Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Tuesday, September 7, 2021

A Beautiful Mind

 SPOILER ALERT! The plot will be discussed.

 

The title of the film, A Beautiful Mind (2001), takes on depth as the story of mathematician John Nash unfolds. His mental abilities produced Nobel Prize winning insights. But, he also was schizophrenic, so the same mental powers that engendered brilliant rational breakthroughs also created damaging hallucinations. He was someone who was searching for insights in the abstract realm of numbers to be applied to the real world, but he also many times had no connection with reality. 

 

The opening speech from Professor Helinger (Judd Hirsch) at Princeton University in 1947 to new graduate students places a great deal of pressure on the entering class. There is a focus on using the science of mathematics to fight enemies, breaking codes and building the atomic bomb. Nash sits in the back of the room, his eyes avoiding contact, already establishing himself as an outsider. Helinger’s outlook may have contributed to what John Nash (Russell Crowe, excellent here) thought was his purpose and which fueled his paranoia about foreign adversaries targeting him. 

 At the outside reception, Nash observes patterns in what he sees as lines forming geometric shapes light up as he perceives them. This camera imagery is often used by director Ron Howard (winning the Oscar for Best Director here) and his fellow filmmakers to highlight mental epiphanies, according to IMDb. They are flashes of light, showing how his creativity is drawing things together. The scene also illustrates Nash’s witty humor when he says to a fellow student that “there could be a mathematical explanation for how bad your tie is.” Martin Hansen (Josh Lucas) asks Nash to get him a drink, saying Nash presented himself as a waiter, probably because he is wearing a bow tie. Again, Nash’s sarcasm surfaces as he says, “Imagine you’re getting quite used to miscalculation.” He goes on to say that Hansen’s publications lack anything noteworthy. Rivalry obviously exists between these two winners of the Carnegie Scholarship. Nash has nervous, quirky hand movements which show his awkward social skills. Nash also meets Sol (Adam Goldberg) and Bender (Anthony Rapp) here who become Nash’s friends. 

 

In his room, Nash encounters his supposed English major British roommate, funny (“Officer, I know who hit me, it was Johnny Walker) Charles Herman (Paul Bettany), who is suffering from a hangover. While he cracks jokes, Nash writes mathematical equations on his dorm window, a sort of metaphor for how his mental powers shed light on his numerical exploits. In answer to Charles’s questions about him, Nash says he is “well-balanced” because he has “a chip on both soldiers.” It is comical, but it also reveals Nash feeling that he must battle adversity. Charles points out that Nash is better with “integers” than people. Nash adds that he had a teacher who said he had “two helpings of brain but only half a helping of heart.” This discussion points to Nash’s lack of emotional connection to others. He admits that he doesn’t like people and they don’t care for him. He is impatient to bypass personal relationships so he will not waste time on his quest to map out “the governing dynamics” of existence and find a “truly original idea” so, as Charles says, he will “matter.” The two are drinking on a roof, which is fitting as Nash looks down literally and figuratively on the other students, calling them, “lesser mortals.” 

 

Hansen challenges Nash to a board game, and he is astonished that he loses to Hansen. Nash feels his “play was perfect.” Competition is at the center of Nash’s drive to succeed. It is here that he starts to investigate “game theory,” which will lead to the idea of those “governing dynamics” that will be applied to economics and for which Nash will be most known. 

 

Nash approaches a blonde at a bar and is unsocial to the point of insensitivity. He says he is not sure what he is supposed to say so that they can have sexual intercourse, so maybe they should skip right to having sex. She slaps him and walks out. He must learn to try a different tactic, as we soon discover. 

At present, Nash hasn’t been attending classes, which he sees as being just derivative, and not aiding invention. He doesn’t have a topic for his doctoral dissertation, and hasn’t published anything. Helinger tells him that he can't be recommended for placement in a post-graduate position. Nash sees recognition and accomplishment as the same thing, showing his need for validation as the talented outsider. 


 Nash becomes upset, telling Charles he must follow “their” rules in order to get ahead instead of taking the road less traveled. This idea is symbolized by his pushing his desk away from the window, on which he scribbles his equations, where the light of inspiration shines upon him. Charles counters that argument by telling Nash he must follow his passion outside the walls of the educational institution, and he pushes the desk through the window, watching it fall to the ground. The act shows the need for Nash to break through traditional restrictions on his “beautiful mind.”


 Nash is in the bar again. But now he employs a version of game theory when he tells the other math men that if they avoid making a play for the blonde who is present, then they will not have to compete for her, and the other women there will not consider themselves as second choices. That way, they can all “get laid,” and thus win. He says that economist Adam Smith's idea of everyone doing what’s best for himself will automatically be good for the group is “incomplete.” Crowe does a hand gesture with the fingers of one hand curled up touching his forehead. It seems to signify that he has some idea to communicate, but it also shows his shyness, a way of not looking directly at someone. Nash says that the individual must do what’s good for himself and take into account what’s good for the group. He is devising a plan that allows for participation in a goal-oriented strategy that does not have to produce a loser but instead allows each person in the group to win. He goes back to his desk, restored to its spot near the illuminating window, and he begins writing his equations, revising 150 years of economic theory. Helinger is impressed with Nash’s work and gives him the go-ahead to develop his theories on “governing dynamics.” He chooses his pals, Sol and Bender, to be part of his team.

