The mtt2k discussion started by math professors John Golden and David Coffey has found many videos by Khan Academy that are poorly done. This goes beyond just errors to sloppiness and fuzziness in the lessons themselves. The KA videos are thus not a test of crisp procedural lessons but of fuzzy, sloppy, sometimes error prone procedural lessons. Moreover, in the flipped classroom, this is what students see first in learning a topic.
The flipped classroom with poor quality procedural videos as the first encounter with each topic for a year has not been studied formally very likely. However, the impact of such an approach would be valuable. Would parents consent to using poor procedural videos in a flipped classroom experiment for an entire school year? Apparently, they already have with Khan Academy.
The blog link above has an article by Susan Brown, Antoinette Seidelmann, and Gwendolyn Zimmermann on procedural v conceptual learning.
The following excerpt has us understand the potential impact of poor quality procedural Khan Academy videos in a flipped classroom.
Procedural or Conceptual: Which Comes First?
Several studies have shown that learning procedures actually interferes with the development of meaningful knowledge if a solid conceptual foundation has not been laid (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998; Mack, 2001; Pesek & Kirshner, 2000). In the study by Pesek and Kirshner, students were divided into two groups for a unit on area and perimeter. One group (PF) studied procedures first, learning to use and compute with formulas. Next, this group of students received conceptually oriented instruction, with an emphasis on developing relationships. No formulas were given. Instead, students were assisted in developing their own methods to solve problems. The second group of students (CO) received only the conceptually oriented instruction, spending less than half the time that the PF group did studying area and perimeter.
The study found that the CO group, whose instruction focused exclusively on relationships and concepts, demonstrated better understanding of area and perimeter than the PF students, who were taught formulas first and then the concepts. The PF students could not articulate the difference between area and perimeter in the context of a real life situation. When asked which measurement, area or perimeter, would be needed to determine the amount of wallpaper needed to decorate a room, most of the PF group chose perimeter because, as one student explained, “Walls don’t have area because they go around” (p. 535). However, all students in the CO group understood that area was needed to solve the wallpaper problem. When students were asked to explain how area and perimeter formulas work, the PF students spoke about the processes in terms of computations. They could not explain why the formulas worked, but many in the CO group were able to make sense of the formulas even though they had not received explicit instruction on them.
Compared to their weaker classmates, stronger students in the PF group were more easily able to overcome the negative effects of learning procedures first. Thus, the evidence shows that the direct instruction method that traditionalists think will help weaker students is actually handicapping them! This finding has important implications for the classroom as we strive to reach more students with mathematics.
The Pesek and Kirshner study clearly illustrates the potentially detrimental effect of teaching for procedural knowledge prior to teaching for conceptual understanding. Procedural knowledge does not easily transfer to conceptual knowledge. In contrast, by working with meaning and concepts first, students form a much richer knowledge base. Moreover, they are able to apply that knowledge and give meaning to the formulas and computations.
What does this mean for us as teachers? It tells us how to approach instruction. On the one hand, if we want our students to have a conceptual understanding of mathematics, we need to teach conceptually. On the other hand, a rapid push for procedural skill will actually do more harm than good. In other words, if skills are taught before the concepts are developed and connected to the understanding students bring to the topic, students are likely to struggle to develop conceptual knowledge.
==The references cited by the 3 authors are
- Allen, F. (1996). A program for raising the level of student achievement in secondary school mathematics. Retrieved June 27, 2001, from Mathematically Correct Web site. http://www.mathematicallycorrect.com
- Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3-20.
- Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 25-41). Dordrecht: Kluwer.
- Fey, J. (1999). Standards under fire: Issues and options in the math wars. Retrieved June 27, 2001, from the University of Missouri, Show-Me Center Web site. http://www.showmecenter.missouri.edu/showme/perspectives/keynote1.html
- Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.
- Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., Oliver, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
- Kouba, V. L., Zawojewski, J. S., & Strutchens, M. E. (1997). What do students know about numbers and operations? In P. A. Kenney & E. Silver (Eds.), Results from the sixth mathematics assessment of the National Assessment of Educational Progress (pp. 87-140). Reston, VA: National Council of Teachers of Mathematics.
- Mack, N. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267-295.
- National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
- Pesek, D. D., & Kirshner, D. (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31(5), 524-540.
- Ross, D. (2001). The math wars [Electronic version]. Navigator, 4(5).
- Skemp, R. (1987). The psychology of learning mathematics. Hillsdale, NJ: Erlbaum.
- Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.
- Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Thinking and Learning (pp. 495-511). New York: Macmillan.
- U.S. Department of Education. (1996). Pursuing excellence: A study of U.S. eighth-grade mathematics and science teaching, learning, curriculum, and achievement in an international context. Washington, DC: U.S. Government Printing Office.
- Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 23, 14-19, 50-52.
Note about the authors: Susan Brown, Antoinette Seidelmann, and Gwendolyn Zimmermann currently teach high school mathematics in the Chicago area. Gwendolyn Zimmermann has a Ph.D. in Mathematics Education. Susan Brown and Antoinette Seidelmann are Ph.D. candidates in Mathematics Education at Illinois State University. They collectively have 56 years of teaching experience spanning grades three through college.
==End of excerpt.
Weaker students are the ones that could be harmed the most in a flipped classroom using poor quality procedural videos. This is particularly true if this was the approach for an entire school year. Using badges to further push the poor quality procedural approach would likely accentuate the negative impact on weaker students that the above authors discuss.
==Summer Sale on e-book
Pre-Algebra New Math Done Right Peano Axioms
is on sale for 2.99 at e-book vendors.
The Peano Axioms are the foundations of the concepts and structure of arithmetic and beginning algebra. So to teach concepts right from the start, one has to understand the Peano Axioms and the proofs of basic laws like the associative and commutative laws of addition for natural numbers. This is done at below college level in the book linked to above. This is a major breakthrough in education at below college level in math.