Month: July 2025

Families and caregivers can boost children’s confidence and interest in science, technology, engineering and mathematics while school is out for summer. heshphoto/Getty Images

The Trump administration is reshaping the pursuit of science through federal cuts to research grants and the Department of Education. This will have real consequences for students interested in science, technology, engineering and mathematics, or STEM learning.

One of those consequences is the elimination of learning opportunities such as robotics camps and access to advanced math courses for K-12 students.

As a result, families and caregivers are more essential than ever in supporting children’s learning.

Based on my research, I offer four ways to support children’s summer learning in ways that feel playful and engaging but still foster their interest, confidence and skills in STEM.

Find a problem

To support STEM learning outside of school, encourage children to find and solve problems. kali9/Getty Images

Look for “problems” in or around your home to engineer a solution for. Engineering a solution could include brainstorming ideas, drawing a sketch, creating a prototype or a first draft, testing and improving the prototype and communicating about the invention.

For example, one family in our research created an upside-down soap dispenser for the following problem: “the way it’s designed” − specifically, the straw − “it doesn’t even reach the bottom of the container. So there’s a lot of soap sitting at the bottom.”

To identify a problem and engage in the engineering design process, families are encouraged to use common materials. The materials may include cardboard boxes, cotton balls, construction paper, pine cones and rocks.

Our research found that when children engage in engineering in the home environment with caregivers, parents and siblings, they communicate about and apply science and math concepts that are often “hidden” in their actions.

For instance, when building a paper roller coaster for a marble, children think about how the height will affect the speed of the marble. In math, this relates to the relationship between two variables, or the idea that one thing, such as height, impacts another, the speed. In science, they are applying concepts of kinetic energy and potential energy. The higher the starting point, the more potential energy is converted into kinetic energy, which makes the marble move faster.

In addition, children are learning what it means to be an engineer through their actions and experience. Families and caregivers play a role in supporting their creative thinking and willingness to work through challenging problems.

Spark curiosity

Spontaneous learning moments can lead to deep engagement and learning of STEM concepts. cglade/Getty Images

Open up a space for exploration around STEM concepts driven by their interests.

Currently, my research with STEM professionals who were homeschooled talk about the power of learning sparked by curiosity.

One participant stated, “At one time, I got really into ladybugs, well Asian Beatles I guess. It was when we had like hundreds in our house. I was like, what is happening? So, I wanted to figure out like why they were there, and then the difference between ladybugs and Asian beetles because people kept saying, these aren’t actually ladybugs.”

Researchers label this serendipitous science engagement, or even spontaneous math moments. The moments lead to deep engagement and learning of STEM concepts. This may also be a chance to learn things with your child.

 Facilitate thinking

In my research, being uncertain about STEM concepts may lead to children exploring and considering different ideas. One concept in particular − playful uncertainties − is when parents and caregivers know the answer to a child’s uncertainties but act as if they do not know.

For example, suppose your child asks, “How can we measure the distance between St. Louis, Missouri, and Nashville, Tennessee, on this map?” You might respond, “I don’t know. What do you think?” This gives children the chance to share their ideas before a parent or caregiver guides them toward a response.

Bring STEM to life

Overhearing or participating in budget talks can help children develop math skills and financial literacy. SeizaVisuals/Getty Images

Turn ordinary moments into curious conversations.

“This recipe is for four people, but we have 11 people coming to dinner. What should we do?”

In a recent interview, one participant described how much they learned from listening in on financial conversations, seeing how decisions got made about money, and watching how bills were handled. They were developing financial literacy and math skills.

As they noted, “By the time I got to high school, I had a very good basis on what I’m doing and how to do it and function as a person in society.”

Globally, individuals lack financial literacy, which can lead to negative outcomes in the future when it comes to topics such as retirement planning and debt.

Why is this important?

Research shows that talking with friends and family about STEM concepts supports how children see themselves as learners and their later success in STEM fields, even if they do not pursue a career in STEM.

My research also shows how family STEM participation gives children opportunities to explore STEM ideas in ways that go beyond what they typically experience in school.

In my view, these kinds of STEM experiences don’t compete with what children learn in school − they strengthen and support it.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Amber M. Simpson*

Prime numbers are numbers that are not products of smaller whole numbers. Jeremiah Bartz

A shard of smooth bone etched with irregular marks dating back 20,000 years puzzled archaeologists until they noticed something unique – the etchings, lines like tally marks, may have represented prime numbers. Similarly, a clay tablet from 1800 B.C.E. inscribed with Babylonian numbers describes a number system built on prime numbers.

As the Ishango bone, the Plimpton 322 tablet and other artifacts throughout history display, prime numbers have fascinated and captivated people throughout history. Today, prime numbers and their properties are studied in number theory, a branch of mathematics and active area of research today.

A history of prime numbers

Some scientists guess that the markings on the Ishango bone represent prime numbers. Joeykentin/Wikimedia Commons, CC BY-SA

Informally, a positive counting number larger than one is prime if that number of dots can be arranged only into a rectangular array with one column or one row. For example, 11 is a prime number since 11 dots form only rectangular arrays of sizes 1 by 11 and 11 by 1. Conversely, 12 is not prime since you can use 12 dots to make an array of 3 by 4 dots, with multiple rows and multiple columns. Math textbooks define a prime number as a whole number greater than one whose only positive divisors are only 1 and itself.

