What do children understand about calculators



What do students understand about calculators?

Interview questions:

Explain purpose is to find out what students your age know about calculators and will be taped (but names will not be used).

1. Have you used a calculator before? What kind? In school? Out of school – establish past experience.

2. Have you ever received any direct instruction on how to use any particular calculator?

3. Why do you use a calculator…in what circumstances/situations? Are there types of problems you don’t use a calculator for?

4. Pretend I’ve never seen a calculator before – I know how to add, subtract, multiply, and divide on paper, but not using a calculator. Explain to me how to use a calculator (and perhaps what it’s good for).

5. How do you think a calculator works? (i.e. how does it know what it knows?)

6. Do calculators ever make mistakes? Can you show me an example of a mistake using a calculator?

7. Are there limitations on the types of numbers you can use with a calculator (i.e. big/small)?

1. Try adding 123456789 + 123456789.

2. Try 2/7 (Why are there different answers?)

3. Now multiply by 7 again …what do you think the answer should be?

8. I have two calculators here…I want you to try calculating the following using the 10-key calculator, and then the graphing calculator. 5 + 10 ÷ 2 – 3 · 4

4. What do you get using the first calculator?

5. The graphing calculator? Are they the same? What’s the difference? Should they be the same?

6. Try doing the problem by hand.

9. In general, do you get a different answer using different calculators? Should you?

[Well developed protocol. There is a direction and purpose implied by the questions.]

Purpose:

The purpose of the study is to find out what students use calculators for, what they know about their operation, and if they know how to evaluate the calculator’s answers to see if it is reasonable. All students have used calculators by the time they reach high school, but many are not aware of their limitations or have not been properly instructed as to their use. Also, there have been reports of increased reliance on calculators to the point that students virtually cannot do simple addition, subtraction, multiplication, and division by hand or in their heads. Many teachers think that students see the calculator as a magical black box that churns out correct answers no matter how it is used. If true, then the use of calculators could be a significant stumbling block to students as they reach higher mathematics. Also, blind reliance on a calculator as a source of perfect information may lead to incorrect application in the real world. While the difference between correct and incorrect answers may be only 1 millionth of a unit, this can still lead to vastly incorrect answers if the project is financial projections or launching a shuttle into space, or building a bridge. If students don’t first master the concept and application of basic mathematical concepts before moving on to using a calculator, the result can be that the calculator become a crutch for students to lean on.

This area has sparked a huge debate between educators and parents on both sided of the issue. An article published by Education World describes the debate this way:

The philosophical war rages. On one side: the accused "kill and drillers," dedicated to times tables and long division, preaching the gospel of repetition and memorization. On the other side: alleged "fuzzy math" reformers preaching concept over content, insight over "right." Between them: the most visible symbol of the continuing conflict -- the classroom calculator. Depending on one's perspective, the use of calculators at the elementary school level is seen as either the solution to or the cause of many of the problems affecting math education in this country. What's the truth? Even the experts disagree. (EW, 2002)

The use of calculators has become so contentious in some communities that their use has been banned. The Nevada State Board of Education voted to ban the use of calculators on the high school exit exam in 1997 due to parent complaints, and controversy over whether allowing calculators on the test discriminates against certain populations of students. (Friess, 1997).

What are they key issues in the debate? Education World described the positions of proponents and opponents of widespread calculator use as follows:

Calculator proponents claim that calculators

allow students to spend less time on tedious calculations and more time on understanding and solving problems.

help students develop better number sense.

allow students to study mathematical concepts they could not attempt if they had to perform the related calculations themselves.

allow students who would normally be turned off to math because of frustration or boredom to increase their mathematical understanding.

simplify tasks, while helping students determine the best methods for solving problems.

make students more confident about their math abilities.

