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11.3: Careful Use of Language in Mathematics- = - Mathematics

11.3: Careful Use of Language in Mathematics- = - Mathematics



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The notion of equality is fundamental in mathematics, and especially in algebra and algebraic thinking. The symbol “=”’ expresses a relationship. It is not an operation in the way that + and are × operations. It should not be read left-to-right, and it definitely does not mean “… and the answer is …”.

For your work to be clear and easily understood by others, it is essential that you use the symbol = appropriately. And for your future students to understand the meaning of the = symbol and use it correctly, it is essential that you are clear and precise in your use of it.

Let’s start by working on some problems.

Problem 7

Akira went to visit his grandmother, and she gave him $1.50 to buy a treat.

He went to the store and bought a book for $3.20. After that, he had $2.30 left.

How much money did Akira have before he visited his grandmother?

Problem 8

Examine the following equations. Decide: Is the statement always true, sometimes true, or never true? Justify your answers.

  1. $$5+3=8 ldotp$$
  2. $$frac{2}{3} + frac{1}{2} = frac{3}{5} ldotp$$
  3. $$5 + 3 = y ldotp$$
  4. $$frac{a}{5} = frac{5}{a} ldotp$$
  5. $$n + 3 = m ldotp$$
  6. $$3x = 2x + x ldotp$$
  7. $$5k = 5k + 1 ldotp$$

Problem 9

Consider the equation

[18 - 7 = \_\_\_ ldotp]

  1. Fill in the blank with something that makes the equation always true.
  2. Fill in the blank with something that makes the equation always false.
  3. Fill in the blank with something that makes the equation sometimes true and sometimes false.

Problem 10

If someone asked you to solve the equations in Problem 8, what would you do in each case and why?

Think / Pair / Share

Kim solved Problem 7 this way this way:

Let’s see:

[2.30 + 3.20 = 5.50 - 1.50 = 4,]

so the answer is 4.

What do you think about Kim’s solution? Did she get the correct answer? Is her solution clear? How could it be better?

Although Kim found the correct numerical answer, her calculation really doesn’t make any sense. It is true that

[2.30 + 3.20 = 5.50 ldotp]

But it is definitely not true that

[2.30 + 3.20 = 5.50 - 1.40 ldotp]

She is incorrectly using the symbol “=”, and that makes her calculation hard to understand.

Think / Pair / Share

  • Can you write a good definition of the symbol “=”? What does it mean and what does it represent?
  • Give some examples: When should the symbol “=”’ be used, and when should it not be used?
  • Do these two equations express the same relationships or different relationships? Explain your answer. $$x^{2} - 1 = (x + 1)(x - 1) ldotp$$$$(x+1)(x-1) = x^{2} -1 ldotp$$

This picture shows a (very simplistic) two-pan balance scale. Such a scale allows you to compare the weight of two objects. Place one object in each pan. If one side is lower than the other, then that side holds heavier objects. If the two sides are balanced, then the objects on each side weigh the same.

Think / Pair / Share

In the pictures below:

  • The orange triangles all weigh the same.
  • The green circles all weigh the same.
  • The purple squares all weigh the same.
  • The silver stars all weigh the same.
  • The scale is balanced.

1. In the picture below, what do you know about the weights of the triangles and the circles? How do you know it?

2. In the picture below, what do you know about the weights of the circles and the stars? How do you know it?

3. In the picture below, what do you know about the weights of the stars and the squares? How do you know it?

Problem 11

In the pictures below:

  • The orange triangles all weigh the same.
  • The green circles all weigh the same.
  • The purple squares all weigh the same.
  • The scale is balanced.

How many purple squares will balance with one circle? Justify your answer.

Problem 12

In the pictures below:

  • The orange triangles all weigh the same.
  • The green circles all weigh the same.
  • The purple squares all weigh the same.
  • The silver stars all weigh the same.
  • The scale is balanced.

How many purple squares will balance the scale in each case? Justify your answers.

