# Awesome things from 2019

### How to prepare for Math Competitions?

Since it is that time of the year with all the high school math competitions around the corner, I am sharing the tips and what worked for me.

### How to prepare?

The best way to prepare is to practice, practice, practice. There are no real shortcuts to prepare for any math competitions that involve problem solving. If you are lucky and you can find a group (or even one more problem solver) that helps a lot as you always learn from others on different ways to solve a problem or parts of a problem. Solving problems from older tests is always helpful and solving problems from Art of Problem Solving books can assist greatly as well. Most old contests can be found on the web or can be ordered from MAA websites for AMC competitions. Most of the AMC contests including solutions can be found on Art of Problem Solving (AoPS) website. AoPS website has AMC 8, AMC 10, AMC 12, AIME and many other competition problems. Practicing problems will help reinforces concepts and help you with creative and beautiful solutions.. I also found that making mistakes and learning from them has helped me understand the higher level concepts much better. I highly recommend taking systematic approach and gradually building up skills in problem solving. I always ordered books from MAA and AoPS. Besides going gradually through Introduction and Intermediate Series of AoPS books, my favorites were Competition Math for Middle School, AoPS Volume 1: The Basics, and AoPS Volume 2: and Beyond.

MAA published this awesome book for AIME and I used Problem Solving Strategies that help me prepare for both AIME and USAMO.

Here are some basic tips:

1. Don’t be discouraged if you cannot solve all the problems. As you practice more, over time, you will become better for sure. You have to start somewhere. I was able to solve only a few problems (below 10) initially for both AMC 8 and AMC 10/12. I would practice with past tests every weekend (one test per week) during the school year and a few weeks in the summer.
2. You can try challenging yourself by trying to solve AMC 12 while you prepare for AMC 10.
3. There are almost always multiple ways to solve problems. You will learn the beauty of math and learn how to apply concepts better as you practice consistently.
4. It is totally okay to get stuck. It happens to everyone. Come back the next day or next week, and try solving that problem again.
5. There are many online courses, summer camps, classes during school years that can help you with new or advanced concepts. I found (and many of my friends) AwesomeMath summer camps very helpful.
6. Most previous competitions have solutions online. Try to understand the solution when you cannot solve a problem. Even if you have solved a problem, see the solution as they may have solved it using different methods.
7. There is no shortage of problems or competitions that happen throughout the year. Start participating in one and slowly start participating in more as you get more confidence. Not all competitions are created equally. Do not get discouraged when you do not perform well in a particular one. I did horrible in AMC 10 in 8th grade but did a lot better in 9th grade and 10th grade.
8. Check if your school has math club, math teams, or other organizations that meet regularly and participate in individual or team competitions. If not, attempt the problems and ask your math department if they can help especially if you get stuck on a concept.
9. AoPS has a very active and mostly supportive community. You can always become a member and members usually respond when you have a questions.
10. Practice-Be persistent-Don’t be discouraged-Practice again.
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### Gender similarities in the brain during mathematics development

CNN recently published this article based on the research done by Dr. Jessica Cantlon and her team at Carnegie Mellon University. The full report can be found here. I am really encouraged by such research which continues to debunk the myth that boys are better than girls at math.

(CNN)Several studies have already debunked the myth that boys are innately better at math than girls, but those are largely based on analysis of test scores.

Now, researchers also have brain imaging that proves young children use the same mechanisms and networks in the brain to solve math problems no matter their gender. The study was published Friday in the journal Science of Learning.
To answer this question, Cantlon and her team got 104 kids between the ages of 3 and 10 to perform cognitive tests and watch videos of engaging math lessons while in an MRI scanner. It’s the first study to use neuroimaging to evaluate biological gender differences in the math aptitude of young children.
“We looked at which areas of the brain respond more strongly to mathematics content in the videos and tasks, compared to non-math content like reading or the alphabet. So you can define the math network that way by looking at regions that respond more strongly,” she said.
“When we do that in little girls, we see a particular network of the brain (respond), and when we do that same analysis in boys we see the exact same regions. You can overlay the network from girls on top of the network from boys and they are identical,” she added.
What Cantlon’s study doesn’t answer is why the belief that boys are stronger in STEM subjects than girls still persists. The stereotype is so pervasive that one research team even issued a consensus statement clarifying that “no single factor,” including biology, “has been shown to determine sex differences in science and math.”
Cantlon said she thinks society and culture are likely steering girls and young women away from math and STEM fields.
Previous studies show that families spend more time with young boys in play that involves spatial cognition, while teachers also preferentially spend more time with boys during math class, she said. Also, children often pick up on cues from their parent’s expectations for math abilities.
“Typical socialization can exacerbate small differences between boys and girls that can snowball into how we treat them in science and math,” Cantlon said. “We need to be cognizant of these origins to ensure we aren’t the ones causing the gender inequities.”
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### Data Science or Algebra 2 in High School?

