Some Common Alternative Conceptions (Misconceptions)
Earth Systems, Cosmology and Astronomy
The correct conception of seasonal change is that it is caused by the tilting of the earth relative to the sun’s rays. As the Earth goes around its orbit, the Northern hemisphere is at various times oriented more toward or more away from the Sun, and likewise for the Southern hemisphere. Seasonal change is explained by the changing angle of the Earth’s rotation axis toward the Earth’s orbit, which causes the alteration in light angle toward a concrete place on the Earth.
A major misconception about seasonal change, held by school students and adults (university students — and teacher trainees and primary teachers — Atwood & Atwood, 1996; Kikas, 2004; Ojala, 1997) is known as the “distance theory.” In this theory, seasons on the Earth are caused by varying distances of the Earth from the Sun on its elliptical orbit. Temperature varies in winter and summer because the distance between the Sun and the Earth is different during these two seasons. One way to see that this reasoning is erroneous is to note that the seasons are out of phase in the Northern and Southern hemispheres: when it is Summer in the North it is Winter in the South. (see Atwood & Atwood, 1996; Baxter, 1995; Kikas, 1998; 2003, 2004; Ojala, 1997). Correct scientific theory on the earth’s shape posits a spherical shape of the earth.
Knowledge about the Earth
Misconceptions: Elementary school children (1st through 5th grades) commonly hold misconceptions about the earth’s shape. Some children believe that the earth is shaped like a flat rectangle or a disc that is supported by the ground and covered by the sky and solar objects above its “top.” Other children think of the earth as a hollow sphere, with people living on flat ground deep inside it, or as a flattened sphere with people living on its flat “top” and “bottom.” Finally, some children form a belief in a dual earth, according to which there are two earths: a flat one on which people live, and a spherical one that is a planet up in the sky. Due to these misconceptions, elementary school children experience difficulty learning the correct scientific understanding of the spherical earth taught in school. It appears that children start with an initial concept of the earth as a physical object that has all the characteristics of physical objects in general (i.e., it is solid, stable, stationary and needing support), in which space is organized in terms of the direction of up and down and in which unsupported objects fall “down.” When students are exposed to the information that the earth is a sphere, they find it difficult to understand because it violates certain of the above-mentioned beliefs about physical objects. (See Vosniadou, 1994; Vosniadou & Brewer, 1992; Vosniadou et al. 2001.)
The correct explanation for the day/night cycle is the fact that the earth spins. Elementary school children (1st through 5th grades) show some common misconceptions about the day/night cycle.
Misconception #1: The earliest kind of misunderstanding (initial model) is consistent with observations of everyday experience. Clouds cover the Sun; day is replaced by night; the Sun sets behind the hills.
Misconception #2: Somewhat older children have “synthetic” models that represent an integration between initial (everyday) models and culturally accepted views (e.g., the sun and moon revolve around the stationary earth every 24 hours; the earth rotates in an up/down direction and the sun and moon are fixed on opposite sides; the Earth goes around the sun; the Moon blocks the sun; the Sun moves in space; the Earth rotates and revolves). (See Kikas, 1998; Vosniadou & Brewer, 1994)
The correct understanding of plants is that plants are living things.
Misconception: Elementary school children think of plants as nonliving things (Hatano et al., 1997).
Path of blood flow in circulation
The correct conception is that lungs are involved and are the site of oxygen-carbon-dioxide exchange. Also, there is a double pattern of blood flow dubbed the “double loop” or “double path” model. This model includes four separate chambers in the heart as well as a separate loop to and from the lungs. Blood from the right ventricle is pumped into the lungs to be oxygenated, whereas blood from the left ventricle is pumped to the rest of the body to deliver oxygen. Hence, one path transports de-oxygenated blood to receive oxygen, while the other path transports oxygenated blood to deliver oxygen.
Misconceptions: Yip (1998) evaluated science teacher knowledge of the circulatory system. Teachers were asked to underline incorrect statements about blood circulation and provide justification for their choices. Most teachers were unable to relate blood flow, blood pressure, and blood vessel diameter. More experienced teachers often had the same misconceptions as less experienced teachers.