Problem 13

In the pictures below:

  • The orange triangles all weigh the same.
  • The green circles all weigh the same.
  • The purple squares all weigh the same.
  • The scale is balanced.

What will balance the last scale? Can you find more than one answer?

Problem 14

In the pictures below:

  • The orange triangles all weigh the same.
  • The green circles all weigh the same.
  • The purple squares all weigh the same.
  • The scale is balanced.

  1. Which shape weighs the most: the square, the triangle, or the circle? Which shape weighs the least? Justify your answers.
  2. Which of the two scales is holding the most total weight? How do you know you’re right?

Think / Pair / Share

What do Problems 11–14 above have to do with the “=” symbol?


Math 305 - Introduction to Advanced Mathematics Fall 2010

For 8/30: Section 1.1, problems A2 - A6, B2, B16, B17, B20, B41, C1.

For 9/8: Section 1.2, problems A1, A5, A21, A29, A49, B5, B14, B21, B45 Section 1.3, problems A1, A17 - A20, class problem.

For 9/13: Section 1.3, problems A27, A29, A33, B1, B4, B10, B22, C6, class problem.

For 9/20: Section 1.4, problems A10, A17, A20, B9, B21, C5 Section 1.5, problems A1, A19, A33, A42, B3, B12, C1.

For 9/27: Section 1.6, problems A2, A17, A52, B8, B25, B41, B43 Section 2.1 problems A16(a,c), B20, READ problems C23-C24 (you don't have to turn anything in) Section 2.2, problems A3, A17, class problem.

For 10/4: Section 2.2, problems B4, B19, B25, B39, B62(a,b), B87, C11. Section 2.3, problems A14, A19, A32(a,d), B22, B60, B87, C3. Section 2.5, problems B23, B37, C2 (you may assume that there exists at least one irrational number).

For 10/11: Section 3.1, problems A7, A9, B4, B16, B40 (in some students' books, this seems to be B43 the one about the square root of x+y), C1. Section 3.2, problems A29, B27 (not by cases). Section 3.3, problems A4, A19, A22, B3, B36, B47.

For 10/27: Section 3.4, problems B17, C4, C5. Section 3.5, problems A7, B9. Section 3.6, problems B23, B24, B25. Section 4.2, problems A9, A10, A26.

For 11/3: Section 4.2, problems A47, A48, B3, B31, B34, B67, B69. Section 5.1, problems A18, B21, B27, B40, C16.

For 11/10: Section 5.1, problems A14, A16, B2, B19. Section 5.2, problems A3, A10, B6, B15, B23. Section 5.3, problems A6, B8, B9.

For 11/17: Section 5.3, problem C3. Section 6.1, problems A1, A3, A7, B15, B17, B29, B36, C1.

For 11/24: Section 6.2, problems A9, B2, B14, B28. Section 6.3, problems A2, A4, A5, B5, B52. Section 6.4, problems A4, B18, B21.

For 12/3: Section 6.5, problems B1, B4, B9, B11, B12, B15. Probability problems.


ORIGINAL RESEARCH article

Karin Kucian 1,2,3 * , Isabelle Zuber 1 , Juliane Kohn 4 , Nadine Poltz 4 , Anne Wyschkon 4 , Günter Esser 5 and Michael von Aster 1,2,3,6
  • 1 Center for MR-Research, University Children's Hospital, Zurich, Switzerland
  • 2 Children's Research Center, University Children's Hospital, Zurich, Switzerland
  • 3 Neuroscience Center Zurich, University of Zurich, ETH Zurich, Zurich, Switzerland
  • 4 Department of Psychology, University of Potsdam, Potsdam, Germany
  • 5 Academy for Psychotherapy and Intervention Research, University of Potsdam, Potsdam, Germany
  • 6 Clinic for Child and Adolescent Psychiatry, German Red Cross Hospitals, Berlin, Germany