 

When we catch up with Nash later he has his doctorate and has gained that recognition he sought, appearing on the cover of Fortune magazine. The Pentagon calls him to attempt to decipher code they believe the Russians are transmitting. After looking at a wall of numbers, the digits illuminate for Nash, and he says that there are latitudes and longitudes among the figures that relate to routing places in the United States. When he asks about what the Russians may be planning, he is basically told to leave as he is not authorized access to further information. He looks up and he sees a man on a walkway, but the others there do not acknowledge this person. 

 

While in a car, Sen. Joseph McCarthy’s voice is heard, the man who promulgated the “red scare,” and paranoia about communism. The belief that the enemy was among us plays into Nash’s personal feelings of persecution and the urgency being promulgated that mathematics must be used to fight political adversaries. Nash now works at the defense labs at the Massachusetts Institute of Technology. He complains to Sol and Bender that the Russians have the hydrogen bomb, the Nazis found sanctuary in South America, and the Chinese are gaining force. But he complains that he is being underused by being assigned to study stress problems in a dam. He has taken Helinger’s mission that he first heard as a graduate student very much to heart.

 

Nash must still teach classes as part of the deal to keep his research projects going. At the class, he throws the textbook into the trash, showing his disdain for tradition. He sees the class as a waste of his time. He puts an equation on the board as a kind of test to weed out who is worthy of his teaching efforts. In the class is Alicia Larde (Jennifer Connelly, winning a Best Supporting Actress Oscar for this role). By using her attractive attributes, she is able to persuade the construction workers outside to be quiet during the class, winning Nash’s admiration for problem solving.


 Just as Nash has complained that he isn't getting a chance to show his abilities to fight America’s enemies, the man who Nash called “Big Brother” at the Pentagon observing from above shows up to give Nash a chance at more recognition. He calls himself William Parcher (Ed Harris), and he says he supervised security over J. Robert Oppenheimer’s atomic project. When Parcher brings up how many lives can be lost in the pursuit of weapons, Nash is rather cold, saying progress requires the need for sacrifice. Parcher says Nash’s “lone wolf” life will be advantageous, most likely for performing covert activities. Thus, Parcher’s existence justifies Nash’s anti-social nature and his desire for recognition. As they walk into a “secure” area, Parcher says they know him there, so he doesn’t have to show credentials. Therefore, he does not interact with the guard. The above details become important later.

 

Parcher takes Nash into what were supposed to be abandoned warehouses. Inside, he sees many men in white coats operating rows of computers. Parcher tells Nash he now has top secret clearance, and calls him the best natural codebreaker around. Parcher’s function then is to bestow the acknowledgement of Nash’s talents that the mathematician believes he deserves. Parcher says that the Russians took a portable atomic bomb that the Nazis had developed, and the locations that Nash identified earlier at the Pentagon are places the Russians are exploring to explode the bomb. Parcher says that the Russians are placing coded messages in newspapers and magazines, and that’s why they need Nash’s help. Parcher makes an interesting observation when he says, “Man is capable of as much atrocity as he has imagination.” His comment adds irony to the title of the film by showing the underside of a brilliant “beautiful mind.” Parcher’s team supposedly puts an implant in Nash’s arm so he will have access to a drop point where he can deliver his findings.

 

Alicia comes to his office with what Nash calls an “elegant” proof of the problem he wrote on the board, but she made assumptions which didn’t solve the problem. She asks him to dinner, but he says he usually eats alone, and he says he is like Prometheus with a bird circling above. His humorous image reveals that he sees himself as a rebel, like the mythological personage, who gave fire to earthlings. Nash sees himself as a godlike entity who will endure tribulations to bring mental illumination to those incapable of such achievement. 


 Nash takes Alicia to a formal university affair. She is able to navigate his social awkwardness by engaging in his peculiar humor. He draws objects in the night sky by connecting stars with his hand. He sees patterns, maybe where there are none except what we impose on them. The exercise again shows his desire to find form and insights in the universe. However, at the event, he believes men there are observing him, which shows how imagination can warp reality into something ominous. He later drops his sealed classified findings into a lockbox at a gated house by using the changing codes on his implant that are illuminated under a black light. It is done at night and there is a dangerous feel to the place as a car slows down to observe him there.

