Category: mathematics

If You take a look at the square numbers (n^2, n a positive integer), you’ll notice plenty of patterns in the digits. For example, if you look at just the last digit of each square, you’ll observe the repeating pattern 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, … If you construct a graph of “last digit” vs n (like the one below, built with Falthom), the symmetry and period of this digit pattern is apparent.

Why does this happen? The periodic nature of the pattern is easy to understand – when you square a number, only the digit in the ones place contributes to ones place of the product. For example, 22*22 and 32*32 are both going to have a 4 as their last digit – the values in the tens place (or any other place other than the ones) do not affect what ends up as the last digit.

The reason for the symmetry about n=5 is a little less obvious. To see what is going on, it is helpful to use modular arithmetic and to realize that ” last digit of n” is the same as “n mod 10”. Considering what 10-n looks like mod 10 after it is squared, we have the equation below.

This tells us that the last digit of (10-n)^2 is the same as the last digit of n^2, because everything else that is different about these two numbers is divisible by 10.

If you look at the last two digits of the square numbers, you see another repeating pattern that has similar symmetries.

This is a nice looking graph – the period is 50 with a line of symmetry at n=25. You can think about it in the same way as the one-digit case, this time the symmetry is understood by looking at (50-n)^2 mod 100. (Looking at numbers mod 100 tells us their last two digits.)

If you decide to investigate patterns in cubes or higher powers, you’ll see somewhat similar results. Using the binomial theorem and modular arithmetic, you can see why even powers give symmetry similar to the n^2 case, while odd powers do not (although all are periodic).

This graph shows the pattern in the last digit of n^3.

This last graph shows the pattern for the last two digits of n^4.

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*Credit for article given to dan.mackinnon*

The number 33 has surprising depth

Add three cubed numbers, and what do you get? It is a question that has puzzled mathematicians for centuries.

In 1825, a mathematician known as S. Ryley proved that any fraction could be represented as the sum of three cubes of fractions. In the 1950s, mathematician Louis Mordell asked whether the same could be done for integers, or whole numbers. In other words, are there integers k, x, y and z such that k = x³ + y³ + z³ for each possible value of k?

We still don’t know. “It’s long been clear that there are maths problems that are easy to state, but fiendishly hard to solve,” says Andrew Booker at the University of Bristol, UK – Fermat’s last theorem is a famous example.

Booker has now made another dent in the cube problem by finding a sum for the number 33, previously the lowest unsolved example. He used a computer algorithm to search for a solution:

33 = 8,866,128,975,287,528³ + (-8,778,405,442,862,239)³ + (-2,736,111,468,807,040)³

To cut down calculation time, the program eliminated certain combinations of numbers. “For instance, if x, y and z are all positive and large, then there’s no way that x³ + y³ + z³ is going to be a small number,” says Booker. Even so, it took 15 years of computer-processing time and three weeks of real time to come up with the result.

For some numbers, finding a solution to the equation k = x³ + y³ + z³ is simple, but others involve huge strings of digits. “It’s really easy to find solutions for 29, and we know a solution for 30, but that wasn’t found until 1999, and the numbers were in the millions,” says Booker.

Another example is for the number 3, which has two simple solutions: 1³ + 1³ + 1³  and 4³ + 4³ + (-5) ³ . “But to this day, we still don’t know whether there are more,” he says.

There are certain numbers that we know definitely can’t be the sum of three cubes, including 4, 5, 13, 14 and infinitely many more.

The solution to 74 was only found in 2016, which leaves 42 as the only number less than 100 without a possible solution. There are still 12 unsolved numbers less than 1000.

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*Credit for article given to Donna Lu*

While adding up your grocery bill in the supermarket, you’re probably not thinking how important or sophisticated our number system is.

But the discovery of the present system, by unknown mathematicians in India roughly 2,000 years ago – and shared with Europe from the 13th century onwards – was pivotal to the development of our modern world.

