? Worse? Pretty much the same? Is there some clear difference in the likelihood that one of these will help students gain full mastery of those tables over time?
I’d mention the research into calculator use and its impact on students’ mathematical learning in higher grades, but of course that all gets dismissed by calculator foes except for the studies that show problems. That has been a consistent problem in the Math Wars, one that we see carried over into the Education Reform/Deform Wars of today: it’s hard to find agreement about what research should be heeded. Reading about the battles over phonics-centered reading instruction vs. “whole language” approaches in the ’80s and ’90s, I’m seeing that some highly questionable research results were used to shift the entire direction of literacy instruction in this country. Of course, those who favor phonics over all other approaches will disagree.
Not a lot gets settled, unfortunately, and when there’s a strong swing in a particular direction, it generally means that political forces have successfully pushed policy in that direction, as happened with reading under George W. Bush and Reid Lyon.
So now we have battles about the Common Core, but the oddest thing seems to be taking place: people who disagree about a host of basic issues in math and/or literacy education are finding themselves on the same side of this latest policy initiative. Ze’ev Wurman and I aren’t on one another’s holiday card list, yet we both oppose the Common Core. Go figure. 😉
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If the child really understand the concept of the Place Value System….they would be able to convert 5 + 5 to the Base 10 to any Base…with ease….
Base 2…..11 base 10 = 1011 Base 2
Base 3…..11 Base 10 = 102 Base 3
etc
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The problem I ran into with calculator use arose in teaching middle school LD students. They were used to doing everything with a calculator. The problem is they had no idea what or why they were doing. Through the years, well meaning teachers had handed them a calculator when they fell too far behind the rest of the class. It did not matter if they understood what they were doing; it only mattered if they could get the right answer. The problem was that if they accidentally punched the wrong key they could not tell that the answer was wrong. They did it on the calculator, so it must be right! We spent a lot of time on estimation skills, so they could at least recognize if they were in the ball park. It broke their math phobia because they didn’t have to have the exact answer. I also made them play calculator games nightly to make retrieval of basic facts automatic. One little girl was diligent in her practice and amazed the others when it became obvious that she was mastering a task that they all thought was beyond them. Her fluency allowed her to focus on a slightly more complex operation without having to spend mental energy or calculator time on simple operations. My job as a special education teacher allowed me to take the time to differentiate instruction for these kids. They did need my services. My concern would be for those students who don’t learn quite as quickly relying on memorizing calculator operations because they had not had the time they needed to internalize the concepts. I saw that regularly in the time I spent supporting students in the mainstream classroom and tutoring. When to allow the use of calculators is not always an easy call. I think your position, Mike, recognizes that ambiguity.
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I would like the constants in a line to be the same for a quadratic equation. This is something that chafes on me day in and day out.
Bx + A and Cx^2 + Bx + A ? or
A + Bx and A + Bx + Cx^2 ?
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Ha, I made that gigantic comment before I clicked to look at the article Diane was taken with, assuming it would be anti-calculator. Silly me. ;^)
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Silly you indeed! 🙂
But not so much because you didn’t check the referent, but because you still think that “Relatively few people have argued the most extreme pro-calculator position: calculators on demand for all kids regardless of age.” Everyday Mathematics argues for availability of calculators from Kindergarten and up, all the time. And it has (had?) the largest classroom penetration in the country. So it is not about “relatively few.”
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Ze’ev, try to keep the personal comments and ill-will in check.
Can you show facts to support your claim about EM’s use? And as many people know, teachers don’t follow textbooks blindly, for the most part. In most, if not all of the elementary classrooms I have coached in where EM was the textbook, teachers chose when to distribute calculators. I don’t claim any more than that, but at least I have been in real K-5 classrooms to observe. How about you?
But the issue STILL is: are students taught to make appropriate decisions about using tools? Isn’t that what we all want? Or is there nothing we all want? Can someone oppose the Common Core, as I do, and still see good points in it, or is the world black or white? I think I am still waiting to see someone from the MC/HOLD camp give a nuanced, mixed review, but maybe you can point me to something I missed.
