# how far do i need to rise to get calculators out of the classroom?

Discussion in 'General Education' started by Bak2Math, Oct 27, 2015.

1. ### Bak2MathRookie

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Oct 27, 2015

somewhat serious question...

I have been a first time board substitute teacher now for about 2 weeks teaching math. Exactly how many generations have been 'crippled' by using calculators too early and too much? I have otherwise intelligent kids in the class, 10th 11th and 12th grade algebra 2, who can not recognize that the cube root of 27 is 3 and a few kids in higher 'maths' who are struggling with simple equations. It's so reflexive that when told they can leave the answer in fraction form, even without simplifying, they insist on using the calculator which not only wastes their time but possibly leads to a wrong answer if they make a simple mistake.

So if this continues to bug me is this something that could be (or should be) changed at the school level? county level? state level? federal level?

Am I the only one that feels like this? It seems like the other teachers, who have been great, may feel the same way as me but have accepted it.

I'm gonna go to sleep and see if I can figure out if the cats smile remains after it's gone... thanks for listening...

3. ### ImmeritoRookie

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Oct 27, 2015

A calculator is a tool. Children and adults need to know now to use tools wisely.

If a child doesn't understand the concepts, campaigning to get the tool out won't solve that problem.

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4. ### GTB4GTCohort

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Oct 28, 2015

I feel the same way as you. In (my) perfect world, a student would not have access to a calculator until his/her junior or senior year. When working with students individually, I will ask them a simple question...like "what is 2 + 3?" Most will reach for their calculators. The worst case is when I asked a young lady "what is -1 + 1?". She typed that in to get the answer!!! This makes factoring extremely difficult as many students struggle to identify all the factors of 64 for example.
While the calculator is a useful tool, it appears that over reliance on them at an early age seems to inhibit a sound mathematical foundation. This is all just my opinion as I am sure there has been research done on this topic.

My other observation that is perhaps pertinent to my school only. We have no structure in place regarding calculators. So the students show up with all makes and models...with no formal training in how to use them. For simple arithmetic, this is no issue. But in higher level math, each make/model is unique in the way it accepts input as they all will handle order of operations a bit differently. I am of the opinion that maybe 50% of the "math errors" that I see are really operator input errors (i.e. failure to use the calculator correctly). i would standardize calculator usage to one model (maybe the TI-30 or equivalent) beginning in the junior year and spend some instructional time on its use. I waste time trying to troubleshoot calculator issues due to the variety that my students bring to the classroom.

5. ### otterpopPhenom

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Oct 28, 2015

GTB4GT, I agree that factoring is way harder due to this issue! Once upon a time, I was teaching remedial math to high schoolers, and we were working on factoring... man, is that rough if they don't know their facts.

I don't teach math, but I know that in elementary school, kids often take "mad minutes" or similar quizzes to see how many facts they can answer in a minute. While it may not be possible to eliminate calculators, these types of quizzes could be used even with older kids to encourage them to know basic facts such as the one the OP mentions.

6. ### a2zVirtuoso

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Oct 28, 2015

Mad minutes do little to teach those who struggle. They actually make it harder for those who struggle. I've seen the biggest problem is that the concept of multiplication is taught then the instruction of the facts and learning process is released to the students whereas schools used to include repetition and memorization of facts as part of every single school day. That is long gone. A practice at my local school was to send flash cards home the first day the facts were introduced and have kids learn the facts by using flash cards. That is a terrible idea for a large group of kids because it is constantly guessing. There was no other instruction other than mad minutes and doing problems on their own.

7. ### otterpopPhenom

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Oct 28, 2015

I think this could definitely still benefit students. There are a lot of practices that once were common place that are unfortunately not around anymore.

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8. ### mathmagicEnthusiast

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Oct 28, 2015

I wonder if this is a conversation you could bring up amongst your middle/high school math peers - I think the outright banning until a certain age would not necessarily be the best fix, but as others said, a solid education on when and how to utilize the calculator as a tool. If all of the teachers are on the same page with how they plan on utilizing and teaching the usage of calculators, I think you'd see a decrease in "grabbing it for the easy parts" and an increase in conceptual understandings, too. As an added benefit, they'd be able to more efficiently and accurately solve problems.

Much like you align other curriculum (or it's already aligned) so that students are building upon previously learned foundations...getting to that point with the calculator (technology) use would be beneficial.