Now, what if our “decimal” arithmetic, often called the Indo-Arabic system, had been discovered earlier? Or what if it had been shared with the Western world earlier than the 13th century?

First, let’s define “decimal” arithmetic: we’re talking about the combination of zero, the digits one through nine, positional notation, and efficient rules for arithmetic.

“Positional notation” means that the value represented by a digit depends both on its value and position in a string of digits.

Thus 7,654 means:

(7 × 1000) + (6 × 100) + (5 × 10) + 4 = 7,654

The benefit of this positional notation system is that we need no new symbols or calculation schemes for tens, hundreds or thousands, as was needed when manipulating Roman numerals.

While numerals for the counting numbers one, two and three were seen in all ancient civilisations – and some form of zero appeared in two or three of those civilisations (including India) – the crucial combination of zero and positional notation arose only in India and Central America.

Importantly, only the Indian system was suitable for efficient calculation.

Positional arithmetic can be in base-ten (or decimal) for humans, or in base-two (binary) for computers.

In binary, 10101 means:

(1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + 1

Which, in the more-familiar decimal notation, is 21.

The rules we learned in primary school for addition, subtraction, multiplication and division can be easily extended to binary.

The binary system has been implemented in electronic circuits on computers, mostly because the multiplication table for binary arithmetic is much simpler than the decimal system.

Of course, computers can readily convert binary results to decimal notation for us humans.

As easy as counting from one to ten

Perhaps because we learn decimal arithmetic so early, we consider it “trivial”.

Indeed the discovery of decimal arithmetic is given disappointingly brief mention in most western histories of mathematics.

In reality, decimal arithmetic is anything but “trivial” since it eluded the best minds of the ancient world including Greek mathematical super-genius Archimedes of Syracuse.

Archimedes – who lived in the 3rd century BCE – saw far beyond the mathematics of his time, even anticipating numerous key ideas of modern calculus. He also used mathematics in engineering applications.

Nonetheless, he used a cumbersome Greek numeral system that hobbled his calculations.

Imagine trying to multiply the Roman numerals XXXI (31) and XIV (14).

First, one must rewrite the second argument as XIIII, then multiply the second by each letter of the first to obtain CXXXX CXXXX CXXXX XIIII.

These numerals can then be sorted by magnitude to arrive at CCCXXXXXXXXXXXXXIIII.

This can then be rewritten to yield CDXXXIV (434).

(For a bit of fun, try adding MCMLXXXIV and MMXI. First person to comment with the correct answer and their method gets a jelly bean.)

Thus, while possible, calculation with Roman numerals is significantly more time-consuming and error prone than our decimal system (although it is harder to alter the amount payable on a Roman cheque).

History lesson

Although decimal arithmetic was known in the Arab world by the 9th century, it took many centuries to make its way to Europe.

Italian mathematician Leonardo Fibonacci travelled the Mediterranean world in the 13th century, learning from the best Arab mathematicians of the time. Even then, it was several more centuries until decimal arithmetic was fully established in Europe.

Johannes Kepler and Isaac Newton – both giants in the world of physics – relied heavily on extensive decimal calculations (by hand) to devise their theories of planetary motion.

In a similar way, present-day scientists rely on massive computer calculations to test hypotheses and design products. Even our mobile phones do surprisingly sophisticated calculations to process voice and video.

But let us indulge in some alternate history of mathematics. What if decimal arithmetic had been discovered in India even earlier, say 300 BCE? (There are indications it was known by this date, just not well documented.)

And what if a cultural connection along the silk-road had been made between Indian mathematicians and Greek mathematicians at the time?

Such an exchange would have greatly enhanced both worlds, resulting in advances beyond the reach of each system on its own.

For example, a fusion of Indian arithmetic and Greek geometry might well have led to full-fledged trigonometry and calculus, thus enabling ancient astronomers to deduce the laws of motion and gravitation nearly two millennia before Newton.

In fact, the combination of mathematics, efficient arithmetic and physics might have accelerated the development of modern technology by more than two millennia.