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This is illustrative: I teach 5th grade, we are trying to solve for mean on a science experiment. The division was with 4 trials into a 3-digit number, which we are past in math class, so I used it as practice for math to NOT use calculators. When it came to finding the class mean, the calculators came out. It gives good opportunities for learning. But I do think learning to solve by using brain power does develop brain power. I also agree that even a reliance on algorithms has problems, such as the example of 20x 30 given on a comment here. Pet peeve of mine too.
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If one of my students reaches for their calculator to sketch the graph of, say, 2^x +3, I give them a hard time because they should be able to do that with their understanding of transformations. However, if we are finding all solutions of a polynomial, why would they want to test all possible rational roots when we can simply find any rational roots by graphing and then use synthetic division to find remaining real and non real roots? My point is, technology is absolutely appropriate in some cases and quite inappropriate in others. Our challenge as math teachers is to help our students determine when it is appropriate or not. I often have a calculator portion and a non-calculator portion to my assessments. I want my kids to understand that calculators definitely have their place, but being a capable mathematician requires a certain amount of mental fluency.
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Calculators are tools. The proper use of tools needs to be taught. The aim of math class should be to get students to think. It is the teacher’s job to ask students questions to probe their thinking. In the above example, the teacher needs to ask questions about the interpretation of slope and Y-intercept which is the important concepts when graphing.
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Absolutely
Meaning of Slope….not just find the slope..
Easy real world problems….
Snowing 3 inches per hour(We wish)
Melting 2 in per hour….etc…
Understanding the concept is mental math.
Make the first problem real world and discovering the positive and the negative rate of change…..
then count…. vertical change..horizontal change…all with easy but meaningful computations…
So when do we use the calculator???
After we understand the mental process..
I would definitely want my brain surgeon to have exact measurements when performing an operation….calculator needed..
If estimating the time when my 6 foot snowman will be Frosty in a Puddle…..no calculator needed…
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I would go with Wolfram Alpha, not a calculator.
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I did not allow my children to use calculators until Algebra I. (Of course, there were a couple of exceptions along the way, such as when a teacher would want them to provide a decimal approximation for a square root.) I didn’t base this decision on theoretical arguments, but rather on the unfortunate outcomes reported by friends and relatives with older kids. Perhaps calculators COULD be used to good effect in the early grades, but all too often they are used in ways that are detrimental for budding math students. When my kids reached high school, they thanked me for having protected them from the dreaded “calculator brain” they have witnessed but not experienced.
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The so-called “principles” that children are supposedly not learning when they use calculators are mostly just arbitrary algorithms. The principles of mathematics are many many years in the future for children, and few ever encounter them.
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I agree with this, and I think that is a problem.
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As a high-school maths and statistics teacher of ten years experience, I find myself in agreement with most of what Michael Paul Goldenberg says. My first shock when I started teaching was to find classes of high school students (i.e. ages 13 to 18 in New Zealand) none of whom could (or at least, would attempt) to calculate 9 x 7 without reaching for a calculator. The second shock was to find senior high school maths classes who at eighteen years of age were still practising basic calculation skills without using calculators because of the belief that until they acquired these skills no further progress was possible. (These poor students had been tortured in this way for years, and had developed a healthy loathing for anything mathematical as a result).
Over the years I have tried to develop a ‘middle ground’ approach to this question. There is no age or stage of learning at which it becomes acceptable to introduce calculator use. It is a question of how they are used. The extreme pro- and anti-calculator camps usually fall into the error of assuming that learning maths is a simple linear progression: you can’t get from A to C until you have got past B first. It’s just not that simple.
The analogy I like to use is driving a car. A car can be a useful means of transportation. You need to know how to drive it, and you need to know where you’re going. But you don’t need to know all the details of how the internal combustion engine works in order to use it, preferable though that might be. At the same time, over-reliance on the car, when walking would be a better means of transport, can also be a problem.
