How Do I Get My Students Over Their Alternative Conceptions (Misconceptions)?

When teachers provide instruction on concepts in various subjects, they are teaching students who already have some pre-instructional knowledge about the topic. Student knowledge, however, can be erroneous, illogical, or misinformed. These erroneous understandings of students are termed alternative conceptions or misconceptions (or intuitive theories). Alternative conceptions (misconcep-tions) are not unusual. In fact, they are quite common in children and adults. They are a normal part of the learning process. We quite naturally form ideas from our everyday experience, but obviously not all the ideas we develop are cor-rect with respect to the most current evidence and scholarship in a given disci-pline. Moreover, some concepts in different content areas are simply very difficult to grasp. They may be very abstract or counter-intuitive or quite complex. Hence our understanding of them is flawed. In addition, things we have already learned through instruction can sometimes not be helpful in learning new concepts/theo-ries. This occurs when the new concept or theory is inconsistent with previously learned material. Accordingly, as noted, it is very typical for students (and adults) to have misconceptions in different domains (content knowledge areas). Indeed, researchers have found that there is a common set of alternative conceptions (misconceptions) that most students typically exhibit. There is one class of alter-native theories (or misconceptions) that is very deeply entrenched. These are “ontological misconceptions,” which relate to ontological beliefs (i.e., beliefs about the fundamental categories and properties of the world).

Description of some of these common alternative conceptions (misconceptions) and ontological misconceptions in different subject areas.

Alternative conceptions (misconceptions) can really impede learning for several reasons. First, students themselves are generally unaware that the knowledge they have is wrong. Moreover, misconceptions can be very entrenched in student thinking. In addition, new experiences are interpreted through these erroneous understandings, thereby interfering with correctly grasping the new experiences. Also, alternative conceptions (misconceptions) tend to be very resistant to in-struction because learning entails replacing or radically reorganizing student knowledge. Hence, conceptual change has to occur for learning to happen. This puts teachers in the very challenging position of needing to bring about significant conceptual change in student knowledge. Generally, ordinary forms of instruction, such as lectures, labs, simple discovery learning, or simply reading texts, are not very successful at overcoming student misconceptions. For all these reasons, misconceptions can be hard nuts for teachers to crack. However, several in-structional strategies have been found to be successful at achieving conceptual change and helping students leave their alternative conceptions behind and learn correct concepts or theories.

DOsInstructional Strategies that Can Lead to Change in Students’ Alternative Con-ceptions (misconceptions) and to Learning of New Concepts and Theories:

1. Present the new concepts or theories that you are teaching in a way that students see these concepts or theories as plausible, of high quality, intel-ligible, and generative.

Presenting new concepts or theories

2. Use students’ correct conceptions and build on those by creating a bridge of examples to the new concept or theory that students are having trouble learning due to having alternative conceptions (misconceptions).

Using Bridging Analogies

3. Use model-based reasoning, which helps students construct new repre-sentations that vary from their intuitive theories.

Using Model-Based Reasoning

4. Use “diverse instruction,” wherein you present a few examples that chal-lenge multiple assumptions rather than a larger number of examples that challenge just one assumption.

Using Diverse Instruction

5. Help students become aware of (raise their metacognition about) their own alternative conceptions (misconceptions).

Raising Student Metacognition

6. Present students with experiences that cause cognitive conflict in the student’s mind. Experiences that can cause cognitive conflict are ones that get students to consider their erroneous (misconception) knowledge (as in strategy 3 above) side-by-side with or at the same time as the correct con-cept or theory.

Creating Cognitive Conflict in Students’ Minds

7. Engage in Interactive Conceptual Instruction (ICI)

Components of ICI

8. Develop students’ epistemological thinking, which is their beliefs and theories about the nature of knowledge and the nature of learning, in ways that will facilitate conceptual change. The more naïve students’ beliefs are about knowledge and learning, the less likely they are to revise their mis-conceptions.

Developing Student Thinking about Knowledge and Learning

9. Help students “self-repair” their misconceptions.Student Self-Repair of Misconceptions

Student Self-Repair of Misconceptions

10. Once students have overcome their alternative conceptions (miscon-ceptions), engage them in argument to help strengthen their new knowl-edge (representations)

Using Argument to Strengthen Newly Acquired Correct Knowledge

DON’Ts for when trying to eliminate alternative conceptions (misconcep-tions):1. Do not rely on lectures2. Do not rely on labs or simply hands-on activities 3. Do not rely on demonstrations 4. Do not rely on having students simply read the text

Presenting new concepts or theories

In presenting new concepts or theories, teachers should be sure to show these theories or concepts as:1. plausible – The new information should be shown to be consistent with other knowledge and able to explain the available data. Learnes must see how the new conception (theory) is consistent with other knowledge and a good explanation of the data.

2. high quality – Of course, the theory/concept to be taught is of high quality from a scientific point of view since it is a correct theory. However, the theory should be a better account of the data that students have available to them. For example, the instructor should deal with the problem from the perspective of the students (e.g., students for whom a “flat earth” theory provides a better account of the data available than does a “spherical earth” theory). Hence, the quality of the new theory must be considered along with the kind of data that students know about.

3. intelligible – Teachers should do what they can to increase the intelligibility of the new theory. Learnes must be able to grasp how the new conception works. To increase intelligibility, teachers can use methods such as use of: - analogies (see Chiu & Lin, 2004)- models (both pictorial conceptual and physical) -(See Clement, 1993 with high school students; Mayer & Gallini, 1990 with college students; Mayer, 1993; Vosniadou et al., 2001 with 5th and 6th graders) - direct exposition (see Klahr & Nigam, 2004).

4. generative/fruitful – Teachers should show that the new concept/theory can be extended to open up new areas of inquiry. Learners must be able to extend the new conception to new areas of inquiry. Teachers might accomplish this by illustrating the application of the new concept/theory to a range of problems. These problems can include familiar ones and new ones.

(See Chinn & Brewer, 1993; Mayer, 2008; Posner et al., 1982).