Misconception #1: The most common misconception is the “single loop” model, wherein the arteries carry blood from the heart to the body (where oxygen is deposited and waste collected) and the veins carry blood from the body to the heart (where it is cleaned and re-oxygenated) (Chi, 2005).
This conception differs from the correct conception in three ways:
It does not assume that lungs are involved, but assumes that lungs are another part of the body to which blood has to travel.
It does not assume that the site of oxygen-carbon-dioxide exchange is in the lungs; instead, it assumes such exchange happens in the heart
It does not assume there is a double loop (double paths), pulmonary and systemic, but instead assumes that there is a single path of blood flow and the role of the circulatory system is a systemic one only.
“Single loop” misconceptions contain five constituent propositions:
Blood flows from the heart to the body in arteries.
Blood flows from the body to the heart in veins.
The body uses the “clean” blood in some way, rendering it unclean.
Blood is “cleaned” or “replenished with oxygen” in the heart.
Circulation is a cycle.
Misconception #2: There is a “heart-to-toe” path in answer to the question of “What path does blood take when it leaves the heart?” (8th and 10th graders) (Arnaudin & Mintzes, 1985; Chi 2005)
Categories of Misconceptions (Erroneous Ideas) (See Pelaez, Boyd, Rojas, & Hoover, 2005)
The groups of blood circulation errors detected among prospective elementary teachers fell into five categories:
Blood pathway. These are common conceptual errors about the pathway a drop of blood takes as it leaves the heart and travels through the body and lungs. A typical correct answer explains dual circulation with blood from the left side of the heart going to a point in the body and returning to the right side of the heart, where it is pumped to the lungs and back to the left side of the heart.
Blood vessels. A correct response has blood traveling in veins to the heart and arteries carrying blood away from the heart, and the response recognizes that arteries feed and veins drain each capillary bed in an organ.
Gas exchange. A correct response indicates that a concentration gradient between two compartments drives the net transport of gases across cell membranes.
Gas molecule transport and utilization. A correct response explains that oxygen is transported by blood to the cells of the body and carbon dioxide is transported from the cells where it is produced and eventually back to the lungs.
Lung function. A correct response explains that lungs get oxygen from the air and eliminate carbon dioxide from the body.
Force and Motion of Objects
The correct conception of force, which is based on Newtonian physics (Newtonian theory of mechanics), describes force as a process used to explain changes in the kinetic (caused by motion) state of physical objects. Motion is the natural state that does not need to be explained. What needs to be explained are changes in the kinetic state. Force is a feature of the interaction between two objects. It comes in interactive action-reaction pairs (e.g., the force exerted by a table on a book when the book is resting on the table) that are needed to explain, not an object’s motion, but its change in motion (acceleration). Force is an influence that may cause a body to accelerate. It may be experienced as a lift, push or pull upon an object resulting from the object's interaction with another object. Hence, static objects, such as the book on the table, can exert force. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction ceases, the two objects no longer experience the force. Forces only exist as a result of an interaction. Two interacting bodies exert equal and opposite forces on each other. Force has a magnitude and a direction. (See Committee on Science Learning, Kindergarten through Eighth Grade, 2007)
Misconception #1: Motion/velocity implies force. One of the most deeply held misconceptions (or naive theories) about force is known as the pre-Newtonian “impetus theory” or the “acquired force” theory and it is typical among elementary, middle and high school students (see Mayer, 2003; McCloskey, 1983; Vosniadou et al., 2001) and among adults (university students — Kikas, 2003; and teacher trainee and primary teachers — Kikas, 2004). It is erroneously believed that objects are kept moving by internal forces (as opposed to external forces). Based on this reasoning, force is an acquired property of objects that move. This reasoning is central to explaining the motion of inanimate objects. They think that force is an acquired property of inanimate objects that move, since rest is considered to be the natural state of objects. Hence, the motion of objects requires explanation, usually in terms of a causal agent, which is the force of another object. Hence force is the agent that causes an inanimate object to move. The object stops when this acquired force dissipates in the environment. Hence force can be possessed, transformed or dissipated.
