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• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
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Unit 2: Solving equations & inequalities

About this unit.

There are lots of strategies we can use to solve equations. Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions.

Linear equations with variables on both sides

• Why we do the same thing to both sides: Variable on both sides (Opens a modal)
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• Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
• Equation with variables on both sides: fractions (Opens a modal)
• Equation with the variable in the denominator (Opens a modal)
• Equations with variables on both sides Get 3 of 4 questions to level up!
• Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

Linear equations with parentheses

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• Equations with parentheses Get 3 of 4 questions to level up!
• Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!
• Reasoning with linear equations Get 3 of 4 questions to level up!

Analyzing the number of solutions to linear equations

• Number of solutions to equations (Opens a modal)
• Worked example: number of solutions to equations (Opens a modal)
• Creating an equation with no solutions (Opens a modal)
• Creating an equation with infinitely many solutions (Opens a modal)
• Number of solutions to equations Get 3 of 4 questions to level up!
• Number of solutions to equations challenge Get 3 of 4 questions to level up!

Linear equations with unknown coefficients

• Linear equations with unknown coefficients (Opens a modal)
• Why is algebra important to learn? (Opens a modal)
• Linear equations with unknown coefficients Get 3 of 4 questions to level up!

Multi-step inequalities

• Inequalities with variables on both sides (Opens a modal)
• Inequalities with variables on both sides (with parentheses) (Opens a modal)
• Multi-step inequalities (Opens a modal)
• Using inequalities to solve problems (Opens a modal)
• Multi-step linear inequalities Get 3 of 4 questions to level up!
• Using inequalities to solve problems Get 3 of 4 questions to level up!

Compound inequalities

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• A compound inequality with no solution (Opens a modal)
• Double inequalities (Opens a modal)
• Compound inequalities examples (Opens a modal)
• Compound inequalities review (Opens a modal)
• Solving equations & inequalities: FAQ (Opens a modal)
• Compound inequalities Get 3 of 4 questions to level up!

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A Guide to Problem Solving

When confronted with a problem, in which the solution is not clear, you need to be a skilled problem-solver to know how to proceed. When you look at STEP problems for the first time, it may seem like this problem-solving skill is out of your reach, but like any skill, you can improve your problem-solving with practice. How do I become a better problem-solver? First and foremost, the best way to become better at problem-solving is to try solving lots of problems! If you are preparing for STEP, it makes sense that some of these problems should be STEP questions, but to start off with it's worth spending time looking at problems from other sources. This collection of NRICH problems  is designed for younger students, but it's very worthwhile having a go at a few to practise the problem-solving technique in a context where the mathematics should be straightforward to you. Then as you become a more confident problem-solver you can try more past STEP questions. One student who worked with NRICH said: "From personal experience, I was disastrous at STEP to start with. Yet as I persisted with it for a long time it eventually started to click - 'it' referring to being able to solve problems much more easily. This happens because your brain starts to recognise that problems fall into various categories and you subconsciously remember successes and pitfalls of previous 'similar' problems." A Problem-solving Heuristic for STEP Below you will find some questions you can ask yourself while you are solving a problem. The questions are divided into four phases, based loosely on those found in George Pólya's 1945 book "How to Solve It". Understanding the problem

• What area of mathematics is this?
• What exactly am I being asked to do?
• What do I know?
• What do I need to find out?
• What am I uncertain about?
• Can I put the problem into my own words?

Devising a plan

• Work out the first few steps before leaping in!
• Have I seen something like it before?
• Is there a diagram I could draw to help?
• Is there another way of representing?
• Would it be useful to try some suitable numbers first?
• Is there some notation that will help?

Carrying out the plan STUCK!

• Try special cases or a simpler problem
• Work backwards
• Guess and check
• Be systematic
• Work towards subgoals
• Imagine your way through the problem
• Has the plan failed? Know when it's time to abandon the plan and move on.

