In response to Claire's question, I'd like to share my research findings on the following terms which are misunderstood and frequently result in incorrect work by many students:
1. the minus symbol
Many students are unaware that the minus sign changesmeaning subtly as the curriculum develops. This presents a challenge that some may see as a minor difficulty to be overcome, while others see conflict and increasing disaffection.Initially the minus sign is encountered in whole number arithmetic as the operation of subtraction, or 'take away.' The minus sign in an expression such as 5–3 means 'start with 5 and take away 3.' The result is 2 and is visibly less than the initial quantity 5. This is consistent with the practical idea that 'take away' leaves less.
The second meaning of the minus sign occurs when signed numbers are encountered. Here the symbol 3 refers to the negative number –3 which can be represented as a point on the number line. Now numbers are of three distinct kinds: positive numbers where +3 is the same as the familiar number 3, negative numbers such as –3 and the single number 0 which is neither positive nor negative. While 'subtract 3' is an operation, the negative number '–3' is an object that can be visualized as a fixed point on the number line as the set of points to the left of the origin on a horizontal number line or below zero on a vertical number line.
A third meaning occurs in algebra where –x is the additive inverse of x. It is the number which, when added to x, gives zero. Prior knowledge of the minus symbol to indicate subtraction, while supportive for whole numbers and positive fractions, becomes problematic for many students when they encounter negative numbers. It becomes even more so in the context of algebra to indicate the additive inverse of an unknown, (–
x). The belief held by many students is that a minus sign in front of it indicates that it has a negative value – a belief that results in increasing difficulties in subsequent math courses.
This meaning is different from that in dealing with a number such as –3, for the latter carries with it the meaning that the symbol is a negative number. However, when the variable x is given a particular numerical value, the value of –x need no longer represent a negative number. If the value of x is positive, then –x is negative, if the value of x is negative, then –x is positive, and if x is zero, then –x is also zero. The expression –x may be called 'negative x', 'minus x' or 'the additive inverse of x'. The minus symbol when indicating subtraction, refers to a binary operation (requiring 2 inputs. Used in front of a number, it is an object and, preceding a variable, indicates the additive inverse or opposite of the variable's original value. Research findings document the many errors that can be attributed to students' confusion about the meaning of the minus symbol in a given situation.
Data on students' difficulties evaluating – 3
2 and (–3)
2 have been reported in my various published research studies. Pre- and post surveys administered to 516 community college and university students enrolled in a developmental algebra course showed that, at the completion of the course, 81% (418/516) of the students correctly evaluated (–3)
2 but only 49% (251/516) of the students could correctly evaluate
– 3
2. The problem may become more complicated when evaluating a quadratic expression such as f(x) = –x^2 - 3x + 5, and asked to determine
f(–3).
Many students are confused as to whether the minus sign in (
x – c) represents subtraction or is the sign attached to
c, They say "The value of
c is negative because of the – sign in front of
c.
c will subtract from any number that comes before the "–" symbol."
Students faced with making sense of function notation also have to interpret the minus symbol in expressions such as f(–x) and –f(x). Many students believe that f(–x) represents "f of negative x," or "a negative input value" and that –f(x) represents "negative f of x" or "a negative output value". Some students interpret –f(x) as "the entire function is negative" and f(–x) as "only the x is negative." There are also students who interpret function notation as indicating multiplication: –f(x) means –f times x; and f(–x) as f times –x.
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Mercedes McGowen
Harper College (Retired)
[Streamwoo] [IL]
[630 320-4343]
[
mercmcgowen@sbcglobal.net]
mercmcgowen@sbcglobal.netmercmcgowen@sbcglobal.net
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Original Message:
Sent: 05-10-2021 07:29:12
From: Claire Wladis
Subject: IMPACT in Action - Symbolic Structure Sense
After reading the blog, can you think of any core concepts or terminology that we use frequently in our instruction that we do not tend to address explicitly? What are some ways that we could adapt our instruction to be more explicit about what particular commonly-used words or concepts mean?
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Claire Wladis
Asst. Professor
Borough of Manhattan CC
New York NY
(212) 220-1363
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