Application Knowledge II

Application knowledge is a crucial component in students' ability to solve problems [1]. Students often possess a rudimentary procedural knowledge of how to carry out certain procedures, but do not posses the application knowledge about when to carry out those procedures when solving problems. For example, possessing application knowledge of the physics generalizations for diffusion, photosynthesis and velocity is necessary in solving many chemical, biological and physical problems [2]. Application knowledge of the mathematics generalization for the addition principle in probability is necessary in solving many mathematics problems [3]. It is analogous in playing chess to knowing how to properly move chess pieces on a board (which novices have), but not possessing an application knowledge of numerous key playing positions and the most appropriate next move or chess pieces for each position (which experts have) [4].

A classic study of novices and experts in college physics demonstrates the relationship between application knowledge of physics generalizations and ability in problem solving [5]. Undergraduate physics majors (novices) and college faculty in physics (experts) were individually given 24 cards showing physics problems they had not seen before and were asked to sort the problems into six or more separate categories in any way they liked, and to do so in less than a minute. The problems were depicted as drawings on the cards, such as a drawing of two weights joined by a rope that ran over a pulley. When they finished sorting they were asked to explain the rationale for their classifications. The members of both groups were alike in that they created about 8 categories, and finished in about 40 seconds. But they were unlike in the kinds of categories they created, and in the explanations they gave. Experts sorted the cards into categories corresponding to the physics generalizations exemplified in the problems. Consequently, some problems with pulleys were placed in a pile with some problems having inclined planes if they were examples of the same physics generalization. Experts utilized their application knowledge of the physics generalizations in sorting the problems into categories. Novices, on the other hand, sorted the cards into categories corresponding to their physical characteristics. They placed all cards with pulleys into one pile and all cards with inclined plans in another, and so forth. The novices did not possess the application knowledge needed in recognizing the drawings as examples of specific generalizations. They had to fall back on their application knowledge of shapes. The novices would have been unable to solve the problems shown on the cards unless they were told which equations to use. They are like the chess novices who know the basic procedures for moving chess pieces properly but are unable to recognize examples of key chess positions. Whatever instruction the novice physics students received for the physics generalizations was not effective in developing an application knowledge of them. The novices likely possess only recall knowledge of the physics generalizations: meaning that they are able to recall definitions of the generalizations, but are unable to recognize novel examples of them, such as with the drawings of physics problems.

Students often face severe difficulties in learning application knowledge because many generalizations are abstract and complex [6]. Teachers frequently fail to appreciate that. Duckworth noted that: "Teachers are often, and understandably, impatient for their students to develop clear and adequate ideas, but putting ideas in relation to each other is not a simple job. It is confusing, and that confusion does take time. All of us need time for our confusion if we are to build the breath and depth that gives significance to our knowledge." [7]

Students often intuitively acquire application knowledge of faulty generalizations, and that hinders their learning new application knowledge [8]. Examining studies of students' learning of faulty generalizations provide us a clearer understanding of the implicit cognitive processes involved in application knowledge.

In a study at the high school level a researcher identified students in a physics class who were able to calculate correctly the speed and position of moving objects when given the appropriate equation. Those students were given an assessment task for application knowledge. They were asked to draw the path of a ball that was kicked off a cliff. They had not previously talked about such a situation. Most of them showed the ball going straight out from the cliff and then falling straight down. They drew this path in spite of the fact that it clearly was not a correct example of the physics generalizations about motion. According to the physics generalizations about motion the ball should have fallen off the cliff in a graceful parabola [9]. Those students clearly did not possess an application knowledge of the physics generalizations about motion; although they might have had a recall knowledge of it, meaning that they could retrieve a definition from their long-term memories.

Many young adults go through high school and university physics courses without ever giving up their application knowledge of faulty pre-Newtonian generalizations about motion [10]. For example, the teacher in this study has been instructing college students for an application knowledge of the generalization about Newton's law of inertia. It states, in part, if an object is in motion it continues in a straight line unless it is acted upon by physical forces. Following instruction, students were given an assessment task for application knowledge. The assessment task (a curved pellet shooter) is shown below. They have not seen this curved pellet-shooter example before. They were asked to draw the path of a pellet when it emerged from the curved tube though which it had been shot. Their two most common performances are shown. The correct performance is A [11].

Another study reported that after two months of instruction for an application knowledge of the physics generalization acceleration in an introductory physics course, only 40 percent of the students could correctly perform application knowledge assessment tasks for the generalization [12].

Research indicates that students of all ages, preschool through Ph.D., and in many subject areas, hold onto their application knowledge of faulty generalizations tenaciously even when the evidence that initially produced the generalizations is discredited. They hold on because they believe the faulty generalizations were initially derived from "real life" experiences [13]. For example, many students believe the faulty generalization that heavier objects displace more water than lighter objects. Based on the faulty generalization, they predict that the heavier of two metal cubes of equal size would displace more water in a container than would the lighter of the two metal cubes. They place the cubes in the container of water one at a time and measure the height of the water before and after each cube is placed in the container. They are surprised when both cubes raise the water to an equal level, but they are undaunted. A student explains, "Your tricked us. You brought magic water." Students were willing to violate their understanding about the nature of water in order to defend their faulty generalization about weight determining the displacement of water [14].