Looking back

• Have I answered the question?
• Sanity check for sense and consistency
• Check the problem has been fully solved
• Read through the solution and check the flow of the logic.

Throughout the problem solving process it's important to keep an eye on how you're feeling and making sure you're in control:

• Am I getting stressed?
• Is my plan working?
• Am I spending too long on this?
• Could I move on to something else and come back to this later?
• Am I focussing on the problem?
• Is my work becoming chaotic, do I need to slow down, go back and tidy up?
• Do I need to STOP, PEN DOWN, THINK?

Finally, don't forget that STEP questions are designed to take at least 30-45 minutes to solve, and to start with they will take you longer than that. As a last resort, read the solution, but not until you have spent a long time just thinking about the problem, making notes, trying things out and looking at resources that can help you. If you do end up reading the solution, then come back to the same problem a few days or weeks later to have another go at it.

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1.1: Introduction to Problem Solving

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• Michelle Manes
• University of Hawaii

The Common Core State Standards for Mathematics ( http://www.corestandards.org/Math/Practice ) identify eight “Mathematical Practices” — the kinds of expertise that all teachers should try to foster in their students, but they go far beyond any particular piece of mathematics content. They describe what mathematics is really about, and why it is so valuable for students to master. The very first Mathematical Practice is:

Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.

This chapter will help you develop these very important mathematical skills, so that you will be better prepared to help your future students develop them. Let’s start with solving a problem!

Draw curves connecting A to A, B to B, and C to C. Your curves cannot cross or even touch each other,they cannot cross through any of the lettered boxes, and they cannot go outside the large box or even touch it’s sides.

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it).

• What did you try?
• What makes this problem difficult?
• Can you change the problem slightly so that it would be easier to solve?

Problem Solving Strategy 1 (Wishful Thinking).

Do you wish something in the problem was different? Would it then be easier to solve the problem?

For example, what if ABC problem had a picture like this:

Can you solve this case and use it to help you solve the original case? Think about moving the boxes around once the lines are already drawn.

Here is one possible solution.

Problem Solving in Mathematics

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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History Daily

New Orleans teens solve 2,000-year-old math problem

Posted: May 11, 2024 | Last updated: May 11, 2024

Some ingenuity

Teacher Michelle Blouin Williams initiated a math competition with a bonus question tasking students to develop a new proof for the Pythagorean theorem using trigonometry, without anticipating that anyone would successfully tackle the challenge.  

She said, “I was just looking for some ingenuity.”   

Williams' expectations

Calcea Johnson and Ne’Kiya Jackson exceeded Williams' expectations by successfully solving the challenge in 2023.   

St. Mary's Academy

These two teenagers, who were seniors at St. Mary's Academy in New Orleans, a renowned Catholic school for girls with impressive college acceptance and graduation rates, were featured on CBS News' "60 Minutes" to discuss their accomplishment.   

$500 reward

Initially enticed by the math competition's $500 reward, their determination to complete the task intensified as they delved into the complex bonus question.   

Over a period of two months, these high school seniors dedicated themselves to completing their proof.  

Pages and pages

During an interview with "60 Minutes," CeCe Johnson, the mother of Calcea, said “It was pages and pages and pages of, like, over 20 or 30 pages for this one problem.”  

The garbage can

Her father, Cal Johnson, added, “Yeah, the garbage can was full of papers, which she would, you know, work out the problems and — if that didn’t work she would ball it up, throw it in the trash.”  

Upon completion

Upon completion, the teachers at St. Mary's Academy acknowledged the significance of Calcea and Ne'Kiya's achievement and submitted their proof to the American Mathematical Society for recognition at a conference in March 2023, where the students presented their findings.  

The Pythagorean theorem

The Pythagorean theorem, at its core, states that knowing the lengths of two sides of a right triangle allows you to determine the length of the third side using the formula a² + b² = c².   

While commonly attributed to the Greek mathematician Pythagoras, historical evidence suggests its existence in Babylon and Iron Age India. This theorem finds practical applications in construction, architecture, navigation, and surveying.  

