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37 people found this helpful

ByBrandon J. Porteron January 28, 2011

Those above with negative things to say are only one viewpoint. How about a review from someone who has used this text (2nd, 3rd, and now 4th edition)?

I LOVE this geometry text. As a high school math teacher, I have used Serra's text for 10 years now, with great results. It teaches students how to think inductively, which is a greatly lacking skill among high school students. It also helps students to develop into independent thinkers and great problem solvers. In addition, it teaches students how to work with others, a skill that anyone who works with people can attest is lacking. It also develops students into better readers, as they have to read written instructions on a daily basis to succeed in the course. All of these are great benefits of the book. BUT...

If the facilitator/teacher doesn't use proper constructivist techniques in the classroom then this approach will fail. The teacher must be on his/her toes, must enforce cooperative problem solving norms in the classroom, must have rules for group work, social skills that students are to adhere to, etc... I suspect that those who have taught using this text and have had a hard time with it have experienced such because they aren't using it correctly.

This text will revolutionize your geometry class. It will transform your class from a traditional, 150 year old teacher-centered classroom to progressive, modern, discovery learning, student-centered classroom. It places the responsibility of the learning on the student--where it should be.

So what if there's no glossary? It's misleading of people to say that this is a bad text because there's no glossary. This is, once again, a product of using the wrong method with this text. Students discover and write their own definitions for terms, and keep a notebook to add definitions to each day. They also investigate geometric relationships and add conjectures (theorems) to their notebooks daily. If the students don't know terms or theorems, it's because they aren't doing the work (again, responsibility is theirs).

So what if there are no answers to the problems in the back of the book? This allows students time to explore and apply and critically think about problems without the crutch of the answer (most math books also don't offer solutions to problems, only answers, so they're not that helpful anyway). I make a copy of the answer key for each group in my class, and they sit as a group and discuss homework problems at the beginning of each class, so they have a chance to peer-tutor every day--another great skill to have.

Again, the nay-sayers aren't being fair. This is a great text, and certainly adds interest and fun to what is otherwise an often poorly taught, boring class.

One thing I love about this text is that it shows students that geometry is a mathematical system, and as the instructor you can facilitate some great discussions about the validity of Euclidean geometry being based on Euclid's assumptions. Obviously, they're not fool-proof as there are other types of geometry whose authors disagreed with some of Euclid's propositions.

Lastly, 99% of your students will not grow up to become mathematicians or engineers. Why do they have to engage in a class that forces proofs too early, asks them to think and act like mathematicians when they are not, and is often downright boring? Who cares if this text waits too long to introduce proof? How many of you did proof anywhere besides geometry? (By the way, the 4th edition has a proof strand throughout, and slowly allows students to get used to the idea of what a proof is and how to develop one.)

Traditional texts are too difficult for most students. Research suggests that more than 90% of students who take HS geometry never really understand proofs. Again, the nay-sayers need to read up on the recommendations from NCTM. This text is exactly what organizations like the NCTM recommend and endorse. They recommend more investigative, discovery learning because they realize that in the long run it is better for students and helps them to become better problem solvers. This text does exactly that.

I LOVE this geometry text. As a high school math teacher, I have used Serra's text for 10 years now, with great results. It teaches students how to think inductively, which is a greatly lacking skill among high school students. It also helps students to develop into independent thinkers and great problem solvers. In addition, it teaches students how to work with others, a skill that anyone who works with people can attest is lacking. It also develops students into better readers, as they have to read written instructions on a daily basis to succeed in the course. All of these are great benefits of the book. BUT...

If the facilitator/teacher doesn't use proper constructivist techniques in the classroom then this approach will fail. The teacher must be on his/her toes, must enforce cooperative problem solving norms in the classroom, must have rules for group work, social skills that students are to adhere to, etc... I suspect that those who have taught using this text and have had a hard time with it have experienced such because they aren't using it correctly.

This text will revolutionize your geometry class. It will transform your class from a traditional, 150 year old teacher-centered classroom to progressive, modern, discovery learning, student-centered classroom. It places the responsibility of the learning on the student--where it should be.

So what if there's no glossary? It's misleading of people to say that this is a bad text because there's no glossary. This is, once again, a product of using the wrong method with this text. Students discover and write their own definitions for terms, and keep a notebook to add definitions to each day. They also investigate geometric relationships and add conjectures (theorems) to their notebooks daily. If the students don't know terms or theorems, it's because they aren't doing the work (again, responsibility is theirs).