Many children show negative emotions related to mathematics and some even develop mathematics anxiety. The present study focused on the relation between negative emotions and arithmetical performance in children with and without developmental dyscalculia (DD) using an affective priming task. Previous findings suggested that arithmetic performance is influenced if an affective prime precedes the presentation of an arithmetic problem. In children with DD specifically, responses to arithmetic operations are supposed to be facilitated by both negative and mathematics-related primes (=negative math priming effect).We investigated mathematical performance, math anxiety, and the domain-general abilities of 172 primary school children (76 with DD and 96 controls). All participants also underwent an affective priming task which consisted of the decision whether a simple arithmetic operation (addition or subtraction) that was preceded by a prime (positive/negative/neutral or mathematics-related) was true or false. Our findings did not reveal a negative math priming effect in children with DD. Furthermore, when considering accuracy levels, gender, or math anxiety, the negative math priming effect could not be replicated. However, children with DD showed more math anxiety when explicitly assessed by a specific math anxiety interview and showed lower mathematical performance compared to controls. Moreover, math anxiety was equally present in boys and girls, even in the earliest stages of schooling, and interfered negatively with performance. In conclusion, mathematics is often associated with negative emotions that can be manifested in specific math anxiety, particularly in children with DD. Importantly, present findings suggest that in the assessed age group, it is more reliable to judge math anxiety and investigate its effects on mathematical performance explicitly by adequate questionnaires than by an affective math priming task.


11.3: Careful Use of Language in Mathematics- = - Mathematics

A Russian translation of this article can be found here.

XSLT's full support of XPath's math capabilities lets you do all the basic kinds of arithmetic and a little more. Let's look at a stylesheet that demonstrates these capabilities by using the values from this document:

Lines A through N of the stylesheet each make (or attempt to make) a different calculation. These calculations use the numbers in the document above and some other numbers that are either hardcoded in the stylesheet or retrieved from functions that return numbers.

Before we talk about what each line is doing, let's look at the result of applying the stylesheet to the numbers document.

The stylesheet has a single template rule for the source tree's numbers element. This template has a series of xsl:value-of instructions whose select attributes use the values of the numbers element's x, y, and z child elements to do various kinds of math. Mathematical expressions like these can use the full power of XPath to say which element or attribute has a number they need this stylesheet, however, is more concerned with demonstrating the range of mathematical operations available than with using fancy XPath expressions to retrieve elements and attributes from odd parts of a document.

Line A of the template adds the value of x (4) to the value of y (3.2) and puts their sum, 7.2, in the result tree. It's simple, straightforward, and shows that you're not limited to integers for stylesheet math.

Line B subtracts 4 from 3.2 for a result of -0.8. Negative numbers shouldn't pose any difficulties for XSLT processors.

Warning With some XSLT processors, the use of decimal numbers may introduce a tiny error. For example, the "3.2 - 4" in this example comes out as "-.7999999999999998" on some processors. While being off by .0000000000000002 isn't much, being off at all shows that math is not XSLT's strong point.

Line C multiplies 4 and 3.2, using the asterisk as the multiplication operator, for an answer of 12.8.

Line D divides the value of the z element, (11), by 3.2, showing an XSLT processor's ability to perform floating-point division. Although most programming languages traditionally use the slash character to represent mathematical division, XPath already uses the slash to separate the steps in an XPath location path (for example, wine/vintage to represent the vintage child element of the wine element). Accordingly, XPath and XSLT use the string "div" for division.

Lines E and F show how parentheses have the same effect on operator precedence that they have in normal math notation: without them, multiplication happens before addition, so that 4 + 3.2 * 11 = 4 + 35.2. With the parentheses around the "4+3.2", that happens first, so that (4 + 3.2) * 11 = 7.2 * 11.

Line G demonstrates the mod operator, which shows the remainder if you divide the first term by the second. The example shows that 11 mod 4 equals 3, because 4 goes into 11 twice with 3 left over. This operator is great for checking whether one number divides into another evenly just check whether the larger number mod the smaller equals zero.

Line H demonstrates the sum() function. With a node list as an argument, it sums all the numbers in the list. In the example the asterisk means "all the children of the context node" -- the numbers element's x, y, and z children.

