David Wees, a mathematics edublogger, posted the following earlier this week. My comments to it follow below. In my opinion, math teachers must help student’s learn math, not erect barriers by insisting on manual manipulations when a calculator is more effective, and frankly, the way the world operates in the 21st century. Read David’s posting, too.

### COMPUTERS SHOULD TRANSFORM MATHEMATICS EDUCATION

Stephen Shankland **posted an interesting article on CNET today**. Here is an exerpt from his article, which you should read in full. He says:

*Clearly, children need some understanding on their own of math, and reliance on a computer has a lot of drawbacks. But computers can also aid those who otherwise would fall by the mathematical wayside, or let people with more advanced abilities bypass drudgery and move on to the challenging material. Graphing calculators can let many students explore curves and functions that realistically they’d more likely ignore if they had to plot them by hand.*

I wholeheartedly agree, David. Students should take an exploratory set of courses in basic numeracy skills in primary school to develop familiarity with the fundamental operations of mathematics. However, once they have observed and experimented with the fundamentals, regardless of whether they mastered the more complex ones or not, they should be free to use technology resources such as calculators, even on standardized tests. Requiring students to continue manually manipulating fractions, performing long division, and pondering other foundational mathematical operations best left to calculators or computers is an archaic and ineffective use of time, talent, and treasure; we stopped using slide rules a while back for a reason.

In this country, we waste considerable amounts of scarce resources (time, young minds, and money) in a senseless devotion to the past. Due to a mathematics curricula imposed mostly by academics, rather than a mixture of academia and industry, we are crippling our nation’s intellectual capacity. It is time these outdated approaches are shown for what they are: loadstones around the neck of our nation.

For those that will lament not doing math the old-fashioned way, each of the procedures for manual mathematical manipulations can be taught in a history of math class, or as a dedicated segment of another class, to illustrate how our “ancestors” used less effective means to perform mathematical operations. Those “skills” are simply prerequisites for manipulating data, a means to an end. However, with their inclusion in today’s mathematics curricula and standardized tests as well the mandate by “old school” math teachers not to rely on technology, these manual operations serve more as a roadblock to students developing higher order mathematical thinking.

Most students get stuck procedurally, or make a minor math error transposing digits, dropping a negative sign or some other lower-order task, and equate that with a lack of mathematical expertise or capability. We have created tens of millions of mathphobic citizens with this approach. In spite of this, we continue to instill fear and loathing of math today in students of all ages. When will we wake up, let go of the past, and embrace our future? When will we push aside the gatekeepers who insist on outdated procedure over unleashing the creative potential of our youth? We do not have to remain on this tragic trajectory. Success is within our grasp, if we embrace in our educational system that which industry recognized over forty years ago: leverage technology to unleash efficiency, creativity, and innovation.

Technological efficiency exponentially improves effectiveness. We can overcome the results of our mistaken insistence on drudgery as the path to mathematical might if we embrace technology and strive for higher-order thinking rather than lower-order procedural prowess. Allow students to leverage their iPhone, iPod, calculator, home computer, or whatever other technology that is accessible to them; and for those without access, make it so. Let them start using technology to assist in problem solving as most in industry do. The time has come. We can no longer afford to constrain our national intellectual capacity due to a well-intentioned, but outdated insistence on doing things as they were before the Internet age.

The only real debate is whether to use a conventional or RPN calculator! Or for computers, which OS: Mac OS X, Windows, or Linux?

Dave,

I totally agree with you – there is far too much “busy time” in Math as it is currently taught. The more advanced kids, as you say, spend time they could be learning new concepts. In doing calculations with fractions/percentages/algebra there is no loss, as I can see, to the child in either keeping the numbers simple or introducing a calculator (no effect on learning the concepts). It also makes it soooooo tedious. Math could be such a fun and relevant subject.

Then for both the less mathematically talented kids (who will need this skill in everyday life) and the more advanced (to check their answers are in the ball park) they should be taught how to do approximate mental math. Here, I am constantly dividing numbers by 46 (exchange rate $ to rupee) – and have my method of doing a quick approx in my head. This is the sort of things all the kids should be taught to do. I’m sure the fear of math holds so many people back – take “algebra” – a word that fills some with dread – when most of life is spent solving unknowns. We use basic algebraic conventions day in, day out.