So what if there are no answers to the problems in the back of the book? This allows students time to explore and apply and critically think about problems without the crutch of the answer (most math books also don't offer solutions to problems, only answers, so they're not that helpful anyway). I make a copy of the answer key for each group in my class, and they sit as a group and discuss homework problems at the beginning of each class, so they have a chance to peer-tutor every day--another great skill to have.

Again, the nay-sayers aren't being fair. This is a great text, and certainly adds interest and fun to what is otherwise an often poorly taught, boring class.

One thing I love about this text is that it shows students that geometry is a mathematical system, and as the instructor you can facilitate some great discussions about the validity of Euclidean geometry being based on Euclid's assumptions. Obviously, they're not fool-proof as there are other types of geometry whose authors disagreed with some of Euclid's propositions.

Lastly, 99% of your students will not grow up to become mathematicians or engineers. Why do they have to engage in a class that forces proofs too early, asks them to think and act like mathematicians when they are not, and is often downright boring? Who cares if this text waits too long to introduce proof? How many of you did proof anywhere besides geometry? (By the way, the 4th edition has a proof strand throughout, and slowly allows students to get used to the idea of what a proof is and how to develop one.)

Traditional texts are too difficult for most students. Research suggests that more than 90% of students who take HS geometry never really understand proofs. Again, the nay-sayers need to read up on the recommendations from NCTM. This text is exactly what organizations like the NCTM recommend and endorse. They recommend more investigative, discovery learning because they realize that in the long run it is better for students and helps them to become better problem solvers. This text does exactly that.

69 people found this helpful

Byalacarteon June 28, 2008

I had the misfortune of learning geometry from this textbook as a student, and now I have the misforture of teaching from it. I remember hating math as a high school student, and textbooks like these were the culprit. In high school, math was always presented as a set of problem-solving techniques that I had to memorize and apply. I was generally able to solve whatever problems came my way, but it always seemed like a trivial and pointless exercise. Luckily, I had some great college professors who made me realize that math was much more than memorizing algorithms, but a comprehensive logical system grounded in deductive reasoning.

Geometry is the only math course in which rigorous deductive reasoning can be made accessible to high school students -- and not surprisingly, it was the first area of mathematics to be axiomatized (by Euclid). Unlike algebra or calculus, almost all of the theorems and formulas in geometry can be systematically obtained from postulates in a way that is intelligible to high school students; contrast this with algebra, where it would not be reasonable to expect students to understand a proof of Cramer's Rule or the Binomial Theorem. The fact that geometry introduces students to a different, mathematical way of thinking is the only justification for maintaining geometry as a standalone math course, rather than integrating it into algebra courses. Otherwise, the "facts" of geometry are nothing remarkable in themselves. So what if opposite sides of a parallelogram are congruent? It wouldn't be too difficult to teach students this "fact" in an algebra class when they're learning about slopes of parallel lines. But what's important is that students understand and see how this fact derives systematically from already known facts.

What does all this have to do with the book at hand? "Discovering Geometry" reduces geometry to the same collection of facts and algorithms that students have been doing in every math class since elementary school. While the problems that Michael Serra devises are occasionally interesting and even clever, he completely misses the point of geometry -- to understand WHY those "facts" are true.

Unlike many critics of this book, I do not have any inherent qualms with the investigative approach to learning geometry. Investigation plays a central role in mathematics, and I applaud the author for giving inductive reasoning its fair shake in this book. But investigation has become more of an ideology than a pedagogical tool in this book. Even my weakest students groan at having to do some of the investigations, whose results they deem obvious. There are simply too many unnecessary investigations, many of which exist only to aggrandize the author's educational philosophy.

As a student, I used the second edition of this book. The author has clearly made significant improvements for the third edition, but there are still serious pedagogical flaws. While Chapter 13 is a valiant attempt at introducing students to the deductive method of geometry, it is too little, too late. Geometry classes rarely reach the last chapter, and separating the proofs from the theorems themselves feels artificial and contrived. The author makes another questionable pedagogical decision to cover area and volume into nonconsecutive chapters, Ch. 8 and 10 -- just so he can prove the Pythagorean Theorem using area in Ch. 9. But if he would only introduce the concept of similarity (Ch. 11) before the Pythagorean Theorem, he would be able to prove the Pythagorean Theorem using similar triangles in a much more elegant and motivated way.