A mathematical proof

A mathematical proof is a logical argument that demonstrates the truth of a mathematical theorem. American mathematician Daniel Kane likens proofs to essays, but rooted in mathematical concepts.  

Using trigonometry

As per the "60 Minutes" segment, “there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible.”  

Mathematician Elisha Loomis

Back in 1927, mathematician Elisha Loomis made a statement in his book "The Pythagorean Proposition." Loomis contended that a trigonometric proof of the theorem was impossible as it would create a circular argument.  

Stuart Anderson

Stuart Anderson, a retired mathematics professor from Texas A&M University-Commerce, mentioned to Scientific American, “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem.”  

Trigonometric functions

Loomis contended that attempting to use trigonometric functions to prove the Pythagorean theorem would result in a circular reasoning loop, as the functions themselves are built upon the theorem. This, according to Loomis, would constitute a fundamental mathematical mistake.  

The law of sines

As reported by Scientific American, the teenagers challenged this notion during their presentation in 2023 and stated that “a trigonometric identity called the law of sines didn’t depend on the Pythagorean theorem and that they could use it to prove the theorem.”  

Final peer review

Calcea and Ne'Kiya are now part of a select few who have achieved a similar milestone, such as mathematician Jason Zimba, who developed a new proof in 2009. They submitted their proof for the final peer review earlier this year and are actively working on crafting additional proofs.  

In recognition

In recognition of their accomplishment, the teenagers were honored with the keys to the city of New Orleans and received commendations from the governor of Louisiana, along with other public acknowledgments.  

Widespread recognition

Despite the widespread recognition of their accomplishment which “blew up,” as Ne'Kiya expressed it, the two students maintain their humility and even chuckled at being labeled as geniuses.  

Surprised and skeptical

Upon the revelation of their achievement, there were individuals who appeared surprised and skeptical, dismissing the news as untrue, as mentioned by St. Mary's president Pamela Rogers during the interview.  

African Americans

“They were saying, ‘Oh, they could not have done it. African Americans don’t have the brains to do it.’ ... People — have a vision of who can be successful. And — to some people, it is not always an African American female. And to us, it’s always an African American female.”  

The significant reaction

When questioned by interviewer Bill Whitaker about their thoughts on the significant reaction, Ne’Kiya said, “Probably because we’re African American, one. And we’re also women. So I think — oh, and our age. Of course our ages probably played a big part.”  

Great mathematical achievement

“I’d like to actually be celebrated for what it is. Like, it’s a great mathematical achievement,” she continued.  

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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Jim Simons, Math Genius Who Conquered Wall Street, Dies at 86

Using advanced computers, he went from M.I.T. professor to multibillionaire. His Medallion fund had 66 percent average annual returns for decades.

By Jonathan Kandell

Jim Simons, the prizewinning mathematician who abandoned a stellar academic career, then plunged into finance — a world he knew nothing about — and became one of the most successful Wall Street investors ever, died on Friday in his home in Manhattan. He was 86.

His death was confirmed by his spokesman, Jonathan Gasthalter, who did not specify a cause.

After publishing breakthrough studies in mathematics that would play a seminal role in quantum field theory, string theory and condensed matter physics, Mr. Simons decided to apply his genius to a more prosaic subject — making as much money as he could in as short a time as possible.

So at age 40 he opened a storefront office in a Long Island strip mall and set about proving that trading commodities, currencies, stocks and bonds could be nearly as predictable as calculus and partial differential equations. Spurning financial analysts and business school graduates, he hired like-minded mathematicians and scientists.

Mr. Simons equipped his colleagues with advanced computers to process torrents of data filtered through mathematical models, and turned the four investment funds in his new firm, Renaissance Technologies , into virtual money printing machines.

Medallion, the largest of these funds, earned more than $100 billion in trading profits in the 30 years following its inception in 1988. It generated an unheard-of 66 percent average annual return during that period.