If you quickly review the Algebra 2 curriculum at Khan Academy, you see topics such as Polynomials, Complex Numbers, Rational Exponents, Logarithms, Function Transformations, Trig, etc. Whereas when you look at typical introductory courses in Data Science, they tend to focus on charts, histograms, functions, groups, joins, causality, confidence intervals, designing experiments, iterations, etc. The focus tends to be more on Statistics and Probability concepts than classic Algebra. In the following article authored by my favorite Dr. Boaler and Freakonomics author Dr. Levitt, they recommend that modern high school math should focus on Data Science instead of Algebra 2. The arguments are interesting and very sound. However, I believe that one should not have to choose between one or the other. Algebra 2 concepts are still helpful in real life problem solving whether in the financial services industry, healthcare, technology, or government. A solid background in Algebra makes one a better student of data science and Algebra 1 concepts and learnings are too basic to achieve proficiency at Data Science. Here is the article referred to that was published in the LA Times:

Thanks to the information revolution, a stunning 90% of the data created by humanity has been generated in just the past two years.

Yet the math taught in U.S. schools hasn’t materially changed since Sputnik was sent into orbit in the late 1950s. Our high school students are taught algebra, geometry, a second year of algebra, and calculus (for the most advanced students) because Eisenhower-era policymakers believed this curriculum would produce the best rocket scientists to work on projects during the Cold War.

It has been 50 years since the U.S. reached the moon, almost 30 years since the Berlin Wall fell. Technology has advanced to the point that tiny powerful computers are routinely carried around in pockets and purses. Times have changed, and so has the math people use in everyday life.

We surveyed 900 “Freakonomics” podcast listeners — a pretty nerdy group, we must admit — and discovered that less than 12% used any algebra, trigonometry or calculus in their daily lives. Only 2% use integrals or derivatives, the foundational building blocks of calculus. In contrast, a whopping 66% work with basic analytical software like Microsoft Excel on a daily basis.

When was the last time you divided a polynomial? If you were asked to do so today, would you remember how? For the most part, students are no longer taught to write cursive, how to use a slide rule, or any number of things that were once useful in everyday life. Let’s put working out polynomial division using pencil and paper on the same ash heap as sock darning and shorthand.

What we propose is as obvious as it is radical: to put data and its analysis at the center of high school mathematics. Every high school student should graduate with an understanding of data, spreadsheets, and the difference between correlation and causality. Moreover, teaching students to make data-based arguments will endow them with many of the same critical-thinking skills they are learning today through algebraic proofs, but also give them more practical skills for navigating our newly data-rich world.

Data-based math courses allow students to grapple with real-life problems. They might analyze issues about the environment, space travel or nutrition. Students can examine the threat of wildfires or the ways social media is tracking their data, learning how to apply math to real-world issues.

Other countries are moving much faster than the U.S. in instituting such a curriculum. Over the last 50 years, statistics and data science have become an integral part of the United Kingdom curriculum. Canada’s educational system, which is ranked highly internationally, also incorporates statistics and data.

In addition, the Program for International Student Assessment, or PISA, measures how effectively countries are preparing students for the mathematical demands of the 21st century. Last week, PISA released a mathematics framework that guides the assessments. Data literacy is central to the framework. In contrast, U.S. high school students learn algebra and geometry — and are woefully underprepared for the modern world.

The Los Angeles Unified School District is leading the way in updating the way math is taught. In 2013, the LAUSD secured approval from the University of California to recognize data science as a statistics course that students can substitute for Algebra 2 in the college pathway. Over 2,000 students are taking advantage of this option. The classroom we observed was full of critical thinkers who see data everywhere and appear comfortable interpreting, analyzing and questioning it.

Modernizing math at a national level will require an intensive effort from educators, policymakers and high school counselors — as well as from students and parents who will need to advocate for it. Some states are already exploring changes to their mathematics frameworks, while a fair number of innovative teachers across the country are independently developing their own data-focused lesson plans.