The unorthodox ordering of topics to which I have previously alluded creates problems for even the author. There are many practice problems that require concepts or techniques from later chapters. For example, students are asked to construct a square in Chapter 3 given a diagonal, before either the properties of quadrilaterals (Ch. 5) -- or even the properties of triangles (Ch. 4) -- have been introduced! How students are supposed to "guess" that the diagonal of a square bisects the angles -- I do not know. Furthermore, the first proof in the text is a paragraph proof that the perpendicular bisectors of a triangle are concurrent. I can only imagine the horrified looks on the faces of Serra's students. And these are supposedly students who are having too much trouble with traditional two-column proofs!

There are outright mistakes in the textbook as well besides the usual typos. On page 333, Serra defines an irrational number as a number whose "decimal form never ends" and a transcendental number as a number whose "pattern of digits does not repeat." So according to his (incorrect) definition, 1/3 would be an irrational number, and sqrt(2) would be a transcendental number -- the former false for obvious reasons, the latter false because sqrt(2) satisfies the polynomial equation x^2 - 2 = 0. Moreover, this is something that a reasonably bright high schooler would know -- much less an ostensibly expert math teacher!

In his manifesto "Tracing Proof in Discovering Geometry," Serra attacks two-column proofs, saying that "so many students fail to master two-column proofs that some teachers are skeptical of claims that all students can learn geometry." While I agree that two-column proofs misrepresent mathematics and can proofs unnecessarily complicated, I'll gladly take them over "Discovering Geometry" any day.

Addendum: Some commenters have asked about alternative geometry textbooks. While I (sadly) have yet to find a textbook that lives up to the promise, the study guide E-Z Geometry (I used an older edition which was called "Geometry the Easy Way") does an excellent job of presenting the material in a coherent, logical fashion, replete with proofs, and comes with a collection of excellent exercises. It's also cheap, although I'm not sure how it would fare as a classroom textbook, since all solutions are at the back. I have also heard excellent things about Harold Jacobs' Geometry: Seeing, Doing, Understanding, although I have not had the fortune to look over the book myself.

Geometry is the only math course in which rigorous deductive reasoning can be made accessible to high school students -- and not surprisingly, it was the first area of mathematics to be axiomatized (by Euclid). Unlike algebra or calculus, almost all of the theorems and formulas in geometry can be systematically obtained from postulates in a way that is intelligible to high school students; contrast this with algebra, where it would not be reasonable to expect students to understand a proof of Cramer's Rule or the Binomial Theorem. The fact that geometry introduces students to a different, mathematical way of thinking is the only justification for maintaining geometry as a standalone math course, rather than integrating it into algebra courses. Otherwise, the "facts" of geometry are nothing remarkable in themselves. So what if opposite sides of a parallelogram are congruent? It wouldn't be too difficult to teach students this "fact" in an algebra class when they're learning about slopes of parallel lines. But what's important is that students understand and see how this fact derives systematically from already known facts.

What does all this have to do with the book at hand? "Discovering Geometry" reduces geometry to the same collection of facts and algorithms that students have been doing in every math class since elementary school. While the problems that Michael Serra devises are occasionally interesting and even clever, he completely misses the point of geometry -- to understand WHY those "facts" are true.

Unlike many critics of this book, I do not have any inherent qualms with the investigative approach to learning geometry. Investigation plays a central role in mathematics, and I applaud the author for giving inductive reasoning its fair shake in this book. But investigation has become more of an ideology than a pedagogical tool in this book. Even my weakest students groan at having to do some of the investigations, whose results they deem obvious. There are simply too many unnecessary investigations, many of which exist only to aggrandize the author's educational philosophy.

As a student, I used the second edition of this book. The author has clearly made significant improvements for the third edition, but there are still serious pedagogical flaws. While Chapter 13 is a valiant attempt at introducing students to the deductive method of geometry, it is too little, too late. Geometry classes rarely reach the last chapter, and separating the proofs from the theorems themselves feels artificial and contrived. The author makes another questionable pedagogical decision to cover area and volume into nonconsecutive chapters, Ch. 8 and 10 -- just so he can prove the Pythagorean Theorem using area in Ch. 9. But if he would only introduce the concept of similarity (Ch. 11) before the Pythagorean Theorem, he would be able to prove the Pythagorean Theorem using similar triangles in a much more elegant and motivated way.

