For the past three years, Math for America has invited Michael Serra to facilitate workshops in New York City. We invited Michael because many teachers were anxious about finding engaging and accessible ways to introduce geometric concepts, particularly that of geometric proof, to their students. From my own experience as a classroom teacher, I knew that Discovering Geometry often succeeded in engaging students in thinking about the ideas behind the mathematics where other books failed.
In our workshops, Michael has worked with nearly two-hundred teachers from schools all across New York City. Whether he is presenting on proof, problem solving, or games that encourage critical thinking, participants are consistently positive about the workshops. They appreciate the way that Michael engages them in doing problems, the way he is able to frame the activities in the context of a real classroom, and the perspective, humor and sensitivity he brings to the material as a teacher himself.
As one participant stated, “I liked that Michael Serra has been a teacher for many years and understands what it takes to make a classroom run well. I liked the chance to play with the materials and to learn some new ways of looking at math.”
Former Director, NY Program
Math for America
Michael Serra was my math teacher at George Washington High School in San Francisco. His imaginative and humorous teaching style gave me the tools to explore creative problem solving techniques. He inspired me to embrace challenges I thought were daunting and impossible. For problems I’d never seen before, I learned to look at them from many points of view, consider all the possibilities, and pursue them in succession until I found the answer. After taking his class, I learned sometimes what looked like the most unlikely approach is the one that worked. As the faculty sponsor for the Backgammon Club, Mr. Serra also taught me to factor in the probabilities of success and ramifications of my choices and to plan for long term strategy.
Exploration and the use of games and puzzles are a powerful means of teaching skills and are applicable to more than just classroom math problems. As a software engineer who works with companies from large corporations to small startups on internet-based enterprise applications, I've applied those logistical skills to successfully complete projects on time and within budget. I’ve used everything I learned to adapt and confidently clear obstacles and move projects forward.
Most math books are written with the steps for deriving the theorems (conclusions) spelled out and handed back to us so we can plug in numbers at the end of the lesson. Life doesn't come all neatly mapped out that way and by teaching me to observe and develop my own conclusions (math theorems for life), Mr. Serra gave me a great start to solving real life problems that don't have answers in any book.
For students who are open to humor, creative team work, and learning to solve problems on their own, Michael Serra's math resources are a valuable part of the life long adventure of learning.
Enterprise Software Engineer
Please be patient. Due to the graphics, some of them may take more time to download. Handouts are also listed under Presentations and Workshops by event.
In this section I’ve included a number of forms, outlines, and letters that I’ve used with my classes. At the beginning of the year I pass out the description of how the class will be conducted, including the class rules, social skills, homework, and grading policies. A letter is sent home to parents explaining all of this. During the first few days of the school term I pass out the semester schedule of class activities, homework assignments, and last year’s final exam to the students. The students know what material they are going to experience and when they are going to receive it. With last year’s final exam in front of them, they see their year end objective. Samples of these classroom handouts can be downloaded below. They are titled:
• Introductory Outline for our Geometry Course CMOutlineGeoCourse (PDF)
• Geometry Class Policies and Expectations CMClassPolExpect (PDF)
• Geometry Semester Calendar CMGeoSemesterCal (PDF)
• Homework Sheet Spring Grading Period S.3 CMHomeworkSheet (PDF)
A very important part of my geometry curriculum is the Student Notebook. The Student Notebook is a 3-ring binder with six sections: Notes/Homework, Chapter Summaries, Definitions, Investigations, Conjectures, Journal, and for my honors class, a seventh section, Postulates and Theorems. After a large number of geometry vocabulary terms have been developed for van Hiele level 1 understanding, I give a vocabulary qualifying exam (VQE) . When a student passes the VQE they have earned the right to use their notes for all tests for the rest of the year.
Grading notebooks is very time consuming and takes valuable energy away from the students. Whenever you can turn a task of yours into a learning experience for your student, it helps everyone. I create detailed rubrics for assessing the notebooks and the students do most of the notebook evaluations. Samples of these classroom handouts can be downloaded below. They are titled:
• Vocabulary Qualifying Exam CMVocabQualExam (PDF)
• Notebook Evaluation Sheet CMGeoNotebookEval (PDF)
I do not take homework home to grade. I turn the task into a learning experience for the students by having them do the lion’s share of the homework evaluation. I created a homework log sheet so students become responsible for monitoring the amount of homework attempted by members of their group. A sample of the Homework Log is available here. CMHomeworkLog (PDF)
 The vocabulary qualifying exam is a 48-item exam. On one side of the exam paper there are 24 terms. The student is to sketch and correctly label the figure that matches each term. On the other side of the exam paper there are 24 geometric figures. The student is to give the term that correctly matches with the figure.
