Two studies give us a glimpse into our ancestors’ microbiome — you know, those trillions of bacteria that live in the human gut.

Image:  Maria Fabrizio for NPR

And the take-home message of the studies is clear: Western diets and modern-day hygiene have wiped a few dozen species right out of our digestive tracts. One missing microbe helps metabolize carbohydrates. Other bygone bacteria act as prebiotics. And another communicates with our immune system.

In other words, Americans’ digestive tracts look like barren deserts compared with the lush, tropical rain forest found inside indigenous people.

“The concern is that we’re losing keystone species,” says microbiologist M. Gloria Dominguez-Bello, at the New York University School of Medicine. “That’s a hypothesis, but we haven’t proved it.”

Dominguez-Bello and her colleagues are the first to characterize the gut bacteria of people completely isolated from modern medicine, food and culture.

In 2009, her colleagues and a medical team with the Venezuelan government took a helicopter to a remote Yanomami tribe at the border of Venezuela and Brazil. Members of the tribe have lived as hunter-gatherers for more than 11,000 years in a mountainous area of the Amazon rain forest.

The visit was the first time that particular tribe had direct contact with modern society. “They knew about us, but we didn’t know about them,” Dominguez-Bello says. “They had names [in their language] for our helicopters and planes.”

Dominguez-Bello’s colleagues took samples from 12 of the villagers’ fecal matter. Back in New York City, the team used DNA analysis to figure out which species thrived in the hunter-gatherers’ guts.

The biggest surprise was how many different species were present in the Yanomami’s microbiome. The tribe had about 50 percent more ecological diversity than the average American has, Dominguez-Bello and her colleagues reported Friday in the journal Science Advances.

Continue to Article by  MICHAELEEN DOUCLEFF for NPR

by Adriana Marais , Betony Adams , Andrew K. Ringsmuth , Marco Ferretti , J. Michael Gruber , Ruud Hendrikx , Maria Schuld , Samuel L. Smith , Ilya Sinayskiy , Tjaart P. J. Krüger , Francesco Petruccione  and Rienk van Grondelle Published:14 November 2018

Abstract

Biological systems are dynamical, constantly exchanging energy and matter with the environment in order to maintain the non-equilibrium state synonymous with living. Developments in observational techniques have allowed us to study biological dynamics on increasingly small scales. Such studies have revealed evidence of quantum mechanical effects, which cannot be accounted for by classical physics, in a range of biological processes. Quantum biology is the study of such processes, and here we provide an outline of the current state of the field, as well as insights into future directions.

1.  Introduction

Quantum mechanics is the fundamental theory that describes the properties of subatomic particles, atoms, molecules, molecular assemblies and possibly beyond. Quantum mechanics operates on the nanometre and sub-nanometre scales and is at the basis of fundamental life processes such as photosynthesis, respiration and vision. In quantum mechanics, all objects have wave-like properties, and when they interact, quantum coherence describes the correlations between the physical quantities describing such objects due to this wave-like nature.

In photosynthesis, respiration and vision, the models that have been developed in the past are fundamentally quantum mechanical. They describe energy transfer and electron transfer in a framework based on surface hopping. The dynamics described by these models are often ‘exponential’ and follow from the application of Fermi’s Golden Rule [1,2]. As a consequence of averaging the rate of transfer over a large and quasi-continuous distribution of final states the calculated dynamics no longer display coherences and interference phenomena. In photosynthetic reaction centres and light-harvesting complexes, oscillatory phenomena were observed in numerous studies performed in the 1990s and were typically ascribed to the formation of vibrational or mixed electronic–vibrational wavepackets. The reported detection of the remarkably long-lived (660 fs and longer) electronic quantum coherence during excitation energy transfer in a photosynthetic system revived interest in the role of ‘non-trivial’ quantum mechanics to explain the fundamental life processes of living organisms [3]. However, the idea that quantum phenomena—like coherence—may play a functional role in macroscopic living systems is not new. In 1932, 10 years after quantum physicist Niels Bohr was awarded the Nobel Prize for his work on the atomic structure, he delivered a lecture entitled ‘Light and Life’ at the International Congress on Light Therapy in Copenhagen [4]. This raised the question of whether quantum theory could contribute to a scientific understanding of living systems. In attendance was an intrigued Max Delbrück, a young physicist who later helped to establish the field of molecular biology and won a Nobel Prize in 1969 for his discoveries in genetics [5].