The unorthodox ordering of topics to which I have previously alluded creates problems for even the author. There are many practice problems that require concepts or techniques from later chapters. For example, students are asked to construct a square in Chapter 3 given a diagonal, before either the properties of quadrilaterals (Ch. 5) -- or even the properties of triangles (Ch. 4) -- have been introduced! How students are supposed to "guess" that the diagonal of a square bisects the angles -- I do not know. Furthermore, the first proof in the text is a paragraph proof that the perpendicular bisectors of a triangle are concurrent. I can only imagine the horrified looks on the faces of Serra's students. And these are supposedly students who are having too much trouble with traditional two-column proofs!

There are outright mistakes in the textbook as well besides the usual typos. On page 333, Serra defines an irrational number as a number whose "decimal form never ends" and a transcendental number as a number whose "pattern of digits does not repeat." So according to his (incorrect) definition, 1/3 would be an irrational number, and sqrt(2) would be a transcendental number -- the former false for obvious reasons, the latter false because sqrt(2) satisfies the polynomial equation x^2 - 2 = 0. Moreover, this is something that a reasonably bright high schooler would know -- much less an ostensibly expert math teacher!

In his manifesto "Tracing Proof in Discovering Geometry," Serra attacks two-column proofs, saying that "so many students fail to master two-column proofs that some teachers are skeptical of claims that all students can learn geometry." While I agree that two-column proofs misrepresent mathematics and can proofs unnecessarily complicated, I'll gladly take them over "Discovering Geometry" any day.

Addendum: Some commenters have asked about alternative geometry textbooks. While I (sadly) have yet to find a textbook that lives up to the promise, the study guide E-Z Geometry (I used an older edition which was called "Geometry the Easy Way") does an excellent job of presenting the material in a coherent, logical fashion, replete with proofs, and comes with a collection of excellent exercises. It's also cheap, although I'm not sure how it would fare as a classroom textbook, since all solutions are at the back. I have also heard excellent things about Harold Jacobs' Geometry: Seeing, Doing, Understanding, although I have not had the fortune to look over the book myself.

ByBrandon J. Porteron January 28, 2011

Those above with negative things to say are only one viewpoint. How about a review from someone who has used this text (2nd, 3rd, and now 4th edition)?

I LOVE this geometry text. As a high school math teacher, I have used Serra's text for 10 years now, with great results. It teaches students how to think inductively, which is a greatly lacking skill among high school students. It also helps students to develop into independent thinkers and great problem solvers. In addition, it teaches students how to work with others, a skill that anyone who works with people can attest is lacking. It also develops students into better readers, as they have to read written instructions on a daily basis to succeed in the course. All of these are great benefits of the book. BUT...

If the facilitator/teacher doesn't use proper constructivist techniques in the classroom then this approach will fail. The teacher must be on his/her toes, must enforce cooperative problem solving norms in the classroom, must have rules for group work, social skills that students are to adhere to, etc... I suspect that those who have taught using this text and have had a hard time with it have experienced such because they aren't using it correctly.

This text will revolutionize your geometry class. It will transform your class from a traditional, 150 year old teacher-centered classroom to progressive, modern, discovery learning, student-centered classroom. It places the responsibility of the learning on the student--where it should be.

So what if there's no glossary? It's misleading of people to say that this is a bad text because there's no glossary. This is, once again, a product of using the wrong method with this text. Students discover and write their own definitions for terms, and keep a notebook to add definitions to each day. They also investigate geometric relationships and add conjectures (theorems) to their notebooks daily. If the students don't know terms or theorems, it's because they aren't doing the work (again, responsibility is theirs).

So what if there are no answers to the problems in the back of the book? This allows students time to explore and apply and critically think about problems without the crutch of the answer (most math books also don't offer solutions to problems, only answers, so they're not that helpful anyway). I make a copy of the answer key for each group in my class, and they sit as a group and discuss homework problems at the beginning of each class, so they have a chance to peer-tutor every day--another great skill to have.

Again, the nay-sayers aren't being fair. This is a great text, and certainly adds interest and fun to what is otherwise an often poorly taught, boring class.

One thing I love about this text is that it shows students that geometry is a mathematical system, and as the instructor you can facilitate some great discussions about the validity of Euclidean geometry being based on Euclid's assumptions. Obviously, they're not fool-proof as there are other types of geometry whose authors disagreed with some of Euclid's propositions.

Lastly, 99% of your students will not grow up to become mathematicians or engineers. Why do they have to engage in a class that forces proofs too early, asks them to think and act like mathematicians when they are not, and is often downright boring? Who cares if this text waits too long to introduce proof? How many of you did proof anywhere besides geometry? (By the way, the 4th edition has a proof strand throughout, and slowly allows students to get used to the idea of what a proof is and how to develop one.)