Discovering Geometry began in my classroom over 35 years ago. During my first ten years of teaching I did not use a textbook, but created my own daily lesson plans and classroom management system. I believe students learn with greater depth of understanding when they are actively engaged in the process of discovering concepts and we should delay the introduction of proof in geometry until students are ready. Until Discovering Geometry, no textbook followed that philosophy.
I was also involved in a Research In Industry grant where I repeatedly heard that the skills valued in all working environments were the ability to express ideas verbally and in writing, and the ability to work as part of a team. I wanted my students to be engaged daily in doing mathematics and exchanging ideas in small cooperative groups.
The Discovering Geometry package includes the student textbook, online and teacher resources, supplemental materials, and the Discovering Geometry Video Series in english (and soon in spanish). Check out DG FAQ and contact us if you have any questions about subscription services.
The latest edition of Discovering Geometry includes new hands-on techniques, addresses Common Core State Standard's mathematical practices, and technologies that enhance my vision of the ideal geometry class. I send my heartfelt appreciation to the many teachers who contributed their feedback during classroom use. Their students and future students will help continue the evolution of Discovering Geometry. BUY NOW at the Kendall Hunt website.
Institutes are one to five full-day sessions. This format is ideal for full immersion into Inquiry-based Learning. At Michael’s institute, you will be engaged in exciting hands-on activities from start to finish. Participants, working in a cooperative-learning environment, will perform investigations, discuss results with group members, make conjectures, test them, and use them to solve interesting problems. In addition to the traditional geometry tools of compass, straightedge, ruler, and protractor, participants will have extensive experiences with Patty Paper and The Geometer’s Sketchpad. Some recent examples can be seen below.
Each institute is custom designed to meet your needs. Institutes can be set up at host school district sites or arrangements can be made for a full immersion geometry institute in San Francisco. Pricing is based on length of time, the number of visits, topics requiring special equipment and materials, and amount of travel involved. For more information please contact us at firstname.lastname@example.org.
One Day Institute
One Day Institutes are ideal for teachers/school districts that want one day of PD covering a few content strands or a quick overview of the Discovering Geometry textbook and its approach. If the client wishes to focus on only one topic, such as Cooperative Learning, then a One Day Institute may be sufficient. A One Day Institute provides approximately five to six hours of instruction.
Two Day Institute
Two Day Institutes are ideal for teachers/school districts that want a more in-depth look at Inquiry-based Learning or a more detailed look at the Discovering Geometry textbook. In addition to a closer look at Discovering Geometry, topics may include Cooperative Learning, Discovering Learning, and Classroom Management. A Two Day Institute provides about 10 to 12 hours of instruction.
Three Day Institute
A Three Day Institute provides about 15 to 18 hours of instruction.
Four Day Institute
A Four Day Institute provides about 20 to 24 hours of instruction.
Five Day Institute
A Five Day Institute provides about 25 to 30 hours of instruction.
"Cooperative and Inquiry based Learning, Proof in HS mathematics, New Investigations"
Presented by Michael Serra
Monday and Tuesday, May 17-18, 2010
Full day sessions
How well does DG align to the Common Core State Standards for Geometry?
Discovering Geometry is an inquiry based learning program and developed before the CCSS were mandated yet relevant today at every level. When DG was first introduced, traditional textbooks taught about proof without investigations. DG is unique because geometric proofs are not introduced until students are ready. Students experience geometry through a series of engaging hands on activities and develop their own conclusions to support proofs. Students work in small cooperative groups on projects using take out cartons, patty paper, and string eventually using compass and straightedge and software. They must defend their positions and are presented with investigations which may or may not be true. When reviewing the eight standards, compare them to our FAQ below. More information can be found here CommonCore_Correlation_DM.pdf.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
How does DG teach vocabulary?
In Discovering Geometry we begin with students discovering their vocabulary. Students perform investigations that lead not only to the definitions of the basic terms but do so with depth of understanding. For example, students observe examples of figures that are labeled trapezoids and selectively chosen figures that are not trapezoids. Then students discuss with their group members what makes for a trapezoid being a trapezoid. The group creates their definition and then tests and fine-tunes their definition against the examples and non-examples until confident of their definition. The groups then present their definitions to the rest of the class and the class reaches consensus. Each definition along with its illustration is then added to the student’s vocabulary list in their notebook.