Great article,

Thanks

Helen

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I agree that technology should have a place in the education of our students as well. What concerns me is that many students are using technology as a “crutch” to replace true understanding of mathematical procedures and not in the way intended by well-meaning individuals. I believe this is happening at a very early age (or at least has in the past); I teach high school math and the “thought” processes displayed by some of the students I’ve worked with over the years is shocking. Having said that, I’ve also seen some very well prepared students over the years and I’m certain that both “versions” have shared the same classroom before.

I believe it is very important to show patterns and to represent concepts & procedures geometrically wherever possible; this approach will solidify understanding, allowing for technology to step in and handle the “grunt” work. Technology and true understanding don’t have to be mutually exclusive.

ps. have a look at some James Tanton video lessons on YouTube.

Thanks David.

Earl

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Agree, Earl, that technology and true understanding do not have to be mutually exclusive. At the same time, I do not believe in-depth understanding, or complex manipulations, of lower-order operations is necessary for most students today. This is especially so for repetitive work such as worksheets or homework where 10 to 20 variants of some operation is required. That is mind numbingly BORING. It is also not necessary as a means to demonstrate proficiency solving (unnecessarily) complex equations; offload that to technology. If that is viewed as a crutch, so be it. I do not write emails using binary digits or worry about transmission protocols like TCP/IP, even though that is how they get sent over the internet.

I am not making excuses for students. They need to develop some proficiency with numeracy, an a rudimentary understanding so they know how to use technology properly. Let’s just not force them to repeat certain lower-order procedures by hand over and over or to solve them manually where mistakes reign supreme and inhibit understanding.

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Definitions of “complex manipulations” can vary significantly and I don’t believe them to be synonymous with true understanding. Some would say my notes on facebook are “complex”; I beg to differ. My son will be writing his final exam in Math 30P this coming week. Over the course of this past semester, I’ve watched him work through problems both with pencil and paper AND with his graphing calculator. On numerous occassions, I’ve shown him a different way of looking at a procedure that was NOT dependant on the technology. He claimed to prefer my approach because he could “reason” his way through the problem and not rely on memorizing a sequence of keys strokes……….I certainly hope you’re not advocating the memorization of key stroke sequences!!! I’m sure you are not!

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Good math, like good Science Fiction, is about ideas. The procedure for factoring a quadratic ax^2 + bx + c = 0, a NE 1, is no more meaningful to students than a sequence of key-strokes. For students, a lot of math falls into that category, i.e. equivalent to key-strokes. In fact, ask senior students, or for that matter the student in Math 30P, “Why do you factor?” My bet is that not one will say the purpose is to solve equations of the form f(x)=0. Yet, those students have been factoring for years! By all means solve for x, including by rote method, but not at the expense of employing technology to assist us in solving.

You know, I recall the method of partial fractions in integrating rational polynomial functions, and I can honestly say that while it was an interesting technique, it didn’t really extend to any bigger ideas in modern mathematics. I would have benefited none the less if a computer had performed the calculation for me, getting me to the next decision point in the problem.

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Hey Mike…….you’re painting with an awfully wide brush here. Try develop some meaningful scenarios involving quadratic and linear functions; set them up as systems of equations and THEN find the roots. My bet is that the vast majority of students WILL understand what they’re solving for. I agree that some of these procedures can become rather laborious at times but what is important is the meaning behind the solution. Having said that, there are many little tricks that make calculations very easy and add meaning at the same time……….I’ll be posting more examples of such things within 20 minutes on my facebook site; I have that linked to my blog and will be posting a notice on twitter shortly.

Earl

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How about the factoring question? That’s a litmus test of the job we’re doing, when we spend so much time and effort on factoring polynomials by hand. Students could be forgiven for thinking that multiplying binomials is an end in itself. And of course let’s only ask them factoring questions that have whole-number or what’s worse, rational roots. So for students, “Why do you factor?”

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What about the factoring question? Have I said I never use technology??? I don’t recall that. I like to show students how to factor with pencil and paper FIRST and explain exactly what they are finding. I will give an example here; I know it isn’t related directly to anything tangible but the same procedure CAN be used later with examples that DO relate to “real-life” scenarios……….and I always look at solving equations from a “systems” point of view. Here it is………..

Find where the following functions intersect: y=2x^2 and y=7x-5

………values of the two functions are equal at point of intersection…….. so we set the functions equal to one another…………… 2x^2=7x-5

We want to set this equation equal to zero now……….2x^2-7x+5=0.