It is clear from history that without mathematics, real progress in science and technology is not possible (try building a mobile phone without mathematics). But it’s also clear that mathematics alone is not sufficient.

The prodigious computational skills of ancient Indian mathematicians never flowered into advanced technology, nor did the great mathematical achievements of the Greeks, or many developments in China.

On the other hand, the Romans, who were not known for their mathematics, still managed to develop some impressive technology.

But a combination of advanced mathematics, computation, and technology makes a huge difference.

Our bodies and our brains today are virtually indistinguishable from those of ancient times.

With the earlier adoption of Indo-Arabic decimal arithmetic, the modern technological world of today might – for better or worse – have been achieved centuries ago.

And that’s something worth thinking about next time you’re out grocery shopping.

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*Credit for article given to Jonathan Borwein (Jon)*

The purpose of education is to ensure that students acquire the skills necessary for succeeding in a world that is constantly changing. Self-assessment, or teaching students how to examine and evaluate their own learning and cognitive processes, has proven to be an effective method, and this competence is partly based on metacognitive knowledge.

A new study conducted at the University of Eastern Finland shows that metacognitive knowledge, i.e., awareness of one’s cognitive processes, is also a key factor in the learning of mathematics. The work is published in the journal Cogent Education.

The study explored thinking skills and possible grade-level differences in children attending comprehensive school in Finland. The researchers investigated 6th, 7th and 9th graders’ metacognitive knowledge in the context of mathematics.

“The study showed that ninth graders excelled at explaining their use of learning strategies, while 7th graders demonstrated proficiency in understanding when and why certain strategies should be used. No other differences between grade levels were observed, which highlights the need for continuous support throughout the learning path,” says Susanna Toikka of the University of Eastern Finland, the first author of the article.

The findings emphasize the need to incorporate elements that support metacognitive knowledge into mathematics learning materials, as well as into teachers’ pedagogical practices.

Self-assessment and understanding of one’s own learning help to face new challenges

Metacognitive knowledge helps students not only to learn mathematics, but also more broadly in self-assessment and lifelong learning. Students who can assess their own learning and understanding are better equipped to face new challenges and adapt to changing environments. Such skills are crucial for lifelong learning, as they enable continuous development and learning throughout life.

“Metacognitive knowledge is a key factor in learning mathematics and problem-solving, but its significance also extends to self-assessment and lifelong learning,” says Toikka.

In schools, metacognitive knowledge can be effectively developed as part of education. Based on earlier studies, Toikka and colleagues have developed a combination of frameworks for metacognitive knowledge, which helps to identify students’ needs for development regarding metacognitive knowledge by offering an alternative perspective to that of traditional developmental psychology.

“This also supports teachers in promoting students’ metacognitive knowledge. Teachers can use the combination of frameworks to design and implement targeted interventions that support students’ skills in lifelong learning.”

According to Toikka, the combination of frameworks enhances understanding of metacognitive knowledge and helps to identify areas where individual support is needed: “This type of understanding is crucial for the development of metacognitive knowledge among diverse learners.”

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Credit of the article given to University of Eastern Finland

Elliptic curves have a long and distinguished history that can be traced back to antiquity. They are prevalent in many branches of modern mathematics, foremost of which is number theory.

In simplest terms, one can describe these curves by using a cubic equation of the form

where A and B are fixed rational numbers (to ensure the curve E is nice and smooth everywhere, one also needs to assume that its discriminant 4A³ + 27B² is non-zero).

To illustrate, let’s consider an example: choosing A=-1 and B=0, we obtain the following picture:

At this point it becomes clear that, despite their name, elliptic curves have nothing whatsoever to do with ellipses! The reason for this historical confusion is that these curves have a strong connection to elliptic integrals, which arise when describing the motion of planetary bodies in space.

The ancient Greek mathematician Diophantus is considered by many to be the father of algebra. His major mathematical work was written up in the tome Arithmetica which was essentially a school textbook for geniuses. Within it, he outlined many tools for studying solutions to polynomial equations with several variables, termed Diophantine Equations in his honour.