All living systems are made up of molecules, and fundamentally all molecules are described by quantum mechanics. Traditionally, however, the vast separation of scales between systems described by quantum mechanics and those studied in biology, as well as the seemingly different properties of inanimate and animate matter, has maintained some separation between the two bodies of knowledge. Recently, developments in experimental techniques such as ultrafast spectroscopy [6], single molecule spectroscopy [7–11], time-resolved microscopy [12–14] and single particle imaging [15–18] have enabled us to study biological dynamics on increasingly small length and time scales, revealing a variety of processes necessary for the function of the living system that depend on a delicate interplay between quantum and classical physical effects.

Quantum biology is the application of quantum theory to aspects of biology for which classical physics fails to give an accurate description. In spite of this simple definition, there remains debate over the aims and role of the field in the scientific community. This article offers a perspective on where quantum biology stands today, and identifies potential avenues for further progress in the field.

2.  What is quantum biology?

Biology, in its current paradigm, has had wide success in applying classical models to living systems. In most cases, subtle quantum effects on (inter)molecular scales do not play a determining role in overall biological function. Here, ‘function’ is a broad concept. For example: How do vision and photosynthesis work on a molecular level and on an ultrafast time scale? How does DNA, with stacked nucleotides separated by about 0.3 nm, deal with UV photons? How does an enzyme catalyse an essential biochemical reaction? How does our brain with neurons organized on a sub-nanometre scale deal with such an amazing amount of information? How do DNA replication and expression work? All these biological functions should, of course, be considered in the context of evolutionary fitness. The differences between a classical approximation and a quantum-mechanical model are generally thought to be negligible in these cases, even though at the basis every process is entirely governed by the laws of quantum mechanics. What happens at the ill-defined border between the quantum and classical regimes? More importantly, are there essential biological functions that ‘appear’ classical but in reality are not? The role of quantum biology is precisely to expose and unravel this connection.

Fundamentally, all matter—animate or inanimate—is quantum mechanical, being constituted of ions, atoms and/or molecules whose equilibrium properties are accurately determined by quantum theory. As a result, it could be claimed that all of biology is quantum mechanical. However, this definition does not address the dynamical nature of biological processes, or the fact that a classical description of intermolecular dynamics seems often sufficient. Quantum biology should, therefore, be defined in terms of the physical ‘correctness’ of the models used and the consistency in the explanatory capabilities of classical versus quantum mechanical models of a particular biological process.

As we investigate biological systems on nanoscales and larger, we find that there exist processes in biological organisms, detailed in this article, for which it is currently thought that a quantum mechanical description is necessary to fully characterize the behaviour of the relevant subsystem. While quantum effects are difficult to observe on macroscopic time and length scales, processes necessary for the overall function and therefore survival of the organism seem to rely on dynamical quantum-mechanical effects at the intermolecular scale. It is precisely the interplay between these time and length scales that quantum biology investigates with the aim to build a consistent physical picture.

Grand hopes for quantum biology may include a contribution to a definition and understanding of life, or to an understanding of the brain and consciousness. However, these problems are as old as science itself, and a better approach is to ask whether quantum biology can contribute to a framework in which we can repose these questions in such a way as to get new answers. The study of biological processes operating efficiently at the boundary between the realms of quantum and classical physics is already contributing to improved physical descriptions of this quantum-to-classical transition.