How does DG support students that may be lacking in pre-requisite skills?
Careful slow development of proof
Algebra review lessons
1. 1. Condensed lessons for Make-up work (English/Spanish)
2. 2. Practice Your Skills with Answers
3. 3. DG Investigation videos (coming soon)
4. More Projects
How does DG incorporate manipulatives into Geometry?
Almost every lesson involves students working in groups performing geometric investigations with manipulatives. The investigations use a wide range of readily available manipulatives including compasses, straightedges, rulers, protractors, scissors, heavy stock paper, graph paper, patty paper, square dot paper, isometric dot paper, glue stick, tape, string, wooden or plastic cubes, mirrors, and pattern blocks, to name the most common.
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We recommend commercially available geometric solids for discovering volume formulas and surface area formulas for prisms, pyramids, cylinders, cones, and spheres. Students typically use graphing calculators and geometry software such as The Geometer’s Sketchpad for investigations and extra explorations. Projects and Explorations in the text and supplementary materials also encourage the use of hands-on materials. Students build arches out of Chinese take-out cartons, build kaleidoscopes out of paper towel tubes, and build clinometers and hypsometers out of cardboard and straws.
How does DG move students from concrete models to abstract ideas?
Moving students from the concrete to the abstract is at the heart of Discovering Geometry. We begin with students discovering all the basic vocabulary and then all the basic properties of geometry through hands-on investigations. Students work with the tools of geometry, build models, perform geometric investigations, then make and test conjectures about their experiences. As they become more and more experienced with the terms and concepts of geometry, gaining confidence, we gradually begin to ask them to explain what they did and why they think what they discovered is true. See comments on van Hiele levels and development of proof here.
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How does DG teach construction with a straight edge and compass?
Right from the start (Chapter 0 Geometric Art) students use the ruler and compass to create geometric art. Then on an almost daily basis in Chapters One and Two students use the ruler, protractor, and compass to perform investigations to discover geometric properties (vertical angles and the properties of parallel lines). In Chapter Three we introduce the classic Euclidean compass and straightedge constructions. This is done inductively with students discovering the construction methods rather than the constructions being a set of rules or steps to memorize. The Euclidean constructions are blended with paper folding constructions (patty paper geometry) to reinforce the properties being discovered. We also recommend that teachers use The Geometer’s Sketchpad at this time as a third way to reinforce the concepts they are learning. The use of the compass and straightedge as tools for construction exercises as well as tools for discovery continue throughout the remainder of the text.
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What is DG’s philosophy on teaching students to write proofs?
Discovering Geometry's primary focus is getting students to work together to discover and discuss geometry concepts in a student-centered environment. In Discovering Geometry students are guided to discover on their own not only the basic vocabulary of geometry but even the construction techniques they are going to use to discover their own geometric properties.
At the same time students are inductively discovering geometric properties we are carefully developing a full breadth of reasoning skills, both inductive and deductive. There is a heavy emphasis in Discovering Geometry on using higher order thinking as opposed to memorization and drill. In this investigative approach to geometry, reasoning, concepts, and skills are developed with two basic principles in the background – the research of the van Hieles (van Hiele Model) and the work of Michael de Villiers and others on purposes of proof.
At the beginning of the course we assume a low level of geometric reasoning in students. (van Hiele Research suggests that over 70% of all high school students enter geometry at level 0 or 1 on the van Hiele scale.) Through this concrete hands-on discovery approach we attempt to move them up the stages to higher levels of reasoning over the course of the term.
At the same time we recognize that one of the traditional purposes of a proof in geometry, to convince someone that a particular property is true, is not the best use of proofing at the early stages. With most beginning geometry students, conviction comes upon seeing and doing not through proof. The proof does not convince. In fact it often creates more doubt. (See Rethinking Proof by Michael de Villiers.) What we see as a better purpose for proof, in the early stages, is proof as a means of explaining why something they discovered through hands-on investigations is indeed true. Proof as a means of explanation is our initial and primary focus. First students are asked to explain their reasoning to their group members orally then in short paragraphs. In addition to the more typical exercises where students apply the concepts discovered, we also utilize three unique types of exercises that ask students to explain their reasoning.
One type of exercise we call Angle Chase (In DG3 see page 188 exercise #14, page 224 exercise #20, and page 292 Exercise #25, in DG4 see page 190 exercise #14, page 226 exercise #20, and page 297 Exercise #25). In these exercises, which we recommend that students work on in pairs in class, students take turns finding the measure of an angle and then explaining why that is the measure.