If students in my classes need to calculate 4.7 x 10 in a hurry, and I see them reaching for a calculator, I make a big drama about it. There are lots of dramas of this kind in my classes. By the time they leave, they know how to say “Estimate it first! Estimate it first!” if nothing else.
On the other hand, teaching of Statistics has been transformed by the use of statistical packages. In my day, high school Statistics was unbelievably boring – because I never got past the tedious graph-drawing and calculations to see the statistical meaning. These days, you can throw up a scatterplot in an instant, and add the regression line in two seconds. I felt a little nervous about this at first, and made sure I spent an hour or so making sure they understood exactly how the least-squares regression line is produced. These days I don’t both with that. For the students who like to know where these things come from I point to a useful source, and for the rest I consider it is just not important. The important thing is for them to read the statistical meaning in the scatterplot.
As a result, the statistical thinking these students produce is on a higher plane altogether than anything I ever did at high school. And that is even true of those students among them whose impulse would still be to reach for a calculator when they need to calculate 9 x 7.
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One of the full professors in my department tells a story of a student asking for a calculator to simplify a fraction just to make sure that he was correct. The fraction was 120/1.
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That is exactly what has happened..
Try to get them to add a few numbers without a calc….
Impossible…
Some students can not figure 1/3 of a dollar………no kidding..
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In a nutshell!!!
Estimation = Mental Math = needs to be developed before someone is handed a calculator..
Calculators do serve their purpose..
As I said….a brain surgeon needs to be absolutely exact…..as do the scientists who puts the rovers on Mars….
Mental Math should be in the curriculum…but it is not….nowhere…
These people (who have been on a classroom of the top 2% who can figure it out for themselves and have never taught a low performing student) need to bow out of the Core Math Standards..
They do not know what the H*ll they are doing…
Give me a minute and I will have all of you figuring it out in your heads.,….even the logarithms…
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Calculators do not think! Do you know there are 31 possible places to make a mistake in A 3-digit x3-digit problem! If a child calculates that 8*7=54, three times on the same test, does he/she get 3 different problems wrong?
It is the student who must think through a math procedure and enter the information. This is the heart of Mathematics, not the memorization of facts. Yes, teach what the four operations mean , and the concept behind them, but don’t hold our kids hostage to the “tables”.
Math phobics are the products of poor Math teaching.
Chris Cain, 30 year elementary Math.
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You are kidding??
The multiplication facts are important……..very……..
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It all depends, of course. We all want children to be fluent with their math “facts” (though why 2×3=6 is a fact and .2x.3=.06 is not leads to a fruitful discussion). On the other hand, we do not need to drill fluency on decimal division with 3 and 4 digit divisors. No employer–of a clerk, an accountant, or a civil engineer–would dream of allowing the unverified use of such a calculation done by hand. Humans, even well-trained, just aren’t as reliable as calculators.
As a math teacher for 30 years, my job was to help students gain fluency with the decimal number system (which includes memorizing more “facts” than required for the older Roman number system) AND to become competent at using–and interpreting–mechanical aids to computation such as calculators. Whether we used calculators varied with the task at hand.
The bottom line is that TEACHERS need to decide–day to day–when kids should have the opportunity to use calculators. Telling me thety can NEVER use them, or that they MUST use them, lessens my ability to teach, and robs the students of opportunities to learn.
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As a high school science teacher, I am firmly against calculators. When I have 11th graders who can’t do 35 divided by 7 without a calculator, I’m just sad. Not all memorization is bad.
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Having only taught mathematics in grades k-3, I am of the opinion that children should not be using calculators. As others have already stated, kids in these grades need to develop their number sense. Addition and subtraction using ten-frames and number racks, place value blocks (hundreds, tens, and ones) and the contraction of arrays using manipulatives help children develop a deeper understanding of the four operations. Later on calculators may be appropriate, but I will defer to math teachers of the upper grades on that one.
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Memorizing times tables 1 thru 10 is 55 combinations to remember. Compared to memorizing words meanings or spellings, or the Chinese alphabet, it is relatively little to commit to memory by comparison.
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8 x 7 = 7 x 8
Only learn it once…
Good Post..