More immediately, quantum biology promises to give rise to design principles for biologically inspired quantum nanotechnologies, with the ability to perform efficiently at a fundamental level in noisy environments at room temperature and even make use of these ‘noisy environments’ to preserve or even enhance the quantum properties [19,20]. Through engineering such systems, it may be possible to test and quantify the extent to which quantum effects can enhance processes and functions found in biology, and ultimately answer whether these quantum effects may have been purposefully selected in the design of the systems. Importantly, however, quantum bioinspired technologies can also be intrinsically useful independently from the organisms that inspired them.

3.  Quantum mechanics: an introduction for biologists

At the beginning of the twentieth century, the success of classical physics in describing all observable phenomena had begun to be challenged in certain respects. In 1900, as a means to explain the spectral energy distribution of blackbody radiation, Planck introduced the idea that interactions between matter and electromagnetic radiation of frequency ν are quantized, occurring only in integer multiples of hν, where h is the fundamental Planck constant. Five years later, Einstein further developed the notion of energy quantization by extending it to include the photon, a quantum of light. This concept is illustrated by the photoelectric effect where light incident on a material leads to the emission of electrons. It is, however, not the intensity of the light that determines this emission but rather its frequency. Even low-intensity light of a suitable frequency will lead to electrons being emitted whereas high-intensity light below this threshold frequency will have no effect. Einstein explained this by proposing that in this instance light behaves as a particle rather than a wave, with discrete energies hν that can be transferred to the electrons in a material. Bohr’s 1913 model of the hydrogen atom, with its discrete energy states, and Compton’s 1923 work with X-rays all contributed to the beginning of a new era in modern physics. These ways of explaining blackbody radiation and the photoelectric effect, as well as atomic stability and spectroscopy, led to the development of quantum mechanics, a theory that has proved extremely successful in predicting and describing microphysical systems [21,22].

Whereas Planck and Einstein began the quantum revolution by postulating that radiation also demonstrates particle-like behaviour, de Broglie, in 1923, made the complementary suggestion that matter itself has wave-like properties, with a wavelength related to its momentum through Planck’s constant. This hypothesis suggested that matter waves should undergo diffraction, which was subsequently proved by experiments that demonstrated that particles such as electrons showed interference patterns. Schrödinger built on this observation in his formulation of quantum mechanics, which describes the dynamics of microscopic systems through the use of wave mechanics. The formulation of quantum mechanics allows for the investigation of a number of important facets of a quantum state: its mathematical description at any time t, how to calculate different physical quantities associated with this state and how to describe the evolution of the state in time [21,22].

July 2, 2013 by Emerging Technology from the arXiv

Teleportation is one of the more extraordinary phenomena in the quantum world. It allows a quantum object, such as a photon or electron, to travel from one location to another without passing through the space in between.

Teleportation is a standard procedure in any decent quantum mechanics laboratory. Physicists use it on a daily basis for quantum communication and quantum computation.

If that sounds exotic, you ain’t seen nothing yet; teleportation is about to get a whole lot weirder. That’s because until now, physicists have only been able to teleport single particles, one at a time. Today, Christine Muschik at the Mediterranean Technology Park in Barcelona and a bunch of mates say they’ve worked out how to teleport quantum stuff continuously.

That will allow them to manipulate one quantum particle while watching the effects occur in another particle elsewhere. That’s essentially quantum remote control.

The basic set up is a straightforward extension of traditional teleportation. This is possible because of a strange quantum phenomenon known as entanglement, which occurs when two quantum particles are so deeply linked that they share the same existence.

In mathematical terms, both particles are described by a single wave function. So any manipulation of one particle automatically influences the other instantly, regardless of the distance between them.

Teleportation occurs when the first of these entangled particles interacts with another quantum particle, let’s call it X. This interaction immediately influences the state of the second entangled particle.

The trick that physicists have perfected is to arrange this interaction so that the second entangled particle ends up in the same state as the quantum particle called X. This isn’t just a very similar state, it is an identical state—no measurement could distinguish this particle from the original X. When this happens, X has been teleported.

