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Three Comprehension Tests and three sets of discussion questions are provided for use with this film.

Index to Answer Keys:

This test assumes that students have been given the information in the Helpful Background sections entitled Ideas Move Across Continents and Oceans and We Can Get A Taste of Calculus in Finding the Area of a Circle. For a printable version of this test suitable to distribute to a class (without suggested answers) click here.

1.   What is the great advantage of place-value numbers? Suggested Response: Place-value numbers are important because they allow for easy expression of large numbers and easy computation with those numbers.

2.   Describe in your own words the concept of place-value numbers. After giving your description, demonstrate it by describing what the numerals 5 and 8 mean in the number 545,680. Suggested Response: Our place-value, ten based numbering system works as follows. The term numeral refers to an Arabic numeral (1,2,3,4,5,6,7,8,9 or 0) within a larger number. The location of an Arabic numeral within the number indicates the power of the base to which it relates. Because we use a base of 10, the numeral refers to its value multiplied by the power of ten that corresponds to its position within the number. Thus, in the number 545,680 the Arabic numeral "8" is multiplied by 101. This is 8 X 10 or 80. Therefore 80 is added to the value of the number. The numeral 5 appears twice in the number. The first time it appears in the thousands place, which is three places to the left of the one's place. It is therefore multiplied by 103 (or 1,000). The second time it appears it is located in the hundreds of thousands place, five places to the left of the ones place. The numeral 5 is therefore multiplied by 105 or 100,000 and adds 500,000 to the value of the number. In total, the Arabic numeral "5" adds 505,000 to the value of the number.

3.   Why was it so hard for people to come up with the concept of nothing as a number? When did the concept of the zero come to Europe?Suggested Response: The reason that zero was so hard to conceptualize was that numbers were developed for the purpose of counting things that existed, like farm produce, cattle, or sheep, and to measure the distance of lines. People didn't need zero for this. Zero doesn't come naturally when counting objects that exist. The concept of zero was not known in Europe until about the 11th century, C.E. when the Europeans learned it from the Arabs.

4.   Name the first two civilizations that used both the concept of zero as a number in computation and place-value numbers. Give the approximate dates of the development. Suggested Response: The Hindus (Indians) were using the concept by about 600 C.E. and the Maya hundreds of years before.

5.   Name two advanced civilizations in Europe that flourished after the concept of place-value numbers was created in Babylon but which did not know this concept. Suggested Response: Ancient Greece and Ancient Rome.

6.   Describe the movement of the concept of zero and place-value numbers to the Americas after the Mayan civilization declined. Suggested Response: Place-value numbers were used in Ancient Babylon beginning about 2500 years ago. The concept of zero as a unit for mathematical calculation arose in India which used the numerals 1,2,3,4,5,6,7,8,9, and 0 by about 600 C.E. The Indians also had place-value numbers. There was a lot of trade between India and Arabia. From India, these concepts traveled on the trade routes to Arabia and from there to Europe. Place-value numbers and zero as a number for computation were used in Europe beginning in the 11th century, 1000 - 1099 C.E. From Europe the use of zero and place-value numbers traveled across the Atlantic to America with the European colonists.

7.   Name four advanced characteristics of culture developed by the Maya. Suggested Response: Here are seven: (1) calendars; (2) advanced astronomy; (3) the concept of zero and place-value numbers; (4) large stone buildings; (5) large cities; (6) fine metalwork in gold and silver; (7) hieroglyphics, an early form of writing.

8.   Trace the movement of the insult of giving someone "the finger" from the earliest evidence that it was used through to 20th century United States. Suggested Response: Greece to Rome to Europe to North America. The last step could have been directly from Europe to North America or from Europe to Central or South America and from there to North America.

9.  Explain how Archimedes, the ancient Greek mathematician, found the area of a circle and how that relates to calculus. Draw diagrams if it would help in your explanation. (The answer to this question counts for two points.) Suggested Response: Archimedes knew the equation for the area of a triangle and, since a regular polygon can be divided into several triangles, he could calculate the area of a regular polygon. He added sides to his regular polygon so that it approached closer and closer to the curve of the circle. While he could never get his polygon to eliminate all of the area between the edge of the polygon and the curve of the circle, he found that the values for the area of his many sided polygon approached π times the square of the radius (π r2). This then is a good approximation for the area of a circle. Calculus permits mathematicians to determine the area under a curved line. It does this by finding the area of polygons with small sides that approach the shape of the curve, just as Archimedes made the polygons inscribed within the circle approach the shape of the circle.