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An interesting calculator story: my husband & I were at a popular steakhouse chain restaurant that was on its last day of a 20% off special. Naturally, the restaurant was packed with us last minute bargain hunters. As it was 7:30 PM, many had eaten and stood at the cash register to pay. Just then, the register conked out. The cashier became frantic–she sought out the manager to figure out the final bills for each customer. Guess what? The manager–a young man, say, in his twenties, brought his calculator. Uh-oh! The calculator didn’t work–dead batteries! Facing an ever-growing check-out line, the manager simply gave up, allowing each guest to tabulate his./her own bill (i.e., figuring out the 20% off tab). Some whipped out their calculators and figured it out. Those of us w/o them simply did the math. Bottom line, though–the management was forced to rely on the honesty of the crowd and, hopefully, didn’t lose any money that night. Like “2old2 teach,” I taught Math to LD students (Grades 4-8), and I worked them through the more basic calculations, encouraging them to use charts and other modifications that they could handle for the basic Math skills (+, -, X & division–which was the most troublesome). We played LOTS of visual & tactile games which involved thinking skills, making it more fun. My students were allowed to use calculators for more involved problems (such as convoluted story problems, 3-step problems, or any otherwise complicated problems).
Surprisingly, there was very little whining and moaning…and they LEARNED Math!
In fact, the majority of students met (or exceeded–1-2 Gifted L.D. students, for 2-3 years) passing scores on the state assessments every year.
And they LOVED the restaurant story!
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It was…used to be that no child got our of a class without knowing how to compute 10% mentally (and they understood the concept totally)
From there 5%
then 20% then 15% etc etc..
ALL using mental math and all low performers who were not really low performers just held back because of the Push Button Math…
Fractions …same as decimals…just different wardrobe..
Use it or lose it..
My students came back year after year thanking me for teaching them (HS students) to use their brain to estimate and to use the calculator to exactamate….(new word)
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I teacher ‘upper’ level mathematics – Calculus (upper level for high school but lowest level for college). I am a mathematician – not an arithmetician! Mathematics is very deep, complex, and often abstract. Those ‘concepts’ would be lost if a student had to get entangled in rigorous calculating. If it is going to take you 20 minutes to do a bunch of manipulating of variables across multiple disciplines (say algebraic and trigonometric) just to get a ‘number’ but the MEANING of what that number represents is lost – then there is no understanding.
By the time a students begins their real journey into mathematics in calculus, they will need a tool to graph (a picture is worth a thousand words) what they are studying and get numeric ‘answers’ so they can get to a deeper understanding of the ‘what’ and ‘why’ of the concept they are studying.
That being said… the use of these tools should start around the Precalculus level. Certainly elementary students should be learning without a calculator and gathering their number sense. The graphing calculator should be their first tool and they should get it introduced around Algebra I (still going back and forth between ‘by hand’ and ‘verify on the calculator’) and then increase it’s use as the depth of what they are studying increases.
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Push Button Math……There is a time ……but only after…..
Before students learn to read..
They learn their A-B-C’s
Before students learn to write
They learn the words from site
Before students understand
They must have the tools in hand
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“sight”…ooops..not site
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I know we never used them in school, and I did three college levels of calculus without them, but I use the pretty heavily at work (even for simple operations). Sometimes I feel like I start to forget some of the easier math I do because of using the calculators though. Technology will be an ever increasing presence in our lives and children do need to learn and use it, but I still think they would be served well by a minimalists approach to calculator use in the classroom.
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Calculators are tools and should be used wisely, consistently, and appropriately. Conceptual understanding is the key. Does the calculator hinder or help the conceptual development?
I am working on a doctorate in Elem Math. I have read and seen all sides of this argument. My short answer: it should be available from the middle elem grades (typically 3rd and above), and the teacher needs to balance and stress conceptual development and number sense, using the calculator to support understanding, not supplement it. (This also begs the question… do elementary teachers really understand how to do this?)