Using Bridging Analogies

One of the best ways that teachers can correct misconceptions is by a strategy called “using bridging analogies.” This strategy attempts to bridge pupils’ correct beliefs (called “anchoring conceptions”) to the new concept/theory (target) by presenting a series of intermediate similar or analogous examples between the students’ initial correct conception and the new concept or theory (target) to be learned. (see Brown, 1992; Brown & Clement, 1989; Clement, 1993; Minstrell, 1982; Yilmaz, Eryilmaz, & Geban, 2006)

Example 1

Another similar strategy teachers can try is the use of the “bridging representa-tion.”Example 2

Using Bridging Analogies: Example 1

Many high school students hold a classic misconception in the area of phys-ics, in particular, mechanics. They erroneously believe that “static objects are rigid barriers that cannot exert forces.” 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 miscon-ception 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. At the same time that students hold the misconception about static objects, they also believe that a spring pushes up on one’s hand when the hand is pushing down on the spring. Physicists understand that these 2 situations – book on table and hand pressing on a spring - as equivalent. The Bridging Strategy establishes analogical connec-tions between situations that students initially view as not analogous as a means to get students to extend their valid intuitions (the spring) to initially counterin-tuitive target situations (the table). The use of bridging analogies entails use of concrete examples for a connected sequence, starting from an anchor (situation in which most students believe there is upward force), through an intermediate example(s), to a target situation (book on table).

(A) ANCHOR Ex. (B) BRIDGING Ex. (C) BRIDGING Ex (D) TARGET Ex.

Hand on spring Book resting on-Flexible foam pad

Book resting onflex-ible Board

Bookon table

Using Bridging Analogies: Example 2

In physics instruction, use of the SRI (symbolic representation of interactions) diagram has been found to be helpful. The SRI emphasizes forces as interac-tions and makes identification of the mechanical interaction between pairs of objects explicit. It is contrasted with the free-body diagram that concentrates on the forces acting on one target object. The pedagogic function of the SRI is to provide a bridge, referred to as a “bridging representation.” (See Savinainen, Scott, & Viiri (2004)

Using Model-Based Reasoning

(See Committee on Science Learning, Kindergarten through Eighth Grade, 2007)Effective science learning often requires that students construct new representa-tions that vary in important ways from ones used in everyday life. Science entails new ways of seeing data in terms of idealized representations or models. Sci-ence generally entails mathematical relations, physical intuitions, and sensorimo-tor action schemes in these models. Teachers should teach these idealization techniques, such as thought experiments and limiting case analyses. These techniques are integral to constructing those abstract representations that can facilitate student recognition of deep analogies between superficially different phenomena. A thought experiment, in the broadest sense, is the use of a hypothetical scenario to help us understand the way things actually are. There are many different kinds of thought experiments. All thought experiments, however, employ a methodol-ogy that is a priori, rather than empirical, in that they do not proceed by observation or physical experiment. Scientists tend to use thought experiments in the form of imaginary, “proxy” experiments which they conduct prior to a real, “physical” experiment. In these cases, the result of the “proxy” experiment will often be so clear that there will be no need to conduct a physical experiment at all. Scientists also use thought experiments when particular physical experiments are impossible to conduct.

Newton’s cannonball was a thought experiment that Isaac Newton used to hypothesize that the force of gravity was universal, and it was the key force for planetary motion.

Newton's cannonball

In this experiment Newton visualizes a cannon on top of a very high mountain.If there was no force of gravitation the cannonball should follow a straight line away from Earth.So long as there is a gravitational force acting on the cannon ball, it will follow dif-ferent paths depending on its initial velocity.

1. If the speed is low, it will simply fall back on Earth. (A and B)2. If the speed equals some threshold orbital velocity it will go on circling

around the Earth along a fixed circular orbit just like the moon. (C)3. If the speed is higher than the orbital velocity, but not high enough to leave

Earth altogether (lower than the escape velocity) it will continue rotating around Earth along an elliptical orbit. (D)

4. If the speed is very high, it will indeed leave Earth. (E)

Using Diverse Instruction

Diverse instruction simultaneously challenges at least two erroneous beliefs that underlie a misconception (alternative conception). It is based on a literature that shows adults and children to draw stronger inductive inferences from information that impacts on diverse aspects of their underlying beliefs (see Hayes, Good-hew, Heit, & Gillan, 2003 for review). Hayes et al. extend the diversity principle to conceptual change and propose that shifts in intuitive theories or alternative conceptions (misconceptions) are more likely to occur when people encounter new information that challenges several features or assumptions of these mod-els. Conceptual change is thought to be more likely if students are presented with a few examples that challenge multiple assumptions than with a larger number of examples that challenge just one assumption.

Example 1

Using Diverse Instruction:Example 1

The effect of diverse instruction on children’s understanding of the shape of the earth was studied by Hayes et al. (2003). Children’s erroneous beliefs about the earth (their non-belief in a spherical earth) can be linked to two more general misconceptions (Vosniadou & Brewer, 1992). One is the belief that the earth appears flat to an observer on the ground. The second is a poor understanding of gravity and failure to understand the influence of gravity on objects on differ-ent parts of the earth’s surface. Indeed, in considering the earth’s surface, when students think that unsupported objects fall, they are likely to construct either a “disk” model of the earth or a “dual earth” model (with a round earth located in space co-existing with a flat earth where people live).

Hayes et al. (2003) randomly assigned 6-yr-old children to either of 3 conditions: control (no training), single-belief training (all four instructional videos focused on either the relative size of the earth or the effects of gravity), dual-belief train-ing (4 instructional videos where 2 focused on the relative size of the earth and 2 focused on the effects of gravity). Results showed that only children receiving instruction about two core beliefs showed an increased rate of acceptance of a spherical earth model at posttest.

Raising Student Metacognition

Student metacognitive abilities may be critical to achieving conceptual change (Beeth, 1998; Beeth & Hewson, 1999; Case, 1997; Chinn & Brewer, 1993; Gel-man & Lucariello, 2002; Inagaki & Hatano, 2002; Minstrell, 1982,1984). Metacog-nition entails a range of processes, including monitoring, detecting incongruities or anomalies, self-correcting, planning and selecting goals, and reflecting on the structure of one’s knowledge and thinking (Gelman & Lucariello, 2002).

There are several good methods to get students to think metacognitively:

A. Engage students in representing their thinking externally as in verbal expression of ideas through interactive discussion and open exchange and debate of ideas. To help students increase their metaconceptual awareness (awareness of their own cognition), it is important to create learning environments that make it pos-sible for them to express their knowledge, including misconception knowledge. This can be done in environments that facilitate group discussion and the verbal expression and debate of ideas. The learning environment should allow for stu-dents to express their knowledge and compare it with that of others. Such activi-ties assist students in becoming aware of what they know and what they need to learn. (See Kuhn, 2006, Minstrell (1982, 1989), Savinainen & Scott, 2002; Vosniadou et al., 2001)

