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What can you do to make students understand High school geometry?

If you read the first part of this article, you can already see that the measures to take happen generally before high school. The best approach involves changing how math and especially geometry is taught BEFORE high school. The main points to consider are:

• Most importantly, improve geometry teaching in the grade school years so that students' van Hiele levels are brought up to at least to the level of abstract/relational.
• Include more justifications and informal 'proofs' into math teaching during the school years in general.
• In general make students think and reason and use their brain via different school tasks (not just math).
• Make sure the children's diet has all the nutrients for proper brain development even all through the teen years. For example, essential fatty acids are often totally lacking in typical American diets. Or, girls who diet and try to lose weight are at risk of iron deficiency leading to reduced IQ levels. Junk food can lead to "junky brains".

This article will concentrate only on the first point.

Understanding geometry concepts

You can expect kids up through first grade to be in the first van Hiele level - visual. (Geometric figures are recognized based on their appearance - not based on their properties.) During the next several grades (grade 2 and on) kids SHOULD get enough experience with geometrical shapes to be on the second van Hiele level - descriptive/analytic - where they can identify properties of figures and recognize them by their properties.

And if all goes well, somewhere during the 6th thru 9th grades the student would proceed to the third level - abstract/relational where they can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes. And thus, the student would be prepared for the formal proofs and deductive reasoning in high school geometry.

Experiments have shown that this is indeed possible with the right kind of geometry teaching. The key is to emphasize the geometrical concepts and providing the kids lots of hands-on activities like drawing the figures and working with manipulatives - instead of only memorizing formulas and definitions, and merely calculating areas, perimeters etc. See below some examples of activities that will help children and young people to develop their geometric thinking.

To develop correct understanding of a single concept:

• When studying a concept, show correct AND incorrect examples, and in different ways or representations (rotate the pictures upside down etc!). Students are asked to distinguish between correct and incorrect examples. This will help prevent misconceptions.
• Aks students to draw correct and incorrect examples of a concept.
• After these activities, ask the students to provide a definition for a concept. This can get them to thinking about which properties in the definition are really necessary and which are not.
• Allow the students to experiment, investigate, or "play" if you allow me, with geometrical ideas and figures. For this you could use manipulatives, lots of drawing, and computer programs (more on them below).
• Have each student make his/her own geometry concepts notebook, with examples, nonexamples, definitions and other notes or pictures.

My Geometry 1 ebook for grades 3-5 contains many exercises that are NOT your typical "use the formula and calculate" type of stuff found in school books. The main of the book is to prepare a child's understanding of geometry in a way as will be needed later in high school.

One interesting possibility for a middle school geometry course might be RightStart Geometry. It's a hands-on course where much of the work is done with a drawing board, T-square, and triangles.

Computers and interactive geometry

Computer can really help in geometry teaching, since it allows a dynamic, interactive manipulation of a figure. A child can move, rotate, or stretch the figure, and observe what properties stay the same.

For example, say you are teaching the concept of isosceles triangle. You could even simply use the Drawing Toolbar in Microsof Word, which has the AutoShape for isosceles triangle (as well as for right triangle and parallelogram). Let kids draw one or two and then tell them to drag it from the white handles to make it bigger/smaller, and also to rotate the figure. Ask, "What changes? What does not change? What stays the same? Can you draw this kind of thing on paper?"

Better yet, try Mathsnet interactive shape - a free online tutorial about basic shapes and geometry concepts, their patterns and properties. You will be able to change the pictures yourself using the mouse and see the effect these changes have. Uses Java applets which load slowly, but it is definitely worth waiting and free!

There exists also commercial software specifically designed to teach geometry in an interactive investigational way. Such programs have been used in research experiments and in schools with good results. After you see what can be done, it is very easy to fall in love with such a program - the idea is just great!

• Cabri
• Geometer's Sketchpad
• Cinderella
• GeoGebra

And what if your student is already studying geometry in high school and is having problems? The only reasonable thing to do, I feel, is still go back and try to fill in the holes in his/her understanding. Hopefully you can take a 'time-out' for a few weeks and study the geometry concepts themselves first without the proving stuff.

And even with good preparation, high school geometry and the proofs can still be difficult. All in all, there is no quick and easy answer to the difficulties in this course. Remember that even math teachers in schools struggle with this problem of getting students to understand and construct proofs. Maybe the explanations on Ask Dr. Math: FAQ About Proofs and the ideas in this article can be of some help. If you have other ideas, write them in using the comment form below.

Hyperbolic geometry can help with proofs/theorems/axioms

Hyperbolic geometry is almost like normal plane (Euclidean) geometry, but in it, there can be several lines through a point that are parellel to a given line.

In high school geometry, students can often have difficulty understanding what is an axiom or what is a theorem, or the necessity for a proof. Students may think, "I can SEE that parallel lines are equidistant from each other" or, "I can SEE that there are two equal angles in a isosceles triangle - why prove it?"

Studying hyperbolic geometry can help students understand the difference between axiom and theorem, and the basic idea of proving theorems - because in it, things look different and so a student cannot rely on just what they see with their eyes.

There are several models for hyperbolic geometry, and it is especially easy to investigate and explore hyperbolic geometry in a Poincare disk, or in the upper plane, because there exist totally free computer applets or software for that.

Just check out NonEuclid - the site has a Java applet and an exploration guide with activities all ready for you.