Critics say calculators

produce students who can't perform basic tasks without a calculator.

encourage students to randomly try a variety of mathematical computations without any real understanding of which is appropriate or why.

prevent students from discovering and understanding underlying mathematical concepts.

keep students from benefiting from one of the most important reasons for learning math -- to train and discipline the mind and to promote logical reasoning.

inhibit students from seeing the inherent structure in mathematical relationships.

give students a false sense of confidence about their math ability.

The proper use of technology (such as calculators) is one of the major standards of the National Council of Teachers of Mathematics. Also, students are allowed to use calculators on certain state and national tests (such as the SAT II and CIM/CAM testing). The NCTM official statement on the use of technology states, “The appropriate use of instructional technology tools is integral to the learning and teaching of mathematics and to the assessment of mathematics learning at all levels.” (NCTM 2000) They also recommend that “every student should have access to an appropriate calculator.” While this is a good goal for incorporating the use of technology in the classroom, there is no mention of teaching students about how to use calculators in a way that minimizes their use of the technology as a crutch, and enables them to critically evaluate the results given by any calculation generated calculation. [I think this is generally correct. Also, instruction does not typically consider how to make students LESS dependant upon instruction over a course. Independence is verballized but not taught.]

The NCTM does make mention of these important aspects in its position paper entitled “Calculators and the Education of Youth,” stating that:

Appropriate instruction that includes calculators can extend students' understanding of mathematics and will allow all students access to rich problem-solving experiences. Such instruction must develop students' ability to know how and when to use a calculator. Skill in estimation, both numerical and graphical, and the ability to determine if a solution is reasonable are essential elements for the effective use of calculators. (NCTM, 2000)

While these are certainly worthy and necessary goals, there is no mention of how teachers are to incorporate these ideas into their curriculum. Indeed, the NCTM technology standards themselves do not make any mention of teaching for or assessing students’ ability to critically use calculators.

This is not surprising, given the fact that most math teachers are already under the gun to cover a wide spectrum of content are topics. Given the choice between taking the time to instruct and assess students on appropriate use of calculators, most teachers tend to concentrate on teaching core material.

There may be good reasons if students aren’t able to understand the limitations and appropriate use of calculators. For example, tracing the errors made by a calculator to show students how it makes certain errors requires that students can do somewhat complicated and/or tedious calculations accurately by hand. If students use a calculator because they don’t understand the mathematics behind it, then the odds of that student understanding what the calculator did wrong are nearly zero.

Source Used

• Education World, 2002. “Educators Battle Over Calculator Use: Both Sides Claim Casualties.” Last accessed 12/10/02

• Friess, Steve. 1997. Las Vegas Review-Journal, July 25. Last accessed 12/10/02.

• National Council of Teachers of Mathematics, 2000. E-standards – “Calculators and the Education of Youth.” Last accessed 12/10/02

The Concept:

The main function of a calculator is to perform numerical calculations that would be difficult and/or time consuming to do by hand. Although there are a variety of types of calculators, they all function in basically the same way at a basic level. The user inputs the expression into the calculator by pressing its buttons in particular order. The calculator stores the numbers using its electronic circuitry as binary strings, and then performs on those strings using instructions that are hard-encoded in its circuitry. The result is returned to its temporary memory and converted to a numerical output which is displayed on its screen for the user.

The calculator has a finite memory, and is therefore necessarily limited in the number of digits it can store in its RAM, which is why most calculators will only allow numbers expressed by up to 8 to 16 digits in base 10. Answers are generally returned using the maximum number of digits allowed by the calculator’s working memory, which is different for each calculator. Four function and 10 key calculators generally return answers in standard decimal notation, using as many digits as possible (but counting leading zeros on a decimal answer) i.e 2/987 returns 0.0020263 (using all eight digits). Scientific calculators will usually return answers by eliminating leading zeros and expressing the answer in scientific notation (i.e. 2/987 returns 2.02634245187 x 10-3. Graphing calculators usually have more memory and can therefore store more digits of accuracy, but its calculating memory is still limited to about 12 digits of accuracy, so while you can enter almost arbitrarily large numbers, it will still only display answers using the first 12 significant digits.