This next part takes care in explaining………..since we want to solve for x^1, our goal is to now express the left side of this equation as the PRODUCT of 2 linear functions; we want the product of these two linear functions to equal 0……….this happens if one or the other is equal to “0”…………THIS is why I factor……….break into product of linear functions, the product must equal zero, therefore one of those functions must equal zero to satisfy the given condition. …….. (2x-5)(x-1)=0…………therefore x=5/2 or x=1.

Then I like making a small change in the original functions that tie in with my notes on transformations………you can view those by visiting my blog and following the links to facebook: http://samuelsonmathblog.blog spot.com

Here is what I do………..replace “x” in each of the two original functions with “(x+3)”…. this, as you should know, will move everything to the left by 3 units (including the solutions). Those solutions should therefore be x=-1/2 and x=-2.

Now we can verify………..this is a valuable exercise to go through with kids as it allows them to visualize and use REASON to funds solutions or at the very least, to predict what should happen.

ps. we need to teach transformations anyways……..why not incorporate some of that here while we’re at it????? Let’s solve the new system:

y=2(x+3)^2 y=7(x+3)-5

2x^2+12x+18=7x+16…==> 2x^2+5x+2=0…==> (2x+1)(x+2)=0…=>x=-1/2 & x=-2

Represent everything graphically and connect the algebraic representation of each to the corresponding geometric version……

I don’t spend weeks on this type of thing; in fact, this goes very quickly. Once these basic notions are understood by students, I DO use graphing technology; they do NOT have to be mutually exclusive. I’m certain we all want the same for our students and there will always be more than one effective approach. I think the approach I’ve shown here and in my notes on facebook still have value in helping kids have a better understanding of math.

Thank you for reading this.

Earl

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Sorry you feel so frustrated, Nathan. BTW, I considered not accepting your comment since you made an ad hominem attack on students writ large. However, I will chalk that up to accumulated frustration from your three plus decades of service without the opportunity to engage in meaningful discussion about it. More to my post, and in my opinion, use of a calculator before learning arithmetic basics might limit understanding, but its use is not crippling, and its use after primary school is liberating, with the potential to unleash creativity, not squelch the minds of the majority of our students simply because they bore of computations best left to a calculator or computer. I personally believe our entire approach to mathematics instruction is antiquated and does more to harm our nation’s strategic future than has ever been considered. Nostalgia is not the best method to discover the future. Finally, I look forward to a properly designed and tested machine determining proper dosage over the recall ability of a human any day of the week; the machine has much greater potential to set proper dosing based on age and weight, detect possibly lethal overdosing, take into account potential drug interactions, etc.

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[Nathan:] I think you and I disagree about the definition of math itself. Is moving symbols around on a piece of paper, with limited comprehension of the process, mathematics? Is computation mathematics? Or is the purpose of math to learn how to problem solve, to find patterns, to think critically, and to use the calculations one has learned in these processes? I’d argue that computations and calculations are a tool mathematicians use to do mathematics, and are not the actual mathematics themselves.

If you look at this map of numeracy levels across Canada (http://www.ccl-cca.ca/cclflash/numeracy/map_canada_e.html), which I’m sure is similar to numeracy levels across the US, you’ll see a disturbing pattern. Almost everyone, everywhere, lacks functional numeracy, as defined by the ability to apply and understand mathematics in their lives. Given that many, many of the people in this map are products of the a system which taught them rote calculations, then I can hardly see how returning that system would improve math instruction.

Should kids know how to do some computations in their head, or on paper? Absolutely, as you point out, experience with numbers and computations is useful for learning numeracy. However I don’t see how experience working with number operations necessarily corresponds to memorizing addition & multiplication tables. Isn’t it more critical that students learn different ways of manipulating numbers, and learn applications of those manipulations? Shouldn’t they be learning more of the beauty of mathematics and less of the drudgery?

I think your last comment, by the way, is highly insulting, and if you want to promote a healthy dialog about this issue, you would do well to treat your adversaries with more respect.

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Agreed on all accounts! I think it boils down to whether we want to teach our students about arithmetic or mathematics. Personally, I am teaching students at a senior high school level, and I believe that the skills of arithmetic should be second nature to these students (yes, I know about the dangers of assuming :D). However, some students are unable to experience the beauty and wonder of the mathematics I am trying to teach them at the high school level because they get so caught up in the arithmetic.

While students do need to understand these mathematical processes – this is something that can be taught when these concepts are introduced. As students progress in their mathematical careers, we should be giving them the tools to simplify previous knowledge and allow them to focus on the more complex material.