Lines I and J demonstrate the floor() and ceiling() functions. If you pass either of these an integer, they return that integer. If you pass floor() a non-integer, it returns the highest integer below that number. In the example, floor(3.2) is 3. The ceiling function returns the smallest integer above a non-integer number in the example, ceiling(3.2) equals 4.

Line K's round() function rounds off a non-integer by returning the closest integer. When 3.2 is passed to it, it returns 3 passing 3.5 or 3.6 to it would cause it to return 4.

Line L incorporates another XPath function, count(), which returns the number of nodes in the set passed to it as an argument. XPath offers several functions that, while not explicitly mathematical, return numbers and can be used for any calculations you like: count(), last(), position(), and string-length(). Line M demonstrates string-length(), which returns the length of a string.

Line N demonstrates what happens when you try to perform math with something that isn't a number: when 11 gets added to the string "hello", the result is the string "NaN", an abbreviation for "Not a Number." When you pull a number out of an element's content or attribute value and then use it for a calculation, you can't always be sure that what you pulled is really a number, so XSLT's clearly defined behavior for the unworkable case makes it easier to check for and cope with in your code.

XSLT is about manipulating text, not numbers, but you can build on the mathematical operations provided as part of XSLT to perform more complicated calculations. For example, the following stylesheet, which accepts any document as input, computes the value of pi. The precision of the result depends on the value of the iterations variable.

The repetition is implemented using a recursive named template. With the iterations setting shown, the stylesheet creates this result:

With that many iterations, the answer is only accurate up to the first four digits after the decimal. Of course, if you want to compute the value of pi seriously, there are many more appropriate languages, but it's nice to know that you can push XSLT to do some fairly complex math when necessary.


Putting Math in Javascript Strings¶

If your are using javascript to process mathematics, and need to put a TeX or LaTeX expression in a string literal, you need to be aware that javascript uses the backslash ( ) as a special character in strings. Since TeX uses the backslash to indicate a macro name, you often need backslashes in your javascript strings. In order to achieve this, you must double all the backslashes that you want to have as part of your javascript string. For example,

This can be particularly confusing when you are using the LaTeX macro , which must both be doubled, as . So you would do


In this study, the term “textbooks” is used broadly to refer to conventional print textbooks, e-textbooks, and other curriculum resource books for teaching and learning.

In Shanghai, primary school education consists of 5 years from Grade 1 (at age of 6) to Grade 5 (at age of 11). In England, it consists of 6 years from Year 1 (at age of 5) to Year 6 (at age of 11).

The first and third authors were working at the Southampton Education School of the University of Southampton while the main stage of this study was undertaken.

It appears arguable whether Joan, supposedly a girl, has £200 at her disposal, and whether two dictionaries cost £79 are realistic and to what extent for learners in England. Nevertheless, as aforementioned, a further discussion about how the adaptation can be improved is beyond the scope and intention of this study.

The number was changed to 6341 in the latest edition of the series.

According to available statistics, there were 861 million active monthly QQ users in China http://technode.com/2017/08/07/wechats-older-sibling-qq-plans-to-stay-forever-young. Accessed 15 Oct 2017.


Exploratory Courses

Philosophy concerns itself with fundamental problems and examines the efforts of past thinkers to understand the world and people's experience of it.

Science, Technology, and Society

Provides students with an interdisciplinary framework through which to understand the complex interactions of science, technology and the social world.


Summing up the Philosophical Problems

We can now summarize the philosophical problems that have emerged in our historical account of the technology–mathematics relationship.

First, we have the technology-dependence of mathematical knowledge. We noted in Section 2 that from its very beginnings, human knowledge of mathematics has depended on aide-mémoires such as notches on a stick, pebbles on a counting board, or symbols on paper. We need notation not only to remember numbers but also to keep track of the successive steps of a computation, derivation, or proof. As we saw in Section 5, we now depend increasingly on more advanced technological devices, namely computers, not only to record but also to perform the steps of mathematical operations. Since mathematical knowledge is usually considered to be non-empirical, this creates problems for mathematical epistemology.

