High school geometry: why is it so difficult?It is not any secret that high school geometry with its formal (two-column) proofs is considered hard and very detached from practical life. Many teachers in public school have tried different teaching methods and programs to make students understand this formal geometry, sometimes with success and sometimes not. Of course it is even more difficult for a homeschooling parent This article explores the reasons why a typical geometry course in high school is so difficult for many students, and what could a teacher possibly do to help the situation. Lack of proof and proving in earlier school yearsSince high school geometry is typically the first time that a student encounters a formal proof, this can obviously present some difficulties. It can also lead kids to think that two-column proof is the only kind of proof there is - yet that form of proof is almost never used by practicing mathematicians. It could be easier, if children encountered informal 'proofs' in earlier school years, and were required to justify their statements and reasoning. This of course would not be on such formal level as it is in high school, but simply a mindset of teaching mathematics where mathematical statements and truths are justified, there are explanations of where things come from, why something works - and the child also is asked to provide explanations and justifications. Lack of understanding of geometry conceptsTwo Dutch researchers, Dina van Hiele-Geldof and Pierre van Hiele, suggest that students' geometrical understanding progresses through various levels, which cannot be skipped. These levels are now known as van Hiele levels. Other research supports their theory, and has found that most students enter high school geometry with a low Van Hiele level of understanding. Thus they cannot possibly understand the teaching, since writing formal proofs requires at least a van Hiele level 4. The levels of van Hiele are are (note they can also be numbered from 0 to 4):
This theory is not perfect but based on other research, it seems to model the progress of geometrical thinking. The important point is that a lot of the geometry taught before high school does NOT foster students into higher level of geometrical thinking. A lot of the geometry problems in text books are just of the type, "Calculate the area/circumference/perimeter/radius etc. of this figure." It's too much calculating and using formulas - and not enough of analyzing concepts, making conjectures about the properties, testing them, studying lots and lots of figures and shapes experimentally. More on this later. Student's cognitive developmentThis point ties in with the previous one, but has more to do with general cognitive development instead of just geometrical reasoning. According to the psychologist Jean Piaget's theory about one's cognitive development, a person needs to achieve a certain level (called formal operational stage) to be able to reason formally and understand and construct proofs. If a high school student has not achieved that, then it will be very hard to understand the geometry course. Sadly, there is some research suggesting that even most college students have not achieved that level (Ausubel, Novak, and Hanesian 1968).
Continue to part 2: What can be done to make high school geometry less of a pain? Sources and resourcesVan Hiele levels and learning geometry notes Research Sampler 8. Students' Difficulties with Proof by Keith Weber Geometry and Proof by Michael T. Battista and Douglas H. Clements What is proof? Do you need proof before high school? - my article.
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