This “impetus theory” misconception is evident in the following problems taken from Mayer (2007) and McClosky, Caramaza and Green (1980):
The drawing on the left — with the curved line — is the misconception response and reflects the impetus theory. This is the idea that when an object is set in motion it acquires a force or impetus (e.g., acquired when it went around through the tube and gained angular momentum) that keeps it moving (when it gets out of the tube). However, the object will lose momentum as the force disappears. The correct drawing on the right — with the straight path — reflects the Newtonian concept that an object in motion will continue until some external force acts upon it.
Misconception #2: Static objects cannot exert forces (no motion implies no force). Many high school students hold a classic misconception in the area of physics, in particular, mechanics. They erroneously believe that “static objects are rigid barriers that cannot exert force.” The classic target problem explains the “at rest” condition of an object. Students are asked whether a table exerts an upward force on a book that is placed on the table. Students with this misconception will claim that the table does not push up on a book lying at rest on it. However, gravity and the table exert equal, but oppositely, directed forces on the book thus keeping the book in equilibrium and “at rest.” The table’s force comes from the microscopic compression or bending of the table.
Misconception #3: Only active agents exert force. Students are less likely to recognize passive forces. They may think that forces are needed more to start a motion than to stop one. Hence, they may have difficulty recognizing friction as a force.
On the correct understanding of gravity, falling objects, regardless of weight, fall at the same speed.
Misconception: Heavier objects fall faster than lighter objects. Many students learning about Newtonian motion often persist in their belief that heavier objects fall faster than light objects (Champagne et al., 1985).
There is one class of alternative theories (or misconceptions) that is very deeply entrenched. These relate to ontological beliefs (i.e., beliefs about the fundamental categories and properties of the world). (See Chi 2005; Chinn & Brewer, 1998; Keil, 1979).
Some common mistaken ontological beliefs that have been found to resist change include:
beliefs that objects like electrons and photons move along a single discrete path (Brewer & Chinn, 1991)
belief that time flows at a constant rate regardless of relative motion (Brewer & Chinn, 1991)
belief that concepts like heat, light, force, and current are a material substance (Chi, 1992)
belief that force is something internal to a moving object (McCloskey, 1983; See section on physics misconceptions).
Other Misconceptions in Science
Belief that rivers only flow from north to south.
Epistemological Misconceptions about the Domain of Science Itself (its objectives, methods, and purposes)
Many middle school and high school students tend to see the purpose of science as manufacturing artifacts that are useful for humankind. Moreover, scientific explanations are viewed as being inductively derived from data and facts, since the hypothetical or conjectural nature of scientific theories is not well-understood. Also, such students tend not to differentiate between theories and evidence, and have trouble evaluating theories in light of evidence (See Mason, 2002 for review).
A correct understanding of money embodies the value of coin currency as noncorrelated with its size.
Misconception: At the PreK level, children hold a core misconception about money and the value of coins. Students think nickels are more valuable than dimes because nickels are bigger.
Correct understanding of subtraction includes the notion that the columnar order (top to bottom) of the problem cannot be reversed or flipped (Brown & Burton, 1978; Siegler, 2003; Williams & Ryan, 2000).
Misconception #1: Students (age 7) have a “smaller-from-larger” error (misconception) that subtraction entails subtracting the smaller digit in each column from the larger digit regardless of which is on top.
Misconception #2: When subtracting from 0 (when the minuend includes a zero), there are two subtypes of misconceptions:
Misconception a: Flipping the two numbers in the column with the 0. In problem “307-182,” 0 – 8 is treated as 8 – 0, exemplified by a student who wrote “8 ” as the answer.
Misconception b: Lack of decrementing; or not decrementing the number to the left of the 0 (due to first bug above, wherein nothing was borrowed from this column.) In problem “307-182,” this means not reducing the 3 to 2.
Correct understanding of multiplication includes the knowledge that multiplication does not always increase a number.
Misconception: Students have a misconception that multiplication always increases a number. For example, take the number 8:
3 x 8 = 24
5 x 8 = 40
This impedes students’ learning of the multiplication of a (positive) number by a fraction less than one, such as ½ x 8 = 4.