The new technique works in a similar way. First, physicists create a pair of entangled particles. They then place one particle in a varying magnetic field to influence its state.

The new trick that they’ve discovered is to arrange this experiment so that manipulation of the first particle causes the state of its entangled partner to change in the same way.

In other words, they use the magnetic field in one region of space to continuously control the state of a particle somewhere else in space. Or as Muschick and co put it: “We show how the ability to perform quantum operations continuously and deterministically can be leveraged for inducing nonlocal dynamics between two separate parties.”

By Robert Coolman - Live Science Contributor September 26, 2014

Quantum mechanics is the branch of physics relating to the very small.

It results in what may appear to be some very strange conclusions about the physical world. At the scale of atoms and electrons, many of the equations of classical mechanics, which describe how things move at everyday sizes and speeds, cease to be useful. In classical mechanics, objects exist in a specific place at a specific time. However, in quantum mechanics, objects instead exist in a haze of probability; they have a certain chance of being at point A, another chance of being at point B and so on.

Quantum mechanics (QM) developed over many decades, beginning as a set of controversial mathematical explanations of experiments that the math of classical mechanics could not explain. It began at the turn of the 20th century, around the same time that Albert Einstein published his theory of relativity, a separate mathematical revolution in physics that describes the motion of things at high speeds. Unlike relativity, however, the origins of QM cannot be attributed to any one scientist. Rather, multiple scientists contributed to a foundation of three revolutionary principles that gradually gained acceptance and experimental verification between 1900 and 1930. They are:

Quantized properties: Certain properties, such as position, speed and color, can sometimes only occur in specific, set amounts, much like a dial that “clicks” from number to number. This challenged a fundamental assumption of classical mechanics, which said that such properties should exist on a smooth, continuous spectrum. To describe the idea that some properties “clicked” like a dial with specific settings, scientists coined the word “quantized.”

Particles of light: Light can sometimes behave as a particle. This was initially met with harsh criticism, as it ran contrary to 200 years of experiments showing that light behaved as a wave; much like ripples on the surface of a calm lake. Light behaves similarly in that it bounces off walls and bends around corners, and that the crests and troughs of the wave can add up or cancel out. Added wave crests result in brighter light, while waves that cancel out produce darkness. A light source can be thought of as a ball on a stick being rhythmically dipped in the center of a lake. The color emitted corresponds to the distance between the crests, which is determined by the speed of the ball’s rhythm.

Waves of matter: Matter can also behave as a wave. This ran counter to the roughly 30 years of experiments showing that matter (such as electrons) exists as particles.

In 1900, German physicist Max Planck sought to explain the distribution of colors emitted over the spectrum in the glow of red-hot and white-hot objects, such as light-bulb filaments. When making physical sense of the equation he had derived to describe this distribution, Planck realized it implied that combinations of only certain colors (albeit a great number of them) were emitted, specifically those that were whole-number multiples of some base value. Somehow, colors were quantized! This was unexpected because light was understood to act as a wave, meaning that values of color should be a continuous spectrum. What could be forbidding atoms from producing the colors between these whole-number multiples? This seemed so strange that Planck regarded quantization as nothing more than a mathematical trick. According to Helge Kragh in his 2000 article in Physics World magazine, “Max Planck, the Reluctant Revolutionary,” “If a revolution occurred in physics in December 1900, nobody seemed to notice it. Planck was no exception …”

Planck’s equation also contained a number that would later become very important to future development of QM; today, it’s known as “Planck’s Constant.”

Quantization helped to explain other mysteries of physics. In 1907, Einstein used Planck’s hypothesis of quantization to explain why the temperature of a solid changed by different amounts if you put the same amount of heat into the material but changed the starting temperature.