Math historian Peter S. Rudman suggests that Greek mathematicians were likely the first to understand the concept of prime numbers, around 500 B.C.E.

Around 300 B.C.E., the Greek mathematician and logician Euclid proved that there are infinitely many prime numbers. Euclid began by assuming that there is a finite number of primes. Then he came up with a prime that was not on the original list to create a contradiction. Since a fundamental principle of mathematics is being logically consistent with no contradictions, Euclid then concluded that his original assumption must be false. So, there are infinitely many primes.

The argument established the existence of infinitely many primes, however it was not particularly constructive. Euclid had no efficient method to list all the primes in an ascending list.

In the middle ages, Arab mathematicians advanced the Greeks’ theory of prime numbers, referred to as hasam numbers during this time. The Persian mathematician Kamal al-Din al-Farisi formulated the fundamental theorem of arithmetic, which states that any positive integer larger than one can be expressed uniquely as a product of primes.

From this view, prime numbers are the basic building blocks for constructing any positive whole number using multiplication – akin to atoms combining to make molecules in chemistry.

Prime numbers can be sorted into different types. In 1202, Leonardo Fibonacci introduced in his book “Liber Abaci: Book of Calculation” prime numbers of the form (2p – 1) where p is also prime.

Prime numbers, when expressed as that number of dots, can be arranged only in a single row or column, rather than a square or rectangle. David Eppstein/Wikimedia Commons

Today, primes in this form are called Mersenne primes after the French monk Marin Mersenne. Many of the largest known primes follow this format.

Several early mathematicians believed that a number of the form (2p – 1) is prime whenever p is prime. But in 1536, mathematician Hudalricus Regius noticed that 11 is prime but not (211 – 1), which equals 2047. The number 2047 can be expressed as 23 times 89, disproving the conjecture.

While not always true, number theorists realized that the (2p – 1) shortcut often produces primes and gives a systematic way to search for large primes.

The search for large primes

The number (2p – 1) is much larger relative to the value of p and provides opportunities to identify large primes.

When the number (2p – 1) becomes sufficiently large, it is much harder to check whether (2p – 1) is prime – that is, if (2p – 1) dots can be arranged only into a rectangular array with one column or one row.

Fortunately, Édouard Lucas developed a prime number test in 1878, later proved by Derrick Henry Lehmer in 1930. Their work resulted in an efficient algorithm for evaluating potential Mersenne primes. Using this algorithm with hand computations on paper, Lucas showed in 1876 that the 39-digit number (2127 – 1) equals 170,141,183,460,469,231,731,687,303,715,884,105,727, and that value is prime

Also known as M127, this number remains the largest prime verified by hand computations. It held the record for largest known prime for 75 years.

Researchers began using computers in the 1950s, and the pace of discovering new large primes increased. In 1952, Raphael M. Robinson identified five new Mersenne primes using a Standard Western Automatic Computer to carry out the Lucas-Lehmer prime number tests.

As computers improved, the list of Mersenne primes grew, especially with the Cray supercomputer’s arrival in 1964. Although there are infinitely many primes, researchers are unsure how many fit the type (2p – 1) and are Mersenne primes.

By the early 1980s, researchers had accumulated enough data to confidently believe that infinitely many Mersenne primes exist. They could even guess how often these prime numbers appear, on average. Mathematicians have not found proof so far, but new data continues to support these guesses.

George Woltman, a computer scientist, founded the Great Internet Mersenne Prime Search, or GIMPS, in 1996. Through this collaborative program, anyone can download freely available software from the GIMPS website to search for Mersenne prime numbers on their personal computers. The website contains specific instructions on how to participate.

GIMPS has now identified 18 Mersenne primes, primarily on personal computers using Intel chips. The program averages a new discovery about every one to two years.

The largest known prime

Luke Durant, a retired programmer, discovered the current record for the largest known prime, (2136,279,841 – 1), in October 2024

Referred to as M136279841, this 41,024,320-digit number was the 52nd Mersenne prime identified and was found by running GIMPS on a publicly available cloud-based computing network.

This network used Nvidia chips and ran across 17 countries and 24 data centers. These advanced chips provide faster computing by handling thousands of calculations simultaneously. The result is shorter run times for algorithms such as prime number testing.

New and increasingly powerful computer chips have allowed prime-number hunters to find increasingly larger primes. Fritzchens Fritz/Flickr

The Electronic Frontier Foundation is a civil liberty group that offers cash prizes for identifying large primes. It awarded prizes in 2000 and 2009 for the first verified 1 million-digit and 10 million-digit prime numbers.

Large prime number enthusiasts’ next two challenges are to identify the first 100 million-digit and 1 billion-digit primes. EFF prizes of US$150,000 and $250,000, respectively, await the first successful individual or group.

Eight of the 10 largest known prime numbers are Mersenne primes, so GIMPS and cloud computing are poised to play a prominent role in the search for record-breaking large prime numbers.

Large prime numbers have a vital role in many encryption methods in cybersecurity, so every internet user stands to benefit from the search for large prime numbers. These searches help keep digital communications and sensitive information safe.

This story was updated on May 30, 2025 to correct the name of the Greek mathematician Euclid and to correct the factors of 2047.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jeremiah Bartz*