B. Elicit student predictions on the topic followed by a teacher-led demon-stration which tests those predictions. Discussion is then used to arrive at a common observation and then to reconcile differences between predic-tion and observation.Keep in mind that students (or anyone) can be biased by the ideas (in this case, misconceptions) they already have when observing things and this can actu-ally interfere with their observing events correctly. Chinn & Malhotra (2002) have noticed “theory bias at the observation stage.” For example, only about 26% of children who predicted that the heavy or light rock would actually hit the ground first (instead of the correct prediction that they would hit at the same time) actu-ally observed that they in fact did hit at the same time (cited in Mayer 2007). An important instruction implicated of this is to try and make the data to be observed so obvious so as to minimize their incorrect observation by students (Mayer, 2008). “Predict-Observe-Explain” Teaching Strategy(See Kuhn, 2006; Champagne et al. 1985; Gunstone, Robin Gray, & Searle 1992: Use of Predict-Observe-Explain (P-O-E); Mayer, 2008; Minstrell, 1982)

C. Provide opportunities for reflective inquiry and assessment (White & Frederickson, 1998)White and colleagues designed a computer-based microworld “Thinker Tools (TT)” (1993; White & Frederiksen 1998). This is a middle school science cur-riculum that engages students in learning about and reflecting on the processes of science inquiry as they construct increasingly complex models of force and motion phenomena. The TT Inquiry Curriculum centers around a metacognitive model of research, called the Inquiry Cycle, and a metacognitive process, called Reflective Assessment, in which students reflect on their own and each other’s inquiry.

Description of Inquiry Cycle

“Predict-Observe-Explain” Teaching Strategy

One such effective teaching strategy is the “Predict-Observe-Explain (P-O-E)” method. In the P-O-E strategy, the teacher plans/presents a demonstration or example that s/he will subsequently conduct/explicate. The topic or issue of the demonstration or example should be one that relates to possible student miscon-ceptions and the demonstration/example should be specifically designed to be effective at eliciting such misconceptions. Before conducting the demonstration, pupils are to predict what will occur. The teacher then conducts the demonstra-tion (explicates the illustration/example) and the students observe this. After the demonstration (illustrative example), the students must explain why their obser-vations conflicted with their predictions.

The P-O-E strategy does not entail the traditional hands-on laboratory work done by students themselves. The idea is to allow students greater intellectual activity in the POE structure, as compared with traditional laboratory work. Hence, gen-erally the demonstration component is conducted by the teacher (not the stu-dents) and such has been found to be successful at helping students eliminate misconceptions. When the teacher does the demonstration it allows students to focus more of their resources on the conceptual issues. The strategy of “eliciting predictions” has been employed by many and has often been incorporated into larger instructional practices.

Description of Inquiry Cycle

The Inquiry Cycle guides students’ research and helps them understand what the research process is all about. It begins with formulating an investigable QUES-TION. It moves to a PREDICT phase, wherein students generate alterative hy-potheses and predictions with respect to the question. Next comes the EXPERIMENT phase, wherein students design and carry-out experiments in the real world and on the computer. Students then move to the MODEL phase, wherein they analyze their data to construct a conceptual model that includes scientific laws that would predict and explain their findings. Finally, comes the AP-PLY phase, wherein they apply their model to different situations to investigate its utility and limitation. The latter raises new questions and the cycle begins again. Students go through the Inquiry Cycle for each research topic in the curriculum. They engage in Reflective Assessment at each step in the Inquiry Cycle and after each completion of the Cycle.

The Reflective Assessment component provides students with “Criteria for Judg-ing Research.” These include goal-oriented criteria, such as “understanding the science,” and process-oriented criteria, such as “being systematic” and “reason-ing carefully,” and socially-oriented-criteria, such as “communicating well.”The TT curriculum was implemented by 3 teachers in 12 urban classes (across 7th to 9th grade). The sample included many low-achieving and disadvantaged students.

Student’s learning was found to be greatly facilitated by the Reflective Assess-ment component.

Creating Cognitive Conflict in Students’ Minds

The idea that cognitive conflict or disequilibrium can lead to learning is rooted in Piagetian theory. Piaget proposed that cognitive conflict or “disequilibrium” arises when students encounter experiences that are not able to be assimilated or that are incongruous with their current cognitive structures/conceptions. Cognitive conflict can lead to conceptual change or accommodation of current cognitive concepts.

There are a variety of ways that teachers generate cognitive conflict in the mind of the student:

A. Present students with anomalous data (data that do not accord with their misconception)This strategy is thought to be a major means of eliciting cognitive conflict and get-ting students to change or abandon their current erroneous theories and adopt new ones. However, just presenting anomalous data is not sufficient. Students have been found to ignore or reject such data, profess uncertainty about its validity, and reinterpret it, among other things (Chinn & Brewer, 1998). There are certain optimal ways to repsent such data.

Best Ways to Present Anomalous Data

B. Present students with refutational texts (texts wherein a misconception is explicitly refuted by presenting contrasting information). Present refu-tational texts alone or in combination with discussion, conducted under teacher guidance. The discussion, which can be conducted with peers, should require students to articulate and support their views with evidence from the text.A refutational text introduces a common misconception, refutes it, and offers the new (alternative) theory that is shown to be more satisfactory. These texts are defined as ones wherein a misconception is explicitly refuted by presenting con-trasting information. In this way, refutational texts are a means to create cognitive conflict. The following text from Hynd (2001) is an example of refutational text:: Despite the fact that many people think that a ball will slow or stop on its own, this will not happen….Moving objects will keep moving at a constant rate unless they are slowed or stopped or their direction is changed because of an outside force such as friction. See Diakidoy, Kendeou, & Ioannides, 2003; Guzzetti, Snyder, Glass, & Gamas, 1993; Guzzetti, 2000; Hynd, 1997; Maria & MacGinitie, 1987)

C. Present students with typical expository (nonrefutational) text, which presents the newtheory or concept, and at the same time use teacher strat-egies or activities that elicit the students’ erroneous knowledge and make

students consider such in relation to the text information. At elementary level, having narrative structure in the expository text is helpful, where such seems unnecessary at secondary level.

Activities to Use in Combination with Text

D. Conduct Conceptual Change Discussions (See Eryilmaz, 2002) Protocol for conceptual change discussion from Eryilmaz (2002)

Best Ways to Present Anomalous Data

Of course, students might not accept the anomalous data and hence not change their minds. Teachers can increase the chances of anomalous data being ac-cepted and leading to conceptual change by:

1. making the anomalous data credible - This can be done by in a few ways. Teachers can make it clear that the data were collected according to accepted principles. In addition, live demonstrations and hands-on experiences may also increase the credibility of the anomalous data. Also teachers can appeal to the real-world data that students already know about (as in the use of anchoring con-ceptions as described earlier in discussing the bridging analogies strategy).

2. avoiding ambiguous data – Teachers can accomplish this by choosing data that are perceptually obvious. Also, if teachers are aware of the specific miscon-ceptions their students have, they can choose data, in light of that, that will be unambiguous to their students.