For this revolution to be carried out across the country, decision makers will need to hear from parents and other interested parties who recognize that our children deserve math instruction that is relevant to their lives.

Jo Boaler is a professor of mathematics education at Stanford University and author of “Limitless Mind.” Steven D. Levitt is a professor of economics at the University of Chicago and co-author of “Freakonomics.”

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### Math + culture = Gender Gap?

I have been really encouraged and inspired by all the work done by Dr. Hyde who is Evjue-Bascom Professor of Psychology and Gender & Women’s studies. Her research has focused on Biology of Brain, Clinical, Cognitive Neuroscience, Developmental, and Perception. There has been a lot of work done in both Psychology and Philosophy fields related to gender performance discrepancy in mathematics. Beth Azar has done a nice job in this article summarizing multiple research studies in this discipline which was published in July/August 2010.  I have covered some of these topics in my earlier blogs on why cultural/societal factors prevent girls from pursuing advanced mathematics.

Researchers have all but debunked the idea that girls are innately worse at math than boys. But psychologists have identified other factors that might set girls back.

Most experts agree that if gender differences do exist, they are small and likely to affect specific areas of math skill at the highest end of the spectrum — and there’s no indication that women cannot succeed in mathematically demanding fields. Still, women continue to be underrepresented in math, science and engineering-related careers, and there’s evidence that girls can lose ground in math under certain circumstances.

One factor inhibiting girls is self-confidence, says University of Wisconsin psychologist Janet Hyde, PhD. “Even when girls are getting better grades, boys are more confident in math. It’s important to understand what might be sapping girls’ confidence.”

And that lack of self-assurance likely stems from culture, research suggests. After reviewing decades of research on gender differences, Cornell University psychologists Steven Ceci, PhD, and Wendy Williams, PhD, conclude that while there’s probably some genetic basis for small differences between the sexes in math and spatial ability, culture plays by far the bigger role in men and boys’ higher interest and achievement.

“If you look at the students scoring in the top one in 10,000 in mathematics in 1983, there were 13 boys for every girl,” says Ceci. “Since then, until 2007, that gap has shrunk to somewhere between 2.8 and four boys for every girl.

So if the difference were just in the genome, there would not be that improvement. Rather, shifts like that are due in large part to increases in the number of girls who take higher level math courses in high school, where girls traditionally began falling behind boys. They appear to be taking more math courses because changing cultural norms make it more acceptable.

Research by Hyde supports that idea. In a January article in Psychological Bulletin (Vol. 136, No. 1), she and her colleagues found that the more gender equity a country had — measured by school enrollment, women’s share of research jobs and women’s parliamentary representation — the smaller its math gender gap.

“When girls see opportunities for themselves in science, technology, engineering and math, they’re more likely to take higher math in high school and more likely to pursue those careers,” says Hyde.

In fact, women in the United States now earn 48 percent of bachelor’s degrees in mathematics and 30 percent of the doctorates, says Hyde. “If they can’t do math, how are they doing this? They can do math just fine.”

That doesn’t mean, however, that just because girls and women can do the math, they want to. When Vanderbilt University psychologist David Lubinski, PhD, and his colleagues interviewed a group of more than 5,000 intellectually precocious girls and boys they’d followed from childhood into their mid-30s, they noticed that while men and women earned equal proportions of advanced degrees, there were gender differences in the areas people decided to study.

He found that just as many women as men started college planning to go into physical sciences and math. However, women more than men later switched to humanities and social science majors. Every one of these study participants had the ability to succeed in math-related careers, but many of them were more likely to choose law school or medicine, Lubinski says.

“The sexes are making different choices,” he says. “But when we look at how satisfied these people are with their career choices, they’re equally satisfied and equally successful.”

Ceci and Williams posit that girls are more attracted to a variety of careers because they tend to have both strong math and verbal skills. “Boys who are really good at math say, ‘This is who I am, I’m a mathematician,’” says Ceci. “Girls who are really good at math are more likely to be really good at verbal skills, too, and they ask themselves, ‘I wonder what I want to do?’”

It doesn’t help that the corporate culture of many math-centered careers speaks more to boys’ well-documented tendency to be interested in “things” than girls’ tendency to be interested in working with people, says Hyde. “Engineering portrays itself as being about things,” she says. “Maybe if engineering professors made better connections to how engineering helps people, women would be more enticed.”