Another group of students had acquired an application knowledge of a faulty generalization about heavier objects falling faster than lighter objects. On the basis of that faulty generalization they predicted that if a heavier metal ball and a lighter wooden ball of the same size were dropped at the same time from the same height, the metal ball would hit the ground first. When they tested out their prediction by dropping the two balls at the same time, and watching the balls hit at the same time, they refused to accept the contradictory information. Some claimed they observed the metal ball hitting first. Others claimed the balls must be the same weight even though the scales showed different weights [15].

A teacher tried to change students' application knowledge of the faulty generalization that electricity wears out as it travels from a battery around a circuit containing a light bulb. Students predicted that if the teacher placed two ammeters to measure current on each side of the light bulb, they would show less current on one side. When the ammeters were in place and the current turned on, the ammeters showed the current on each side of the bulb was the same. But instead of changing their faulty generalization, students tended to reject the data on methodological grounds. They claimed the ammeters were not accurate, the light bulb was bad, and the battery did not work [16].

Students' difficulties solving science problems can frequently be traced to their an application knowledge of faulty generalizations. One study reported that 80 percent of the university-level engineering students it focused on were proficient in solving word problems only when they were given the algebraic equations that were appropriate for solving the problems. They were like the novice chess players who knew how to move chess pieces properly, but lacked application knowledge of key playing positions. The difficulties engineering students were having were traced, in part, to their lack of a very basic application knowledge of what "X" means in algebraic equations, such as "6X + 5 = 17 [17].

Many of the difficulties students have in carrying out multi-digit addition and subtraction with regrouping have been traced to their inadequate application knowledge of generalizations about place value [18]. In a large national study it was found that less than 50 percent of third graders possess application knowledge of place value in the hundreds digit, and only 64 percent in the tens digit. One-third of third graders gave incorrect answers on two-digit subtraction problems involving regrouping, and half did so for three-digit problems involving regrouping [19]. Students' severe deficiencies in their application knowledge of key mathematics generalizations, and the resulting inadequate performance in arithmetic computation, have been documented by extensive research [20]. The majority of students in elementary school acquire the ability to "run off" arithmetic procedures, but few acquire an application knowledge of the generalizations that are the basis of those procedures. They consequently have great difficulty finding errors in their work and in solving word problems [21].

A four-year longitudinal study of fifth and eight-graders' acquisition of recall knowledge and application knowledge of such social studies generalizations as freedom and representativeness found that some acquired recall knowledge of the generalizations, but no application knowledge of them [22]. This was evidenced by their ability to recall definitions of the terms, and their inability to recognize novel examples of the terms. Reviews of large-scale studies of students' attitudes and knowledge about government found that students typically exhibit some strong feelings about their government, but possess by little actual recall knowledge of it, and no application knowledge [23]. One study reported that 80 percent of fourth-graders chose the presidency as the most important role for a adult, but less than 25 percent of them could describe any of the president's duties [24].

Studies indicate that basing instruction on current textbooks results in students' inadequate acquisition of application knowledge because the texts have major instructional weaknesses. Examinations of social studies textbooks found that they do not provide the kind of information about generalization examples, both familiar and novel, that are needed in developing students' application knowledge of social studies generalizations [25]. In fact, the authors of the textbooks seem to assume that students reading the texts have already acquired application knowledge of the generalizations that the texts are presumably intended to teach.

A growing realization among teachers of the general ineffectiveness of most traditional textbook-based instruction is causing many of them to provide their students more "hands-on" laboratory activities. But these laboratory activities are unlikely, by themselves, to increase students' application knowledge of generalizations. They are often just confusing. What is needed is more emphasis on "minds-on" activities [26]. The activities should provide students both appropriate application-knowledge instructional tasks to perform actively, and appropriate application-knowledge guidance while they perform those tasks.

Asian children in elementary school consistently out perform children from the U.S. Studies over the last two decades of U.S. and Asian classrooms indicate a reason U.S. children perform poorly is that Asian children receive more instruction for application knowledge while they are learning mathematical procedures. For example, Japanese and Taiwanese first and fifth-graders score much higher on test items measuring application knowledge of place value and of problem solving in multi-digit addition and subtraction with regrouping than do U.S. first and fifth-graders [27]. Korean children aged 6, 7 and 8 carry out multi-digit addition and subtraction with regrouping much more accurately and have a greater understanding of what they are doing than do their U.S. counterparts [28]. Korean, Japanese and Chinese children possess a higher degree of application knowledge of generalizations for place value than do children in the United States [29]. School in Mainland China, Japan, the former Soviet Union and Taiwan all begin instruction for application knowledge of generalizations for place value and procedures in multi-digit addition and subtraction with regrouping earlier than do schools in the U.S., and they complete this instruction earlier [30]. Studies in Japanese, Chinese and U.S. classrooms found that Asian teachers provide instruction for application knowledge while U.S. teachers seldom do. The result is that Asian children understand why a certain mathematical procedure works, how to use it, and how to spot and correct errors [31].