That was a far better long-term performance than famed investors like Warren Buffett and George Soros achieved.

“No one in the investment world even comes close,” wrote Gregory Zuckerman, one of the few journalists to interview Mr. Simons and the author of his biography, “The Man Who Solved the Market.”

By 2020, Mr. Simons’s approach to the market — known as quantitative, or quant, investing — accounted for almost a third of Wall Street trading operations. Even traditional investment firms that relied on corporate research, instinct and personal contacts felt compelled to adopt some of Mr. Simons’ computer-driven methodology.

For much of its existence, Renaissance funds were the largest quant funds on Wall Street, and its style of investing spurred a sea change in the way hedge funds traded and made money for their wealthy investors and pension funds.

By the time he retired as chief executive of the business in 2010, Mr. Simons was worth $11 billion (almost $16 billion in today’s currency), and a decade later his fortune had doubled.

While he continued to oversee his funds as Renaissance chairman, Mr. Simons increasingly devoted his time and wealth to philanthropy. The Simons Foundation became one of the largest private funders of basic science research. And his Flatiron Institute used cutting-edge computational techniques for research into astrophysics, biology, mathematics, neuroscience and quantum physics.

James Harris Simons was born on April 25, 1938, in Cambridge, Mass., the only child of Matthew Simons, the general manager of a shoe factory, and Marcia (Kantor) Simons, who managed the home. A prodigy in mathematics, he did his undergraduate work at the Massachusetts Institute of Technology and was only 23 when he received his doctorate from the University of California, Berkeley.

Beginning in 1964, Mr. Simons taught at M.I.T. and Harvard University while simultaneously working as a breaker of Soviet codes at the Institute for Defense Analyses, a federally funded nonprofit group. But he was fired from the institute in 1968 for publicly expressing strong anti-Vietnam War views.

Over the next decade, he taught mathematics at Stony Brook University on Long Island, part of the State University of New York, and became chairman of its math department. While running the department he won the nation’s highest prize in geometry in 1975.

Then, in 1978, he abandoned his scholarly career and founded Monemetrics, an investment company with offices in a small shopping mall in Setauket, just east of Stony Brook on the North Shore of Long Island. He had never taken a financial course or shown more than a passing interest in the markets. But he was convinced that he and his small team of mathematicians, physicists and statisticians — mainly former university colleagues — could analyze financial data, identify market trends and make profitable trades.

After four roller coaster years, Monemetrics was renamed Renaissance Technologies. Mr. Simons and his growing staff of former scholars initially focused on currencies and commodities. Every conceivable type of data — news reports of political unrest in Africa, bank statistics from small Asian nations, the rising price of potatoes in Peru — was fed into advanced computers to glean patterns that enabled Renaissance to score consistently huge annual returns.

But the real bonanza came when Renaissance plunged into equities, a much larger market than currencies and commodities.

Stocks and bonds were long seen as the purview of Wall Street brokerages, investment banks and mutual fund companies whose young, tireless M.B.A.s analyzed listed companies and turned over their research results to senior wealth managers, who then relied on their experience and instinct to pick market winners. They initially scoffed at the math nerds at Renaissance and their quantitative methods.

A few times, Mr. Simon’s methodology led to costly mistakes. His company used a computer program to buy so many Maine potato futures that it nearly controlled the market. This met with the opposition of the Commodity Futures Trading Commission, the regulatory agency in charge of futures trading. As a result, Mr. Simons had to sell off his investments and miss out on a large potential profit.

But far more often he was so successful that his biggest problem was hiding his trades and research techniques from competitors. “Visibility invites competition, and, with all due respect to the principles of free enterprise — the less the better,” he wrote in a letter to clients.

Business rivals weren’t the only ones eyeing Mr. Simons’s results with envy or suspicion. In 2009, he faced a rebellion from outside investors over the enormous disparity in the performance of different Renaissance Technologies portfolios. The previous year, the Medallion Fund, which was available only to Renaissance present and past employees, registered an 80 percent gain, while the Renaissance Institutional Equities Fund, offered to outside investors, dropped 16 percent in 2008.