3. presenting multiple lines of data when necessary – In presenting anoma-lous data, single experiments are often not convincing. Hence, introducing mul-tiple lines of data, such as use of a series of experiments, should be helpful. If using a single experiment/demonstration, it is useful to be prepared to effectively address student objections.

4. introducing the anomalous data early in the instructional process – This might be helpful because it appears that the more background knowledge in the topic students have, the more their misconceptions impede the acceptance of anomalous data.

5. engaging students in justification of their reasoning about the anoma-lous data.(See Chinn & Brewer 1993 and Posner et al., 1982)

Activities to Use in Combination with Text

Some activities that produce cognitive conflict when used in combination with text are:

Augmented Activation Activities – This activity has two components. One is the activation activity designed to activate or bring to students’ attention their misconception knowledge (e.g., by asking them to recall or reiterate their belief; by reminding them of their belief). Then, the raising of the misconception knowl-edge is supplemented by information to cause cognitive conflict. This information is typically in the form of directing the reader’s attention to contradictory informa-tion in the text or providing illustrative demonstrations that are incongruous with the misconception. This instructional strategy is similar to the Socratic teaching method and involves students in dialogues that compel them to handle counter-examples and face contradictions to their misconceptions.

The Discussion Web- This is a discussion strategy led by teacher. It can entail using a graphic aid to form students’ positions around a central question. Stu-dents are required to take a stance (e.g., on the shape of the earth), defend their positions, and persuade each other with evidence from the text. Direct question-ing helps students rethink their prior conceptions. The Discussion Web fosters cognitive dissonance by requiring students to defend their positions with informa-tion from the text and by provoking interactive discussion in which direct ques-tioning helps students rethink their prior conceptions.

Think Sheets – is a written contrast of student-generated and text-generated ideas of a concept posed as a central question. It is a text based activity that con-trasts learners’ prior conceptions to scientific conceptions from text. Learners are said to self-monitor their prior knowledge in light of information from the text and from the discussion.

See Guzzetti, Snyder, Glass, & Gamas, 1993; Guzzetti, 2000; Hynd, 1997)

Protocol for conceptual change discussion from Eryilmaz (2002)

Components of Interactive Conceptual Instruction (ICI)

ICI, described and studied by Savinainen & Scott (2002,) incorporates several key pedagogical aspects:

1. use of interactive approaches that entail ongoing teacher-student dialogue, which focuses on developing conceptual understandings and wherein students have the opportunity to talk through their understandings with the support of the teacher

2. teacher use of research-based instruments (questionnaires/assessments/inventories) which afford quick and detailed formative assessments of students’ knowledge in a subject-area

3. teacher’s development of a detailed knowledge and understanding of the con-ceptual terrain of the subject area, including knowledge of the canonical informa-tion in the subject, student misconceptions, and the representations (understand-ings) between these two.

Develop Student Thinking about Knowledge and Learning

(See Mason 2002 for review)

Conceptual change is facilitated if students view knowledge as:a. Complex (not simple)b. Uncertain and evolving (not stable and absolute)

Conceptual change is facilitated if students view learning as: a. a gradual, slow process (not as “quick or not at all”) b. an ability that is improvable (malleable) (not fixed or unmodifiable)

Conceptual change is also facilitated by addressing students’ epistemologies about specific domains.

For example, with respect to science, having students reflect on the nature of sci-ence (see Smith, Maclin, Houghton, & Hennessey, 2000) and on the criteria that characterize good research facilitate conceptual change in science.

Using Argument to Strengthen Newly Acquired Correct Knowledge

(See Committee on Science Learning, Kindergarten through Eighth Grade, 2007)

Engaging in argument may be a central way that a student’s new conceptual system becomes strengthened and overtakes a student’s alternative conceptions (misconceptions). Argument entails asking students to evaluate or debate the adequacy of the new system with competing alternative conceptions (misconcep-tions). Students, even in the elementary school years, are sensitive to many of the features that make for a good concept/theory, such as plausibility, fruitfulness, and explanatory coherence. Children seem to actually prefer accounts that explain more, are not ad hoc, are internally consistent, and fit the empirical data. (Samarapungavan, 1992)

See also Duschl & Osborne, 2002, for how to support and promote argumenta-tion discourse.

Student Self-Repair of Misconceptions

If students engage in a process called “self-explanation,” then conceptual change is more likely (Chi, 2000). Self-explanation entails prompting students to explain the text aloud as they read it.

Some Common Alternative Conceptions (Misconceptions)

SCIENCE

EARTH SYSTEMS, COSMOLOGY, AND ASTRONOMY

Seasonal ChangeThe correct conception of seasonal change is that such is caused by the tilting of the earth relative to the sun’s rays. As the Earth goes around its orbit, the North-ern hemisphere is at various times oriented more toward and 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.” On 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 be-cause the distance between the Sun and the Earth is different during these two seasons. One way to see that this reasoning is erroroneous 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)

Knowledge about the Earth Correct scientific theory on the earth’s shape posits a spherical shape of 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 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 which is a planet up in the sky. Due to these misconceptions, elementary school children experience dif-ficulty 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 the larger context of a physical world 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.)

Day/Night CycleThe correct explanation for the day/night cycle is the fact that the earth spins.Elementary school children (1st through 5th grades) show some common mis-conceptions about the day/night cycle.

Misconception# 1 -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 rep-resent 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)

BIOLOGY

Classification of AnimalsThe correct classification of animals defines an animal as any member of the kingdom Animalia, comprising multicellular organisms that have a well-defined shape and usually limited growth, can move voluntarily, actively acquire food and digest it internally, and have sensory and nervous systems that allow them to respond rapidly to stimuli.

Misconception - Many elementary school students, from kindergarten through 8th grade, are confused about how animals are actually classified as such. They do not know what makes an animal an animal. One persistent confusion is the classification of invertebrates (such as jellyfish and octopus) as animals. Children rely on the wrong features, such as habitats, color, appendages in their classifi-cation. See Alexander, 2006 and Stovall and Nesbit (2003)

PlantsThe correct understanding of plants is that plants are living things.

Misconception - Children tend to thinks plants are not alive. Elementary school children think of plants as non-living things. (Hatano et al., 1997).

PhotosynthesisThe correct conception of photosynthesis is as the process by which plants, some bacteria, and some protistans use the energy from sunlight to produce sugar, which cellular respiration converts into ATP, the "fuel" used by all living things. The conversion of unusable sunlight energy into usable chemical energy, is associated with the actions of the green pigment chlorophyll. The process of photosynthesis takes place in the chloroplasts, specifically using chlorophyll, the green pigment involved in photosynthesis.

