Classroom Instruction that Works: Research-Based Strategies for Increasing Student Achievement by Robert J. Marzano, Debra J. Pickering, and Jane E. Pollock identifies 9 categories of instructional strategies that have been shown through research to be effective in the classroom. They base their conclusions on meta-analyses done by researchers at Mid-continent Research for Education and Learning. It is important to realize that there is much overlap in the strategies and the good techniques in one strategy are often used to advantage to enhance the learning effectiveness of other strategies. Below we list the strategies employed in this web activity. The strategies are listed in order of effectiveness as rated by the average effect size (achievement in standard deviation units).

Identifying Similarities and Differences: The strategy of Indentifying Similarities and Differences is at the heart of this activity. In order to create theorems, students need to identify similarities and differences as they change the values of the properties of the parallelogram (for example the value of side1). You can download a table that I created using MSWord to help your students organize their thoughts. You may wish to modify the table, which has an example filled in for one row. I would recommend that students start by changing only one property at a time and grouping their observations for the changes in just that one property. It is important to stress to your students that they are using a powerful technique of identifying similarities and differences to help them create theorems. As an example of using this technique to create a theorem, imagine that a group of students decides to change only side1. They note down the similarities and differences. In each case they still have a parallelogram with the same angles. However, each time the area changes. Looking at their table, they find if side1 doubles, the area doubles. If it triples, the area triples, etc. Then they might create the theorem that if you change side1, the area changes in direct proportion to the change in side 1. Then they might generalize by trying side2 and they would find that their theorem holds for either side1 or side2. They will be exercising their brain muscles.

Summarizing and Notetaking: Students should take notes as they make their observations. If they use the table described above, they will organise their observations. However, an effective note taking structure is to use the left side for notes in text, perhaps an outline, and the right hand sides for drawings and other graphical aids that help organize and clarify their observations. Finally, a summary can be written along the bottom as the groundwork that holds the structure together (download an MSWord version of the note taking structure). This method of notetaking would be effective in this activity if students put all their observations for exploring the effect of changing one property on the same sheet or group of sheets. The summary could be a first draft of their theorem. For the most effective use of this technique, have students discuss and compare their notes and summaries.

Reinforcing Effort and Providing Recognition: Reinforce students positively as they explore, make progress on organizing and recording their observations and when they create a theorem. Have the students present their theorems as they find them so you and the class can recognize their good work. The very best reinforcement and recognition comes from parents, teachers, and other students.

Homework and Practice: You should assign reading about parallelograms and theorems as homework. You could assign worksheets based on the theorems the students create to reinforce the theorems or you could assign parallelogram geometry problems that use the NASA CONNECT Theorem Challenge tools. The students can get plenty of practice creating theorems because there are many different possible ways to state each theorem and several different possible theorems. Many of the theorems your students find can be generalized so one theorem will do where several stood before. That is fine because through practice, they will notice these generalizations and have a higher level learning experience. When they are finished with the activity, assign as homework a brief paper summarizing their efforts and conclusions. High achieving students could be assigned the task of proving their theorems (see Extensions).

Nonlinguistic Representations: This activity is replete with nonlinguistic representations such as graphics and animations. Students will learn more from nonlinguistic actions as they draw parallelograms, change properties and measure areas graphically. If the students deconstruct the Squeak project to find out how it works and construct a modified project using the graphical tiles, they will be working with nonlinguistic representations of ideas (programming commands) and mathematics (arithmetic). The natural integration of these representations enhances the learning experience.

Cooperative Learning: Setting up cooperative learning groups is the recommended way to maximize student learning in this activity. Five defining elements of cooperative learning are: positive interdependence, face-to-face promotive interaction, individual and group accountability, interpersonal and small group skills, and group processing. Reciprocal Teaching is a research-based strategy that can be used effectively with cooperative groups. The four phases are summarizing, questioning, clarifying, and predicting. If you assign groups to work on creating theorems, they can each record different changes of the same property so they have a larger base of cases to draw on to organize their observations into a theorem. Then through discussion the team members can summarize, question and clarify. Students could be asked to predict theorems, but that would be much harder.

Setting Objectives and Providing Feedback: Objectives that are set shouldn't be too narrowly focused or learners tend to miss too much related material. For this activity a good objective would be to understand the process of creating a theorem by consolidating observations (inductive reasoning). The process should be emphasized more than whether they create true theorems. Althought it certainly would be nice if they create true theorems so they can get more positive feedback. Feedback on exams or projects has been shown to enhance learning and the best form is an explanation as opposed to just being given the correct answer. Students could offer feedback to other students through discussions explaining their theorems. If you give your students a test on the activity, research shows that the optimal time is one day after exposure to the material.

Generating and Testing Hypotheses: Both inductive (abstracting a principle from a set of specific observations) and deductive (using a principle to predict a specific result) reasoning can be used to advantage to promote learning. Deductive reasoning activities have been shown to be more effective, but it depends on the circumstance. The division into inductive and deductive is often blurred and the concepts are most valuable when considered as two extremes of reasoning. In this case the inductive reasoning process is made very concrete by implementing it with a highly visual tool like Squeak in hopes that it will be more effective. To close the loop you should have the students use deductive reasoning with their theorems to predict what will happen when certain changes are made in the appropriate values of the parallelogram properties. This could be homework or an in class group assignment. It has also been shown valuable for students to explain their hypotheses and predictions, which they could do as homework or in class or both if time permits.

Cues, Questions, and Advance Organizers: These strategies all take advantage of students' prior knowledge and are good ways to start a lesson. As you give cues and ask questions, keep in mind that higher-order questions are more effective and students are more interested in things they already know something about. For example, a good starting question would be "What is a parallelogram?" Remember that it is important to wait after asking your questions to give the students time to collect their thoughts before they respond - you will have a much better discussion. Advance organizers are a way of giving your students a brief "heads up" before starting a topic - they aren't outlines or summaries. Research shows the most effective advanced organizers are expository, followed closely by skimming. In this case a story involving Norbert and Zot as theorem hunters, where you introduce the concept of a theorem in the context of geometry, would be in order.