9. ### lucybelleConnoisseur

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Oct 28, 2015

I'm a big fan of calculators. I agree that students should be able to know basics, but we have to come to terms with the fact that calculators are always going to be around. I have trouble taking tools away from students when they will always be available to them. For instance- I don't like making my students memorize equations, because if they really needed to know it, they could look it up. And if they need to solve a quick math problem they can pull up a calculator on their phone.

I think there's a healthy balance between the two, but I usually lean towards letting them use it.

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10. ### Bak2MathRookie

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Oct 28, 2015

One of the things I learned while growing up is that by and large most people are not at the extremes of life and odds are that while someone may have it 'easier' than you there is the probability that someone has it harder.

"My other observation that is perhaps pertinent to my school only. We have no structure in place regarding calculators."

My biggest issue coming up may be with differentiating one TI-84 from another... I can't imagine having to deal with something along those lines. by the way... when I was in college Hewlett Packard was king of the calculators. Wish I still had mine. It was way ahead of its time.

I guess I will do what I can and try not to let this get under my skin too much. Another new pet peeve is the loose handling of σ (lower case sigma) and S for standard deviation. I would guess a lot of people who gravitate towards math are concrete thinkers and something like this can torture me (I mean someone) for decades.

It bears repeating. All of the teachers and students I have been around are great.

Ending on a bright note. There is a student who smiled for the first time in two weeks in class today when they understood something.

Sorry I didn't have anything more to add regarding calculators.

"When all you have is a hammer... everything looks like a nail..."
-paraphrased, Abraham Maslow

11. ### RockguykevConnoisseur

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Oct 28, 2015

Should we also get rid of the Internet? Seems like kids should learn to use a library for research. Maybe keyboards so they learn better penmanship? How about dictionaries so they learn Latin roots?

There are simply things that we don't need to keep in memory any more since machines can do it far more efficiently and effectively than we can.

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12. ### czaczaMultitudinous

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Oct 28, 2015

The students we teach are digital learners. Not only should they be taught how and when to use calculators appropriately, but how to use a variety of technology solutions/tools to facilitate learning. I'm not sure exactly what you are seeking to change....

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13. ### mollydollConnoisseur

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Oct 28, 2015

Students should not be using their calculators for basic math. My algebra and algebra II students struggle because they do not know their multiplication tables at all. Since they can use the calculator, they have no motivation to learn to do anything without that crutch. This also means that every problem takes them much longer so these students fall further and further behind.

Calculators were not allowed in any of my college math classes, with the exception of a 4 function calculator for linear algebra and statistics. Actually, the GED doesn't allow calculators for half the math section either.

14. ### AmandaAdministratorStaff Member

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Oct 28, 2015

It seems to me the problem is not the calculator, but that the students have made it through school without ever having to learn basic operations.

15. ### GTB4GTCohort

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Oct 29, 2015

I think you just hit the nail squarely on the head in terms of root cause, the rest of us are merely pondering if the calculator contributes to the problem or not.

16. ### ObadiahGroupie

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Oct 29, 2015

From my informal observations, this problem is also occurring in schools without calculators. I wonder if in the elementary grades math and especially arithmetic are sometimes being kept mostly at the knowledge level of a taxonomy of learning, where "=" means "here is the answer". Are teachers being forced to teach a bubble curriculum to continually raise standardized test scores to keep their jobs and consequentially needing to ignore the fundamentals of math: 1. understanding and applying the concepts, 2. using the algorithm to more easily solve a computation, and 3. memorizing key facts, such as sums and products? In the history of math this is how the algorithms and place value originated, to ease calculation. In my school district we all had the same workshops and similar curriculums, but I noticed I was one of only a handful of teachers using manipulatives for concept development. Students are often quizzed on basic facts, but if they are only studying to pass the quiz, (especially just the night before at home), those facts will be quickly forgotten. The brain only re-enforces memories that have a reason for retention.

This problem might extend back in history prior to standardized test mania, too! Many adults my age cannot do arithmetic! An extreme example: I bought about 16 books at a library sale many years ago. Each book was 25 cents. To make things easy for the librarian, I arranged the books into 4 stacks. The librarian was confused why I did so, counted all the books, then multiplied the cost on paper. I watched her calculate as she "showed all her work" to figure out this equation.