Classroom influences

To explore why girls are less confident than boys in their math abilities, University of Georgia psychologist Martha Carr, PhD, studies first-graders, and has found that girls use different strategies and have different motivations to do math.

Boys, Carr says, tend to use memory to retrieve sums and are motivated by a sense of competition to get the answer fast, even if they sacrifice accuracy. Girls care less about speed than accuracy and more often rely on “manipulatives” — counting on their fingers or a counting board.

“Girls will use manipulatives even when they might be able to retrieve [the answer],” says Carr. “They need an added push that boys don’t need to start using cognitive strategies.”

That’s important because while using manipulatives is an excellent strategy when students first learn math, it slows them down as problems get more difficult. In fact, in a study that followed students from second grade through fourth grade, Carr found that becoming fluent, and therefore faster, at basic math is directly linked to math performance. The study also found that girls were less fluent than boys.

“If we make sure all children are fluent [in math facts], we will eliminate most gender differences,” she says.

But what if girls’ confidence and their interest in becoming “fluent” are influenced by math anxiety among their predominantly female elementary school teachers? A 2010 study (PNAS, Vol. 107, No. 5) by University of Chicago psychologist Sian Beilock, PhD, suggests that this may well be the case for some girls. She and her colleagues started with these facts: More than 90 percent of elementary school teachers are women, and studies show that elementary education majors have higher levels of math anxiety than any other major. The researchers then assessed math anxiety in 17 female first- and second-grade teachers, as well as math achievement and gender stereotypes among 52 boys and 65 girls from their classes. At the start of the school year, the researchers found no link between teacher anxiety and student math achievement. But by school year’s end, the more anxious teachers were about math, the more likely girls, but not boys, agreed with the statement, “Boys are good at math and girls are good at reading.” In addition, girls who accepted this stereotype performed significantly worse on math achievement measures than girls who did not and boys overall.

Interestingly, on average, girls and boys performed the same, says Beilock. Only the girls who endorsed the stereotype showed a drop in math performance. That finding supports work Beilock and others have done on “stereotype threat,” which shows that people perform poorly when a negative stereotype is in play.

It’s also not surprising that girls picked up on their teachers’ anxiety and not boys because research shows that young children are more likely to emulate adults of the same gender.

In the end, though, it’s not just girls who need math help, emphasizes University of Missouri psychologist David Geary, PhD, an expert on mathematical development and author of “Male, Female: The Evolution of Human Sex Differences, Second Edition” (APA, 2009). He believes all the focus on gender distracts from the more serious problem that U.S. math achievement is abysmal compared with that of other countries.

Hyde agrees. “We need to look toward better math instruction for the United States, not specifically for boys or girls.”

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### Does Society need IMO Medalists? A student’s view…

I published this blog orginally on MAA’s MathValues.org in July-2019.

A paper written by Dr. Man Keung Siu, a professor of math at the University of Hong Kong, titled “Does Society need IMO medalists?” poses an important question: how do math competitions fit in with the field of math at large? He starts by describing the International Mathematical Olympiad (IMO), including his experience with the competition, and mentions names of IMO winners that have gone to become famous mathematicians. He also ponders the relevance of the IMO winners to society and considers specifically the impact of the IMO winners on the field of math.

The 2018 US team at the International Mathematical Olympiad.

Winners of the IMO are highly trained in math competitions. As a high school student who competed in math competitions, I read Dr. Siu’s paper with interest and believe he asks an important question. Dr. Siu discusses the training and skill sets gained through preparation for the competitions. He states that training for math competitions allows students to acquire logical thinking, confidence, and “academic sincerity”.

However, he also notes that some of the drawbacks of this type of training include the ways competition problems differ from mathematical research, the potential for overtraining, and the possibility that competitive spirit is sometimes different from passion for the subject.

Personally, I believe that the points Dr. Siu brought up are fair, but based on my own experience and observations of other competitors, it seems that in order to do well in math competitions a genuine passion and drive for the subject is necessary to keep one motivated.

I agree with the point in Dr. Sui’s paper from Dr. Petar Kenderov, a math professor at the Bulgarian Academy of Sciences, who points out that math competitions disfavor students who work “slower”, as most competitions involve time pressure. Time restrictions can inhibit a student’s performance.

Kenderov also says that math competitions miss out on a fundamental aspect of math, which is posing questions and problems. Dr. Siu goes on to say something similar by mentioning that research isn’t just about the answer, but exploring a concept to the full depth.