In my experience, when students are given the impression that the calculator is not allowed, ever, they overuse it later in school. I have seen countless the middle school students who, when they are finally able to use a calculator, use it for everything from 2+2 to 5*7+3 just because they can.
But, students who have had access (appropriate access) to a calculator from early grades to support higher order thinking and application problems that build concepts but who also have a good understanding of number sense, do better at using the calculator as a tool not a crutch.
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Another 2 cents: no one ever proved the value of calculators in improving mathematics education, long term numeracy or employment prospects, any of which might justify their use. Instead, calculators became ubiquitous in the classroom following sustained PR campaigns and marketing pushes by the manufacturers (read: Texas Instruments) that started in the 1970’s and continue to this day–attend any ed conference and TI will have a presence.
Ed “reform” in the US is often driven by the profit motive, not by a genuine concern for students’ welfare.
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Take away their iPhones… and throw them on the ground.
Your full time job isn’t monitoring your social life, your full time job is to learn.
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Education has become ALL about MONEY and selling stupid things to schools and students so that the students are not engaged in their own learning, but caught up in gadgets.
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As a psychologist who primarily spends time observing and evaluating children here is what i have learned. Children who know their basic math facts are more efficient at every level of math computation and application of algorithms. Most children have adequate recall of addition and subtraction facts and multiplication is a mess, because as they are learning those facts they are given calculators. Computation of complex information becomes inefficient, more steps are required to obtain a correct answer, and.students are less aware of their errors when they are tightly wedded to their calculators. Additionally, failure to have rapid, fluent access to basic math facts increases cognitive load and working memory demands, even when the calculators is available. Lastly, I have a horrible memory for just about everything. Though my third grade teacher, Mrs. Knettle, was cruel and unsympathetic, I am forever grateful for her demands that I learn those multiplication facts. It was very difficult for me, but I did it. What I learned from that was not only those important times tables, but also that , with effort, time and persistence, I could learn what I needed to know. Giving children calculators when they need to do some hard work is really depriving them of so much learning. Overall, I would say memorization, in general is given a bad rap. those of us who have taken the time to memorize poetry, or a piece of music know how satisfying it is to have that information readily available.
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To read some of the comments here, you would think that before the widespread use of calculators in elementary math classes, most kids went into middle school with a firm procedural grasp of basic arithmetic of whole numbers, and maybe even a solid understanding and mastery of the arithmetic of signed numbers and rational numbers.
Those of us old enough to remember (there were no calculators in school when I graduated high school in 1968) know perfectly well that things weren’t quite that rosy. Neither were most kids really good spellers before there were word processors, computers, the Internet, and spell-check built into many programs and applications. Nor did whole language retroactively account for the poor spellers of the years before it was practiced.
The nostalgia for golden days that were never nearly as golden as some people recall them or insist they were. My strong suspicion is that many such people are prone to generalize from their OWN experience of school, which may be very much informed by having been in higher-track classrooms where they saw little of the kids who struggled with mathematics all the way until graduation (if they indeed made it that far).
All that said, I still see little compromising going on. Some folks feel, as I do, that calculators (and other powerful tools for crunching numbers, graphing, analyzing data, and general playing around with mathematics over a rather wide range of topics beyond K-5 or 6-12) are tools that kids can be given, taught to use wisely and appropriately, and that intelligent instruction that includes access to these tools, but also a lot of mental arithmetic, estimation, and (dare I say it?) an eye towards understanding the reasons behind what they are learning. That means giving kids room to breathe intellectually, something that seems anathema to some people who insist that elementary mathematics is essentially about memorizing certain basic facts and procedures.
It seems obvious to me that it’s perfectly possible to have, for want of a better term, a balanced approach to teaching mathematics so that “the basics” are not ignored, but they do not become the raison d’etre for doing math in the elementary grades. Rather, they are part of the tool set kids acquire as they learn to think about mathematical situations that are problematic for their age/grade level. When it seems likely that children will benefit from using concrete objects to model an idea they’re exploring, they should have access to them. When even paper and pencil are “too much hardware” (e.g., when working on mental math), students should put such tools away.