17. ### a2zVirtuoso

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Oct 29, 2015

Exaggeration doesn't help your argument in this case because technology is usurping knowledge and learning in some areas. Continually using calculators instead of learning facts in most cases will impede learning number sense. It is important to really understand what makes up the squigglies we have named to represent quantities and how those quantities relate. There is now more and more information regarding how writing (instead of typing or pushing numbers) help learning. Having a tool that give the answer allows you not to think about what is happening. Sure some people will still get the number sense, but a lot won't.

Same goes for word processors. When we have technology fix things for us, we stop learning why something is correct which means we know less and depend on technology more. We have already stopped teaching grammar in many schools and even stopped teaching spelling in many schools and rely on the excuse that kids will have word processors to take care of that lack of knowledge, but can you imagine trying to teach someone our language when not knowing why the word processor is changing what is written (if it gets it right). At what point will our language just fall apart because so few know the rules of our language?

There are times technology is an aid. There are more times it is used as a crutch instead of teaching people. There are other times that technology is used ineffectively to speed things up but ends up giving people a false sense that they know what they are doing. Research on the internet? Well most of what is out there is junk and not well vetted.

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18. ### a2zVirtuoso

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Oct 29, 2015

And when NOT to use technology because it impedes learning.

19. ### RockguykevConnoisseur

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Oct 29, 2015

Never. Never is that point. There is more communication happening today than ever before. I'm not personally upset that we don't speak or write in Old English any longer. Our language will change and adapt but so what? The purpose of language is communication. That isn't going to stop.

Same issue with number sense. What is the value in it? My 7th graders can, and do, work out ridiculously long multiplication and division problems constantly while learning higher math concepts. They are not getting the higher concepts because of the sheer time it takes to do the low level computations. The reason they aren't grasping cube roots is that they are asked to divided out to the xteenth decimal point when finding the square root of 5. They never GET to cube roots. Nobody cares. Nobody does that. The purpose of why we ask kids to learn things should be the forefront of our reasons for doing anything and too often it is ignored.

As far as research goes the Internet is far more accurate than most of the old texts found sitting in a library. Saying "most" of what kids will find is junk is simply false. I want a thinking generation not a robotic one. Let the robots do the robotic part and let the humans do the human part.

20. ### mathmagicEnthusiast

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Oct 29, 2015

I would alter this to say that students should learn their basic math skills before doing the skills on the calculator, so they have the flexibility to do it both ways. However, I completely disagree that they should not be using calculators for basic math at any point. I got my B.S. in Mathematics, and you can bet that throughout junior high, high school, and college, and even now (especially when making keys ), I'll use a calculator for certain basics that would be more efficient on the calculator. That's where it comes down to the importance of teaching technology (calculators are a technology, a tool) skills and strategies.

Funny that this thread came up though, since I joked with my fourth graders that they were allowed to use calculators on their multiplication assessment (they're learning the basics right now), and they were all surprised, but then rolled their eyes when I pulled out my usual turn of the tongue and said that the calculators were in their brains .

21. ### readingrules12Aficionado

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Oct 29, 2015

I agree with many of these posts. Calculators do have their place in high schools. You do have a point though. It might be good to have a day once every 2 weeks or so is calculator free and extra emphasis can be done on mental math. I know in my life, mental math is the most common math I use. It helps me to estimate and see if they are giving me the correct change back, and many times I have had to correct the cashier on this. I can't imagine the 100s or even 1000s of dollars I would have lost without using mental math.

22. ### MikeTeachesMathDevotee

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Oct 29, 2015

Calculators are unnecessary before algebra. Period. I don't let my students in classes below algebra use calculators. Period. They hate me for it at first, but it helps them in the long run. Period.

High school is different. Calculator use should increase every year starting with algebra. Still, 75% of the time, I don't let my students use calculators. I have seen way too many kids crippled because of their reliance on the calculator for simple arithmetic.

I used to be very lenient with calculator use. That's changed drastically. I've had students petition (only once successfully) to be switched to a different teacher specifically because of my calculator use policy.

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23. ### Haywood GilesRookie

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Oct 30, 2015

I prefer to not use calculators also. Most often if the concepts are observed and understood they're not needed.

24. ### gr3teacherPhenom

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Anecdotal evidence... but I think a lot of the problem is practice (or lack thereof). I teach kids that come from 6 different schools, and have had kids that had roughly 50 second grade teachers. It's always easy to tell which teachers spend all their time on conceptual understanding of basic facts vs. giving time both for conceptual and practice. One of my feeder schools uses only Investigations... and good lord do those children have so many tools in their tool box for figuring out 8 + 4, as long as you give them enough time to get there. Other students have fewer tools, but just know that 8 + 4 = 12.