In July 2014, Mr. Simons and his firm drew bipartisan condemnation from the Senate Permanent Subcommittee on Investigations for using financial derivatives to disguise day-to-day trading as long-term capital gains. “Renaissance Technologies was able to avoid paying more than $6 billion in taxes,” asserted Senator John McCain, the Arizona Republican, in his opening statement at the subcommittee hearing.

Both Mr. Simons and his onetime co-chief executive, Robert Mercer, were among the largest financial contributors to politicians and political causes. While Mr. Simons generally backed liberal Democrats, Mr. Mercer was fervently right-wing and became a leading funder of Donald Trump’s presidential campaigns .

In 2017, Mr. Simons, then chairman of Renaissance Technologies, fired Mr. Mercer as C.E.O. because his political activities were provoking other key Renaissance executives to threaten to resign. Mr. Mercer stayed on as a researcher. According to both men, they remained friendly and continued to socialize.

In 2011, his foundation gave $150 million to Stony Brook University, with most of the money going to research in medical sciences. At the time, it was the biggest gift ever bestowed in SUNY’s history.

Last year, the foundation outdid that gift with a $500 million donation to Stony Brook, which called it the largest unrestricted endowment gift to a higher education institution in American history.

As he became older and wealthier, Mr. Simons enjoyed a lavish life style. He purchased a 220-foot yacht for $100 million, bought a Fifth Avenue apartment in Manhattan and owned a 14-acre estate in East Setauket, overlooking Long Island Sound. A chain-smoker, he refused to put out his cigarettes in offices or at conferences and willingly paid fines instead.

His first marriage, to Barbara Bluestein, a computer scientist, with whom he had three children — Elizabeth, Nathaniel and Paul — ended in divorce. He then married Marilyn Hawrys, an economist and former Stony Brook undergraduate who received her doctorate there. They had two children, Nicholas and Audrey.

Paul Simons, 34, was killed in a bicycle accident in 1996, and Nicholas Simons, 24, drowned off Bali, Indonesia, in 2003. His wife and other children survive him, as do five grandchildren and one great-grandson.

Mr. Simons lamented to a friend about the deaths of his sons, according to his biographer, saying, “My life is either aces or deuces.”

Hannah Fidelman contributed reporting.

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How Leaky Datasets Undermine AI Math Reasoning Claims

Questions over tests of ai math abilities suggest we may never know how capable intelligent machines computers can become..

Back in 2019, a group of computer scientists performed a now-famous experiment with far-reaching consequences for artificial intelligence research. At the time, machine vision algorithms were becoming capable of recognizing a wide range of objects with some recording spectacular results in the standard tests used to assess their abilities.

But there was a problem with the method behind all these tests. Almost all the algorithms were trained on a database of labelled images, known as ImageNet. The database contained millions of images which had been carefully described in human-written text to help the machines learn. This effort was crucial for the development of machine vision and ImageNet became a kind of industry standard.

In this way, the computer scientists used a subset of the images to train algorithms to identify a strawberry, a table, a human face and so on, using labelled images in the dataset. They then used a different subset of images to test the algorithms. Over time, computer scientists claimed that their algorithms were becoming increasingly good at recognizing objects in the real world.

Image Recognition

But privately, researchers began to wonder whether this was really true. Because the ImageNet database was becoming so famous, an alternative explanation was that its images, or ones very like them, were leaking into the real world. So AI systems trained on them were just recognizing images they had already seen.

At the time, there was no way to test this because there were no high-quality image databases that hadn’t already been used to train the algorithms.

All that changed when a team from the University of California, Berkeley, created a new dataset of carefully labelled images that they knew the algorithms could not have seen. They then asked the algorithms to identify the objects in the images and found they weren’t as good as everyone had claimed.