The operation of 4 function and 10 key calculators is different from that of graphing calculators. Generally, 4 function and 10 key calculators can only store one number at a time, therefore extended calculations (such as 2+4*7-3/2) must be entered carefully as a series of sub-expressions, carefully following the order of operations. So, to calculate the given expression using the 4 function or 10 key one would first need to perform the multiplication and division (4*7 = 28 and 3/2 = .5) and record those results somewhere by hand. Then, the results of those calculations can be used to finish the calculation by entering 2 + 28 – 0.5 = 27.5. Simply entering the expression as written from left to right will cause these calculators to perform the operations as they are listed from left to right (rather than following the order of operations). So, 2+4*7-3/2 would result in 2+4 = 6, *7 = 42, -3 = 39, /2 = 19.5 – an incorrect answer. In contrast, graphing calculators will accept an entire expression, and calculate the answer following the proper order of operations. Some scientific calculators have single item like the 4 function/10 key calculators, while others can accept an entire expression before calculating.

The calculators generally fall into four classes depending on their functionality. At the bottom of the pile is the four function calculator, which can only add, subtract, multiply, or divide a single number at a time. Generally these calculators are also limited to a small number of digits (about 8). The 10 key is very closely related to the four function, but adds the square root and percent function. This type of calculator is generally used for computing finances, which do not require more advanced functions. The scientific calculator can do every thing the 10 key can, but adds a plethora of additional functions including the ability to compute trig functions (sin, cos, tan etc.), probability, exponents, and can work with numbers with up to about 12 digits. Many of these calculators can also calculate using fractions (although they’re tricky to use), and can often store several values (about 5 is typical). Graphing calculators are at the top of the heap in terms of functionality, as they can do anything the others can do, plus a whole other slew of things. They can accept longer expressions, keep track of a series of expressions, store almost any number of values, execute user programs, and (of course) graph two (sometimes three) variable equations. Many can also simplify and/or solve algebraic expressions, and calculate integrals and derivatives. While these calculators can be hard to use, they offer an incredible variety of mathematical operations that could not be done easily without it.

Subject 1’s concept of calculators.

Subject one had a fairly simplistic view of calculators. They are essentially machines that run on electricity. It receives numbers as input, and spits out the answer. She did not know at all of how the calculator worked, she just knew that it did . This subject did not think that calculators ever make mistakes, although she was somewhat aware of the limitations (in terms of digits) of the calculator. She knew that “bigger” calculators can work with larger numbers, but could not explain or surmise why. Also, other differences in data entry between a 10 key and a graphing calculator were not apparent to the subject. She was also not aware of any of the advanced operations of graphing calculators (such as symbol manipulation or solving equations). She had done some graphing with a graphing calculator in school, but generally thought of them mainly as big versions of a “normal” (i.e. 10 key) calculator. She was aware that some calculators could store answers, and that sometimes you needed a different calculator to do certain math calculations for school (i.e. trig functions), but other than that, didn’t see much use for calculators beyond everyday adding, subtracting, multiplying, and dividing numbers. She generally had an unfavorable view of “complex” calculators, and seemed to prefer the simpler calculators that she had used quite a bit. [It is not clear from this discussion where and if you had students try entering the calculation in the protocol and respond to the result. This is a very good plan, gives students something concrete to examine, replicate, speculate about. How did that work?]

Subject 2’s concept of calculators.