25. ### LisabobisaComrade

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Oct 30, 2015

I teach middle school students with autism. One of my students is great in math and had the basic concepts down. We were dividing decimals which resulted in repeating decimals. After he insisted he had to keep dividing until he couldn't anymore I pulled out the calculator. He only divided to the eighth decimal point instead of the millionth because that's all the calculator could show. He then proceeded to check his work..... by hand..... An eight digit number times a 5 digit number. You better believe that calculator came out again.

I understand how it can be a crutch for some students but I continue to pull out a calculator now and then when I KNOW he understands the concept to speed things up a bit.

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26. ### vickilynMultitudinous

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I have a pretty good mind for math, but my son will never be able to tell you for certain what any of the math facts are. I didn't say that he can't use higher order maths, only that a quirk in his brain has forever stymied him when it comes to multiplication tables, and the so-called simple mental math that many here prize. Does he use a calculator? You bet! He understands that the calculator is the tool he uses to come to the answer using complex reasoning and the appropriate steps. I value that. I also taught him to type and master a computer very early, because his handwriting is so tortuous for him to create. If he can give me a well reasoned argument about this or that, or if he can compose masterful poetry while sitting at the keyboard, what would be gained by telling him that he can only create using a pen or pencil?

I love science, and we know that science loves our tools. My son, and countless others just like him, were once considered impaired if they couldn't produce the multiplication tables in short order. If they can use a tool for that, are they still impaired? I think not. I also believe that this is a wonderful time to be alive if some of the old standbys are not your strengths.

As far as mental math, it's great if you can do it quickly and accurately. If not, grab the handy cell phone and use the calculator function. I do encourage the recognition of common estimates, like 10% off, etc. I don't, however, think that the generation practically born with electronics in their hands is going to be too disadvantaged as long as they know which rules tell them what to do with the calculator. The best tool in the world is useless if you don't know how to use it. I work with students who struggle in math, for a variety of reasons. Would I take away their tools? No. And as I get older and less nimble with the mental math, I hope no one tries to take my phone/calculator away from me, either.

I want my students to learn how to learn, using all of the tools they have access to. Some teachers are resolute that the students must know this fact or another. Well, what is a truth today may very well not be a truth tomorrow, and science is particularly prone to these seismic changes. My job is to teach students how to find the most current and then figure out where it fits with what was and what will be.

Last edited: Oct 31, 2015
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27. ### GTB4GTCohort

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Oct 31, 2015

vickilyn, that is a well thought out post and I am glad for your son that things have worked out well. However I would argue that he is the exception rather than the rule (based only on my observations with my students in my classrooms).. I would say it is extremely rare for a kid who is calculator dependent to master the complex reasoning behind higher level maths (again, this is only from my experience in dealing with over 400 +students at the high school level, which is a statistically significant sample size). They CAN be shown how to follow steps to perform tasks/calculate answers but if the problem varies (i.e they have to use mathematical reasoning to maneuver through/around a problem they are stuck). The analogy is similar to the way my wife operates her car and/or her computer. As long as things are functioning exactly as they should, she can operate the vehicle or the PC. But, heaven forbid, if anything unusual happens or there is any variation in the equipment's performance to norm, she is stuck and is totally dependent on others to fix/diagnose the problems so she can resume driving or operating her computer.

Disclaimer (in case she sees this) - the above doesn't mean I lover her any less!!! In the matrimony lottery, I won the Powerball jackpot). Love you, honey!

28. ### a2zVirtuoso

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Oct 31, 2015

Language is for communication, but even Old English and Middle English can be understood by most when someone knows that certain words have different meanings because the structure of the language still remains. We are losing the structure of our language because students no longer understand subject/verb agreement, verb tenses, pronouns, and how to place words and phrases to modify what the author really wants to modify. Effective communication requires precise language. It requires proper structure.

Our language has been adapting and is getting to the point where people can't communicate meaning. Look at many newspaper articles or opinion pieces in the newspaper. Even the "professionals" are making more and more errors in communication.

I guess we will agree to disagree, but my opinion is that while language may adapt, not every adaptation is for the betterment of communication. Just look at the word peruse. It has two meanings and they are the opposite of one another.