Their experiment became a famous example of the pitfalls of relying on single databases for testing machines. Without careful management of this database, AI systems can seem to be good at a task in general but are really only repeating what they have already learnt.

That brings us to the current generation of AI systems which are good at solving certain types of mathematics problems written out in words. For example, “ James writes a 3-page letter to 2 different friends twice a week. How many pages does he write a year? ”.

The fact that AI systems can answer questions like this suggests they are able to reason. In fact, there is a special database called GSM8K that computer scientists use to test AI system’s reasoning ability. This question is taken from there.

GSM8K is a “dataset of 8.5K high quality linguistically diverse grade school math word problems created by human problem writers.” It consists of some 7500 questions for training an AI system and 1000 questions to test the system.

Over the years, AI systems have become increasingly better at answering these questions. That has led to various claims that AI systems are becoming better at the kind of reasoning needed to solve these problems.

But there is another possibility. This is that GSM8K has become so well known that the test questions have begun to leak into the wild. As a result, AI systems may come across them during their broader benchmark training. So rather than answering them by reasoning, they could just be repeating the answer they saw during their training.

“There is growing concern that some of this performance actually reflects dataset contamination, where data closely resembling benchmark questions leaks into the training data, instead of true reasoning ability,” say Hugh Zhang and colleagues at Scale AI, a start-up based in San Francisco focused on cleaning data for use by AI systems.

Following the lead by the Berkeley researchers, the Scale AI team decided to test this idea by developing their own mathematics test of 1250 questions. They call this GSM1k and have carefully ensured that it closely resembles the GSM8K test but has never been published.

“We took extensive efforts to ensure that GSM1k had a similar distribution of difficulty to GSM8k to ensure an apples-to-apples comparison,” they say. “We ensure that the two benchmarks are comparable across important metrics such as human solve rates, number of steps in solution, answer magnitude, and more.”

They then tested a wide range of AI systems on the GSM1k problems to see how well they performed. And the results make for interesting reading.

It turns out that a large number of AI systems perform significantly worse on the new data set than on the original. “When evaluating leading open- and closed-source LLMs on GSM1k, we observe accuracy drops of up to 13 per cent,” say Zhang and co.

The team point to several systems that seem particularly vulnerable, such as the French AI system Mistral and Microsoft’s smaller AI system, Phi.

Reasoned Response

However, others show little or no drop in performance. These include ChatGPT, Claude and Gemini. Zhang and co say that these models might be better at mathematical reasoning or that their model builders are more careful about data contamination.

The team also ask these systems to generate questions from GSM8K. It turns out that their ability to do this is closely correlated with the difference in their ability to answer GSM1k and GSM8k questions. This strongly suggests the models have partially memorized examples from GSM8k, say Zhang and co.

It’s not all bad news, however, “Many models, even the most heavily overfit families, show strong signs of generalizable mathematical reasoning,” they conclude.

That’s interesting work that reveals the limitations of the benchmarking processes used to test the ability of AI systems. Even though these tests show that there has been significant progress in the reasoning ability of AI systems in recent years, caution is needed in interpreting progress.

The bigger question is how more advanced AI systems can be benchmarked accurately, particularly when the datasets are so difficult to curate and as their abilities become superhuman. It raises the very real possibility that at some point in the future, we will never know the true capability of these machines.

Ref: A Careful Examination of Large Language Model Performance on Grade School Arithmetic : arxiv.org/abs/2405.00332

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Davidson College Goldwater Scholars Apply Math, Immunology Research to Widespread Problems

Izzie Meyers ’25 has committed to studying immunology to find better treatment and potential cures for autoimmune disorders. Kenan Wood ’25 has worked on computer programs to make financial transactions more secure and consumer friendly.

The two Davidson College students are among 438 new  Goldwater Scholarship recipients for the 2024-2025 academic year. More than 5,000 students from colleges across the United States applied for the award, which recognizes undergraduates who plan to pursue research careers in the natural sciences, mathematics and engineering.