Secondly, although the notion of a computation is defined mathematically, it has implications for our understanding of operations performed on physical devices. For instance, if we define computation as a process consisting of a particular type of elementary operations, then a machine performing a computation will have to do so by executing suboperations that can reasonably be understood as representing such elementary operations. A technological device that arrives at the desired result by some other means could not be said to have obtained it by computation. For this and other reasons, we need to clarify the relationship between mathematical and technological computability.

Thirdly, the usefulness of mathematics in technology poses a puzzle that is analogous, but perhaps not identical, to the much more widely discussed puzzle of the usefulness of mathematics in science. How does it come that so many technological problems have been solved with mathematical tools that were invented for purposes unconnected with technology? Is there some underlying connection which we have not grasped?

The purpose of the following three sections is to further introduce these three problems and to show that concepts from the philosophy of technology and the philosophy of mathematics may have to be combined in order to solve them.


Is mathematics a science?

Abstract: Mathematics is not a science, but there are grey areas at the fringes.

1991 Mathematics Subject Classification . 01
Key words and phrases . mathematics, science

Mathematics is certainly a science in the broad sense of "systematic and formulated knowledge", but most people use "science" to refer only to the natural sciences. Since mathematics provides the language in which the natural sciences aspire to describe and analyse the universe, there is a natural link between mathematics and the natural sciences. Indeed schools, universities, and government agencies usually lump them together. (1) On the other hand, most mathematicians do not consider themselves to be scientists and vice versa . So is mathematics a natural science? (2) The natural sciences investigate the physical universe but mathematics does not, so mathematics is not really a natural science. This leaves open the subtler question of whether mathematics is essentially similar in method to the natural sciences in spite of the difference in subject matter. I do not think it is.

A disclaimer is in order. This essay is a "native informant's" opinion: I am a practicing (if mediocre) mathematician, but not a philosopher or student of the practice of science or mathematics. I have a relevant philosophical bias, in that I am a Platonist where mathematical reality is concerned. (3)

The object of the natural sciences is to devise and refine approximate descriptions or models of aspects of the physical universe. The feature distinguishing science from other means of doing so is its characteristic method. Crudely, this consists of asking a question, formulating a hypothesis, testing it, and then, on the basis of the results, rejecting or provisionally accepting the hypothesis. One usually repeats the process after refining the question, the hypothesis, or one's ability to test it. The ultimate arbiter of correctness is the available empirical evidence: a hypothesis which is falsified -- i.e. inconsistent with good data -- is not acceptable. (A hypothesis which could not be falsified by any empirical data is not scientific.) Note that a scientific theory or hypothesis is (at best) only provisionally acceptable at any given time, because a new piece of evidence may force it to be modified or rejected outright.

In mathematics, however, the ultimate arbiter of correctness is proof rather than empirical evidence. This reflects a fundamental diffence in what one is trying to achieve: mathematics is concerned with finding certain kinds of necessary truths. For a mathematical statement to be accepted as a theorem, its conclusion must be known to always be true whenever its hypotheses are satisfied. We accept it only when we have a proof: a chain of reasoning demonstrating that the conclusion must follow from the hypotheses. (4) Empirical evidence does, to be sure, play an important part in doing mathematics. Conjectures are usually formed by observing a common pattern in a number of examples, and are often tested on other examples before a proof is attempted. However, such evidence is not sufficient by itself: consider the assertion that every even integer greater than 4 is the sum of two (not necessarily different) odd prime numbers. (5) We have lots of empirical evidence supporting this assertion: 6 = 3+3, 8 = 5+3, 10 = 7+3 and 10 = 5+5, 12 = 7+5, and so on. However, we cannot be sure it is true unless someone finds a proof. Until then, it is conceivable that someone might find a very big even number which is not the sum of two odd prime numbers. (6)

The essential difference in method between mathematics and science, and the weakness of each, is neatly exploited in the following joke:

Some academics relaxing in a common room are asked whether all odd numbers greater than one are prime.