Misconception comes in the form of “division as sharing” (Nunes & Bryant, 1996), or the “primitive, partitive model of division” (Tirosh, 2000). In this model, an object or collection of objects is divided into a number of equal parts or sub collections (e.g., Five friends bought 15 lbs. of cookies and shared them equally. How many pounds of cookies did each person get?). The primitive partitive model places three constraints on the operation of division:
The divisor (the number by which a dividend is divided) must be a whole number;
The divisor must be less than the dividend; and
The quotient (the result of the division problem) must be less than the dividend.
Hence, children have difficulty with the following two problems because they violate the “dividend is always greater than the divisor constraint” (Tirosh, 2002):
“A five-meter-long stick was divided into 15 equal sticks. What is the length of each stick?”
A common incorrect response to this problem is 15 divided by 5 (instead of the correct 5 divided by 15).
“Four friends bought ¼ kilogram of chocolate and shared it equally. How much chocolate did each person get?”
A common incorrect response to this problem is 4 x ¼ or 4 divided by 4 (instead of the correct ¼ divided by 4).
Similarly, children have difficulty with the following problem because the primitive, partitive model implies that “division always makes things smaller” (Tirosh, 2002).
“Four kilograms of cheese were packed in packages of ¼ kilogram each. How many packages contained this amount of cheese?”
Because of this belief they do not view division as a possible operation for solving this word problem. They incorrectly choose the expression “1/4 X 4” as the answer (See Fischbein, Deri, Nello & Marino, 1985).
This “primitive, partitive” model interferes with children’s ability to divide fractions — because students believe you cannot divide a small number by a larger number, as it would be impossible to share less among more.
Indeed, even teacher trainees can have this preconception of division “as sharing.” Teachers were unable to provide contexts for the following problem (Goulding. Rowland, & Barber, 2002):
2 divided by ¼
The correct conception of negative numbers is that these are numbers less than zero. They are usually written by indicating their opposite, which is a positive number, with a preceding minus sign (See Williams & Ryan, 2000).
A Separation Misconception means treating the two parts of the number — the minus sign and the number — separately. In number lines, the scale may be marked: -20, -30, 0, 10, 20...(because the ordering is 20 then 30, and the minus sign is attached afterwards) and later the sequence gets -4 inserted thus: -7, -4, 1,...(because the sequence is read 1, 4, 7 and the minus sign is afterwards attached). Similarly, we can explain: -4 + 7 = -11.
The correct conception of a fraction is of the division of one cardinal number by another. Children start school with an understanding of counting — that numbers are what one gets when one counts collections of things (the counting principles). Students have moved towards using counting words and other symbols that are numerically meaningful. The numbering of fractions is not consistent with the counting principles, including the idea that numbers result when sets of things are counted and that addition involves putting two sets together. One cannot count things to generate a fraction. A fraction, as noted, is defined as the division of one cardinal number by another. Moreover, some counting principles do not apply to fractions. For example, one cannot use counting based algorithms for ordering fractions — ¼ is not more than ½. In addition, the nonverbal and verbal counting principles do not map to the tripartite symbolic representations of fractions (two cardinal numbers separated by a line) (See misconception examples above and Hartnett & Gelman, 1998).
Misconceptions reflect children’s tendency to distort fractions in order to fit their counting-based number theory, instead of viewing a fraction as a new kind of number.
Misconception #1: Student increase the values of denominator maps in order to increase quantitative values. This includes a natural number ordering rule for fractions that is based on cardinal values of the denominator (See Hartnett & Gelman, 1998).
Misconception #2: When adding fractions, the process is to add the two numerators to form the sum’s numerator and then add the two denominators to form its denominator. Example: Elementary and high school students think ¼ is larger than ½ because 4 is more than 2 and they seldom read ½ correctly as “one half.” Rather, they use a variety of alternatives, including “one and two, ” “one and a half,” “one plus two, ” “twelve,” and “three.” (See Gelman, Cohen, & Hartnett, 1989, cited in Hartnett & Gelman, 1998),
Example ½ +1/3 = 2/5 (See Siegler, 2003).
The correct understanding of the decimal system is of a numeration system based on powers of 10. A number is written as a row of digits, with each posi¬tion in the row corresponding to a certain power of 10. A decimal point in the row divides it into those powers of 10 equal to or greater than 0 and those less than 0, i.e., negative powers of 10. Positions farther to the left of the decimal point correspond to increasing positive powers of 10 and those farther to the right to increasing negative powers, i.e., to division by higher positive powers of 10.