Finally, allowing students to use technology in the classroom for their course work is also extremely beneficial. When a student needs to compute something outside of the school, they probably aren’t carrying a calculator! They will have a cell phone, an iPod, or some type of wireless device. Why not teach them to use these devices effectively and responsibly when working with math. Yes, they can find information you may normally expect them to memorize, but it forces teachers to ask questions which require more than googling the answer.

I’m all for technology. I’d be more hesitant to use it while students are first learning their skills, but once the arithmetic is done and you’re moving onto the mathematics, why not use all the tools you have!?

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Nathan,

I have a Ph.D. in mathematics and I am internationally known for my research in mathematics education, as well as being a math (full) professor, and a career as a (full) professor of math education in the UK and the US, and Chair of the Department of Mathematics at my current institution.

I used to be opposed to calculator use but have changed my mind after seeing how students genuinely use them usefully. My university insists that students use calculators, even for math majors, from freshman year on.

I think your assertions are just that: assertions.

I asked my distinguished numerical analysis colleague if she had used calculators extensively and productively in a formative stage of her career. She replied she had, and said she largely used a calculator to numerically investigate conjectures and “play”.

I feel we need to be open and broad minded about this issue.

I, for one, haves tremendous respect for David Wees, and I am always willing to listen to him (though have been known to disagree on minor issues).

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Thanks for sharing your perspective, Gary. My 25+ years in high-tech, using calculators, helped me serve as a more productive engineer, program manager, product manager, and business development manager. In fact, I find Excel to be the most effective tool for all but the most esoteric of applications, and one that most students should master in high school.

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Hi Nathan. I can appreciate your passion for mathematics, and your view that the way you learned it was meaningful and helpful. However, I cannot accept your rigid belief that you hold the truth to all things regarding mathematical pedagogy, especially technology such as calculators. I first used a calculator in high school, after developing a somewhat sound sense of number, and it only helped me efficiently solve problems not only in high school, but for my undergraduate degree, in electrical engineering, and two graduate degrees, MBA and MA in Education – Teaching of Mathematics; they were from fairly prestigious universities, too. It also provided a great launching pad for investigating formal logic; learning computer programming, both machine and high-level programming languages; using circuit and systems modeling programs, spreadsheet applications, and other computationally intensive applications; as well as learning and applying statistics. In fact, I would go so far as to state that these devices helped expand my understanding and application of mathematics.

I must mention that my original posting mentioned the following, which you may not have digested fully. “Students should take an exploratory set of courses in basic numeracy skills in primary school to develop familiarity with the fundamental operations of mathematics. However, once they have observed and experimented with the fundamentals, regardless of whether they mastered the more complex ones or not, they should be free to use technology resources such as calculators, even on standardized tests. Requiring students to continue manually manipulating fractions, performing long division, and pondering other foundational mathematical operations best left to calculators or computers is an archaic and ineffective use of time, talent, and treasure…”

BTW, your repeated personal attacks do nothing to further what comes off as a conspiratorial perspective (e.g. “SO PEOPLE CAN KNOW THE TRUTH”); the same goes for your broad-brushed lambasting of anyone opposing your perspective as unscrupulous, uninformed, or worse.

I continue allowing your postings to further dialog about this all too important pedagogical and curricular matter.

Dave

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As one of my professors once advised us, “Never make absolute statements.” (And when he did so, he was clearly unaware of the irony.)

I disagree with both sides of this argument, at least partially. Dave, I have some concern about your use of “rudimentary,” which you used in one of your earlier comments: “They need to develop some proficiency with numeracy, an [sic] a rudimentary understanding so they know how to use technology properly.” Not all definitions of rudimentary are equal, and not all mathematical situations are equal, and a “faster to tech is better” approach will not always be the right one. For example, I think more of us agree that it’s better to have a rudimentary understanding of the manual finding of square roots to 5 decimal places than it is to have a rudimentary understanding of 2-digit multiplication. Nathan, I have concerns that you’re believing in a body of research that doesn’t exist (which might explain why you didn’t cite it). Research on the use of calculators in mathematics has been ongoing for about 35 years, dating back to George Immerzeel’s work (and that of others) in the mid-1970s. When the National Math Panel released their 2008 report they found limited or no impact of using calculators, and only one study in the past 20 years met their criteria for inclusion in the report. (The Panel was, perhaps justifiably, criticized for being too selective with the studies they selected for review.) And your statement, “There is not one mathematician or truly successful person (nor will there ever be) who has advanced very far in math by using a calculator,” to me sounds equivalent to saying, “There is not one writer who has ever got very far in writing by using spell check.” Or, “My accountant uses a calculator so he must be bad at math.” Again, this is not a debate for absolutes.