29. ### ObadiahGroupie

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Concerning GTB4GT's point above, " if the problem varies...they are stuck," I have heard from several workshops and seen research on this that suggest that the format of elementary textbooks might lead to this phenomenon. The students answer the same format of problems. Even "story problems", (usually 2 story problems at the end of a page), are predictable. Students locate the significant numbers in the story and either use the same operation of that day's lesson or locate key words to guess what operation to use. I can see where calculators could reinforce this situation if the students are just punching in numbers and operations on the text page and copying down the answer. As one workshop leader once commented, outside of school, we aren't given a math book page to solve life's problems. Daily tasks aren't designed to fit a certain curriculum's format.

I'm kind of wishy-washy in the calculator argument. I'm not sure I'd agree with a total ban, and I especially see the advantage of spreadsheet use on computers for some lessons, but on the other hand, I see much advantage in controlling their use so that students are equally successful without calculators and computers for basic math and arithmetic. (Also, my school also had a student who, due to his learning difference, was better off using a calculator in all situations--so I agree, there are exceptions that should be considered). Overall, they are a power tool, but an analogy can be drawn here--if a construction worker doesn't know how to build, the power tools are useless.

30. ### Bak2MathRookie

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Nov 2, 2015

I do not know how this strayed into comparing the use of calculators in algebra to the use of word processors for writing. Granted, I am in the process of preparing my first lesson plan on symbolic logic but I am pretty sure that's a 'straw man argument'.

Is there a need to learn how to find the square root of 5 to 5 decimal places without the use of a calculator? I think not. (Not in high school algebra at least.) Is there a need to know how to completely factor numbers like 6, 27, 36, 64 etc etc, without the use of a calculator, when learning the ins and outs of factoring, multiplying/dividing factors and adding/subtracting factors? Absolutely. Even the kids who could do this without a calculator are reflexively reaching for it to check their math. Not knowing how to do these problems without the use of a calculator drastically increases the time to do a problem and will put the student at risk for not doing well in the class and possibly not passing the SOL.

just my two cents... though our grad schools in engineering and math seem to have no problems with filling their seats with foreign grads... whose reliance on calculators may not be as strong as ours... but correlation does not necessarily equal causation...

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31. ### ObadiahGroupie

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Nov 3, 2015

On the flip side, I was reminded of my first class in classroom measurements and testing back when I was in college in the 70's. A friend of mine from another university who was from Japan was shocked to learn that my professor banned the use of calculators in this class. At this time, I had never seen a "scientific calculator", and I was so amazed when he pressed a button or two to do the same calculations I was figuring out with a pencil. Of course, I'm kind of assuming that my university has changed their policies in the 21st century.

32. ### SpecialPreskooModerator

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Nov 3, 2015

Do any of you use calculators? I mean, we are college educated people. We should be calculator free, too. Just saying.

33. ### mollydollConnoisseur

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I know my basic math facts. I've never used a calculator when I was first learning a concept, but have used them (or computer programs) extensively for various applications.

In my classrooms I've seen precious little evidence that calculators help kids. They do not seem to know basic math facts. This makes science insurmountably difficult. My algebra one kids can't complete more than 3 problems in 30+ minutes because they need to use the calculator to find factors. They also make a lot of errors inputting, but can't catch them. This is killing my chemistry students too.

34. ### RockguykevConnoisseur

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Nov 4, 2015

And, as I said, mine can't because it takes forever to multiply huge numbers. In both cases we are letting low-level math skills get in the way of real mathematical thinking - that's the problem. A calculator is a terrible tool for finding factors. They are using a saw to drive in a nail. Of course they are struggling. If they were taught to use the tools properly they'd have plenty of time to do higher-level thinking.

http://www.wolframalpha.com/input/?i=what+are+the+factors+of+256

When kids do skill-based questions in science they have their lab materials right there with them. When they do skill-based questions in history they have their documents and research right there with them. To expect math to be different just because just doesn't make sense to me in the least.

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35. ### otterpopPhenom

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Nov 4, 2015

I actually thought about this thread last night as I was grading papers.

I wanted to find the letter grade equivalent for scores that were out of 50.

Instead of pulling out my calculator or computer, I decided to mentally double the score out of 50 (48/50=96%, A). It's not what I would always choose but it was just the most convenient. Having that choice is nice.

When we teach kids both mental math and how to use a calculator, we are giving them a more versatile skill set.