One interesting aspect of the article is that Dr. Siu gives three examples of math problems that are solved in two different ways, to generalize that there are two fundamental methods of solving math problems. One method is the standard, longer approach of systematically solving the problem, which is typically taught in school and classroom settings, and the other is a method of finding clever ways to solve the problem in a nonconventional way, which he believes is the methodology taught in math competitions.

He states that both are crucial to the subject of math, but notes that schools don’t typically teach students math using math competition problems, which narrows most students’ horizons into thinking of math in one way. He claims that for a deeper understanding of the subject of math, all aspects should be explored. I strongly agree with Dr. Siu’s statement as the education system traditionally focuses on math solely in a way that is procedural, and it is crucial for students to see math in all facets.

To respond to Dr. Siu’s point that math competitions lack some aspects fundamental to mathematical research, like posing unique questions. However, it’s nearly impossible for tests to include all the components that provide students with a sufficient background in research. Although math competitions don’t comprehensively provide students background needed for a research project, math competitions do allow students  to experience creativity in math that most wouldn’t get exposure to otherwise. That creativity is a crucial component for mathematical research.

At the end of the paper, he shifts the focus to say that “society needs friends of mathematics.” By saying this, he denotes that it is essential to have people in the world who don’t necessarily pursue math but understand the significance of mathematics to the world at large. He claims that there does not exist a substantial amount of people that not only support the field, but also comprehend the value of mathematics. I think Dr. Siu makes a phenomenal point at the end, as lack of support undermines the value of making mathematical breakthroughs because  people aren’t cognizant of math as a field that impacts society.

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### Math Anxiety: Research

As I have written before, mathematics is becoming critical in today’s scientific and technological age. Advanced mathematics is helping with research in understanding our genes, diagnosing and preventing killer diseases, artificial intelligence, and driving many advances in computing. Math proficiency always helps with higher education, a professional career, and even something as basic as managing personal finances. I believe that more research should be done related to mathematics anxiety (MA) because if not understood and addressed, MA can be a significant barrier in learning basic and advanced mathematics. Mathematics anxiety has been defined by Dr. Richadson and Dr. Suinn as “a feeling of tension and anxiety that interferes with the manipulation of numbers and the solving of mathematical problems in … ordinary life and academic situations”

Dr. Dreger and Dr. Aiken were the first ones to publish a study on what they coined ‘number anxiety’ in 1957. The main purpose of this study was to detect presence of emotional reactions to arithmetic and mathematics among 700 college students at Florida State University. They concluded that number anxiety was separate from general anxiety, was not related to general intelligence, and impacted math grades when it was high.

Dr. Ann Dowker is a University Research Lecturer at Oxford. Dr. Dowker and others have done extensive research on individual differences in arithmetic in both children and adults, and on the phenomenon of ‘mathematics anxiety’. Dr. Dowker is also the lead researcher on the Catch Up Numeracy project, which is an individualized intervention program for primary school children with low performance achievements in math. Catch Up Numeracy project has been implemented in 45 local authorities in the UK and is being extended to Ireland and Australia. Dr. Dowker with other researchers at Oxford published the latest compilation on MA, Mathematics Anxiety: What Have We Learned in 60 Years? This report does a nice job of defining MA, and its distinction from other forms of anxiety. I really liked how the research tries to identify the possible factors such as genetics, age, gender, and culture that could influence varying levels of MA.

I came across this another research from one of the co-authors of previous study where the research team used voxel-based morphometry (VBM) to identify the structural brain correlates of MA in 79 healthy children in Spain aged 7–12 years. MA is believed to develop in later years of primary education, and the study identified that “increased MA was associated with reduced attention, working memory and math achievement.”

There are various tips and tricks to reduce MA whether you are an educator or a student.

[Image Credit: Bob Staake, The NY Times]

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### IMO Medalists and their contributions

I wrote this blog which was originally published on mathvalues.org

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International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for high school students. The first IMO was held in 1959 in Romania and only 7 countries participated that year. Now, the competition has expanded to over 100 countries spanning major regions of the world. The team from the United States won the first place title in the most recent IMO, the 2018 International Mathematical Olympiad and also won the title in both 2015 and 2016.

Dr. Viorel Barbu, a participant in the first IMO, who has become President of the Mathematics Department at the Romanian Academy brilliantly wrote that “Mathematics has always been a fresh and dynamical field of human creativity and a fundamental science to the benefit of scientific knowledge and technical achievements. It is the role and duty of young mathematicians to bring and develop new ideas and to construct new bridges between mathematics and other scientific fields.”

