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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 years

Since high school geometry is typically the first time that a student encounters formal proofs, this can obviously present some difficulties. It can also lead students 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 would be easier for students if they had encountered informal proofs and were required to justify their statements and reasoning in earlier school years. This of course would not be on such a formal level as in high school, but simply a mindset of teaching mathematics where mathematical statements and truths are justified, the teacher explains where things come from or why something works—and also the child is asked to provide explanations and justifications.

Lack of understanding of geometry concepts

Two 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 (note they can also be numbered from 0 to 4):

• Level 1 - visual. Geometric figures are recognized based on their appearance - not based on their properties. For example, a rectangle is "something that looks like a door", and not a figure with four sides and four right angles. A student in this level would not recognize a rectangle that is rotated so it's 'standing on its corner', or a very scalene triangle that does not look like the 'prototype' triangle kids are often shown - the equilateral triangle. See also Van Hiele Levels - level 0: Visual.
   
• Level 2 - descriptive/analytic. Students can identify properties of figures and recognize them by their properties. They cannot tell a difference between the necessary and sufficient defining properties of a shape, and extra properties of a shape. For example, a student might not understand that for a definition of rectangle, it is enough to say it has four sides and right angles. Instead, the student might include in the definition also the fact that the opposite sides are equal and parallel. Also, student cannot categorize shapes hierachically; and cannot understand why a square is also a rectangle. See also Van Hiele Level 1: Analysis.
   
• Level 3 - abstract/relational. Students can now understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes. They can, for example, tell that all rectangles are parallelograms, but not vice versa. Students can justify their reasoning informally but not yet construct formal proofs. A student needs to be at least on this level BEFORE taking high school geometry course.
   
• Level 4 - formal deduction. Students can reason formally using definitions, axioms, and theorems. They can construct deductive proofs starting from the givens, and producing statements that ultimately justify the statement they are supposed to prove. This is the level that a typical high school geometry course is taught.
   
• Level 5 - rigor/metamathematical. Students can reason formally and compare different axiomatic systems. This level is needed in college mathematics.

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 calculations of the type, "Calculate the area/circumference/perimeter/radius etc. of this figure." Textbook problems concentrate too much on calculating and using formulas, and not enough on analyzing concepts, making conjectures about the properties, testing them, and studying lots and lots of figures and shapes experimentally. More on this later.

Student's cognitive development

This point ties in with the previous one, but has more to do with the general cognitive development instead of just geometrical reasoning. According to the psychologist Jean Piaget's theory about 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 resources

Van Hiele Levels

Van Hiele model from Wikipedia

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.

Book Reviews

I have reviewed several geometry books:

Geometry: Seeing, Doing, Understanding by Harold Jacobs

Dr. Math geometry books - these are inexpensive companions to middle and high school geometry courses.