 

Nash tells Alicia that his directness has not been socially successful, so it is an effort for him to adapt to the rules of society. She encourages him to say what he wants to say. He concedes that even though he finds her attractive and wishes to have intercourse as soon as possible with her, he feels he must go through “platonic” romantic rituals to reach that outcome. He then says he expects a slap across the face, as he experienced earlier at the bar. Instead, she kisses him passionately, showing that with her his honesty is rewarded, and how the two are compatible.

 

Charles reunites with Nash and brings his niece, Marcee (Vivien Cardone), who Charles says he has taken custody of since the death of his sister. Charles says he is close by at Harvard. IMDb notes that when Marcee runs through a field full of birds, they do not scatter, suggesting that she doesn’t exist. Nash tells Charles about Alicia and wonders how he can be sure that asking her to marry him is the correct move. Charles says, “Nothing’s ever for sure, John. That’s the only sure thing I do know.” That unpredictability comes from a person who supposedly studies literature, an art form, and it offsets Nash’s longing for mathematical certainty.

 

Nash meets Alicia at a restaurant and says he needs “proof” and “verifiable data” that would indicate that they can be in a long-term relationship. He says this while bending on one knee. It is like a mathematical marriage proposal. In response, Alicia says she must modify her notions of romance to accommodate his data-driven inquiry. She asks how big the universe is, and he says it is infinite. But, he concedes that impression can’t be proven, so he just believes it. She says it’s the same with love, one can’t prove it, but somehow just believes it. Alicia is able to show that not all things, such as emotions and ideas, can be proven, but they still exist within people.


 The two get married, but on his wedding day Nash sees Parcher with a disapproving look, since the lack of attachments to others supposedly justified his service. This image shows the conflict within Nash. When Nash drops a report off, Parcher drives by, tells Nash the spot is compromised, and they are being followed. They speed away as Parcher fires on the approaching vehicle. The terrified Nash sees the enemy vehicle eventually end up in the water. Nash is distant when he goes home to Alicia, and locks the door of his room behind him, as he is now suspicious of everyone, which reinforces his detachment from others. He looks at his students and out through windows and doors as if everyone is a threat. His warped view of reality paints him as a victim of other forces which are trying to destroy him. 

 

Parcher visits Nash in his office, and his presence is there to prevent Nash from rejecting his immersion into his world of paranoia. Nash asserts that he has a wife and will soon be a father, and wants to quit so as to shift his focus to positive things, away from his preoccupation with fear. But Parcher is here to assert that feeling of dread by threatening Nash, saying if he doesn’t continue his work, Parcher will not protect him from the Russians. 

 

There are shadows on the walls of Nash’s house as he keeps watch through his blinds. They appear to be real, but shadows are just optical illusions, like Nash’s fears. Alicia begins to realize there is something wrong with her husband as he acts irrationally, suspicious of why she turned on the light at night, and then ordering her to leave for her sister’s place. She looks at the telephone and it seems she is about to seek help for Nash.

 

As Nash gives a lecture he sees men entering from the back of the room that he thinks are enemy agents who are after him. He says to his students that one can’t assign values to variables, which shows that Nash can't even find sanctuary among the predictability of mathematics. He runs out of the lecture hall and he is pursued, but not by enemy agents. Dr. Rosen (Christopher Plummer) approaches him and says he is a psychiatrist. Nash punches Rosen and tries to flee. Rosen injects him with a sedative as Nash sees Charles and his niece observe what transpires.

 

Nash has been admitted to a psychiatric facility where he is in restraints. Nash addresses Charles who he sees there, and he believes that Charles betrayed him by delivering him to the Russians. But, Rosen says there is no one there. If we haven’t already deduced it, Nash has hallucinations. Rosen informs Alicia that Nash is schizophrenic, which many times involves paranoia, and her husband’s belief that he is working to discover conspiracies is a symptom of his mental disorder. Moreover, Nash’s occupation allowed these delusions to go on without being discovered. Rosen gets Alicia to admit she never met Charles, saw a photo of the man, or talked to him on the telephone. Nash said that Charles was his roommate, but Rosen discovered that he lived alone at Princeton. Rosen says he must make Nash distinguish between what is real and what are illusions generated by his otherwise beautiful mind.


 When Alicia gets into Nash’s college office she sees how extreme his activity has been, cutting up magazines and placing pictures and articles all over the walls. Sol and Bender, knowing how offbeat Nash is, gave him a great deal of leeway and did not question his covert activities. Sol followed Nash once to the drop site and now Alicia goes to the estate where Nash was supposed to deliver his findings. The place has been abandoned for some time, with the drop box a broken mailbox and the gate opener busted. 

 

At the hospital, Nash adapts the “facts” as he sees them to fit his beliefs, as most conspiracy obsessed people do. He says the Russians can’t kill him because he is too well known, so they are confining him. Alicia tells her husband that she found out there is no Parcher and no conspiracy. She shows him all the unopened envelopes he placed in the mailbox. Of course, he walks out on her because the truth will upend his universe, and he can’t tolerate that. 