A number written in the decimal system is called a decimal, although sometimes this term is used to refer only to a proper fraction written in this system and not to a mixed number. Decimals are added and subtracted in the same way as in¬tegers (whole numbers), except that when these operations are written in columnar form, the decimal points in the column entries and in the answer must all be placed one under another. In multiplying two decimals, the operation is the same as for integers except that the number of decimal places in the product (i.e., digits to the right of the decimal point) is equal to the sum of the decimal places in the factors (e.g., the factor 7.24 to two decimal places and the factor 6.3 to one decimal place have the product 45.612 to three decimal places). In division, (e.g., 4.32|12.8), a decimal point in the divisor (4.32) is shifted to the extreme right (i.e., to 432.) and the decimal point in the dividend (12.8) is shifted the same number of places to the right (to 1280), with one or more zeros added before the decimal to make this possible. The decimal point in the quotient is then placed above that in the dividend, i.e., 432|1280.0 and zeros are added to the right of the decimal point in the dividend as needed. The division proceeds the same as for integers.
Misconception #1: Students often use a “separation strategy,” whereby they separate the whole (integer) and decimal as different entities. They treat the two parts before and after the decimal point as separate entities. This has been seen in pupils (Williams & Ryan, 2000), as well as in beginning preservice teachers (Ryan & McCrae, 2005).
Division by 100:
300.62 divided by 100
Correct Answer = 3.0062
Misconception Answer = 3.62
Example: When given 7.7, 7.8, 7.9, students continue the scale with 7.10, 7.11.
Misconception #2: This relates to the ordering of decimal fractions from largest to smallest (Resnick et al., 1989; Sackur-Grisvard, & Leonard, 1985). This misconception is also seen in primary teacher trainees (Goulding et al., 2002). Here is an example of a mistaken ordering:
0.203 2.35 X 10-2; two hundreths 2.19 X 10 -1; one fifth
A lack of connection exists in the knowledge base between different forms of numerical expressions AND difficulties with more than two decimal places.
Misconception a: The larger/longer number is the one with more digits to the right of the decimal point, i.e. 3.214 is greater than 3.8 (Resnick et al., 1989; Sackur-Grisvard & Leonard, 1985; Siegler, 2003). This is known as the “whole number rule” because children are using their knowledge of whole number values in comparing decimal fractions (Resnick et al., 1989). Whole number errors derive from students’ applying rules for interpreting multidigit integers. Children using this rule appear to have little knowledge of decimal numbers. Their representation of the place value system does not contain the critical information of column values, column names and the role of zero as a placeholder (see Resnick et al., 1989).
Misconception b: The “largest/longest decimal is the smallest (the one with the fewest digits to the right of decimal).” Given the pairs 1.35 and 1.2, 1.2 is viewed as greater. 2.43 judged larger than 2.897 (Mason & Ruddock, 1986; cited in Goulding et al., 2002, Resnick et al., 1989; Sackur-Grisvard & Leonard, 1985; Siegler, 2003; & Ryan & McCrae) This is known as the “fraction” rule because children appear to be relying on ordinary fraction notation and their knowledge of the relation between size of parts and number of parts (Resnick et al., 1989). Fraction errors derive from children’s attempts to interpret decimals as fractions. For instance, if they know that thousandths are smaller parts than hundredths, and that three-digit decimals are read as thousandths, whereas two-digit decimals are read as hundredths, they may infer that longer decimals, because they refer to smaller parts, must have lower values (Resnick et al., 1989). These children are not able to coordinate information about the size of parts with information about the number of parts; when attending to size of parts (specified by the number of columns) they ignored the number of parts (specified by the digits).
Misconception c: Students make incorrect judgments about ordering numbers that include decimal points when one number has one or more zeros immediately to the right of the decimal point or has other digits to the right of the decimal point. Hence, in ordering the following three numbers (3.214, 3.09, 3.8), a student correctly chooses the number with the zero as the smallest, but then resorts to “the larger number is the one with more digits to the right” rule (i. e., 3.09, 3.8, 3.214) (Resnick et al., 1989; Sackur-Grisvard & Leonard, 1985). This is known as the “zero rule” because it appears to be generated by children who are aware of the place-holder function of zero, but do not have a fully developed place value structure. As a result, they apply their knowledge of zero being very small to a conclusion that the entire decimal must be small (See Resnick et al.,1989).