Misconception - Even after studying photosynthesis, many students persist in their pre-instructional belief that plants obtain food from the soil (Chinn & Brewer, 1998).

DigestionThe correct conception of the digestion process views digestion as sepa-rated into four distinct processes:1. Ingestion: placing food into the mouth,2. Mechanical digestion & chemical digestion: mastication, the use of teeth to

tear and crush food, and churning of the stomach. Addition of chemicals (acid, bile, enzymes, and water) to break down complex molecules into simple structures,

3. Absorption: movement of nutrients from the digestive system to the circulatory

and lymphatic capillaries through osmosis, active transport, and diffusion,4. Egestion: Removal of undigested materials from the digestive tract through

defecation.

The digestive tract -- also called the gastrointestinal tract or alimentary canal -- provides the pathway through which foods move through the body. During this process, foods are broken down into their component nutrients to be available for absorption.

Digestion actually begins in the mouth, as the enzymes in saliva begin to break down carbohydrate (starch). As food is chewed, it becomes lubricated, warmer, and easier to swallow and digest. The teeth and mouth work together to convert each bite of food into a bolus that can readily move into the esophagus ("the food pipe"). After the bolus is swallowed, it enters the esophagus where it continues to be warmed and lubricated as it moves toward the stomach.

The acidic environment of the stomach and the action of gastric enzymes convert the bolus into chyme, a liquefied mass that is squirted from the stomach into the small intestine. Carbohydrates tend to leave the stomach rapidly and enter the small intestine; proteins leave the stomach less rapidly; and fats linger there the longest.

The small intestine is the principal site of digestion and absorption. There, en-zymes and secretions from the pancreas, liver, gallbladder, and the small intes-tine itself combine to break down nutrients so that they can be absorbed. The pancreas is a veritable enzyme factory, supplying enzymes to digest proteins, fats, and carbohydrates. Intestinal cells also supply some enzymes. The liver pro-duces the bile required for the emulsification of fat, and the gallbladder stores the bile until it is needed. The absorption of nutrients in the small intestine is facilitat-ed by tiny projections called villi, which provide more surface area for absorption. The nutrients pass through the intestinal membranes into the circulatory system, which transports them to body tissues. Nutrients are then absorbed into the cells, where they are used for growth, repair, and the release or storage of energy. The overall process -- called metabolism -- is highly complex.

Undigested chyme proceeds from the small intestine into the large intestine (colon), where it becomes concentrated, as liquid is absorbed in preparation for excretion.

Misconceptions about digestion include:Misconception #1 - pupils fail to realize that it is a contained system Misconception #2 - pupils see the stomach as located anywhere beneath the ribs Misconception #3 - different types of food go to different places in the body Misconception #4- food eaten is somehow modified and then comes out again

in a semi-solid mass.

Path of blood flow in CirculationThe correct conception is that lungs are involved and are the site of oxygen-car-bon-dioxide exchange. Also, there is a double pattern of blood flow dubbed the “double loop” or “double path” model. This model includes four separate cham-bers 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 teach-ers often had the same misconceptions as less experienced teachers.

Misconception #1 – (Chi 2005) – The most common misconception is the “sin-gle loop” model on which it is believed that 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 reoxygenated)

This conception differs from the correct conception in 3 ways:a. does not assume that lungs are involved but assumes that lungs are another part of the body to which blood has to travel.b. does not assume that the site of oxygen-carbon-dioxide exchange is in the lungs; instead it assumes such exchange happens in the heartc. does not assume there is a double loop (double paths), pulmonary and sys-temic; 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 contains 5 constituent propositions:1. blood flows from the heart to the body in arteries2. blood flows from the body to the heart in veins3. the body uses the “clean” blood in some way, rendering it unclean4. blood is “cleaned” or “replenished with oxygen” in the heart5. circulation is a cycle

Misconception #2 – heart-to-toe” path in answer to 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.

1. Blood pathway.These are 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.

2. 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. These are errors about the veins, arteries, and capillaries.

3. Gas exchange.A correct response indicates that a concentration gradient between two compart-ments drives the net transport of gases across cell membranes. These are errors about the dependence of gas exchange on diffusion between the blood capillary and alveolar or cell space.

4. 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. These are errors about the mass transport of oxygen by the blood to the cells of the body or about the production of carbon dioxide by the cells and transport to the lungs.

5. Lung function.A correct response explains that lungs get oxygen from the air and eliminate car-bon dioxide from the body. These are errors with functions other than gas exchange assigned to the lungs or lung functions assigned to other organs.

PHYSICS (Newtonian)

Force and Motion of ObjectsThe correct conception of force, which is based on Newtonian physics (Newto-nian 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 chang-es 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, a push, or a 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. Moreover, on Newton’s Third Law in moving objects, 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)

Misconceptions about Motion of Objects and Force:

Misconception #1 – motion/velocity implies force. An intuitive meaning of forces is of active pushes or pulls that needed to explain the motion of objects. 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). On the impetus (or acquired force) force theory, it is erroneously believed that objects are kept moving by internal forces (as opposed to external forces); force is some-thing internal to a moving object. On this reasoning, force is an acquired property of objects that move. Hence, “motion implies a force.” This force is known as the force of impetus, missile, or inertia. 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 naïve concept of force belongs to the ontological category matter. However, the Newtonian concept belongs to a different ontological category, constrained based processes.

This “impetus theory” misconception is evident in the following problems taken from Mayer 2007; McClosky, Caramaza, & 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 mo-tion 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 cor-rect 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 forces.” 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 ForceStudents 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 dif-ficulty recognizing friction as a force. (Taking Sc to School)

Action/Reactive PairsMisconception- Greater Mass Implies Greater Force

Misconception – Most active agent produces greatest force (Heavier, Moving Objects Exert a Greater Force than Lighter Stationary Objects).

GravityOn the correct understanding of gravity, falling objects, regardless of weight, fall at the same speed.

Misconception - Heavier Objects Fall Faster than Lighter Objects. Many stu-dents learning about Newtonian motion often persist in their belief that heavier objects fall faster than light objects (Champagne, Gunstone, & Klopfer, 1985).