A second novel type of exercise that provides an opportunity for students to explain their reasoning (develop proof skills) is the “What’s Wrong With This Picture?” exercises (In DG3 see page 251 exercise #26, page 323 exercise #16, and page 436 exercise #21, In DG4 see page 253 exercise #26, page 328 exercise #16, and page 452 exercise #21). In these exercises students are asked to explain why the figure is incorrect.
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In the third type of exercise that helps students develop their deductive reasoning we simply ask students to explain why something is true (In DG3 see exercises page 436 exercise #20, page 466 exercise #18, page 472 exercise #22, (In DG4 see exercises page 452 exercise #20, page 482 exercise #18, page 488 exercise #22). In the early instances of these exercises we ask them to explain why something is true for a particular case (numbers are provided as in exercise #20 page 436 in DG3) later we begin to ask students to explain their reasoning in more general cases (letters rather than numbers are provided as in DG3 exercise #22 page 472, DG4 exercise #22 page 488).
Proof will also be developed as a means of communication and discovery as student’s progress through Discovering Geometry. Eventually as students move to more sophisticated reasoning abilities we introduce geometry as a mathematical system in the final chapter. Systematization then becomes the purpose of proof in Chapter 13.
In Discovering Geometry we develop reasoning skills in four general ways:
1. First, when students perform investigations and make conjectures in the form of formulas (e.g., a formula for the number of diagonals of an n-gon or a formula for the sum of the measures of the interior angles of an n-gon) students must use inductive reasoning while doing geometry. This occurs throughout Discovering Geometry. Chapter Two: Reasoning In Geometry introduces this type of reasoning skill but this type of reasoning continues throughout the text. It is the type of reasoning mathematicians and scientists primarily do. It is the type of reasoning essential to an investigative discovery approach to learning. It is a critical life skill.
2. The second general way in which reasoning is developed in Discovering Geometry is in the development of logical arguments and proof, or deductive reasoning. This development is based on classroom experiences and research, workshops on reasoning and proof, and particularly the research of the van Hieles, and de Villiers.
• Deductive reasoning is also introduced in Chapter Two: Reasoning In Geometry and it begins informally with exercises that ask students to “explain why.” This occurs in numerous examples, exercises, and special activities mentioned above and in sections called “Take Another Look.”
• Examples of algebraic, paragraph, and flow chart proofs are modeled early. The concepts of conditional statements, converses and counterexamples are emphasized and reinforced. (It is recommended however that students in a regular geometry course not be graded on their proofing abilities at this early stage of geometric reasoning.)
• By Chapter Four, Discovering and Proving Triangle Properties, students are asked to follow the reasoning in paragraph proofs and flow chart proofs. They are eventually asked to create their own flow chart proofs, testing the level of their reasoning.
As students move through the traditional synthetic geometry they move to higher levels of geometric reasoning and critical thinking. Their “explaining why” becomes more fully developed. They are expected to present more detailed paragraph and flow chart proofs. At the same time students are developing a firmer grasp of the content of geometry.
• To mark a transition from discovering geometric properties through inductive reasoning and informally explaining why they think their conjectures are true, more formal deductive reasoning is introduced through a series of three Logic Explorations. These explorations begin with the logical reasoning found in Sherlock Holmes stories. This is followed with four properties of symbolic logic: Modus Ponens, Modus Tollens, Law of Syllogism, and Law of the Contrapositive. Students then create direct, conditional and indirect proofs using these logic properties.
• Finally students develop geometry as a mathematical system (de Villiers Systematization). After establishing definitions, properties of algebra and properties of congruence, and the postulates of geometry, students begin to use these premises to establish theorems and to see the connections between groups of theorems.
3. The third general way in which higher order thinking is developed in Discovering Geometry is in solving word problems. After students have completed their investigations and made conjectures, they are asked to apply their new properties to not only real applications but also in novel situations. Many problems in the exercise sets come from recreational mathematics or are modeled after SAT type questions and mathematics competitions.
4. Finally, a fourth way in which reasoning is fine-tuned in Discovering Geometry is in the puzzles called Improving Algebra Skills, Improving Reasoning Skills, and Improving Visual Thinking Skills. There are 14 Improving Algebra Skills, 22 Improving Reasoning Skills, and 44 Improving Visual Thinking Skills puzzles that are spread throughout the text. They occur after each lesson in every chapter. These puzzles employ a wide variety of higher order thinking skill and problem solving skills. Students are often found working these puzzles without them even being assigned!
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