One of the main problems Diophantus considered was to find all solutions to a particular polynomial equation that lie in the field of rational numbers Q. For equations of “degree two” (circles, ellipses, parabolas, hyperbolas) we now have a complete answer to this problem. This answer is thanks to the late German mathematician Helmut Hasse, and allows one to find all such points, should they exist at all.

Returning to our elliptic curve E, the analogous problem is to find all the rational solutions (x,y) which satisfy the equation defining E. If we call this set of points E(Q), then we are asking if there exists an algorithm that allows us to obtain all points (x,y) belonging to E(Q).

At this juncture we need to introduce a group law on E, which gives an eccentric way of fusing together two points (p₁ and p₂) on the curve, to obtain a brand new point (p₄). This mimics the addition law for numbers we learn from childhood (i.e. the sum or difference of any two numbers is still a number). There’s an illustration of this rule below:

Under this geometric model, the point p₄ is defined to be the sum of p₁ and p₂ (it’s easy to see that the addition law does not depend on the order of the points p₁, p₂). Moreover the set of rational points is preserved by this notion of addition; in other words, the sum of two rational points is again a rational point.

Louis Mordell, who was Sadleirian Professor of Pure Mathematics at Cambridge University from 1945 to 1953, was the first to determine the structure of this group of rational points. In 1922 he proved

where the number of copies of the integers Z above is called the “rank r(E) of the elliptic curve E”. The finite group Τ_E(Q) on the end is uninteresting, as it never has more than 16 elements.

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*Credit for article given to Daniel Delbourgo*

The third diagonal column in Pascal’s Triangle (r = 2 in the usual way of labeling and numbering) consists of the triangular numbers (1, 3, 6, 10, …) – numbers that can be arranged in 2-dimensional triangular patterns. The fourth column of Pascal’s triangle gives us triangular-based pyramidal numbers (1, 4, 10, 20, …), built by stacking the triangular numbers. The columns further out give “higher dimensional” triangular numbers that arise from stacking the triangular numbers from the previous dimension.

It is not by coincidence that the triangular and higher-dimensional triangular numbers appear in Pascal’s Triangle. If you think about layering of polygonal numbers in terms of equations, you get

In the above equation p^d_(k,n) is the nth k-polygonal number of dimension d. Triangular numbers are the 3-polygonal numbers of dimension 2, square numbers are the 4-polygonal numbers of dimension 2, “square based pyramidal numbers” would be denoted as p^3_(4,n).
from the sum above, you can obtain this equation:

Which looks very much like the Pascal Identity C(n,r) = C(n-1,r-1) + C(n-1,r), except for some translation of the variables. To be precise, if we consider the case where k=3 and use r = d and n‘ = n+d-1 we can translate the triangular numbers into the appropriate positions in Pascal’s Triangle.

Along with the definitions for the end columns, the Pascal Identity allows us to generate the whole triangle. This suggests the following strategy for calculating the higher k-Polygonal numbers: create a modified Pascal’s Triangle whose first column is equal to k-2 (instead of 1), and whose last column is equal to 1 (as usual). This modified Pascal’s Triangle is generated using these initial values and the usual Pascal Identity.

Here is an example with k=5, which sets the first column values equal to 3 (except for the top value, which we keep as 1) and yields the pentagonal numbers (column 3) and the higher pentagonal numbers.

The formula for these modified Pascal Triangles is given by this equation:

If we apply the change of variables mentioned above, we can obtain this general formula for the higher polygonal numbers in terms of combinations:

This formula illustrates how polygonal numbers are built out of triangular numbers. It says that the nth d-dimensional k-polygonal number is equal to the nth d-dimensional triangular number, plus (k-3) copies of the n-1 d-dimensional triangular number. This is a little easier to understand when you forget about the higher-dimensions and look at the regular 2-dimensional polygonal number.