Ontological MisconceptionsThere is one class of alternative theories (or misconceptions) that is very deeply entrenched. These relate to ontological beliefs (i.e., beliefs about the funda-mental categories and properties of the world). (See Chi 2005; Chinn & Brewer, 1998.; Keil, 1979). Ontological categories refer to the basic categories of realities (the kinds existent in the world). These kinds include concrete objects, events, and abstractions, which are distinct ontological categories because they are dif-ferentiated by a mutually exclusive set of plausible attributes (e.g., an entity, such as a bit of glass, can be colored whereas an event, such as a baseball game, cannot be colored) (Chi, 2005). In ontological misconceptions, students assume that an entity or relation belongs to a fundamentally different kind of thing (Com-mittee on Science Learning, Kindergarten through Eighth Grade, 2007). Many misconceptions in the physical sciences are caused by faulty ontological under-pinnings (Chi, 1992). (See also Chinn & Brewer, 1993 for discussion of these common ontological misconceptions).

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)- that force is something internal to a moving object (McCloskey, 1983) (see sec-tion on physics misconceptions)

Other misconceptions in Science:- belief that rivers flow south

Epistemological Misconceptions about the Domain of Science Itself (its objectives, methods, purposes):Many middle school and high school students tend to see the purpose of science as manufacturing artifacts useful for humankind. Moreover, scientific explana-tions are viewed as being derived inductively from data and facts since the hy-pothetical 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).

MATHEMATICS

MoneyCorrect understanding of money is that the value of coin currency is not corre-lated with its size.Misconception. At the pre-K level, children have a core misconception about money and distinguishing the values of coins. Students think nickels are more valuable than dimes because nickels are bigger.

Subtraction (Brown & Burton, 1978; Siegler, 2003; Williams & Ryan, 2000)

Correct understanding of subtraction includes the notion that the columnar order (top to bottom) of the problem cannot be reversed or flipped.

Misconception #1. Students (age 7) have a “smaller-from-larger” error (miscon-ception) that subtraction entails subtracting the smaller digit in each column from the larger digit regardless of which is on top.143 83 -28 -37125 54

Misconception #2 - when subtracting from 0 (when the minuend includes a zero) there are two subtypes of misconceptions:

307 856 606 308 835-182 -699 -568 -287 -217 285 157 168 181 618

Misconception a - flipping the two numbers in the column with the 0In problem “307-182,” 0 – 8 is treated as 8 – 0, and wrote “8” as the answer.

Misconception b - lack of decrementing; 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,” not reducing the 3 to 2.

MultiplicationCorrect understanding of multiplication is that it does not always increase a num-ber.

Misconception - Students have misconception that multiplication always increas-es a number. For example, take the number 8,3 x 8 = 245 x 8 = 40This impedes their learning of the multiplication of a (positive) number by a frac-

tion less than one, such as ½ x 8 = 4.

DivisionMisconception in the form of “division as sharing” (Nunes & Bryant, 1996) or the “primitive, partitive model” of division (Tirosh, 2000). On this model, an object or collection of objects is divided into a number of equal parts or subcollections (e.g., five friends bought 15lb of cookies and shared them equally. How many pounds of cookies did each person get?). The primitive partitive model places 3 constraints on the operation of division:1. the divisor must be a whole number2. the divisor (the number by which a dividend is divided) must be less than the

dividend3. 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 vio-late 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 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 solv-ing this word problem. They incorrectly choose the expression “1/4 X 4” as the answer (see Fischbein, Fischbein, Deri, Nello, & Marino, 1985)

This “primitive, partitive” model interferes with children’s ability to divide fractions – because student believe you cannot divide a small number by a larger number because it is impossible to share less among more.

Indeed, even teacher trainees can have this preconception of division “as shar-ing.”. They were unable to provide contexts for the following problem (Goulding et al., 2002):

“2 divided by ¼”

Negative Numbers (see Williams & Ryan, 2000)The correct conception of negative numbers is that these are numbers which are less than zero. They are usually written by indicating their opposite, which is a positive number, with a preceding minus sign.

Separation Misconception - 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 at-tached afterwards) and later the sequence gets -4 inserted thus: -7, -4, 1,…(be-cause the sequence is read 1, 4, 7 and the minus sign is afterwards attached). Similarly, we can explain: -4 + 7 = -11

Fractions (see misconception examples above) (see Hartnett & Gelman, 1998)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 (applies the counting principles). They have moved toward using count words and other symbols that are numerically meaningful. The numberhood of fractions is not consistent with the counting principles and the idea that numbers result when sets of things are counted and that addition involves putting together two sets. 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).

Misconceptions – reflect children distorting fractions to fit their counting based number theory and not viewing fractions as exemplars of a new kind of number.

Misconception #1 - increasing values of the denominators maps to increasing quantitative values. Natural number ordering rule for fractions based on cardinal value of the denominator (see Hartnett & Gelman, 1998)

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 used 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),

Misconception #2 - when adding fractions that process is to add the two numera-tors to form the sum’s numerator and add the two denominators to form its de-

nominator. Example ½ +1/3 = 2/5 (See Siegler, 2003)

Decimal/ Place-ValueThe 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. For example, 4,309=(4×103)+(3x102)+(0×101)+(9×100)=4,000+300+0+9, and 4.309=(4×100)+(3×10−1)+(0×10−2)+(9×10−3)=4+3/10+0/100+9/1000. 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 are in-tegers (whole numbers) except that when these operations are written in colum-nar 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 where there is a decimal point in the divisor (4.32), the point 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 zeros are added to the right of the decimal point in the dividend as needed, and the division proceeds the same as for integers.

Misconception #1 - the “separation strategy:” whereby students separate the whole (integer) and decimal as different entities; they treat the two parts before and after the decimal point as separate entities. Seen in pupils Williams & Ryan, 2000. Also seen in beginning preservice teachers (Ryan & McCrae).

Example: Division by 100: 300.62 divided by 100Correct Ans = 3.0062Misc Ans = 3.62

Example: When given 7.7, 7.8, 7.9, they continue the scale with 7.10, 7.11

Misconception #2 - relates to the ordering of decimal fractions from largest to smallest: (Resnick et al., 1989; Sackur-Grisvard, & Leonard, 1985). This miscon-ception is also seen in primary teacher trainees (Goulding, Rowland, & Barber 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 nu-merical expressions AND difficulties with more than two places of decimals

Misconception a.- Larger/longer number is one with more digits to right of deci-mal point. 4th-9th grade So 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 val-ues in comparing decimal fractions (Resnick et al., 1989). Whole number errors derive from students’ applying rules for interpreting multi-digit integers. Children using this rule appear to have little knowledge of decimal numbers. Their repre-sentation 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 Resn-ick et al., 1989).

Misconception b. LS “largest/longest decimal is smallest (one with fewer digits to right of decimal)” Misconception. 4th-9th grade. So, given the pair 1.35 and 1.2, 1.2 is viewed as greater. So 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 hun-dredths 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 in-formation 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. Make correct judgment when one or more zeros are immediate-ly to the right of the decimal point in one of the numbers, and otherwise chooses as larger the number with more digits to the right of the decimal point. Hence, in ordering the following three numbers (3.214, 3.09, 3.8), student correctly chooses the number with the zero as then smallest but then resorts to the “larger number is the one with more digits to 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 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 that zero is very small to conclude that the entire decimal must be small. (see Resnick et al., 1989).