 He tries cutting out the implant in his arm, but says that it is already gone. Nash starts to get a glimmer of what has been plaguing him. As Rosen prepares Nash for drug therapy, Rosen says how horrible it is to realize that people an individual thought one knew never existed, and that beliefs one held were completely false. It’s as if the “fake news” that one accused others of propagating was real, and one’s own beliefs were the false ones. 

 

Back at Princeton, Alicia tells Sol that the delusions have passed, but Nash will not show up at Princeton, possibly feeling shame, where his academic competitor Hansen is now department chairperson. She feels an obligation to take care of the man she fell in love with, making sure he takes his medications and encourages him to be active. Sol visits Nash, who tells him not to sit on Harvey, the imaginary rabbit from the film. Nash has kept his sense of humor, saying what’s the point of being “nuts” if one can’t have some fun. But when he hands his indecipherable scribblings to Sol, it is obvious that Nash isn’t capable of functioning efficiently, as he says, due to the effects of the medications he is taking. He still feels that his work is the most important part of his life at this stage. He holds his child while in a stupor, despondent, devoid of any emotional attachment to his family.

 

Alicia has her own paranoia concerning what Nash states, assuming what he says is influenced by his schizophrenia. He tells her he was talking to the garbage collectors, but she says they don’t pick up trash at night. Then she sees the men working outside. They both giggle, and she apologies, probably realizing she too must adjust to what is really happening. But that light moment is followed by a heartbreaking one as Alicia makes sexual overtures in bed and he resists. He admits it’s the medication, which can decrease the libido drastically. She looks devastated, so the implication is that there has been no intimacy for a long time. She goes into the bathroom and throws a glass of water at the mirror, shattering both, and screams her frustration. When Nash takes the shards out to the trash, it is a metaphor for what is broken in their lives.

 

Nash secretly stops taking his pills, most likely so he can provide the intimacy that Alicia wants. But, that brings back his condition as he is confronted by Parcher who has armed soldiers with him. He takes Nash to a large shed on Nash’s property that he staffed with personnel and electronic equipment to pinpoint the location of the nuclear weapon the Russians want to detonate. Nash tries to deny the existence of what he sees, but then submits to the fantasy. The film seems to be saying that delusions which feed our preconceptions are difficult to let go.

 

Alicia hears a radio transmission coming from the dilapidated building on their property, and when she enters she sees that her husband has reproduced his office and plastered the place with newspaper clippings and push pins with thread which try to depict an imagined conspiracy. She runs back into the house where Nash has left their baby alone in a tub filling up with water. He says that Charles was watching over the child, but she can't see him because Charles has “been injected with a cloaking serum.” His own arm implant “dissolved” allowing him to see Charles. The storm that is raging outside mirrors the mental one that is devastating Nash’s mind, tearing down all the sturdy logical ideas that once thrived there. 

 

Parcher appears and tells Nash that he must get rid of Alicia because she is a national security risk. After Alicia sees Nash talking to nobody, Nash then conjures up Charles and his niece, and Charles tells him to do what Parcher said. Nash is mentally at war with himself, wanting to believe what he sees but also protective of his family. Then, his mathematical rationality bursts through with an epiphany that will not allow a delusion to extinguish. He stops Alicia and says that he realizes that Charles’s supposed niece, Marcee, never ages; therefore, she can’t be real. 

 

With Rosen present, Nash still sees Charles and his niece, and Rosen says he must return to the hospital for more treatment. Nash says the medication stops him from working, taking care of his child, and responding to his wife. He says he is a problem solver, and he needs time to figure out the solution. But Rosen points out the dilemma, since Nash’s condition is not like a mathematical mystery, and Nash’s mind can’t be the tool to fix things since the defect is in his brain. 

 

Nash does not want to return to the institution, but fears for how his condition threatens Alicia’s safety. So, he tells her to go to her mother’s place where the baby is already. But, she refuses to leave him, and touches him, saying what’s real is in his feelings, not in his mind. In this way, she expands his sense of reality.

 

Two months later, Nash visits Hansen at Princeton University. Hansen says he is an old friend, and he no longer seems to be a foe. As opposed to feeling that being apart from others allowed him to excel without distractions, Nash now sees being part of a community will help him become mentally healthy. He just wants to be able to hang out in the library. But, he appears outside, fighting his demons, as Parcher resurfaces and harasses him, while others watch as Nash argues with an illusion.

 

But, with Alicia supporting his fight to overcome his symptoms without resorting to extreme medical treatment, Nash goes back to the college library each day, writing equations as he once did, on the windowpanes, letting the real and figurative light shine upon him. He now refuses to speak to his imaginary creations. 

 

Time passes, their son grows up, and there is now a gray-haired Nash in 1978 still working on equations at the Princeton University library. He eventually engages with some students and expresses a desire to teach again. He admits to Hansen that he still sees Parcher, Charles and Marcee, but since he has ignored them, they don’t intrude anymore. Hansen says that they still haunt him, but Nash says. “They are my past. Everyone is haunted by their past.” The problem he must deal with, though, is that Nash’s past just happens to feel like it has more substance in the present than that of others. 