Considered the “preeminent undergraduate award of its type in these fields,” the Goldwater Scholarship was founded by the U.S. Congress in 1986 to honor former U.S. Senator Barry Goldwater. 

Here’s a little bit more about Meyers and Wood. 

Izzie Meyers ’25

When Izzie Meyers’ younger brother Zac started losing weight and energy and was excessively thirsty all the time, the family’s journey with Type 1 diabetes began.

Their mom, Melissa, recognized the symptoms early on and got him to a doctor. That quick medical attention and diagnosis helped protect Zac, then 11, from the potentially life-threatening consequences of his pancreas’s inability to produce enough insulin.

Izzie Meyers remembers how Zac learned to adjust to a routine of monitoring blood sugar, carefully scheduling meals and counting carbohydrates. Melissa Meyers set a 3 a.m. alarm each night to check on him.

Unlocking the puzzles of diabetes and other autoimmune diseases and seeking better treatments and cures has since become a lifelong goal for Izzie Meyers. At Davidson, she majors in biology, with a minor in Hispanic Studies, participated in the Duke-Davidson Immunology Partnership, and is a North Carolina GlaxoSmithKline Foundation Women in Science Scholar. She has impressed professors with her intellect, determination, work ethic and desire to learn more.

“I have worked with Izzie practically from the first day she set foot at Davidson,” said Sophia Sarafova, an associate professor of biology. “Right there and then she declared that she would do immunology research because she wanted to help people with autoimmune diseases. It was a great pleasure to interact with her over the past three years as she made good on her promise and then some. She has been an outstanding addition to my lab, working both hard and smart, which will result in an honors thesis next semester.”

This past January, Meyers received the prestigious Ray Owen Poster Prize, normally awarded to graduate students, at the Midwinter Conference of Immunologists at Asilomar. Her presentation centered on her lab research and discoveries related to T cell development. The crucial cells help protect the body from infection. 

“I love the problem-solving aspect of this kind of research,” she said. “It’s so great to start with this hypothesis, and whether it works or it doesn’t, you learn, that’s a yes, or that’s a no. And if it’s a no, how can I modify this so that it does work?”

Wanting to Know Why

Meyers grew up in suburban Pittsburgh and has three younger brothers, including Zac. She has her own autoimmune issues, including Celiac disease, as well as dairy, egg and peanut allergies. 

“I’m the problem child of the family,” she joked. “And I’m the one who turned to science to figure out why. Is it environmental — some kind of toxin, or a virus? These are all areas I want to research.”

She attended a large public high school in Pittsburgh and came to Davidson because she wanted a smaller college where she could conduct graduate-level research as an undergraduate.

She’s worked in the laboratory at the Rangos Research Center at the University of Pittsburgh Medical Center Children’s Hospital during college breaks. She’ll spend half of this summer at Rangos, and the second half in Zambia with the Davidson in Mwandi program.

After graduation next year, she plans to pursue a dual M.D./Ph.D. in immunology. She’d like to work with children. As a physician-scientist, she will conduct research leading to treatments and cures for autoimmune diseases like Type 1 diabetes. Her other interests are varied, and extensive.

“Don’t make the mistake of thinking about Izzie in just one dimension,” Sarafova said. “Our conversations during long incubation times in the lab went in all directions, especially second language acquisition, the oddities of the half a dozen languages we know between us — (none overlapping other than English) — culinary escapades, and world travel.

“I am looking forward to finding out how she will apply all of her talents for the betterment of society in the fields she chooses to pursue.”

That will involve spending a lot of time in laboratories, which is just fine with Meyers.

“The lab is definitely my happy place at Davidson,” she said. “I’ve loved every minute of it.”

She’s hopeful about future medical breakthroughs, especially when she sees better methods of controlling diabetes. You can look to her brother Zac as proof. As insulin pumps and knowledge about the disease have improved, his life has, too. 

Now 17 and a healthy, active high school athlete, he’s a standout cook and plans to go to culinary school to become a pastry chef.