I have always wondered about the contribution of IMO participants to the field of mathematics and science overall. I came across this fascinating research from Dr. Agarwal and Dr. Patrick Gaule. These researchers analyzed data examining the career and scientific output of participants who competed and performed well in IMO over a 20 years period. This research points to a very positive correlation between the points scored at the IMO and the mathematical knowledge produced, which was measured by the number of mathematical publications and mathematics citations. It also proved that students who performed well on IMO are more likely to become professional mathematicians, measured by getting a Ph.D. in mathematics.

I found some really interesting observations in the research, listed below:

Strong performers at the IMO have a disproportionate ability to produce frontier mathematical knowledge compared to PhD graduates and even PhD graduates from elite schools.

-The conditional probability that an IMO gold medalist will become a Fields medalist is two order of magnitudes larger than the corresponding probability for of a PhD graduate from a top 10 mathematics program.

-Dr. Maryam Mirzakhani, who passed away at a very young age, was an IMO gold medalist with a perfect score, and the first woman to win the Fields medal, the most prestigious award in mathematics. Terence Tao received a gold medal at the 29th IMO and went on to win the Fields medal and is one of the most productive mathematicians in the world.

-Around 22% of IMO participants have a PhD in mathematics; of those, around a third have a PhD in mathematics from a top 10 school (7% of the total IMO participants). 1% of IMO participants became IMC speakers, and 0.2% became Fields medalists.

This research paper clearly articulates the contributions of IMO participants to the field of mathematics. This paper gives strong reason to encourage everyone to participate in math competitions beginning in elementary school, and through college, as problem-solving skills acquired through participating in math competitions have long lasting positive effects that helps you whether you pursue a professional or academic career.

The last time a female qualified for the IMO from the United States was in 2007 and 3 female US students have scored medals at IMO. Their mathematics career and contributions validate the research findings. Sherry Gong represented the United States in 2005 and 2007, winning a Gold Medal in 2007. She famously scored over a 100 in Harvard’s problem solving course, Math 55, and went on to get her Ph.D. at MIT in mathematics. Alison Miller represented the United States in 2004 and also won the Gold Medal. Alison Miller studied mathematics at Harvard and finished her Ph.D. in mathematics at Princeton University. Melanie Wood represented the United States in the 1998 and 1999 IMO and won Silver Medals in both years. She was the first female to qualify for the IMO from United States. She completed her Ph.D. in 2009 at Princeton University and is currently a Vilas Distinguished Achievement Professor of Mathematics at the University of Wisconsin.

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### Summer Math Camps

It’s that time of the year when many of you are researching which middle school summer camps to attend. I have attended a variety of summer camps since 7th grade that were mainly focused on mathematical problem solving and proofs. I frequently get asked if these camps are gender diverse or not, and the short answer is yes. Typically 25 to 30% of the attendees at most of these camps are girls, and I have made some of my best friends through many of these camps. I have met interesting students from all over the country, and there are many international attendees as well at many of these camps from Europe and Asia.

Here are some of the camps for middle school girls:

1. Awesome Math Summer Program: This is one of the best camps for mathematical problem solving training. They usually have 3 different locations every year spanning the East Coast, the Central US, and the West Coast. More information about the locations can be found here. There are very specific course selection guidelines that I followed when registering for the camp and to get the best value out of this summer camp. I have not taken classes in their Year-Round program but I have heard those are good as well for students who cannot commit to Awesome Math’s summer schedules.
2. MathPath: This is a four-week residential summer program for middle school students who are serious about mathematics. The program also provides them a rich social and recreational experience. MathPath encourages applications from female and minority students.
3. Prove it! Math Academy: Even though this program will not be offered in 2019, it is one of the best math summer programs I have attended. It gave me a really solid foundation in proofs. Prove it! acts as a bridge between mathematical problem solving and proof writing, both of which are essential for various mathematical olympiads. This program will return in 2020, and is one of the rare programs that focuses on mathematical proofs, offered in beautiful Colorado.
4. Idea Math: This is a summer math workshop for students in middle schools who wish to expand their mathematical knowledge, also while enhancing their problem-solving skills. Idea Math offers this camp at four different locations in the United States. They have year-round programs as well, which are offered in Boston, NY, TX, and the SF Bay Area.

In addition, please check out the following links for other summer camps including many online programs offered by AoPS.

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