The physicist proceeds to experiment -- 3 is prime, 5 is prime, 7 is prime, 9 doesn't seem to be prime, but that might be an experimental error, 11 is prime, 13 is prime -- and concludes that the experimental evidence tends to support the hypothesis that all odd numbers are prime.

The engineer, not to be outdone by a physicist, also proceeds by experiment -- 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, 13 is prime, 15 is prime -- and concludes that all odd numbers must be prime.

The statistician checks a randomly chosen sample of odd numbers -- 17 is prime, 29 is prime, 41 is prime, 101 is prime, 269 is prime -- and concludes that it is probably true that all odd numbers are prime.

The physicist observes that other experiments have confirmed his conclusion, but the mathematician sneers at "mere examples" and posts the following: 3 is prime. By an easy argument which is left to the reader, it follows that all odd numbers greater than one are prime. (7)

It must be admitted that the difference noted above between science and mathematics is not completely sharp, even aside from the fact that the practice of mathematics does have empirical content. Some of the areas in which mathematics is applied to modelling aspects of the physical universe are very grey indeed. The basic problem is that one can be confident of a fact derived by mathematical methods only to the extent that the mathematical object being considered is an accurate model of the relevant parts of the universe. One can be completely confident this is so in mathematics (where the mathematical object in question is the relevant part of the universe) and quite confident in, for example, computer science (where the physical objects being analysed are made to conform to a mathematically precise pattern) and parts of theoretical physics (where some theories have survived very extensive testing). However, one cannot usually be very confident in, say, long-term economic projections. The moral is that in applying mathematics to problems from the "real" world, one must judiciously temper the use of mathematical knowledge and techniques with empirical knowledge and testing.

With increasing interaction between mathematics and the natural sciences, plus the practical problems involved in finding and checking really long proofs, it is arguable that the grey areas are expanding. It has even been argued that proof and certainty in mathematics are nearly obsolete [4], though most of those who agree that "empirical" mathematics has a place still believe that proofs have an important role ( e.g. [2] and [7]). It is my belief that proofs will remain central for a good while yet.

(1) Which is convenient for mathematicians when grant money is distributed, so don't show this essay to any funding agency !

(2) The problem of showing that mathematics is not a social science is left as an exercise for the reader. One could argue that mathematics ought to be classified with the arts and humanities [3], but it doesn't function like one [6]. There is also the argument that mathematics is "not really accessible enough to be an art and not immediately useful enough to be a science" [1], but this assumes that art is accessible and science is useful.

(3) As for non-mathematical reality, who cares?

(4) Of course, this begs the question of just what constitutes such a chain of reasoning. Philosophers really worry about this, but most mathematicians settle for giving arguments acceptable to most other mathematicians. History suggests that it is a mistake to be too rigid about correctness in mathematics: it took over two centuries, for example, to work out rigorous foundations for calculus.

(5) This assertion is called Goldbach's Conjecture. A prime number is an integer greater than one which is not a product of two smaller positive integers.

(6) If you do either, please publish!

(7) What of the others present in the common room?

The chemist [5] observes that the periodic table gives the answer: 3 is lithium, 5 is boron, 7 is nitrogen, 9 is fluorine, 11 is sodium, . Since elements are indivisible -- nuclear fission being uncommon in chemistry labs --- these are all prime. (The same is true for even numbers!).

The economist notes that 3 is prime, 5 is prime, 7 is prime, but 9 isn't prime, and exclaims, "Look! The prime rate is dropping!"

The computer scientist goes off to write a program to check all the odd numbers. Its output reads: 3 is prime. 3 is prime. 3 is prime. .

The sociologist argues that one shouldn't refer to numbers as odd because they might be offended or as prime because the term implies favouritism, and the theologian concurs since all numbers must be equal before God.

(8) If you're still wondering whether it's true, you haven't paid careful attention. (7)


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