Advice and Recommended Teaching Practices for Courses in Geometry using Volume II of Mathematics for Elementary Teachers, preliminary edition, by Sybilla Beckmann, published by Addision Wesley

This is the advice I give for our geometry content course for prospective elementary teachers here at the University of Georgia:

What Should Students Learn in MATH 5002?

Recommended Teaching Practices

Characteristics of Good Explanations in Mathematics

Writing assignments that I have used

The goal of MATH 5001, 5002, and 5003 (our three math content courses for prospective elementary teachers) is to have a positive impact on the students' future teaching of mathematics. How do we do this? The answers are neither easy nor obvious, and there is certainly not a unique answer. At any given time, we should make a "best guess", observe and reflect on the outcomes, and use our observations and reflections to form a new "best guess".

This is my current best guess for MATH 5002 (our geometry course for prospective elementary teachers).

What Should Students Learn in MATH 5002?

1) Students should know the most fundamental ideas very well and in a very basic, hands-on way that connects as much as possible to intuition and to the physical world. As teachers, they will only be able to teach well what they know well. Also, even material that is known well, but is not seen as connected to their own teaching (in the long run), will probably not have the impact on their teaching that we are hoping for.

This means we need to take a fundamentally different approach in teaching MATH 5002 than we do in teaching calculus or upper level courses, where a majority of the students often develops only a so-so understanding of the material. In MATH 5002, we must try hard to get all the students to understand the most basic and fundamental ideas quite well.

Of course, we can and should ask MATH 5002 students to go beyond the most basic and fundamental material. But we cannot assume that a so-so knowledge of more advanced material ensures a solid knowledge of more basic material.

What is this most basic, fundamental material in MATH 5002? I would say this:

a) What angles are (rotationally and statically!) and how angles arise in real situations.

b) What is a circle---from a mathematical point of view---and how the mathematical definition of circle is used.

c) A sense of space: building solid shapes out of flat shapes, visualizing in space, drawing diagrams to capture certain spacial information (which is often hard, but should be done to the extent possible).

d) Visualizing the rigid moving of plane shapes.

e) The idea of scaling objects and the associated proportional reasoning about lengths.

f) Rigidity and non-rigidity of shapes of fixed side lengths.

g) The concept of measurement and the idea of a unit of measurement. What it means when we say the size of something is so-and-so-many units, especially when area and volume are concerned. The idea that one object can have many measurable attributes.

h) The idea that when we move a shape without stretching, the area/volume doesn't change. The idea that when we combine shapes without overlapping the area/volume is the sum of the areas/volumes. Using these ideas to calculate areas and volumes. That these ideas are the basis for the standard area/volume formulas.

i) That perimeter does not determine area.

j) Overall: a sense of space and objects in space---visualization!; a sense of analyzing by moving around, taking apart, and putting together; knowledge of how to think about size.

2) Students should develop a sense of the continuum of a mathematical idea: they should appreciate that some aspects of an idea can be shown in simple Kindergarten activities (such as cutting and pasting), and that other aspects appear in more sophisticated ways. The point here is that teachers must select activities that are appropriate for the age and skill of their students and that push their students toward deeper understanding of mathematical ideas. I doubt this would be possible without knowing about the continuum of the idea.

3) Students should learn to make sense of mathematics. To do this, I think students need to see math as connected to the real world. They also need to learn to explain why various things in math make sense (for example, why is the standard area formula for a triangle valid?). I believe that rigorous proofs will not help in this. What's needed is something less formal and less general, but still logical. I call this ``explaining why''. My criteria for writing good explanations are attached at the end (these are in volume I of the book). I ask my students to adhere to these when they write explanations. Do they always? No, but I keep pushing for it.

Recommended Teaching Practices

Given the above, what teaching practices might be most effective? My recommendations are these.

1) The students must be active in constructing their own understanding of the material. I recommend a brief lecture to introduce the ideas that will be explored, followed by a class activity in which students work together in groups, or in pairs, or the class as a whole works together, but during which there is time for individual reflection or exploration and for discussions with neighbors. Lately I've been finding the latter most effective. Group work sometimes degenerates into socializing. In any case, it is essential to follow up on the ideas and to ``wrap up'' the discussion by summarizing the developments of the activity.

2) Try to make the material as concrete as possible and to tie it to intuition and daily experience as much as possible. It took me a while to learn to do this, but now I drag all sorts of props to class, pretty much on a daily basis. Rather than just drawing a picture on the board and explaining it, or even having the students explain it, get them to cut it out of paper and explore it physically. Make math real! I really think this helps students' understanding. Of course, make sure you are also asking the students to visualize without the objects. I think you need some of both. Expect to use paper, scissors, tape, string, straws, tape measures, yard sticks, toothpicks, blocks, and the like. I know it feels silly working with stuff like this in a college class, but I think we ought to do what works, silly or not.