Traditional texts are too difficult for most students. Research suggests that more than 90% of students who take HS geometry never really understand proofs. Again, the nay-sayers need to read up on the recommendations from NCTM. This text is exactly what organizations like the NCTM recommend and endorse. They recommend more investigative, discovery learning because they realize that in the long run it is better for students and helps them to become better problem solvers. This text does exactly that.

I LOVE this geometry text. As a high school math teacher, I have used Serra's text for 10 years now, with great results. It teaches students how to think inductively, which is a greatly lacking skill among high school students. It also helps students to develop into independent thinkers and great problem solvers. In addition, it teaches students how to work with others, a skill that anyone who works with people can attest is lacking. It also develops students into better readers, as they have to read written instructions on a daily basis to succeed in the course. All of these are great benefits of the book. BUT...

If the facilitator/teacher doesn't use proper constructivist techniques in the classroom then this approach will fail. The teacher must be on his/her toes, must enforce cooperative problem solving norms in the classroom, must have rules for group work, social skills that students are to adhere to, etc... I suspect that those who have taught using this text and have had a hard time with it have experienced such because they aren't using it correctly.

This text will revolutionize your geometry class. It will transform your class from a traditional, 150 year old teacher-centered classroom to progressive, modern, discovery learning, student-centered classroom. It places the responsibility of the learning on the student--where it should be.

So what if there's no glossary? It's misleading of people to say that this is a bad text because there's no glossary. This is, once again, a product of using the wrong method with this text. Students discover and write their own definitions for terms, and keep a notebook to add definitions to each day. They also investigate geometric relationships and add conjectures (theorems) to their notebooks daily. If the students don't know terms or theorems, it's because they aren't doing the work (again, responsibility is theirs).

So what if there are no answers to the problems in the back of the book? This allows students time to explore and apply and critically think about problems without the crutch of the answer (most math books also don't offer solutions to problems, only answers, so they're not that helpful anyway). I make a copy of the answer key for each group in my class, and they sit as a group and discuss homework problems at the beginning of each class, so they have a chance to peer-tutor every day--another great skill to have.

Again, the nay-sayers aren't being fair. This is a great text, and certainly adds interest and fun to what is otherwise an often poorly taught, boring class.

One thing I love about this text is that it shows students that geometry is a mathematical system, and as the instructor you can facilitate some great discussions about the validity of Euclidean geometry being based on Euclid's assumptions. Obviously, they're not fool-proof as there are other types of geometry whose authors disagreed with some of Euclid's propositions.

Lastly, 99% of your students will not grow up to become mathematicians or engineers. Why do they have to engage in a class that forces proofs too early, asks them to think and act like mathematicians when they are not, and is often downright boring? Who cares if this text waits too long to introduce proof? How many of you did proof anywhere besides geometry? (By the way, the 4th edition has a proof strand throughout, and slowly allows students to get used to the idea of what a proof is and how to develop one.)

Traditional texts are too difficult for most students. Research suggests that more than 90% of students who take HS geometry never really understand proofs. Again, the nay-sayers need to read up on the recommendations from NCTM. This text is exactly what organizations like the NCTM recommend and endorse. They recommend more investigative, discovery learning because they realize that in the long run it is better for students and helps them to become better problem solvers. This text does exactly that.

Byalacarteon June 28, 2008

I had the misfortune of learning geometry from this textbook as a student, and now I have the misforture of teaching from it. I remember hating math as a high school student, and textbooks like these were the culprit. In high school, math was always presented as a set of problem-solving techniques that I had to memorize and apply. I was generally able to solve whatever problems came my way, but it always seemed like a trivial and pointless exercise. Luckily, I had some great college professors who made me realize that math was much more than memorizing algorithms, but a comprehensive logical system grounded in deductive reasoning.

Geometry is the only math course in which rigorous deductive reasoning can be made accessible to high school students -- and not surprisingly, it was the first area of mathematics to be axiomatized (by Euclid). Unlike algebra or calculus, almost all of the theorems and formulas in geometry can be systematically obtained from postulates in a way that is intelligible to high school students; contrast this with algebra, where it would not be reasonable to expect students to understand a proof of Cramer's Rule or the Binomial Theorem. The fact that geometry introduces students to a different, mathematical way of thinking is the only justification for maintaining geometry as a standalone math course, rather than integrating it into algebra courses. Otherwise, the "facts" of geometry are nothing remarkable in themselves. So what if opposite sides of a parallelogram are congruent? It wouldn't be too difficult to teach students this "fact" in an algebra class when they're learning about slopes of parallel lines. But what's important is that students understand and see how this fact derives systematically from already known facts.