No matter how many anecdotes are dragged out about restaurants with broken cash registers and calculators with dead batteries, innumeracy didn’t start in this country with calculators. Neither did math anxiety nor loathing for mathematics. There has been plenty of that to go around for well over a century. Until people begin to approach the issues here more open-mindedly and without the blinders of nostalgia and the tunnel-vision that can come from being in higher-track classrooms, we’ll be reading the same anecdotes and calls for “back to basics” and doing away with anything invented or thought of after about 1950 (although that might not be reactionary enough given that lattice multiplication, target of many complaints from educational conservatives, was in wide use in Europe until the invention of the printing press in the 15th century. You can look it up).
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Thank you for your thoughtful comments and your understanding of students who have always stuggled with memorizing basic math facts and algorithms and over time view themselves as losers and begin to give up or become frustrated and hate going to school. I have taught mathematics to elementary students, middle school students, high school students, and college level students over a period of 44 years and I have changed my opinions several times as technology has evolved and what we expect from our students and work force. I am currently a Co-Chairperson of an IEBC mathematics professional learning community comprised of mathematics teachers at all levels from K through university level professors, including Deans and County level mathematicians. We discuss issues such as the proper use of technology at each grade level and opinions are across the spectrum, but I see a trend toward doing more with technology as a tool for solving complex problems starting at the middle school level and many are now arguing that if a student has not learned their basic math facts by the end of middle school, then why not allow them use of the same technology they will have access to in their real world occupation. They will have computer systems at places like McDonalds or Macy’s if they work as a checkout person and certainly they will use computer systems and other technology if they are engineers. For years, my two brothers who were electrical and chemical engineers asked me why we do not allow high school students the use of graphing calculators. I believe we are finally coming to the conclusion that if students have not mastered the basics by a certain age, then it is not a good use of time and resources to continue trying to force them. The hope is that they can learn to accomplish any necessary future tasks required of them with the support of new technology and thereby, have a career where they can reach their potential as a contributing member of society.
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Thank you, Mark. I saw this comment before, either here or maybe on my blog, but hadn’t gotten around to replying. As you can see, the volume of comments on Diane’s blog has been significant.
I said to someone on my blog that I believe the fault lies not in our calculators but in our mathematics teaching (always like to work Shakespeare in when possible, even if I have to cheat a little to do so). I still think that’s the deeper issue, and I find the calculator debate just one of many sub-controversies that – taken together – reveal a great deal about people’s beliefs about teaching, learning, children, and much else, and not merely in terms of mathematics.
As I continue to argue, an effective program for teaching and learning mathematics in K-12 that is serious (and playful), rigorous (at appropriate levels for a given grade and a given student), and powerful is not only possible with the inclusion of the intelligent and appropriate use of technology, but it seems a bit bizarre at this point not to be seeking that kind of approach. You might find it useful to Google the name “Kirby Urner” for an example of one person who has some intriguing ideas (and actual practice) about these issues. He’s based in Portland, OR. There is also a book by Gary and Maria Litvin about teaching mathematics with Python programming, but it’s pitched a little higher than early elementary students. I wouldn’t be too sure, however, that someone hasn’t tried to extend it to earlier grades.
I wouldn’t wish to put words into other people’s mouths about whether they think we should keep tools out of kids hands until a specific age/grade or not (and by the way, Kirby thinks calculators are way out of date and not worth investing in or teaching; that’s not because he’s more conservative than the anti-calculator crowd, however, but because he thinks that what the Litvins are doing and the sorts of things he advocates with math and computers makes far more sense (I hope I didn’t butcher his ideas there).
I’d also recommend checking the Computer Science Unplugged web site. Lots of free things there, with many activities suitable for kids in elementary grades. I’ve had some experience using some with actual children at some free summer lessons at Notre Dame, and no one died. 🙂
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Public Agenda surveys show that parents and teachers overwhelmingly think kids should “memorize the multiplication tables and learn to do math by hand before they use calculators. . . . ” whereas “Professors of Education” disagree.
http://tinyurl.com/k382xdm
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