We’re missing some middle ground here. Let me propose a series of simple addition problems:

1. 27+10

2. 27+9

3. 27+19

4. 34+29

5. 274+495

Do we want students to know how to use the “standard algorithm?” Yes. Do we want them to focus on it so much that they use it, blindly, to solve the above five problems? No.

Do we want students to know how to leverage technology to their advantage? Yes. Do we want students to use technology to solve the above five problems? No. (At least I don’t.)

Do we want students to understand place value well enough to add 10 without the standard algorithm or technology? Yes. Do we want students to be so familiar with the fact that 9 = 10 – 1 that they see problem #2 as “27 + 10 = 37, and 37 – 1 = 36” and not worry about carrying and borrowing? Yes. Do we want students to understand and expand this “back up one” strategy to solve problem #5 mentally (using 495=500-5), while seeing that the problem 274+879 does not lend itself to that and other mental strategies as easily, and is therefore a more appropriate problem on which to use a calculator? I hope so.

And here’s why I think this debate will remain unresolved: students who become dependent on technology AND students who become dependent on a traditional algorithm are likely to not learn this and other strategies. So we can’t assume a teacher who lets their students use calculators is helping build student understanding any more or less than a teacher who never lets their students use calculators. The issue is more complex than that.

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Thanks for your thoughts, Raymond. I agree your middle ground is a great place for which we should strive, however, not all students may get there. Hence, I do not want their inability to grok the “backup strategy,” as an example, to prevent them from grasping a higher order concept that can just as easily be explored with the calculator handling the lower order arithmetic. I recognize their are trade-offs in every decision, and I am comfortable that my approach allows more students to appreciate, and understand, more math than a “no calculators” approach.

I also agree “rudimentary” may not have been my best word choice. Perhaps, “basic procedural fluency” might have been better, although it, too, is ambiguous regarding specific capabilities. Rather than define the exact cut points for fluency, the gist of my post is: 1) teach arithmetic and problem solving basics in primary school, and 2) while in secondary school, let students use calculators so they are not inhibited by procedural fluency with arithmetic and unable to advance further due to frustration or feelings of incompetence, inadequacy, or what have you. The following excerpt from my original post conveys the essence of my point.

“Students should take an exploratory set of courses in basic numeracy skills in primary school to develop familiarity with the fundamental operations of mathematics. However, once they have observed and experimented with the fundamentals, regardless of whether they mastered the more complex ones or not, they should be free to use technology resources such as calculators, even on standardized tests. Requiring students to continue manually manipulating fractions, performing long division, and pondering other foundational mathematical operations best left to calculators or computers is an archaic and ineffective use of time, talent, and treasure…”

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I missed these earlier comments…interestingly, the IP address, the shared links, and some of the content in the following comments from “Puerto Rican Exec” are identical to another commenter.

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Hi Shanelle. I agree students, ideally, should learn mathematical concepts and procedures manually first, especially arithmetic in primary grades. However, students that come to teachers ill-prepared, or unable to perform simple arithmetic procedures for whatever reason, should not be “punished” and “prevented” from learning higher level concepts, or procedures, because they get stuck, or stumble, on lower level procedures. That is my main point; no amount of forcing them to learn the procedures by withholding access to calculators will make them learn; its kind of like the saying “The beatings will continue until morale improves,” metaphorically speaking, of course. Additionally, they are typically on a non-level playing field with other classmates who have mastered all procedures and concepts up to the current topic.

A calculator does not substitute for learning. However, given the structure of mathematics curriculum today, the existence of accelerated pacing calendars to accommodate high stakes standardized tests and multiple district benchmark tests associated with the standardized tests, the fact that scores of children are subject to social promotion, and the heterogeneous nature of classrooms, it is quite the conundrum for today’s math teacher to teach students with such a diverse range of understanding without allowing accommodations, such as calculators. My point about letting all students use calculators is driven by concerns for equity, access, practicality, effectivity, and motivation.

BTW, you make an excellent point re: helping students interpret data such as the calculator returning a result of 38.925176091. As an added bonus to use of the calculator, even if by a subset of the class, multiple discussions could occur on this one result as to significant figures, precision, accuracy, notation, rounding, truncating, estimation, error, uncertainty, variance, standard deviation, etcetera, depending upon the level of the class and the course.

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Nathan, could you be any more arrogant and pompous in these posts? Surely I have not read a more insulting set of comments in a long time.