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36. ### 3SonsEnthusiast

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Nov 4, 2015

One of my favorite calculator quotes comes from teaching SAT prep for the Princeton Review:

"A calculator helps you get the wrong answer faster"

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37. ### GTB4GTCohort

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this thinking is a bit of a paradox to me. The students failed to learn their multiplication tables so they struggle mightily to factor a quadratic equation. But if we teach them how to use a tool to do the factoring, then we can forego the laborious math and get into really deep level thinking. That runs contrary to what I have seen/experienced in the classroom.

My experience says that if you cannot find the roots of a quadratic equation by factoring (on your own), it is highly unlikely that you can or will understand the theory of relativity. IMO, you cannot skip steps. Of course, I am only speaking from my personal observations in my classroom. I am open to the idea that things may be completely different in your school.

38. ### czaczaMultitudinous

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Nov 6, 2015

Totally agree...Out of 50 is easily done with mental math. Not as much when your test is 28 points or some other number that isn't a factor of 100....and that's where the choice comes in. Our kids need to be able to use a variety of thinking skills, tools and technology in order to be successful.

39. ### Bak2MathRookie

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Nov 6, 2015

It's apparent that anecdotal evidence may not convince people that calculator use too early in the education process is detrimental, at least prior to calculus. But even then if someone can't do simple factoring then something like u substitution would be pointless. Not everyone is meant for calculus though. But I am witnessing kids who managed to get to algebra 2 in the ninth and tenth grade who are giving up on math because they were not given the chance to learn some very basic skills, such as being able to multiply divide add and subtract whole numbers up to 2 digits without the use of a calculator, and are now incapable of factoring in a reasonable amount of time.

Anecdotally... I have witnessed my first full unit being taught and graded. Everyone was able to answer all of the multiple choice questions, not exactly correctly but they answered them. The type you would see on the end of year SOL. The test of course that is meant to assess how well the kids understand the concepts. A significant number of them did not even attempt the free response because they did not know where to start. That in itself would give a pretty good idea how well they understand the concepts, or in their case their complete lack of understanding the process.

And this will be followed up by yet another test administered by the county to determine if the kids are prepared for the SOL. I would like to get my hands on the data that is produced by these multiple choice tests and see if I am right about the correlation between scoring well on the test and understanding the concepts being tenuous at best. But then again I suspect the people using the statistical software to assess the students have very little knowledge of the theory and application of statistics to start with.

Kind of like using a tool (lets say... a calculator) without understanding how it is really supposed to be used. This goes beyond the blind leading the blind. It's the blind actively teaching others how to be blind.

40. ### GTB4GTCohort

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Nov 7, 2015

A question...if you were to correlate test scores vs. understanding concepts, how would you manage to quantify the "understand concepts" variable?

also, I laughed at the "blind teaching others to be blind" comment. Although, like so much of the stuff that we see going on, isn't really funny when you reflect on the impact being done to the students.

41. ### Bak2MathRookie

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Nov 8, 2015

I am brand new to the process of teaching, at this level, and formal teaching in general. I am not new to taking tests and I am not new to having them graded. But I do not ever recall taking a multiple choice test at any point in my schooling except during national standardized tests: PSAT, SAT, SAT math subject text and then AP Calculus (I think it had mixed multiple choice and FRQ). At the time though I never contemplated if these tests accurately reflected my personal abilities as I did well in the classes and I did well on the standardized tests.

Fast forward to now though and what I have seen. While I have been unable to look at every test I have looked at a few and indicated that some student who did not even attempt the FRQ naturally answered all of the multiple choice questions, though I do not know how they did on them yet. These were obviously the students who were not performing well, but some still passing, and clearly did not understand even the first step of the concepts. What about a few students who I do know are performing well, relatively speaking. One of them likely aced every question and even answered more than he had to. However another who is performing at the same level, at least according to his grades until now, is clearly struggling with the concepts. I am not saying this because I think the grading standards should be harder. But if someone passes a test, let alone do well on it, they should at least understand the concepts. Doing well, above basic passing, would mean they can perform the concepts more often than not and so on and so forth.

Getting back to the question of seeing if doing well on a multiple choice math test correlates with understanding how to actually do math. You need people who can grade/evaluate these tests by hand using consistent standards. This would take time and money, essentially resources. Resources that are being wasted at all levels of government but thats another topic. I may be wrong about MC tests and ability though. They may correlate well. I wonder what college math professors think. How many students are they having to remediate who have supposedly passed national and state standardized tests?

I think calculators are only part of the problem.

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