This test assumes that students have been given the information in the Helpful Background section entitled Literary Devices in a Work of Historical Fiction. For a printable version of this test suitable to distribute to a class (without suggested answers) click here.

1.   Compare the characters behavior and appearance at the beginning and the end of the movie. Which character changed the most? Suggested Response: Angel changed the most over the course of the film. The other kids went from poor students to students who mastered the AP Calculus exam. Angel, however, was a gang member. He had further to go than the other students. An alternative choice is Mr. Escalante who went from being an engineer in a high tech industry to being a teacher and then to achieving the status of a master teacher.

2.   Angel is an interesting character, constantly getting into trouble and constantly figuring a way to get out of it. What drove this character? Suggested Response: There is no one correct answer. Here is an example of a good answer: Angel was an angry young man. Through the course of the movie he realized that his anger wouldn't get him anywhere and that learning calculus would.

3.   At the beginning of the movie, Angel is wearing a hair net. What did that mean? Suggested Response: That he was part of a gang and that he was alienated from school and society.

4.   Remember the scene in which Mr. Escalante pinned Angel against a fence and stopped him from joining a gang fight? That action by Mr. Escalante had several meanings for Angel. Name one of them. Suggested Response: The possible meanings include: (1) that Mr. Escalante cared for him; (2) that Mr. Escalante knew what was best for him and that Mr. Escalante was willing to put himself at risk to make sure Angel didn't get into trouble; and (3) that Mr. Escalante was a powerful person.

5.   By the time Angel brought his grandmother to Mr. Escalante's house at Christmas, how had his character changed? Suggested Response: At that point Angel realized that he wanted to be in the class. He was willing to humble himself to stay in. Credit should also be given if the student remarks on Angel's creativity and willingness to go to any lengths to stay in the class.

6.   Just before the class went swimming in the ocean to celebrate the exam being over, what was Angel's clothing like? Suggested Response: It was neatly pressed and fashionable, not like the gang clothes he wore at the beginning of the film.

7.   It was difficult for Angel to decide to join the class when they went swimming to celebrate finishing the exam. The scene begins with Angel fully clothed standing on the shore and the rest of the class in the water. What was the significance of Angel taking off his clothes and jumping into the water to be with his classmates? Suggested Response: This action demonstrates that he was committed enough to the kids in the class to allow them to see him in a vulnerable position.

8.   After the first examination results had been invalidated and Angel had provoked the police, his gang friend challenged him to a fight. Why did Angel walk away from that fight and what did this symbolize? Suggested Response: What the gang boy thought of Angel wasn't important to Angel any more. It symbolizes Angel's separation from the gang life and that his relationship with the gang was no longer important to him.

9.   In this movie there is one character who is a foil for Mr. Escalante and another who is a foil for Angel. Who are these characters and which of their traits make them foils for Mr. Escalante and for Angel? Suggested Response: The foil for Mr. Escalante was the head of the math department who doubted that the students could learn calculus and who believed that they had probably cheated. The foil for Angel was his gang friend who didn't grow out of the gang life.

10.   The movie opens with a long shot of the Los Angeles River, which is a concrete ditch for much of its length. What is the symbolism of this shot? How does it relate to the rest of the film? Suggested Response: The Los Angeles river, like the children in the movie, exists in an urban environment. They are all suffering under the pressures of urban life, but surviving.


This test assumes that students have been given the information in the Helpful Background sections entitled Public Policy and Burdens of Proof (Perhaps the ETS Gave the Students a Break). Students will need to have available for question # 5 the handout entitled Facts that the ETS Knew or Could Have Discovered in 1982. For a printable version of this test suitable to distribute to a class (without suggested answers) click here.

1.   What is a good description for the presumption of innocence when applied to civil cases or private matters? Here is a hint: the term was discussed in the lecture and it has five words. Then describe why we don't just use the term "the presumption of innocence" for civil cases and private matters. Suggested Response: The term is "the presumption of good conduct". The reason that "the presumption of innocence" doesn't always apply in civil cases and private determinations is that these don't determine whether people are guilty of crimes. There is no innocence or guilt in these proceedings. That is reserved for criminal cases.

2.   The chart below mixes up the burdens of proof (on the right) with some of the different situations in which decisions are made in our society (on the left). Match the type of proceeding with the letter assigned to the burden of proof.