I occasionally find it helpful to show a computer demonstration in class. It is also nice for the students to use Geometer's Sketchpad to explore rigid motions in the plane. I was lucky to be able to use one of the calculus labs this semester, which happened to be open during my class time.

3) Try to be guided by the students' understanding of the material. If you do group work, or paired work, or if you can get good discussions going, then you will be able to see more or less where the students are in their understanding, and you can clear up misconceptions and try to push their thinking further. I think this is really hard, and it is one place where it helps to have taught the course several times, because then one is familiar with some common misconceptions or sticky points. The problem with these misconceptions and sticky points is that I don't know how to anticipate them other than from actual experience: things that seem like they would be obvious sometimes aren't, things that seem like they would be harder sometimes aren't. We have to learn to see things that are invisible to us!

4) After being part of the writing intensive program, I'm a big fan of asking students to write about a topic as a way of pulling together their thoughts. Writing can actually be a mechanism for learning. Attached below are some sample writing assignments. In addition, most of the problems in the book ask students to give explanations, or to discuss material. I think this is very good for students. But you have to keep pushing to get the students to write good explanations.

Of course: assign homework on a regular basis and give feedback. Even if you use a grader, it's definitely a good idea to look over the homework and see what's been understood and what hasn't. I like to grade a few papers myself before handing the stack to my grader. I also write a few notes to my grader. This way, the grader knows what I'm looking for, and I get a feel for what the students are doing well and what they need to work on.

5) Finally and significantly, never belittle the course material or the students' understanding of mathematics. Hard as it is, treat all ``why do we need to know this?'' questions seriously and respectfully. This is a legitimate question. Even though you may not know about elementary school teaching, you can refer to the carefully thought out recommendations of respected national organizations. The National Council of Teachers of Mathematics (NCTM) has developed principles and standards for the teaching of mathematics in schools (Pre-K -- 12); see http://standards.nctm.org/. The Conference Board of the Mathematical Sciences (CBMS), has developed recommendations for the preparations of teachers; see http://www.cbmsweb.org/MET_Document/index.htm I believe that the MATH 5001, 2, 3 sequence is very well aligned with these recommendations.

Characteristics of Good Explanations in Mathematics

1) The explanation is factually correct, or nearly so, with only minor flaws (for example, a minor mistake in a calculation).

2) The explanation addresses the specific question or problem that was posed. It is focused, detailed, and precise. There are no irrelevant or distracting points.

3) The explanation is clear, convincing, and logical. A clear and convincing explanation is characterized by the following:

a) The explanation could be used to teach another (college) student, possibly even one who is not in the class.

b) The explanation could be used to convince a skeptic.

c) The explanation does not require the reader to make a leap of faith.

d) Key points are emphasized.

e) If applicable, supporting pictures, diagrams, and/or equations are used appropriately and as needed.

f) The explanation is coherent.

g) Clear, complete sentences are used.

Writing assignments that I have used:

1) Hand in the following write to learn assignment, in which you will use writing as a way to deepen your understanding. Write 1 -- 3 pages on your thoughts or questions about the Pythagorean theorem. You may wish to address such topics as: What does the Pythagorean theorem mean? Do I understand the proof of the Pythagorean theorem that we worked on in class? What are the fundamental ideas that arise in the proof of the Pythagorean theorem? Regardless of what aspect you choose to write about, your writing must show development in your thinking to receive a check.

2) Many people find that writing about something helps them to understand it better. This write to learn assignment is intended to help you clarify and deepen your understanding of the material in sections 8.1 and 8.2. Write a 2 -- 4 page essay on some aspect of the relationship between the transformations we have studied (rotations, reflections, and translations) and the concept of symmetry. Your score will be based on the extent to which your writing is clear and accurate, and shows depth of thought and depth of analysis.

3) Write a 3 -- 4 page essay on the concept of area. Use your writing as a way to develop and consolidate your thinking. Address the following points in your essay:

What is area? What is the most basic information we need to know about area? How is area different from volume?

Does it only make sense to talk about areas of flat shapes in a plane, or is it possible to talk about the area of an object that takes up space? If so, give a specific example; determine its area, explaining your method.

Address one other significant point or question about the concept of area (of your choosing).

Your essay will be scored for its clarity, accuracy, and depth of analysis.