What does all this have to do with the book at hand? "Discovering Geometry" reduces geometry to the same collection of facts and algorithms that students have been doing in every math class since elementary school. While the problems that Michael Serra devises are occasionally interesting and even clever, he completely misses the point of geometry -- to understand WHY those "facts" are true.

Unlike many critics of this book, I do not have any inherent qualms with the investigative approach to learning geometry. Investigation plays a central role in mathematics, and I applaud the author for giving inductive reasoning its fair shake in this book. But investigation has become more of an ideology than a pedagogical tool in this book. Even my weakest students groan at having to do some of the investigations, whose results they deem obvious. There are simply too many unnecessary investigations, many of which exist only to aggrandize the author's educational philosophy.

As a student, I used the second edition of this book. The author has clearly made significant improvements for the third edition, but there are still serious pedagogical flaws. While Chapter 13 is a valiant attempt at introducing students to the deductive method of geometry, it is too little, too late. Geometry classes rarely reach the last chapter, and separating the proofs from the theorems themselves feels artificial and contrived. The author makes another questionable pedagogical decision to cover area and volume into nonconsecutive chapters, Ch. 8 and 10 -- just so he can prove the Pythagorean Theorem using area in Ch. 9. But if he would only introduce the concept of similarity (Ch. 11) before the Pythagorean Theorem, he would be able to prove the Pythagorean Theorem using similar triangles in a much more elegant and motivated way.

The unorthodox ordering of topics to which I have previously alluded creates problems for even the author. There are many practice problems that require concepts or techniques from later chapters. For example, students are asked to construct a square in Chapter 3 given a diagonal, before either the properties of quadrilaterals (Ch. 5) -- or even the properties of triangles (Ch. 4) -- have been introduced! How students are supposed to "guess" that the diagonal of a square bisects the angles -- I do not know. Furthermore, the first proof in the text is a paragraph proof that the perpendicular bisectors of a triangle are concurrent. I can only imagine the horrified looks on the faces of Serra's students. And these are supposedly students who are having too much trouble with traditional two-column proofs!

There are outright mistakes in the textbook as well besides the usual typos. On page 333, Serra defines an irrational number as a number whose "decimal form never ends" and a transcendental number as a number whose "pattern of digits does not repeat." So according to his (incorrect) definition, 1/3 would be an irrational number, and sqrt(2) would be a transcendental number -- the former false for obvious reasons, the latter false because sqrt(2) satisfies the polynomial equation x^2 - 2 = 0. Moreover, this is something that a reasonably bright high schooler would know -- much less an ostensibly expert math teacher!

In his manifesto "Tracing Proof in Discovering Geometry," Serra attacks two-column proofs, saying that "so many students fail to master two-column proofs that some teachers are skeptical of claims that all students can learn geometry." While I agree that two-column proofs misrepresent mathematics and can proofs unnecessarily complicated, I'll gladly take them over "Discovering Geometry" any day.

Addendum: Some commenters have asked about alternative geometry textbooks. While I (sadly) have yet to find a textbook that lives up to the promise, the study guide E-Z Geometry (I used an older edition which was called "Geometry the Easy Way") does an excellent job of presenting the material in a coherent, logical fashion, replete with proofs, and comes with a collection of excellent exercises. It's also cheap, although I'm not sure how it would fare as a classroom textbook, since all solutions are at the back. I have also heard excellent things about Harold Jacobs' Geometry: Seeing, Doing, Understanding, although I have not had the fortune to look over the book myself.

Geometry is the only math course in which rigorous deductive reasoning can be made accessible to high school students -- and not surprisingly, it was the first area of mathematics to be axiomatized (by Euclid). Unlike algebra or calculus, almost all of the theorems and formulas in geometry can be systematically obtained from postulates in a way that is intelligible to high school students; contrast this with algebra, where it would not be reasonable to expect students to understand a proof of Cramer's Rule or the Binomial Theorem. The fact that geometry introduces students to a different, mathematical way of thinking is the only justification for maintaining geometry as a standalone math course, rather than integrating it into algebra courses. Otherwise, the "facts" of geometry are nothing remarkable in themselves. So what if opposite sides of a parallelogram are congruent? It wouldn't be too difficult to teach students this "fact" in an algebra class when they're learning about slopes of parallel lines. But what's important is that students understand and see how this fact derives systematically from already known facts.