 

The next jump is to 1994 and Nash is an elderly man, but he is back to teaching and working on his math projects. Someone from the Nobel Prize committee visits him, saying he is being considered for the award based on his bargaining theories that have numerous economic applications and eventually even biological evolutionary considerations. Nash realizes that the visit is to make sure he doesn’t embarrass the prestigious awards ceremony by acting crazy. He admits to the possibility that he may act out because he still sees things that aren’t there. He takes newer medications, and says he is on a sort of mental “diet” where he does not indulge his fantasies. While entering the faculty dining area for the first time in years the other professors pay Nash tribute, as he saw they did many years ago for another professor, by giving him their pens. He has received the recognition he once sought.


 The story concludes with Nash’s Nobel Prize speech where he says all of his mental explorations have brought him to the conclusion that, “it is only in the mysterious equations of love that any logic or reasons can be found.” Love, a supposedly irrational area, has provided him the most meaning concerning existence. He directly thanks Alicia for all that he has come to really understand about life as a whole.


The next film is Zorba, the Greek.

Thursday, June 14, 2018

Stand and Deliver

SPOILER ALERT! The plot will be discussed.
Many of us have obstacles to overcome in order to live a better life. But, some have more barriers placed in their path that they must ascend than do others. This 1988 film, based on a true story, is one of the most inspirational stories put on film, without being sentimental, that shows the extreme effort needed by some to gain success in the presence of prejudice and the poverty that derives from it.

The movie opens with a scene of flowing water which suggests that we may be ready to visit a lush tropical locale. But it is the river that flows through Los Angeles, and it seems ironic to place nature and concrete side by side. But, the water implies that there can be something that is fertile, life-sustaining, even in an asphalt jungle. Right from the beginning we get a symbol that there can be hope here.
Jaime Escalante (Edward James Olmos, in an Oscar-nominated performance) is heading to the first day of his teaching job at Garfield High School in a Latino section of the city. We are vicariously experiencing the newness of this adventure with him as he takes in the colors, sounds, and diversity of the neighborhood. It doesn’t take long before he encounters his first obstacle. He is supposed to teach computer science, but is told that they do not have computers. The math department chairperson, Raquel Ortega (Virginia Paris) tells him that the computers were supposed to be there already, but it hasn’t happened. It’s like trying to teach carpentry without any tools. So, the bureaucracy already is working against Escalante.

Since he can’t teach computer science, Escalante must switch gears and is assigned to teach mathematics. The next problem is the students. They are unruly, and see no need to learn math. Pancho (Will Gotay) jokingly says, “I don’t need no math. I got a solar calculator with my dozen donuts.” For these young people, the only math they believe they need will be basic calculations done by the machines in fast-food stores.  Escalante uses humor to deal with the situation, trying to employ entertainment to get the class’ attention. For example, when a girl drapes herself over the desk of a boy, he says that there is only one body per desk. But there are many disruptions on his first day. There are students in the class who only speak Spanish. The bell goes off right after class begins, since some students have rigged it to sound haphazardly. His car is vandalized on the first day, a window broken and the radio removed. One student wears a jacket with a picture of Jesus on the back. It can either suggest hope, irony that the students feel that God has forsaken them, or a reminder that God helps those that help themselves.
We find out, through a discussion with a neighbor, that Escalante worked at a computer company, and left a higher paying job because he wanted to help Latino students learn so they can help themselves. He shows up at class dressed as a butcher with a cleaver and places parts of apples on the students’ desks to teach fractions. Pancho ate his apple, and when asked how much he has left, he says he only has a core. Escalante wittily mixes humor and math when he says, “You owe me a hundred percent. And I’ll see you in The People’s court.” A few students come in late, and are intimidating. But, Escalante uses his humor to deflate their threats. One gives him the finger, and the teacher calls him “finger man.” When Angel (Lou Diamond Phillips) reaches for Escalante’s pen, he warns the student that he might lose a finger. Then, he won’t be able to count to ten, which is a mathematical put-down, suggesting that using his hand is the only way he can add. He later shows the students a simple trick to use fingers to do more complicated computations, encouraging them by showing that they don’t have to be geniuses to solve problems. He basically tells them that showing contempt for education just to act cool now won’t help later in life, because “tough guys don’t do math. Tough guys fry chicken for a living.”

He has other effective classroom management techniques. He stands close to a student’s desk, interacting with that person one-on-one, as an individual, not as a member hiding out in a group. This way he can either inhibit bad behavior, or be supportive of positive academic participation. When one student, Lupe (Ingrid Oliu), rebels by refusing to take a test, he separates her from the group, placing her in a chair facing the rest of the class. Instead of his humor and antics providing entertainment, he tells her she is “the show,” and the other classmates ridicule her for being treated like a child in a time-out. Escalante turns the adolescent inclination of making fun of peers as a way to keep students in line.  