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*Credit for article given to dan.mackinnon*

 

An attempt to settle a decade-long argument over a controversial proof by mathematician Shinichi Mochizuki has seen a war of words on both sides, with Mochizuki dubbing the latest effort as akin to a “hallucination” produced by ChatGPT,

An attempt to fix problems with a controversial mathematical proof has itself become mired in controversy, in the latest twist in a saga that has been running for over a decade and has seen mathematicians trading unusually pointed barbs.

The story began in 2012, when Shinichi Mochizuki at Kyoto University, Japan, published a 500-page proof of a problem called the ABC conjecture. The conjecture concerns prime numbers involved in solutions to the equation a + b = c, and despite its seemingly simple form, it provides deep insights into the nature of numbers. Mochizuki published a series of papers claiming to have proved ABC using new mathematical tools he collectively called Inter-universal Teichmüller (IUT) theory, but many mathematicians found the initial proof baffling and incomprehensible.

While a small number of mathematicians have since accepted that Mochizuki’s papers prove the conjecture, other researchers say there are holes in his argument and it needs further work, dividing the mathematical community in two and prompting a prize of up to $1 million for a resolution to the quandary.

Now, Kirti Joshi at the University of Arizona has published a proposed proof that he says fixes the problems with IUT and proves the ABC conjecture. But Mochizuki and his supporters, as well as mathematicians who critiqued Mochizuki’s original papers, remain unconvinced, with Mochizuki declaring that Joshi’s proposal doesn’t contain “any meaningful mathematical content whatsoever”.

Central to Joshi’s work is an apparent problem, previously identified by Peter Scholze at the University of Bonn, Germany, and Jakob Stix at Goethe University Frankfurt, Germany, with a part of Mochizuki’s proof called Conjecture 3.12. The conjecture involves comparing two mathematical objects, which Scholze and Stix say Mochizuki did incorrectly. Joshi claims to have found a more satisfactory way to make the comparison.

Joshi also says that his theory goes beyond Mochizuki’s and establishes a “new and radical way of thinking about arithmetic of number fields”. The paper, which hasn’t been peer-reviewed, is the culmination of several smaller papers on ABC that Joshi has published over several years, describing them as a “Rosetta Stone” for understanding Mochizuki’s impenetrable maths.

Neither Joshi nor Mochizuki responded to a request for comment on this article, and, indeed, the two seem reluctant to communicate directly with each other. In his paper, Joshi says Mochizuki hasn’t responded to his emails, calling the situation “truly unfortunate”. And yet, several days after the paper was posted online, Mochizuki published a 10-page response, saying that Joshi’s work was “mathematically meaningless” and that it reminded him of “hallucinations produced by artificial intelligence algorithms, such as ChatGPT”.

Mathematicians who support Mochizuki’s original proof express a similar sentiment. “There is nothing to talk about, since his [Joshi’s] proof is totally flawed,” says Ivan Fesenko at Westlake University in China. “He has no expertise in IUT whatsoever. No experts in IUT, and the number is in two digits, takes his preprints seriously,” he says. “It won’t pass peer review.”

And Mochizuki’s critics also disagree with Joshi. “Unfortunately, this paper and its predecessors does not introduce any powerful mathematical technology, and falls far short of giving a proof of ABC,” says Scholze, who has emailed Joshi to discuss the work further. For now, the saga continues.

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*Credit for article given to Alex Wilkins*

It could be argued that the square is the ‘nicest’ rhombus, but the rhombus with angles of 60 and 120 degrees seems nicer still. One of the nice things about the 60/120 rhomb are the plane tilings that can be constructed from it. One of these tilings is the ‘tumbling blocks’ tiling shown at the top of the post, in which at some points you see ‘cubes’ (around the degree 3 vertices), while at others you see ‘flowers’ (around the degree 6 vertices). Because of the two different types of vertices, this is known as a quasi-regular rhombic tiling. (Another tiling that uses the 60/120 rhomb is this one.)