Subject 2 had more experience with calculators than Subject 1, however she still did not have a good idea of how calculators worked. She knew that they sometimes messed up, but she didn’t know why. In general, she though that advanced calculators (such as graphing calculators) were more complicated and hard to learn how to use. She preferred to use a simpler calculator, although she did use them all the time (even for simple calculations such as 12 + 5). This student clearly had little confidence in her own mathematical abilities, and relied exclusively on the calculator to perform any necessary calculations. She knew a lot about the differences between type of calculators – for example she knew that advanced calculators could store answers, and that graphing calculators sometimes had games on them. She really expressed little use for a calculator outside of math class (which is ironic, since she seems to be extremely reliant upon the calculator) [Interesting result and good question. Did you follow up on this at all?], and in particular had no way to determine whether the calculator was reasonably correct, or had made an error. She was extremely upset and confused that entering the same mathematical expression in two different calculators (the 10 key and the grapher) could return different results. She thought that the 10 key was more accurate in every case, except in cases where the grapher returned answers with more decimal places. In one case, the grapher rounded a decimal answer to 2, (which was the actual answer) but the subject though that the 10 key answer of 1.99999994 was more accurate (since it had more decimals). What had actually happened is that the grapher used an intermediate number that was much more accurate than the 8 digit number used by the 10 key to make the final calculation. The subject could not conceive of what was going on between the two calculators because she could not make predictions of outcomes without actually using a calculator. She also made a comment that she didn’t see why anyone would need many digits of accuracy (i.e. more than 2, since that can be used for money). She didn’t think that there could ever be a time where she would need to know an answer out to even 3 or 4 decimal places of accuracy. [Of course this is rarely true in most applications but a good question none the less.]

Subject 3’s concept of calculators.

Subject 3 was the oldest of the three subjects, and therefore had much more experience using calculators. She could explain roughly how calculators work. She explained that calculators store the numbers and operators, then computes the result using order of operation on a grapher, but that 10 key or four function calculation require you to do some preliminary calculations (i.e do multiplication/division first). She also explained that calculators were limited by the number of digits they could display, and that advanced calculators could work with bigger numbers because it had a larger memory. She did not have any idea how the numbers were stored, and surmised that the instructions for calculating were stored as a program in the calculator somewhere. She saw the calculator as a mini-computer (although she did not seem to know how either worked in the sense of using circuitry to store info etc.). She was also familiar with graphing and solving equations, although she thought that it was easier to solve equations by hand on paper. She also knew that you could make your own programs on graphing calculators (which explains the mini-computer analogy). She also knew that you had to set the window right in order to see the graphs correctly.

Subject 3 was the only one of the three interview to be able to explain why calculators make mistakes, although she did not appear to understand that it was related to the precision of the calculator. She knew that her teacher had written a program that gave a wrong answer, which was her example of a calculator making a mistake. She did not think at first she could make a simple calculator make a mistake (such as a 10 key). Also, she thought that the answer given by the graphing calculator for 2/7 was 100% accurate because it included so many decimals – she did not realize that 2/7 could not be expressed fully as a decimal (without using bar notation), and thought that any number could be calculated to exact accuracy with a big enough calculator. Even a number like pi she thought could be expressed by a calculator or computer with a big enough memory. [But this is a pretty subtle point. Pushing students on these ideas is fine, but you must provide appropriate scaffolding so that they can feel successful in reasoning about this type of problem.]

She was able to correctly identify the answer of “2” to the problem (2/7)*7 although she expected to get an incorrect answer since she knew that the calculator could only calculate the answer to a certain number of digits. She was not surprised that the 10 key rounded the answer to 1.9999994. Overall, she seemed to have a pretty good sense of where the calculator could make mistakes, although she couldn’t accurate predict when the calculator would make a mistake, nor could she cause the calculator to produce a mistake herself. She could reliably use the calculator to find answers to numerical expressions, and explain answers in scientific notation. She generally saw calculators as useful tools, and had a good attitude toward math. She also explained that she was not allowed to use calculators until she was a freshman, so she was able to perform most simple calculations in her head, and could evaluate the reasonability of calculator answers.