Criminal prosecution                A.   preponderance of the evidence
Civil lawsuit for damages                B.   no accepted burden of proof
but the action must be
reasonable in the
circumstances in light of the
common law obligation to
take reasonable care to
avoid injuring people
Decision or action by a
person, a business,
or a non-governmental
C.   beyond a reasonable doubt

Suggested Response: In a criminal prosecution the burden of proof is C: beyond a reasonable doubt. In a civil lawsuit for damages the burden of proof is A: a preponderance of the evidence. For a decision by a person, a business or a non-governmental organization the correct answer is B, there is no accepted burden of proof. The action should be reasonable in the circumstances in light of the common law obligation to take reasonable care to avoid injuring people.

3.   The determination of the proper burden of proof in different situations depends upon the facts of the situation and considerations of public policy. Briefly describe some of the public policy considerations supporting the general burdens of proof for: (1) criminal prosecutions; (2) civil law suits; and (3) private, non-governmental determinations. There are some exceptions and extraordinary situations in which the burdens of proof are modified. In your answer, don't worry about the exceptions and extraordinary situations, just refer to the general rules. Suggested Response: Here are some of the public policy considerations that should be considered. A good answer will touch on most of them. (1) Criminal proceedings: These involve the most serious potential penalties. The government, which is very powerful and has many resources, wants to put an individual in jail. The individual usually has very few resources to fight off the government. Even in democratic countries the government sometimes uses the criminal process to persecute those who challenge its policies. The evil of wrongfully putting someone in prison is so great that the public (through the Constitution) has made a public policy determination that it is worth letting many criminals go free to prevent one innocent person from being convicted. Therefore, the people have insisted that it be very difficult for the government to imprison someone. One of the protections is a very high burden of proof in criminal proceedings: guilt must be proven beyond a reasonable doubt. (2) Civil law suits for damages: The purpose of these proceedings is to provide compensation if someone commits wrongdoing or is negligent and hurts another person. The stakes are property and money. While these are very important, the importance goes both ways, i.e. defeating an invalid claim and keeping their property and money is important for defendants while obtaining compensation to correct for injuries sustained at the hands of another is also important. Thus, if a plaintiff can demonstrate that it is more likely than not that another person's wrongdoing or failure to take care has caused injury to the plaintiff, the plaintiff can recover damages. Thus we have a lower burden of proof: preponderance of the evidence (more likely true than not true). (3) Private Decisions: With respect to individuals, businesses and non-governmental entities, the usual rule is that they should be free to do what they want. However, there are public policies which intervene to restrict this freedom. These include the public policies that people should obey the law and that people should take care to avoid injuring others. (If they fail to do this they can be sued for damages in a civil suit.)

4.   What burden of proof do you think the ETS should have used in making a determination of whether or not the students cheated? Give your own opinion among the burdens of proof we have discussed in class and briefly describe why you came to that conclusion. Briefly set out the public policies that support your contention. (The answer to this question counts for two points.) Suggested Response: Any of the burdens of proof discussed in the materials is a potentially correct answer. The three possibilities are: (1) "a preponderance of the evidence" (more likely true than not true; this is the standard used in most civil cases tried in court); (2) "clear and convincing evidence" (highly probable that the fact is true; this is the standard used for decisions of certain issues in civil cases; for example, it is often the standard that must be met before punitive damages are assessed) and (3) "beyond a reasonable doubt" (the standard for criminal cases). The justification for applying each of these standards should be well thought out, show that the student has followed the class discussions, and deal with the following two issues: (a) The ETS had to comply with its common law obligation to take reasonable steps to avoid injuring the students under investigation before making a determination that the students had cheated; (b) the ETS had the right to act to uphold the integrity of its test.

An example of a good one paragraph answer is: "I believe that the proper standard is clear and convincing evidence. Being branded a cheater will hurt a student for much of his or her academic career. In addition, these students worked very hard to learn calculus. It is reasonable that the ETS make a determination of cheating only when it is substantially more likely than not that the student cheated. In addition, the ETS is a large organization with all kinds of experts and computers. The interest of the ETS is to prevent cheating. It should be easy for ETS to find out if students cheated using a computerized analysis as well as by looking carefully at the answers."

5.   Apply the burden of proof that you selected in response to the preceding question to the facts known to the ETS when it disqualified the test scores and offered to have the students take the test again. Use the facts found by Mr. Mathews in his investigation. Discuss: (1) whether the burden of proof was sustained with respect to the unusual rate of similar answers to the multiple choice questions and (2) whether the burden of proof was sustained with respect to Free Response Question #6. (The answer to this question counts for two points.) Suggested Response: There is no one correct answer as to the multiple choice questions. A good response will deal with most of the facts set out in the handout Facts that the ETS Knew or Could Have Discovered in 1982 and specifically discuss the explanation that Mr. Escalante drilled and redrilled his students in a specific approach to calculus problems and that this could very well explain the striking similarities in the answers to the multiple choice questions. As to Free Response Question #6, the fact that the students made an identical error in simplifying a fraction is a telling point and would probably meet all of the possible standards, including "beyond a reasonable doubt". In the answer to both parts of this question, students should not take into account the fact that the students passed the second test supervised by ETS monitors. The question refers to the time just after the students had taken the first test and before their answers were disqualified.