He wants his students to reach for something better out of life, so he challenges them. He says, “If the only thing you know how to do is add and subtract, you will only be prepared to do one thing: Pump gas.” He tells them they are going to learn algebra. He goads one of the “tough guys” to show that he can be smart, being able to add positive and negative numbers to get the answer of zero. He then appeals to their cultural background, to instill pride and associate it with math. He tells them, “neither the Greeks nor the Romans were capable of using the concept of zero. It was your ancestors, the Mayans, who first contemplated the zero. The absence of value. True story. You burros have math in your blood.”
At a math teacher meeting, Principal Molina (Carmen Argenziano) says that if the school doesn’t improve it will be put on probation, and will lose accreditation. Chairperson Ortega points out that there is no real help from the outside in terms of resources. One teacher is supposed to be teaching gym, not math. Another, because of better pay, is going into the aerospace industry. But Ortega’s attitude is a defeatist one, always blaming the situation, those obstacles, for the lack of success, and preaches resignation instead of an attempt to rise to the occasion. She even harbors racist inclinations against her own people, as she suggests that they would have to change the demographics of the district for the school to improve. She surrenders to the notion that they can’t teach “logarithms to seventh grade level kids.” She says that they have done all that they can, but, in contrast to this throwing in the towel attitude, Escalante says he can do more. He counters Ortega’s capitulation by saying, “students will rise to the level of expectation.” If they undercut the students confidence by saying they can’t accomplish goals, they will fail.

He tells the students they already have those extra obstacles mentioned above blocking them. In a society that preaches equality but practices prejudice, and assumes they can’t be smart, he says, “You already have two strikes against you: your name and your complexion.” Despite those imposed disadvantages, there is no reason to give up. Knowledge of mathematics will be the “the great equalizer,” leveling the playing field when they must compete for jobs. He wants them to face the reality that an employer does not want, “to hear your problems.” He wants them to understand by working hard, they can still hit a home run despite the “strikes” against them. But, they must have “ganas,” that is, the desire to achieve.

As the students begin to admire Escalante for believing that they can excel in mathematics, they give him the name of “Kemosabe,” which was the term of endearment that the Native American sidekick of the Lone Ranger on the old TV show used for his masked friend. In some interpretations it means “trusted friend.” But, it also has the connotation of someone who has your back when fighting against the enemy, which in this story turns out to be the social system that has placed those “two strikes” against these kids.
Angel begins to realize that math may be his ticket out of poverty and discrimination. But, he has to avoid the peer pressure of the other gang types who think trying to be a good student is selling out to a system that marginalizes them. So, Angel asks Escalante for a math book he can keep at home so it appears that he isn’t taking his studies seriously by carrying his book around. His teacher jokingly comments on the the absurdity of the problem when he says to Angel, “Wouldn’t want anyone thinking you’re intelligent, would you?” When he gives him the book, Angel hides it under his shirt. In this crazy upside down situation, it is not illegal drugs or stolen goods that must be hidden, but instead a book is the incriminating evidence.