What does all this have to do with the book at hand? "Discovering Geometry" reduces geometry to the same collection of facts and algorithms that students have been doing in every math class since elementary school. While the problems that Michael Serra devises are occasionally interesting and even clever, he completely misses the point of geometry -- to understand WHY those "facts" are true.

Unlike many critics of this book, I do not have any inherent qualms with the investigative approach to learning geometry. Investigation plays a central role in mathematics, and I applaud the author for giving inductive reasoning its fair shake in this book. But investigation has become more of an ideology than a pedagogical tool in this book. Even my weakest students groan at having to do some of the investigations, whose results they deem obvious. There are simply too many unnecessary investigations, many of which exist only to aggrandize the author's educational philosophy.

As a student, I used the second edition of this book. The author has clearly made significant improvements for the third edition, but there are still serious pedagogical flaws. While Chapter 13 is a valiant attempt at introducing students to the deductive method of geometry, it is too little, too late. Geometry classes rarely reach the last chapter, and separating the proofs from the theorems themselves feels artificial and contrived. The author makes another questionable pedagogical decision to cover area and volume into nonconsecutive chapters, Ch. 8 and 10 -- just so he can prove the Pythagorean Theorem using area in Ch. 9. But if he would only introduce the concept of similarity (Ch. 11) before the Pythagorean Theorem, he would be able to prove the Pythagorean Theorem using similar triangles in a much more elegant and motivated way.

The unorthodox ordering of topics to which I have previously alluded creates problems for even the author. There are many practice problems that require concepts or techniques from later chapters. For example, students are asked to construct a square in Chapter 3 given a diagonal, before either the properties of quadrilaterals (Ch. 5) -- or even the properties of triangles (Ch. 4) -- have been introduced! How students are supposed to "guess" that the diagonal of a square bisects the angles -- I do not know. Furthermore, the first proof in the text is a paragraph proof that the perpendicular bisectors of a triangle are concurrent. I can only imagine the horrified looks on the faces of Serra's students. And these are supposedly students who are having too much trouble with traditional two-column proofs!

There are outright mistakes in the textbook as well besides the usual typos. On page 333, Serra defines an irrational number as a number whose "decimal form never ends" and a transcendental number as a number whose "pattern of digits does not repeat." So according to his (incorrect) definition, 1/3 would be an irrational number, and sqrt(2) would be a transcendental number -- the former false for obvious reasons, the latter false because sqrt(2) satisfies the polynomial equation x^2 - 2 = 0. Moreover, this is something that a reasonably bright high schooler would know -- much less an ostensibly expert math teacher!

In his manifesto "Tracing Proof in Discovering Geometry," Serra attacks two-column proofs, saying that "so many students fail to master two-column proofs that some teachers are skeptical of claims that all students can learn geometry." While I agree that two-column proofs misrepresent mathematics and can proofs unnecessarily complicated, I'll gladly take them over "Discovering Geometry" any day.

Addendum: Some commenters have asked about alternative geometry textbooks. While I (sadly) have yet to find a textbook that lives up to the promise, the study guide E-Z Geometry (I used an older edition which was called "Geometry the Easy Way") does an excellent job of presenting the material in a coherent, logical fashion, replete with proofs, and comes with a collection of excellent exercises. It's also cheap, although I'm not sure how it would fare as a classroom textbook, since all solutions are at the back. I have also heard excellent things about Harold Jacobs' Geometry: Seeing, Doing, Understanding, although I have not had the fortune to look over the book myself.

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ByElBlufon June 10, 2006

This geometry book has thought provoking problems, but that is all that is good about this book. There are many typos and awkward wordings to be found, and even incorrect answers in the teachers edition (my teacher has been correcting answers in his book all year)! This book is also useless without the only conjectures and vocabulary, something that should have been included in an appendix somewhere in this book! If you want to learn geometry, this is not the book to use.

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ByA. S. Templetonon June 13, 2013

Our local Public Schools board & bureaucrats mandated the "Discover" series, which is loathed by local parents and teachers alike. As a technical professional, I find it simply awful and unsuited to training the minds of those making up the future mathematicians, scientists, and engineers of the "knowledge economy".

The so-called investigative approach it uses is RUBBISH compared with traditional rules-based teaching, and v-e-r-y s-l-o-w at getting to the point, which is destructive in a high school context where a half-dozen other subjects and extracurriculars are competing for non-class time.