Implications for Teaching:

The students interviewed had definitely had different experiences using calculators in their math classes. Subjects 1 and 2 had used calculators mainly as tools to check their answers, like an answer key, while subject 3 had experienced a calculator as a tool to investigate mathematics. Subject 3 had also had the experience of at least one teacher having to explain what a calculator did that was wrong. This subject’s eye’s were apparently opened by this experience to the necessity of evaluation calculator answers critically, and viewed the calculator much more cautiously. That said, subject 3 also had the most positive attitude toward using a calculator. While subjects 1 and 2 simply got frustrated when the calculator “didn’t work,” subject 3 saw it as in interesting anomaly, and proceeded to investigate why the error took place. Students 1 and 2 simply felt helpless, since the calculator’s malfunction was out of their control, and could not have been predicted by them. No wonder these students had poor attitudes toward mathematics. To these subjects, mathematics was a confusing, inconsistent game that only really smart people could understand. Although one of the two students had received direct instruction as to how to use a calculator, neither had ever had a teacher explain anything about how calculators worked, or when it was appropriate to use a calculator. *** The implication for teaching is that students who receive more information about how calculators work seem to be more able to accept the fact that calculators are not perfect, and seem to be able to better explain mathematics in general. It was of course the case the subject 3 had had more advanced math classes than the other 2, but was only an average student in general. The other two were also fairly average students, but neither tended to perform well in either of their math classes. *** Students in general need more instruction about what the calculator is doing, why it does what it does, and what its limitations are if they are to be used in math classes. *** I think that these interviews demonstrate the point that students don’t automatically benefit from the use of calculators, especially if they are allowed to use them indiscriminately. The two students who were allowed to use calculators all the time were overly reliant upon them, and had a hard time explaining and/or understanding even basic mathematical concepts. By contrast the subject who was not allowed to use a calculator until high school had a much better understanding of math (though by no means perfect) and also a greater appreciation for the calculator. [*** You make several good points but don’t really put the depth of your interview results to work. How did students react when you gave them questions to investigate on the functioning of calculators? How would you do this in class? Is this a good idea? What about the value of asking students to express their ideas?]

Personal Observations:

Generally, the subjects that I interviewed showed a serious lack of knowledge about how calculators work, and tended to rely on it exclusively as the source of correct answers to math problems. I was surprised by the absolute inability of subjects 1 and 2 to perform basic mathematics in their hand (although #1 at least attempted on paper). Also, the attitudes toward calculators was exactly opposite of what I expected before conducting the interviews. I expected students who relied on calculators a lot to value them highly (after all, they really wouldn’t be able to perform any mathematics without them), but instead they resented the fact that they were confusing and sometimes gave wrong answers. The subject who was “deprived” of calculators until high school, however, was generally much more appreciative of the abilities of the calculator, seeming almost in awe of the abilities of graphing calculators in particular. It seems like subjects 1 and 2 had unrealistic expectations of calculators (that they would give 100% accurate answers to every problem), while subject 3 did not have an expectation of perfection, and had the background to understand why calculators weren’t 100% accurate all the time. Also, subjects 1 and 2 did not seem as willing to talk about math in general. While calculator use in of itself cannot fully account for this, it certainly seems like an indicator of possible negative attitudes.

All in all, I think I was able to interpret the subject’s answers to questions about calculators and accurately surmise their level of understanding on the spot. I didn’t really hear or realize anything different from doing the transcription that I didn’t catch while conducting the interview. I did however assume that the subjects had a better understanding of calculators from my initial interactions with the subjects than was revealed by the interview on this specific subject. I have a feeling that this would be true on most subjects – students tend to convey a level of understand that does not hold up under direct inspection. That is, students are good at pretending they know everything, even when they don’t. Perhaps they even think of themselves as more skilled than they are, but this may be the norm. I don’t think that this level of understanding of the students’ understanding could be revealed by other methods (even an essay on the subject), since it is likely the student will misinterpret the question being asked. I think that I will try to incorporate more interviews of students during my teaching career in order to better assess where students are at in terms of mathematical knowledge in the future. This doesn’t seem like a practical technique for everyday use (since it does require quite a bit of time), but could work well for pre-assessment (especially at the start of a school year).

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