The film takes time to show the obstacles in the personal lives of these students. Lupe is a surrogate mother for her siblings while her mother must work late in order to provide for her family. Angel has a sick grandmother for whom he watches over. One smart girl, Ana (Vanessa Marquez), tells Escalante that she has to drop out to work in her family restaurant. Escalante has dinner at the restaurant with his wife, Fabiola (Rosanna DeSoto), and tries to convince Ana’s father (James Victor) that his daughter can go to college and become a doctor, which is what she told Escalante she wants to be. The father has some old, sexist attitudes about girls, which are impediments to Ana’s wishes. He thinks that the longer Ana stays in school, the more likely she will become pregnant. He says that the whole family works at the restaurant, so why should she be any different. In this way, the father echoes the school math department chairperson by having low expectations for the children, and thus limiting their changes of realizing their dreams. Escalante does go a bit too far by saying that if Ana drops out, she will get fat and waste her life working in the family eatery. This statement is insulting to the father, who had worked his way up from dishwasher to owner. But, Escalante says he, too, started as a dishwasher, and both he and Ana’s father have done well by working hard for what they wanted. Even though the dad is angry at first, Ana shows up at school again, and shines as a student.
Pancho tells Escalante that they need to know how math works in the world outside of school. So, the teacher takes them on a field trip to where he worked at a computer company. A student asks what type of mathematics is displayed on a computer screen. He is told that it is calculus, and Escalante says that it will be taught to them when they go to college. The worker there tells Escalante that his daughter is taught calculus in high school. This encounter gives Escalante the idea of teaching his students calculus so that they can get college credit by passing the Advanced Placement test. Of course, the obstacles are there trying to prevent him from achieving this goal. The negative chairperson, Ortega, says that even some math teachers can’t pass that test, let alone their students who have a seventh grade reading level. She says they don’t even have the books. Escalante knows that Principal Molina wants to prevent the school from being put on probation, so the math teacher says the only way to turn the school around is from the top down, with the best students modeling success for the rest of the student body. Escalante says either he gets to teach calculus, or he walks. In order to achieve greatness, for him, there can be no compromises. Ortega tells Escalante that his students will lose what “little” (note the condescension) self-respect they have when they fail. Again, she assumes defeat. For Escalante, giving up without even trying is the greater failure. Principal Molina gives him the green light, but he must teach prerequisites such as trigonometry in the summer to be ready for the calculus in the fall of the students’ senior year. Summer school is usually for remedial courses, but Escalante wants to use it not for going backward but moving forward academically. The students toil in the heat as the air-conditioning doesn’t work, just adding another burden to deal with on their difficult road ahead.
When school begins again in the new school year, Escalante wants his students’ commitment in writing. He makes them and their parents sign contracts that guarantee that they will show up before school begins, stay late after classes are over, and even to work with him during vacation breaks. There will be no senior year slacking off. The kids are not thrilled, and one says that Escalante likes scaring them into doing stuff, but that it is getting old. But, Escalante is persistent. One girl, Claudia (Karla Montana), trying to convince her mother of the importance of calculus, tells her that it is fascinating that Sir Isaac Newton invented calculus to understand planetary motion. Her mother, another blocking agent that must be overcome, is reluctant to sign the contract, preaching the sexist idea that boys don’t like smart girls. Claudia tells her she wants to learn so that she doesn’t have to rely on a boy. Her mother, probably understanding the vulnerability of relying on a man, is inspired by her daughter’s desire to be self-sufficient, and signs the contract. Pancho tells Escalante that he has a job offer to work a forklift and he will be making more money than the teacher. They go for a ride in Pancho’s car, and Escalante asks him quickly to decide which way to turn. After the turn, he pulls over and says that if Pancho sticks with the course, he will be able to design machines not just try to operate or repair them. His point is to not just have a short range plan to make money to deal with immediate financial needs, but to work toward a fulfilling future. As Escalante tells Pancho, “You only see the turn. You don’t see the road ahead.”

Escalante sometimes goes too far in pushing his students. Claudia looks outside of the classroom window, watching other youths enjoying the outdoors while she is inside enduring her teacher’s grueling lessons. She gets up and starts to leave, and Escalante says she probably has another date, and has more boyfriends than Elizabeth Taylor. She angrily tells him before leaving that she doesn’t appreciate him using her personal life for the class’ entertainment. He does run after her, though, and she breaks down, saying that she doesn’t see her boyfriend, hasn’t had time to help her mother, or even to take care of personal grooming. (While rushing to catch up with Claudia, Escalante puts his hand over the left side of his chest. The act is a foreshadowing of what is to happen later). She makes Escalante realize that she is sacrificing a great deal of her young life to study. Angel takes his ailing grandmother to a public clinic where they have to wait a long time because they can’t afford decent health care. When he is late for class, Escalante kicks him out, and Angel upends a desk, enraged that his teacher won’t even stop to listen to Angels’ reason for his tardiness. Later, Angel crashes Escalante’s holiday dinner, bringing his grandmother, who explains why Angel was late for class. Angel justifies his action by telling his teacher that he needs the math class to get “a good career.” He may have gone over the line, but he has demonstrated to Escalante that Angel has the “ganas” to succeed.
But, Escalante’s zeal to accomplish a greater good for many others has shortchanged his own family. Besides putting in so many extra hours teaching calculus, he also volunteers to teach English as a Second Language to immigrants, and recently started helping out at junior high schools. His driven nature catches up to him, because the human body has its own way of becoming an obstacle to unlimited accomplishment. Escalante suffers a heart attack while teaching an ESL class. To emphasize how the students run into roadblocks at every turn, with only two weeks to the AP test, their substitute is a music teacher. Escalante knows how little support and time is left, and he returns to the class after only being sidelined for a couple of days. He so inspires his class that even the struggling Pancho excels, rising to meet what was expected of him.