Misconception #3: Multiplication of decimals.
Example: 0.3 X 0.24
Correct Answer = 0.072
Misconception answer: Multiply 3 x 24 and adjust two decimal points. 0.72 (This is seen in the beginning instruction of pre-service teachers as well.)
Misconception #4: Units, tenths and hundredths.
Example: Write in decimal form: 912 + 4/100
Correct Answer = 912.04
Misconception answer = 912.004. 4/100 is ¼ or 100 divided by 4 gives the decimal or 1/25 is 0.25 = 912.25
Overgeneralization of Conceptions Developed for "Whole Numbers" (cited in Williams & Ryan, 2000)
Misconception #1: Ignoring the minus or % sign. Errors such as: 4 + - 7 = -11; -10 + 15 = 25.
Misconception #2: Thinking that zero is the lowest number.
Misconception #1: Incorrect generalization or extension of correct rules. Siegler (2003) provides the following example:
The distributive principle indicates that
a x (b + c) = (a x b) + (a x c)
Some students erroneously extend this principle on the basis of superficial similarities and produce:
a + (b x c) = (a + b) x ( a + c)
Misconception #2: Variable misconception. Correct understanding of variables means that a student knows that letters in equations represent, at once, a range of unspecified numbers/values. It is very common for middle school students to have misconceptions about core concepts in algebra, including concepts of a variable (Kuchemann 78; Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; MacGregor & Stacey, 1997; Rosnick, 1981). This misconception can begin in the early elementary school years and then persist through the high school years.
There are several levels or kinds of variable misconceptions:
Variable Misconception: Level 1
A letter is assigned one numerical value from the outset.
Variable Misconception: Level 3
A letter is interpreted as a label for an object or as an object itself.
At a university, there are six times as many students as professors. This fact is represented by the equation S = 6P. In this equation, what does the letter S stand for?
a. number of students (Correct)
c. students (Misconception)
d. none of the above
Misconception #3: Equality misconception. Correct understanding of equivalence (the equal sign) is the “relational” view of the equal sign. This means understanding that the equal sign is a symbol of equivalence (i.e., a symbol that denotes a relationship between two quantities).
Students exhibit a variety of misconceptions about equality (Falkner, Levi, & Carpenter, 1999; Kieran, 1981,1992; Knuth et al., 2005; McNeil & Alibali, 2005; Steinberg, Sleeman, & Ktorza,1990; Williams & Ryan, 2000). The equality misconception is also evident in adults, like college students (McNeil & Alibali, 2005).
Students do not understand the concept of “equivalent equations” and basic principles of transforming equations. Often, they do not know how to keep both sides of the equation equal. So, they do not add/subtract equally from both sides of the equal sign.
In solving x + 3 = 7, a next step could be
A. x + 3 – 3 = 7 – 3 (Correct)
B. x + 3 + 7 = 0
C. = 7 – 3 (Misconception)
D. .3x = 7
It is assumed that the answer (solution) is the number after the equal sign (i.e., answer on the right)
The correct understanding of poems includes the notion that a poem need not rhyme. Misconceptions are that poems must rhyme.
A correct understanding of language includes the knowledge that language can be used both literally and nonliterally.
The misconception is that language is always used literally. Many elementary school children have difficulty understanding nonliteral or figurative uses of language, such as metaphor and verbal irony. In these nonliteral uses of language, the speaker’s intention is to use an utterance to express a meaning that is not the literal meaning of the utterance. In irony, speakers are expressing a meaning that is opposite to the literal meaning (e.g., while standing in the pouring rain, one says “What a lovely day.”). Metaphor is a figure of speech in which a term or phrase is applied to something to which it is not literally applicable in order to suggest a resemblance, as in “All the world’s a stage." (Shakespeare). Students have difficulty understanding nonliteral (figurative) uses of language because they have a misconception that language is used only literally. (See Winner, 1997.)