That said, the guide for parents and tutors (Discovering Geometry: An Investigative Approach (Condensed Lessons: A Tool for Parents and Tutors)) was found to make some aspects of geometry as-presented marginally less muddled. Worth a try, if you've the time to wade into it and can't afford a paid tutor.

The so-called investigative approach it uses is RUBBISH compared with traditional rules-based teaching, and v-e-r-y s-l-o-w at getting to the point, which is destructive in a high school context where a half-dozen other subjects and extracurriculars are competing for non-class time.

That said, the guide for parents and tutors (Discovering Geometry: An Investigative Approach (Condensed Lessons: A Tool for Parents and Tutors)) was found to make some aspects of geometry as-presented marginally less muddled. Worth a try, if you've the time to wade into it and can't afford a paid tutor.

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Byval321on February 23, 2016

This is the solutions manual that goes with the Discovering Geometry textbook. It is not the teacher's or the student's version of the textbook, those would need to be bought separately (see links below). It contains detailed explanations to questions in addition to the answers. It's not perfect, there are some mistakes, and there are a few places where the method presented to solve the problem is not the best way. However, in total, this is a very valuable resource and I highly recommend it for parents whose kids are using the Discovering Geometry textbook in school.

Discovering Geometry : An Investigative Approach 4TH EDITION

Discovering Geometry: An Investigative Approach, Teacher's Edition

Discovering Geometry : An Investigative Approach 4TH EDITION

Discovering Geometry: An Investigative Approach, Teacher's Edition

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ByEbasco Engineeron December 9, 2013

An awful book, destined to go down in history as the worst possible legacy to a branch of mathematics predicated on the orderly development from definitions to proofs. Any book, every book, is better than this. Go to Free Google books and download a hundred year old textbook and you'll do much (infinitely) better by your students. Euclid is spinning in his grave. Seattle is committed to this series. Go figure -- on a calculator. Can I insert a Mr. Yuk face here?

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ByA Swanky Manon March 5, 2010

This Geometry book is designed for students to learn through investigations of conjectures with hands-on activities. Sounds great...but it's not at all. Many of the activities deal with the use of patty paper, or tracing paper, and heavily emphasizes the use of geometry tools such as compasses and protractors to create types of lines, angles, or polygons. Many of the activities are difficult for students to see the relationships the book wants them to discover. The homework assignments are TERRIBLE! Many of the problems are extremely difficult, or just too simple. The online teacher materials are a joke. They offer a worksheet for each section of the book that contains many of the same problems the book does. The online assessment resources are awful. Even worse, if a student needs to take a book home, and they don't understand - the book doesn't explain the conjectures to the reader, it makes them try to discover (or rediscover) them doing ridiculous patty paper activities. It is a very difficult book for a student to read to get quick information on how to do a problem.

The only reason my school district chose this book is because the publisher's Algebra and Algebra II books are excellent, and they wanted the students to experience a similar format for their Geometry class. But this book is horrible.

The only reason my school district chose this book is because the publisher's Algebra and Algebra II books are excellent, and they wanted the students to experience a similar format for their Geometry class. But this book is horrible.

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ByThanneaon May 2, 2012

I used this text for all three of my homeschooled students. They loved it. Their mastery of geometry showed in their ACT scores. In the math breakdown, they collectively missed only 2 geometry problems! All three went on to earn at least a math minor in college. I am not a math major, but I did find teaching from this text to be very manageable and enjoyable.

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ByTaigen Riggson November 11, 2013

Having taught the "Discovering..." books from Algebra I and Advanced Algebra, I fully expected the Geometry book to be as convoluted as the algebra books such as replacing the traditional slope formula from y = mx + b to y = ax + b for who knows why. But I was pleasantly surprised. As with most Geometry books, it starts at its most basic premise to geometric shapes to simple trig functions. It was well illustrated and the pace (for students new to geometry)was reasonable.

Although I see the need for heuristic methodology in teaching math, it is important to know basic knowledge which this text does well in its presentation. It is clearly not rigorous but it isn't meant to be. I like the book so much that I am buying one for myself.

Although I see the need for heuristic methodology in teaching math, it is important to know basic knowledge which this text does well in its presentation. It is clearly not rigorous but it isn't meant to be. I like the book so much that I am buying one for myself.

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ByV226on June 19, 2013

This book is out of date and does not meet the needs of modern students. It does not give enough examples and explanations for students to follow. Practice problems tend to be difficult.

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byMichael Serra

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