If you want to build a polyhedron that resembles the tumbling block tiling, one method is to reduce the number of petals in your flowers to 5, and then stretch your 60/120 rhombs until they are have angles of 63.435 and 116.565 degrees. Thirty of these rhombs arranged around vertices of degree 3 and 5 produces a quasi-regular polygon known as the rhombic triacontahedron.

The image above is of a model that was built using rhombic units printed onto card stock.

Despite the apparent ugliness of the 63.435 and 116.565 degree angle-measurements, our nice 60/120 rhomb has been, arguably, stretched into an even nicer one – a ‘golden rhombus,’ so called because the ratio of the diagonals is equal to the golden ratio.

The dual of the rhombic triacontahedron is the archemedian polyhedron known as the icosidodecahedron.

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*Credit for article given to dan.mackinnon*

The objects pictured above are interesting structures – they are derived from the prime factorization of a given number n. They can be described in a number of ways – for example, as directed graphs. Because they are nicely structured, they actually form something more special – a lattice. Accordingly, these structures are called factor lattices.
It’s easy to start drawing these by hand following the instructions below.

1. The first node is 1
2. Draw arrows out of this node for each of the prime factors of n.
3. The arrows that you just drew should connect to nodes labled with the prime factors of n.

Now, for each of the new nodes that you drew do the following:

4. Start from a node x that is not equal to n.
5. Draw arrows out of this node for each of the prime factors of n/x.
6. The arrows that you just drew (one for each p = n/x) should connect to nodes labled with the numbers p*x.

7. Now repeat 4,5, and 6 for each new node that you have drawn that is not equal to n.

This process is recursive, and ends when you have the complete lattice. The process is well suited for implementation as a computer program – the images above were created using SAGE using output from a Java program based on the algorithm above.

Manually trying out the steps out for a number like n = 24 goes something like this: First write out the prime factorization of 24, 24=(2*2*2)*3 = (2^3)*3. Starting with 1, draw arrows out to 2 and 3. Now looking at each node and following the algorithm, from the 2 you will get arrows out to 4 and 6. From the 3 you will get an arrow out to 6 as well. From 4 you will get arrows out to 8 or 12. From 6 you will get an arrow out to 12 as well. From 8 and from 12 you get arrows out to 24, and you are done.

In general, the algorithm produces a lattice that can be described as follows. Each node is a factor of the given number n. Two nodes are connected by an edge if their prime factorization differs by a single prime number. In other words, if a and b are nodes, and p = b/a, then there is an arrow p:a–>b.

It’s a good exercise to make the connections between the lattice structure and the prime factorization of a number n.

1. What does the factor lattice of a prime number look like?
2. If a number is just a power of a prime, what does its lattice look like?
3. If you know the factorization, can you find the number of nodes without drawing the lattice.

The answer to the last question (3) can be expressed as:

For example, if n = 24= 2^3*3, then the number of nodes will be (3+1)(1+1) = 8

That these structures can be thought of as “lattices”comes from the fact that you can think of the arrows as an ordering of the nodes, ab. The number 1 is always the least node in the factor lattice for n, while n itself is the greatest node. The property that actually makes these structures a “lattice” is that for any two nodes there is always a lower-bound for any pair of nodes in the lattice, and always an upper-bound for the pair (these are often referred to as meets and joins).

The Wolfram Demonstrations Project has a nice factor lattice demo that will draw factor lattices for a large number of integers for you. There is also a good Wikipedia entry for lattices in general.

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*Credit for article given to dan.mackinnon*

 

For everyone whose relationship with mathematics is distant or broken, Jo Boaler, a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start—to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what WEcall narrow mathematics—a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed, a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math, the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, “Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics,” Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by ‘math-ish’ thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport—these are generally answered with what WEcall “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book WEshare an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2—but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they’re more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. WEthink we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things—all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and We still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

 

WEwonder if people consider the physical representations more appropriate for younger kids.

That’s the thing—elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they’re drawing, they’re sketching, they’re building models, and nobody says that’s elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense.

They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

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Credit of the article given to Stanford University