Misconception #3 - DPI “decimal point ignored” Misconception (Mason & Rud-dock, 1986 cited in Goulding et al., 2002 & Ryan & McCrae)

Misconception #4 - Multiplication of Decimals Example: 0.3 X 0.24 Correct Ans = 0.072;

Misconception Answer: Multiply 3 x 24 and adjust two decimal points. 0.72(seen in beginning preservice teachers as well Ryan & McCrae –s)

Misconception #4 of unit-ths, tenths, hundredthsExample: Write in decimal form: 912 + 4/100Correct Ans = 912.04Misconception Answer = 912.004Misconception 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 % signErrors such as: 4 + - 7 = -11; -10 + 15 = 25

Misconception #2 - Avoid fractions, for ex, by assigning a whole number value to a probability.

Misconception #3 - Think zero is the lowest number

Misconception #4 - Separate treatment of 2 parts of the integer in subtraction of integers:

Given “-2 – (-8) = __” . - 2 refers to process of subtracting the whole number two. They give “-10” as the lowest possible answer or the largest value of “10”

AlgebraMisconception #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 simi-larities and produce:

a + (b x c) = (a + b) x ( a + c)

Misconception #2: Variable MisconceptionCorrect Understanding of Variables is that letters in equations represent, at once, a range of unspecified numbers/values.

It is very common that middle school students have misconceptions about core concepts in algebra, including concepts of a variable (Kuchemann 78; Knuth et al., 2005; MacGregor & Stacey, 1997; Rosnick, 1981). This misconception can begin in the early elementary school years and persists through the high school years.

There are several levels or kinds of variable misconceptions. a. Variable Misconception: Level 1Letter assigned one numerical value from outset

ExampleIf 4x + 4y = 12, then what is the value of x + y? A. 3 (Correct) B. 4 C. Cannot be determined. (Misconception) D. 12

b. Variable Misconception: Level 2 Letter Ignored/Not Assigned a Meaning

Example:Simplify 3m + 5 – 2m + 1: A. 7 (Misconception) B. 10 C. m + 6 (Correct) D. 7m + 8

c. Variable Misconception: Level 3Letter interpreted as a label for an object or as an object itself

Example: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) B. professors C. students (Misconception) D . none of the above

d. Variable Misconception: Level 4Letter considered a specific unknown number

Example:Rita put some hummingbird feeders in her backyard. The table shows the num-ber of hummingbirds that Rita saw compared to the number of feeders.

Number of Feeders (f) Number of Hummingbirds (h)

1 3

2 5

Which equation best describes the relationship between h, the number of hum-mingbirds, and f, the number of feeders? A. h = 11f B. h = 2f + 1 (Correct) C. h = f + 2 (Misconception) D. h = f + 6

Misconception #3 – Equality MisconceptionCorrect Understanding of Equivalence (the equal sign) is the “Relational” view of the equal sign. This is understanding the equal sign as a symbol of equivalence

(i.e., a symbol that denotes a relationship between two quantities).

Students exhibit a variety of misconceptions about equality (Falkner et al., 1999; Kieran, 1981,1992; Knuth et al., 2005; McNeil & Alibali, 2005; Steinberg et al., 90; Williams & Ryan, 2000). The equality misconception is also evident in adults (college students) (McNeil, 20 ).

a. “Operational” View of Equal Sign Interpret equal sign as a command to do something. Hence, fill in blank/variable to right of sign w/completed operation from left hand side.

Example:What should go in the blank?7 boxes + 5 boxes = ______ + 4 boxes

A. 7 boxes B. 8 boxes (Correct) C. 5 boxes D. 12 boxes (Misconception)

b. Do not understand concept of “equivalent equations” & basic principles of transforming equations. Do not know how to keep both sides of equation equal. So do not add/subtract equally from both sides of equal sign.

Example: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

c. Equal sign described in terms of answer and it is assumed that answer (solu-tion) is the number after the equal sign (answer is on the right)

3. If 4(x + 5) = 80, then what is the value of x?

A. 15* (Correct) B. 20 C. 60 D. 80 (Misconception)

LANGUAGE ARTS

PoetryThe correct understanding of poems includes the notion that a poem need not rhyme.

Misconceptions that poems must rhyme

Language A correct understanding of language is that language can be used both literally and nonliterally

Misconception 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, speaker intention is to use an utterance to express a meaning that is not the literal mean-ing 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, saying “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 resem-blance, as in “All the world’s a stage.” (Shakespeare). Students have difficulty understanding nonliteral (figurative) uses of language because they have a mis-conception that language is used only literally.(See Winner, 1997)(back to Common Misconceptions)

SOCIAL STUDIES

Misconceptions about Native Americans.That most or all live in teepees, wear warbonnets, use bow and arrows.

Evidence and Explanation: Why and how do these teaching strategies work?

Students do not come to school as blank slates to be filled by instruction. Chil-dren are active cognitive agents, who arrive at school after years of cognitive growth (Committee on Science Learning, Kindergarten through Eighth Grade, 2007). They come to the classroom with considerable knowledge based on intu-itions, every day experiences, or what they have been taught in other contexts. This pre-instructional knowledge of students is referred to as preconceptions. Since a considerable amount of our knowledge is organized by subject areas, such as mathematics, science, etc., so too are preconceptions. It is important for teachers to know about the preconceptions of their students because learning depends on and is related to student prior knowledge (Brans-ford, Brown, & Cocking, 2000; Gelman & Lucariello, 2002; Piaget; Resnick, 1983). We interpret incoming information in terms of our current knowledge and cognitive organizations. Learners try to link new information to what they already know (Resnick, 1983). This kind of learning is known as assimilation (Piaget). When new information is inconsistent with what learners already know it cannot be assimilated. Rather, the learner’s knowledge will have to change or be altered because of the new information and experience. This kind of learning is known as accommodation (of knowledge/mental structures).