6.   Did these students deserve a chance to take the test a second time? Suggested Response: There is no one right answer to this question and the important concepts conflict. A good answer will be well reasoned and focus on several of the points raised in the class discussion of this issue. These include: the fact that Mr. Escalante drilled them and redrilled them on a specific approach to calculus; the fact that the students passed the test without any help from their "cheat" on FRP # 6; the fact that the calculus program at Garfield High was a good program and it would be hurt if action were taken against the students.

7.   If a group of middle class students took an AP Calculus test and passed but cheated on a question that they got wrong, should they be permitted to take the test a second time? Assume that the students had no history of cheating. Suggested Response: There is no single correct answer to this question but it probably should be the same as the answer to the preceding question. This answer should focus on the points in the answer to the preceding question which apply to the situation posed by this question. Some of the points that could have been mentioned in the last response which don't apply, in addition to the difference in socio-economic group, are that the students described in this question were not under pressure to prove a point (that Latino students from the barrio could learn calculus) and that their success would not have a broad impact. (Note that this question deals with the proper punishment after a determination that the students cheated. On issues of punishment, "Lady Justice" can lift the blindfold a little and consider many things that she could not consider in the determination of the issue of whether the students cheated.)

8.   Does the information that Mr. Mathews developed about whether or not the students cheated on Free Response Question #6 undercut the core messages of the film? Suggested Response: No, the core messages are that inspired students can achieve wonders and that something very good for math education and for the Latino community happened at Garfield High while Mr. Escalante taught there. Whether or not Mr. Escalante's students cheated on one question because of the pressure they were under and their inability to resist temptation, they still learned calculus. It is obvious that they worked hard and mastered the material, given the fact that they passed the examination again when ETS monitors were present. Thus, the evidence that they cheated on Free Response Question #6, does not negate their achievement in mastering calculus.



1.   See Discussion Questions for Use With any Film that is a Work of Fiction.

2.   How does the concept of "ganas" apply to this film? Can you achieve great things without it? Suggested Response: "Ganas" means "desire" or "longing", as in the desire to win. One cannot achieve great things without "ganas".

3.   Do you have "ganas" about anything in your life? Suggested Response: There is no one right answer to this question. A good answer will be heartfelt and meaningful.

4.   What steps did Mr. Escalante and Mr. Gradillas (the principal at Garfield) take to improve the math knowledge of the students who were coming into the calculus class? Suggested Response: The success of Mr. Escalante's class on the 1982 AP Calculus test was the culmination of a six year effort by Mr. Escalante. To help students acquire the background in geometry, algebra II, math analysis, and trigonometry necessary to understand calculus, Mr. Escalante encouraged area junior high schools to offer algebra in their eighth and ninth grade mathematics classes. He also convinced them to upgrade the standard of instruction. Escalante, supported by his principal, Peter Gradillas, also worked to impose higher standards in the pre-calculus math classes offered at Garfield. Mr. Escalante also taught intensive math classes during the summer.

5.   Why isn't the calculus program set up by Mr. Escalante in operation today? Suggested Response: Mr. Gradillas, the principal, went back to school to get his Ph.D. The new principal at Garfield High didn't support the AP Calculus program. Faculty members at the school were upset that so many resources were being devoted to the calculus program. Mr. Escalante left the school when the program would no longer be supported.



1.   Do you think that Mr. Escalante is a role model for a teacher? Suggested Response: Yes. He cared about his students. He was innovative. He held his students to high standards. He knew how to motivate his students. He knew his subject matter. He had a goal which was difficult and he was helpful to other people. He worked hard to accomplish that goal.

2.   The film shows some of Mr. Escalante's mistakes. Can you describe three of them? Suggested Response: They include the following: (1) not listening to Angel's excuse when Angel was late because he was at the emergency room all night with his grandmother; (2) making fun of the personal life of one of the girls in the class; (3) getting angry at the ETS employees and thinking that they were discriminating against his students; (4) threatening one of the ETS investigators; (5) leaving the hospital when the doctors told him that he was not well enough to leave; and (6) not spending enough time with his family.