Kenan Wood ’25

If you worry about the security of financial transactions or getting the best deal on your travel plans, Kenan Wood ’25 is on the case.

Wood, whose early proclivity for mathematical mysteries had him taking high-level courses as a young kid, designs programs to make things such as trading cryptocurrency and buying airline tickets safe from hackers and a better deal for consumers.

His research revolves largely around an area of trading known as expiring assets. Think empty seats on planes or available rooms at resorts. If they’re not booked, everyone’s losing. But with more efficient programming, those seats and rooms get filled, consumers save money, and airlines and resorts still make money.

That involves fine-tuning application systems so that computers everywhere would immediately sync to accurately reflect quickly changing data. 

“How do you get all the computers in the world to agree?,” Wood said. “If I pay for something in cryptocurrency, you have to have everyone agree that it was paid, and here’s how much was paid. You can’t have ambiguity.”

Thinking Outside of the Box

Wood grew up on his grandfather’s cattle farm near Asheville, North Carolina. He developed a deep love for the outdoors, spending time hiking, fishing, hunting and competing in archery contests. 

His mom, Rexann Wood, who’d majored in chemistry and botany in college, homeschooled him and his four younger siblings. He was about five when he developed a strong interest in and excitement about math. His parents enrolled him in an advanced math club, where he excelled through his middle and high school years. 

“That really helped me grow, and learn how to solve problems,” he said. “And it really deepened my interest in research.”

He came to Davidson on a Golden LEAF Scholarship, which supports students from underrepresented areas of North Carolina who attend participating colleges in the state and plan to return to their communities.

In his freshman year, he started research with Jonad Pulaj, assistant professor of mathematics and computer science at Davidson. Pulaj supported Wood’s research with funding through that summer. That fall, Pulaj brought Wood into an ongoing collaboration with Hammurabi Mendes, associate professor of mathematics and computer science, and Maurice Herlihy, a computer science professor at Brown University.

The team’s research centered around Automated Market Makers (AMMs), a type of exchange that uses algorithms to trade digital assets.

Mendes describes how Wood proposed applying geometry concepts to complex programming issues so that computers in large networks would align. He said Wood wrote the mathematical proofs for a paper the team has submitted for publication.

His research “is essentially a mathematical and computational framework for trading assets electronically,” Mendes said.  Impressed with Wood’s research, Pulaj encouraged him to apply for the Goldwater scholarship. “Having closely worked with and overseen Kenan’s research progress over the past two years, I was deeply impressed with his independence, creativity and passion while tackling and making progress in non-trivial problems both in mathematics and computer science,” Pulaj said. “This rare combination of traits, together with a strong sense of intellectual curiosity and a desire to pursue research at the highest levels made Kenan an ideal candidate for the Goldwater Scholarship.”

A recipient of Davidson’s Sophomore Computer Science Award and William D. Vinson Mathematics Award, Wood has delved into research about ensuring computers in a network remain in sync, despite attacks and network errors. A paper on that research has also been submitted for publication.

His long-term goals are to pursue a Ph.D. in distributed computation and continue conducting research on secure distributed systems and provable cryptographic security — and his experience at Davidson has prepared him well for the rigors of graduate study and research.

“Kenan operates as a graduate student: we propose problems and general approaches, and he comes back with detailed analysis and ideas of his own,” Mendes said. “He is very strong technically. He listens very carefully to other people’s comments, engages actively in the conversation, and gives suggestions in a kind, open manner, always maintaining a high technical and intellectual level.

“He is an exemplar, and a gentleman.”

The Goldwater competition is administered at Davidson College through the Office of Fellowships. For more information about the Office of Fellowships or applying for the Goldwater Scholarship, visit  www.davidson.edu/fellowships .

• May 8, 2024
• Mathematics & Computer Science
• Scholarships
• Academic Affairs
• News Headlines
• Mary Elizabeth DeAngelis

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• Christopher Record