Whether learning is a matter of assimilation or accommodation depends on whether student preconceptions are anchoring conceptions or alternative con-ceptions (misconceptions), respectively. Student preconceptions that are consis-tent with concepts in the assigned curriculum are anchoring conceptions. Learn-ing, in such cases, is a matter of assimilation or conceptual growth. It consists in enriching or adding to student knowledge. Assimilation is an easier kind of learn-ing because prior knowledge does not interfere with learning. Rather, prior knowl-edge is a base the learner can rely on to build new knowledge. Student preconceptions that are inconsistent with, and even contradict, con-cepts in the curriculum, are alternative conceptions or misconceptions (or intui-tive theories). Intuitive theories are very typical and children and adults possess them. They develop from the natural effort of the cognitive system to make sense of the world around us. For example, the “distance theory” (a misconception) that explains seasonal/temperature change in terms of different distances be-tween the Earth and the Sun in summer and winter could easily develop from one’s everyday experience with heat sources (Kikas, 2004). Sometimes even textbooks themselves can be the cause of alternative theories. For instance, the diagram of the earth’s orbit, commonly used in textbooks, that renders such as a stretched-out ellipse (although it more closely resembles a circle) can contrib-ute to the erroneous “distance theory” of seasonal change (Kikas, 1998). Hence, intuitive theories or misconceptions are not a reflection of a cognitively deficient

child. Rather, they reflect a child with a cognitively active mind, who has already achieved considerable complex and abstract knowledge. Indeed, young children are not limited to concrete reasoning. Nor should they be viewed as a bundle of misconceptions.

Alternative conceptions (misconceptions) interfere with learning for several reasons. Students use these erroneous understandings to interpret new experi-ences, thereby interfering with correctly grasping the new experiences. Moreover, misconceptions can be entrenched and tend to be very resistant to instruction (Brewer and Chinn, 1991; McNeil & Alibali, 2005). Hence, for concepts or theo-ries in the curriculum where students typically have misconceptions, learning is of a more difficult kind. It is a matter of accommodation. Instead of simply adding to student knowledge, learning is a matter of radically reorganizing or replacing stu-dent knowledge. Conceptual change or accommodation has to occur for learning to happen (Carey, 1985; 1986; Posner, Strike, Hewson, & Gertzog, 1982; Strike & Posner, 1985, 1992). Teachers will need to bring about this conceptual change.

According to conceptual change theory, learning involves 3 steps (see Mayer 2007 for summary):1. recognizing or detecting an anomaly – This refers to an awareness that your current mental model (representation or theory or conception) is inad-equate to explain observable facts. The student must realize that he/she has a misconception(s) that must be discarded or replaced. 2. constructing a new model – This entails finding a better, more sufficient model that is able to explain the observable facts. It involves the students’s replacing one model with another3. using a new model - This refers to students using the new model to find a so-lution when presented with a problem. This reflects an ability to operate with the new model

Hence mental models (representations or theories or concepts) are at core of conceptual-change theory. For example, you are using a mental model when you think the earth is hollow.

Traditional methods of instruction, such as lectures, labs, discovery learning, simply reading text have been found not to be effective at achieving concep-tual change (Chinn & Brewer, 1993; Kikas, 1998; Lee, Eichinger, Andderson, Berkheimer, & Blakeslee 1993; Roth, 1990; Smith, Maclin, Grosslight, & Davis, 1997). Accordingly, recommended teaching strategies are included here.

Frequently Asked Questions

1. Is it normal for students to have misconceptions? Do most students have them?Yes, it is very typical for students to have misconceptions. They acquire or form them through everyday experiences, through instruction on other topics, and because some concepts are very complex to master.

2. Are there some common misconceptions that students have in different subjects?Yes, there are typical misconceptions that students have in the different subjects, such as math and science. Being aware of the typical misconceptions students have in these subject areas can help you focus your instruction carefully to these and address the most common misconceptions.

Common Alternative Conceptions (misconceptions)

Hence, teachers can focus attention on these common misconceptions.

3. Are these strategies to correct student misconceptions applicable to all children?These strategies are general ones that should be effective with most children. However, each is optimally appropriate and effective at specific grade levels. Go to “For whom do these recommendations work” for the details on each strategy for optimal grade(s) use. Moreover, a teacher should use his or her judgment about which strategies might be most effective, given the particular students in the class. For example, for students that have language difficulties (e.g., difficulty reading and processing text and articulating thoughts verbally) the teacher might rely more on the less-verbal strategies (e.g., use of bridging analogies) with those students.

For whom do these recommendations work and under what conditions?

AgeAlmost all these recommendations can be used with students from the elemen-tary grades (beginning at around 5th Grade) through high school. In the case of using bridging analogies (recommendation #2), this strategy is most suitable for high school students.

Individual differences We know very little about how these recommendations might vary by gender or ethnicity. There is good reason to believe, however, that most, if not all, of these recommendations would be generally successful with most students. The little research that has been done with different kinds of children suggests that these strategies would be comparably effective with low-achieving children (as with bet-ter performing children)

Contextual factorsWe know very little about how these recommendations might vary by contextual factors, such as children living in poverty and different kinds of family relations. We do know that misconceptions are quite universal. There is good reason to be-lieve, however that most, if not all, of these recommendations for getting students over their misconceptions would be generally successful with most students. The little research that has been done with urban classes across 7th to 9th grade that had many disadvantaged students suggests that these strategies would be effective for low-SES children. There is no reason to believe that family variables would play any role in the effectiveness of these strategies.

Where teachers can get more information?

Bransford, J. D., Brown, A. L., Cocking, R. R. (Eds.).(2000) How people learn: Brain, Mind, Experience, and School. Washington: National Academy Press.

Brown, J. S., & Burton, R.R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155-192.

Committee on Science Learning, Kindergarten through Eighth Grade (2007). Tak-ing science to school: Learning and teaching science in grades K-8. Washington, DC: National Academies Press.

Duschl, R., & Osborne, J. (2002). Supporting and promoting argumentation dis-course. Studies in Science Education, 38, 39-72.

Eryilmaz, A. (2002). Effects of conceptual assignments and conceptual change discussions on students’ misconceptions and achievement regarding force and motion. Journal of Research in Science Teaching, 39, 1001-1015.

Guzzetti, B. J., Snyder, T. E., Glass, G. V., & Gamas, W. S. (1993). Promot-ing conceptual change in science: A comparative meta-analysis of instructional interventions from reading education and science education. Reading Research Quarterly, 28, 116-159.

Hayes, B. K., Goodhew, A., Heit, E., & Gillan, J. (2003). The role of diverse in-struction in conceptual change. Journal of Experimental Child Psychology, 86, 253-276.

Hynd, C. R. (2001). Refutational texts and the change process. International Journal of Educational Research, 35, 699-714.

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Minstrell, J. (1982). Explaining the “at rest” condition of an object. The Physics Teacher, 20, 10-14.

Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommo-dation of a scientific conception: Toward a theory of conceptual change. Science Education, 66, 211-227.

Resnick, L. B. (1983). Mathematics and science learning: A new conception. Sci-

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