3.   Would you be willing to give up your mornings, afternoons, and weekends, and a good part of your summer to prove to yourself that you could learn calculus and pass the AP Calculus test? Suggested Response: There is no one right answer to this question. Just asking the question is helpful because it stresses how difficult it was for the students to follow Mr. Escalante's program.

4.   Describe the corrosive effects of the loss of self-esteem to the students in this film before they met Mr. Escalante. Suggested Response: There is no one right answer to the question. The basic problem is that low self-esteem prevents people from achieving their full potential and taking the steps necessary for their own happiness.


5.   One of Mr. Escalante's special qualities was that he had high expectations for his students. What is the importance of a teacher's expectations for a class? Suggested Response: Without high expectations from the teacher, few students in the class will work up to their full potential.

6.   Did Mr. Escalante do the right thing in going to the restaurant owned by the family of one of his female students to persuade the girl's father to allow her to attend the extra calculus classes? What is the appropriate role for a teacher when communicating with a student's family? Suggested Response: Mr. Escalante did the right thing. It is legitimate and helpful for teachers to point out to parents a child's special gifts or an unusual opportunity for the child at school. [Then have a discussion based on the following questions, "Would your answer be the same if the teacher had not been Latino but instead Anglo or Asian?" The answer, of course, is yes.]

7.   Describe some of the techniques that Mr. Escalante used to keep his students interested in the class. Suggested Response: The response should show that the students paid attention to the film. They include: the barbecue, the funny language, putting a student on a stool in front of the class for not doing her work, clowning around, making jokes, etc.

8.   Remember the lady who was head of the math department at Garfield High School? This teacher had low expectations for her students and thought that Mr. Escalante's students had cheated. What caused her to have these attitudes? Suggested Response: There is no one right answer. A good answer will refer to the fact that she was basing her conclusions on her experience. She was beaten down by the lack of resources, by the difficult students, and her inability to inspire her students. In this she was not alone. Many other teachers at Garfield High and at schools like it all over the country have suffered similar experiences and many have reactions that are similar to hers.

9.   What does this film tell you about the special contributions that inspiring teachers can make to their students and their community? Suggested Response: They can make an important contribution.


Discussion Questions Relating to Ethical Issues will facilitate the use of this film to teach ethical principles and critical viewing. Additional questions are set out below.


(Be honest; Don't deceive, cheat or steal; Be reliable -- do what you say you'll do; Have the courage to do the right thing; Build a good reputation; Be loyal -- stand by your family, friends and country)

1.   The ETS was faced with strong evidence that the students had cheated. This came from their incorrect responses to Free Response Question #6, in which most of the class applied the same incorrect formula to the problem and made an identical mistake in simplifying a fraction, a type of calculation that they had been doing correctly since the 6th or 7th grade. Assume that the students cheated on Free Response Question #6. Discuss the role of the law of unintended consequences in the outcome of this situation. Suggested Response: They passed the test without getting the benefit of their cheat because they gave an incorrect answer to the question on which they cheated. This meant that if the ETS had not allowed them to take the test again, they would have been denied the goal that they had worked so hard to achieve. They would also have been branded as cheaters. In this case, ironically, the cheating almost denied them the benefits that they deserved because they passed the test without answering FRQ #6 correctly.

2.   Assume that the students cheated on Free Response Question #6. Describe three reasons why cheating at school is not a good idea and how this relates to the situation of the students in Mr. Escalante's class. Suggested Response: Six reasons are set out in the section on Some Problems With Cheating. The best answers will include at least the first two and one other.


(Do what you are supposed to do; Persevere: keep on trying!; Always do your best; Use self-control; Be self-disciplined; Think before you act -- consider the consequences; Be accountable for your choices)

3.   Many teachers and school administrators as well as children and parents have seen this film. Why aren't all children taking the AP Calculus test or engaging in some other specialized effort to excel? What about you? Are you making some special effort in your life to excel? Suggested Response: There is no one correct answer to this question.

4.   Mr. Escalante got angry at Angel and unfairly punished him. The only reason that this did not have disastrous consequences was that Angel was committed and resourceful. How does this incident show the need for self-control by both Angel and Mr. Escalante? Suggested Response: Angel had to control his anger and disappointment with Mr. Escalante. But he was able to step outside of himself, see why Mr. Escalante was angry, see that some of it was justified, and that he could overcome that anger with the abuela stratagem. As for Mr. Escalante, he should have listened to Angel's excuse and if he questioned it, he should have made Angel produce a record from the hospital or a note from his grandmother. However, Mr. Escalante was flexible enough to see that if Angel was willing to resort to the "abuela" stratagem, Angel was interested enough to deserve a third chance.

Last updated September 27, 2008.

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