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2. Schoenfeld, A.H. (1985). Mathematical problem solving. New York: Academic Press.

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4. Ericsson, K.A., Charness, N., Feltovich, P., & Hoffman, R.R. (Eds.). (2006). The Cambridge handbook of expertise and expert performance. Cambridge, UK: Cambridge University Press.

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6. Chi, M.T.H. (2000) Cognitive understanding levels. In A.E. Kazkin (Ed.). Encyclopedia of psychology (pp. 146-151).

7. Duckworth, E. (1987). "The having of wonderful ideas" and other essays on teaching and learning. New York: Teachers College, Columbia University. p. 82.

8. Chi, M.J.H. (2005). Common sense conceptions of emergent processes: Why some misconceptions are robust. Journal of the Learning Sciences, 14, 161-199.

9. Strieley, T. (1988), August/September). Physics for the rest of us. Educational Researcher, 17(6), 7-12.

10. (For example). Champagne, A.S., Klopfer, L.E., & Anderson, J.H. (1980). Factors influencing the learning of mechanics. American Journal of Physics, 48, 1074-1079.

11. Adapted from Green B.F., McCloskey, M., Caramazza, A. (1985). The relationship of knowledge to problem solving, with examples from kinematics. In S.F. Chipman, J.W. Segal, & R. Glasser. (Eds.). Thinking and learning skills (Vol. 2, pp. 127-140).

12. Reif, F. (1987). Instructional design, cognition, and technology: Application to the teaching of scientific concepts. Journal of Research in Science Teaching, 24, 309-324.

13. (For example). Carpenter, T.P., Moser, J., & Romberg, T. (Eds.). (1982). Addition and subtraction: A developmental perspective. Hillsdale, NJ: Erlbaum.

14. Linn, M.C. (1983). Content, context, and process in adolescent reasoning. Journal of Early Adolescence, 3, 63-82.

15. (For example). Eylon, B. & Linn, M.C. (1988). Learning and instructor: An examination of four research perspectives in science education. Review of Educational Research, 58, 251-301.

16. Johusa, S., & Dupin, J.J. (1987). Taking into account student conceptions in instructional strategy: An example in physics. Cognition and Instruction, 4, 117-135.

17. Clement, J., Lockhead, J., & Monk, G. (1979). Translation difficulties in learning mathematics. American Mathematics Monthly, 88 (4), 287-290.

18. Fuson, K.C. (1990). Conceptual structures for multi-digit addition, subtraction, and place value. Cognition and Instruction, 7, 343-403.

19. Kouba, V.L., Brown, C.A., Carpenter, T.P., Lindquist, M.M., Silver, E.A., & Swafford, J.O. (1988). Results of the fourth NAEP assessment of mathematics: Number, operations and word problems. Arithmetic Teacher, 35, (8), 14-19.

20. (For example). Resnick, L.B., & Omanson, S.F. (1990). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3, pp. 41-95). Hillsdale, NJ: Erlbaum.

21. Brown, A.L., & Campione, J.C. (1990). Interactive learning environment and the teaching of science and mathematics. In M. Gardner, J.G. Greeno, F. Reif, A.H. Schoenfeld, & E. Stage (Eds.), Toward a scientific practice of science education (pp. 111-140). Hillsdale, NJ: Erlbaum.

22. Sinatra, G.M., Beck, I.L., & McKeown, M.G. (1992). A longitudinal study of the characteristics of young students' knowledge of their country's government. American Educational Research Journal, 29, 633-661.

23. (For example). McKeown, M.G., & Beck, I.L. (1990). The assessment and characteristics of young learners' knowledge of a topic in history. American Educational Research Journal, 27, 588-627.

24. Greenstein, F.J. (1965). Children and politics. New Haven: Yale University Press.

25. Beck, I.L., McKeown, M.G., & Gromoll, E.W. (1989). Learning from social studies texts. Cognition and Instruction, 6, 99-158.

26. (For example). Smith, E.L., & Anderson, C.W. (1984). The planning and teaching of intermediate science study: Final report. (Research Series No. 147). East Lansing: Michigan State University, Institute of Research on Teaching.

27. Stigler, J.W., Lee, S.Y., & Stevenson, H.W. (1990). The mathematical knowledge of Japanese, Chinese, and American elementary school children. Reston, VA: National Council of Teachers of Mathematics.

28. Song, M., & Ginsburg, H.P. (1987). The development of informal and formal mathematics thinking in Korean and U.S. children. Child Development, 57, 1286-1296.

29. Miura, I.T., Kim, C.C., Chang, C-M, & Okamato, U. (1988). Effects of language characteristics on children's cognitive representation of number: Cross-national comparisons. Child Development, 59, 1445-1450.

30. Fuson, K.C., Stigler, J.W., & Bartsch, K. (1988). Grade placement of addition and subtraction topics in China, Japan, the Soviet Union, Taiwan, and the United States. Journal for Research in Mathematics Education, 19, 449-458.

31. Perry, M. (2000). Explanation of mathematical concepts in Japanese, Chinese, and U.S. first and fifth-grade classrooms. Cognition and Instruction, 18, 181-207.