They take the AP test, and after the exam they jump into the ocean together to regain a sense of freedom and relief from the trying ordeal they have endured. But, the water seems to imply a sort of baptism and rebirth, cleansing themselves of their old life and allowing an opportunity to start again freed from previous restrictions. Pancho is working under a car, grease all over his hands and arms when the test results come in the mail, He tries to open the envelope without sullying it with the grime of his labors. He is joyous as he learns that he has passed which will allow him not to suffer under, but instead climb above the weariness of manual labor.
At a ceremony at the school, Principal Molina tells the gathering that less than two percent of the students in Southern California attempt the AP test in calculus. However, all eighteen students from Garfield High School who took the test passed. The students feel pride maybe for the first time in their lives and they show their gratitude to Escalante by giving him a plaque expressing their thanks.
Their celebrating does last long as the system imposes another obstacle before them. The eighteen who passed the AP test receive notice that they are to be investigated by the Educational Testing Service for possible cheating. On the surface, they are considered suspect because of the uniformity of their answers, so the possibility of copying from each other is raised. Escalante says that the ETS is not taking into account that the students were in one class and taught rigorously as a team by one teacher, which accounts for the uniformity of their test scores. Angel is particularly enraged, but he expresses his anger and despair at being treated unfairly to his friend, while they drive, by using the new knowledge he has learned. He says, “The stars aren’t really there, ese. No, what you’re looking at is where they used to be, man. It takes the light a thousand years to reach the earth. You know, for all we know, they burned out a long time ago, man. God pulled the plug on us. He didn’t tell nobody.” Despite the fact that he is not being recognized for his intelligence and what he has learned, he has enhanced his life because he can comment on it in scientific and religious terms. A police car pulls them over because Angel hangs out of the car and yells at the police as representatives of the system that treats him badly. Angel says to them, “That’s all you know!” It is a simple sentence but it carries meaning. He is pointing out the ignorance of prejudice that assumes a certain ethnic type is worthless, and by making this point, shows his own intelligence. When the police frisk Angel, the only “weapon” he has is a pencil, which negates their assumption that he is dangerous, and points to his immersion in an academic, not criminal, life.
Although the tests are initially screened anonymously, once they are flagged, the ETS discovers the names of the students and then can investigate or not. In this case they send an African American, Dr. Pearson (Rif Hutton), and a Latino, Dr. Ramirez (Andy Garcia, in a very early role). It seems that the ETS is going out of its way to appear fair to minorities by having two men who are from oppressed ethnic backgrounds represent them. They gather the students in a classroom and try to make them confess to cheating, as if it is a foregone conclusion. Ramirez is particularly unsavory when he uses his own background to say he understands why they might feel a need to take a shortcut by cheating. His remarks are insulting and demeaning to his own ethnic group. Outraged by the way they are treated, Angel appears to give the men what they want to hear. He says that they have been caught. He acquired the test ahead of time from the mailman, who he strangled, and whose body is “decomposing in my locker.” The rest of the class laughs hysterically at how Angel satirized the assumption that they are cheaters and criminals.

The math chairperson, Ortega, again assumes the worst concerning the students. Escalante asks her if she thinks that the students cheated. She says that they care for Escalante so much, they would do anything to please him, even cheat. She says that most people who are accused of a crime are guilty. So much for the American legal tenet that people are presumed innocent until proven guilty. Escalante starts to despair. He tells his wife that his students, “lost the confidence in the system they’re finally qualified to be part of. “ But, after his kids borrowed his car so they could detail it and make it look cool, he realizes that his wife was right when she said, “those kids love you.”

Escalante confronts Pearson and Ramirez. They argue that all the students finished the test ahead of time, which to them implies cheating. Escalante, arguing against the assumption of guilt, says that “they should be rewarded, not punished” for the swiftness of their work. He tells them that what no one wants to talk about here is racism, which has blanketed the students’ accomplishment in doubt. Escalante says that if the same results involved youths from Beverly Hills High School, there would be no suspicion of cheating. He tells the ETS men, “Those scores would never have been questioned if my kids did not have Spanish surnames and come from barrio schools.”
Escalante refuses to let his kids be stopped on their road to success by the unfairness of prejudice. The students ask for a retest, but they only have one day for review. Escalante says that this time the test will be harder. They must overcompensate in order not to be considered suspect. They can’t have wandering eyes, or wear clothes with pockets. He praises them, trying to build up their confidence, by saying that they are “the true dreamers” (a word which reverberates with immigrants today), and says they are “the champs.” Late in the evening, while some continue to study at Escalante’s house, he takes on a more dire tone. Claudia asks Escalante if he’s worried they will “screw up royally tomorrow.” His response is, “Tomorrow’s another day. I’m worried you’re gonna screw up the rest of your lives.” He may sound harsh here, but he is trying to tell them the same thing he said to Pancho, which is that the test is only one step on a tough, long journey to realize their full potential.

They take the retest. Anna has to leave early because she has an interview for a college scholarship. She is already changing her life, ready to be the first person in her family to get a higher education. While Escalante waits for the test results, the school secretary tells him, after two years that he has been at the school, they finally received the computers. Escalante satirizes the absurdity of waiting for help from a bogged down educational system by saying, “Yep, that will do it.”
The new test results vindicate the original ones. All of the students receive passing or exemplary scores. The film states at the end that from 1982 to 1987, the number of students passing the AP calculus test at Garfield High School went from eighteen to eighty-seven. As Escalante said, despite numerous obstacles in their path, “students will rise to the level of expectation.”

After a week off, the next film is The Swimmer.