Dave, please do not let this person dissuade you from doing what you think is right for your students.

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I’ll throw in my 2-bits here…………

I wrote a blog post in May addressing my views on technology in education. I’ve pasted a link to that post below rather than duplicate what was written there. I was relatively new to twitter and blogging at that time (and still am) but am trying to be open-minded and accept points of view other than my own.

Over the past several months, I’ve connected with many individuals who are far more knowledgable than I will ever be and I consider myself very fortunate for this. These people have treated me with much respect, for which I am grateful. Through this process, I have gained a different perspective on how to facilitate learning in and out of my classroom; I will strive to employ much of what I’ve learned through this very rich and on-going PD.

Thank you Gary and David for including me in your networks.

http://samuelsonmathxp.posterous.com/technology-integration-in-math-education-good

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Welcome back, Earl, eight months later… 🙂

I know you and David converse often on Twitter and are of similar minds, and respectful in your discourse, unlike others I’ve run across on blogs here and there…But not sure who Gary is…

To be clear, technology is neither a substitute for learning nor a panacea. However, for those who have not yet attained procedural fluency, and may never for whatever reason, calculators offer them the opportunity to continue engaging in material they otherwise may not due to procedural deficiencies which lead to frustration and disengagement. Additionally, those who have established procedural fluency, and ideally conceptual understanding, can be more effective using technology and unleashed to pursue higher order thinking more readily. Consider applications like Mathematica, Wolfram Alpha, Geogebra, Geometer’s Sketchpad, Calculus in Motion as well as various TI, HP, and Casio programs that assign lower order computations to the background while bringing complex, graphical and symbolic representations to the foreground. So, net net, I am not advocating that students never be shown arithmetic properties or procedures. I am saying that those who do not attain proficiency, for whatever reason, have the opportunity to advance with more advanced topics more readily if they are able to rely on calculators, or other suitable supports / tools, to handle the cumbersome aspects of computation; likewise for those who are proficient.

I hope everyone re-reads my original post carefully, maybe reading it two or more times, before posting further comments to ensure my points are clear. I am not advocating abandoning basic math skills for all, or anyone for that matter. I am saying that not everyone will achieve what we desire for them, common core standards, NCLB punishments, discovery-based learning, explicit direct instruction, or whatnot – that is simply life. Yet we can extend a student’s exposure to, and involvement with, math beyond the typical times he or she would normally abandon it if we let go of our insistence that the only way for students to advance with math is if they have deep conceptual understanding and procedural fluency; we can hope for that for all and do our best for all to attain both of those and more. However, as a realist, I plan to offer accommodations that keep all students on the playing field longer, rather than having any of them give up partially, or worse, entirely.

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Thanks for sharing your thoughts, again, Earl. I did not get a chance to thank you in the above reply. Respectful discourse like this is excellent PD.

And if I did not say it before, calculators and nearly everything else have significant downsides to them (e.g. cons along with pros). My view is we need to make trade-offs to help ALL of our students develop as individual, life-long learners using the best methods and practices we know at any moment in time expanding their scope as we advance in our career with experience and PLN’s like ours.

Thanks again!

Dave

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Thank you, Dave, for the original post and allowing me to be part of the discussion. I agree that we need to strive to reach the needs of all students in our charge. Many students ARE lacking some basics upon arrival at the high school level and have an enormous struggle if placed in the “wrong” stream. I do the best job I can to help them through the course; these students tend to gravitate more towards technology than do the more academically-minded students. The latter group of students will pursue studies at the post-secondary level and because of the requirements they will meet with at those institutions, these students strive to understand the concepts/procedures using pencil & paper methods primarily.

Once again, thank you for initiating this discussion.

Earl

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Anytime, Earl. I appreciate your thoughts, suggestions, comments, etc.

Dave

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I, on the other hand, grew up using calculators as a tool and learned how to explore numbers at a much greater rate than I could with pencil and paper…. and Nathan, posting under a pseudonym to throw us off the scent isn’t going to work. No one posting here is stupid enough to believe that two people who use the same type of language structure and posting the same arrogant nonsensical argument would frequent the same post…

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Jan: Why you feel it necessary to equate the use of technology, and or accommodations, in learning math with laziness? I will not dissect your other assertions re: money, learning, and relative smartness of countries, since they have no basis in fact. Technology use enables students (who want to learn) to stay engaged working on problems where they may have given up without one. You also need to recognize that not everyone may see the beauty in mathematics, at least not readily, just as you may not with other subjects.

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