Since the early 1800s, the science of spectroscopy had shown that different elements emit and absorb specific colors of light called “spectral lines.” Though spectroscopy was a reliable method for determining the elements contained in objects such as distant stars, scientists were puzzled about why each element gave off those specific lines in the first place. In 1888, Johannes Rydberg derived an equation that described the spectral lines emitted by hydrogen, though nobody could explain why the equation worked. This changed in 1913 when Niels Bohr applied Planck’s hypothesis of quantization to Ernest Rutherford’s 1911 “planetary” model of the atom, which postulated that electrons orbited the nucleus the same way that planets orbit the sun. According to Physics 2000 (a site from the University of Colorado), Bohr proposed that electrons were restricted to “special” orbits around an atom’s nucleus. They could “jump” between special orbits, and the energy produced by the jump caused specific colors of light, observed as spectral lines. Though quantized properties were invented as but a mere mathematical trick, they explained so much that they became the founding principle of QM.

In 1905, Einstein published a paper, “Concerning an Heuristic Point of View Toward the Emission and Transformation of Light,” in which he envisioned light traveling not as a wave, but as some manner of “energy quanta.” This packet of energy, Einstein suggested, could “be absorbed or generated only as a whole,” specifically when an atom “jumps” between quantized vibration rates. This would also apply, as would be shown a few years later, when an electron “jumps” between quantized orbits. Under this model, Einstein’s “energy quanta” contained the energy difference of the jump; when divided by Planck’s constant, that energy difference determined the color of light carried by those quanta.

With this new way to envision light, Einstein offered insights into the behavior of nine different phenomena, including the specific colors that Planck described being emitted from a light-bulb filament. It also explained how certain colors of light could eject electrons off metal surfaces, a phenomenon known as the “photoelectric effect.” However, Einstein wasn’t wholly justified in taking this leap, said Stephen Klassen, an associate professor of physics at the University of Winnipeg. In a 2008 paper, “The Photoelectric Effect: Rehabilitating the Story for the Physics Classroom,” Klassen states that Einstein’s energy quanta aren’t necessary for explaining all of those nine phenomena. Certain mathematical treatments of light as a wave are still capable of describing both the specific colors that Planck described being emitted from a light-bulb filament and the photoelectric effect. Indeed, in Einstein’s controversial winning of the 1921 Nobel Prize, the Nobel committee only acknowledged “his discovery of the law of the photoelectric effect,” which specifically did not rely on the notion of energy quanta.

Roughly two decades after Einstein’s paper, the term “photon” was popularized for describing energy quanta, thanks to the 1923 work of Arthur Compton, who showed that light scattered by an electron beam changed in color. This showed that particles of light (photons) were indeed colliding with particles of matter (electrons), thus confirming Einstein’s hypothesis. By now, it was clear that light could behave both as a wave and a particle, placing light’s “wave-particle duality” into the foundation of QM.

Since the discovery of the electron in 1896, evidence that all matter existed in the form of particles was slowly building. Still, the demonstration of light’s wave-particle duality made scientists question whether matter was limited to acting only as particles. Perhaps wave-particle duality could ring true for matter as well? The first scientist to make substantial headway with this reasoning was a French physicist named Louis de Broglie. In 1924, de Broglie used the equations of Einstein’s theory of special relativity to show that particles can exhibit wave-like characteristics, and that waves can exhibit particle-like characteristics. Then in 1925, two scientists, working independently and using separate lines of mathematical thinking, applied de Broglie’s reasoning to explain how electrons whizzed around in atoms (a phenomenon that was unexplainable using the equations of classical mechanics). In Germany, physicist Werner Heisenberg (teaming with Max Born and Pascual Jordan) accomplished this by developing “matrix mechanics.” Austrian physicist Erwin Schrödinger developed a similar theory called “wave mechanics.” Schrödinger showed in 1926 that these two approaches were equivalent (though Swiss physicist Wolfgang Pauli sent an unpublished result to Jordan showing that matrix mechanics was more complete).

The Heisenberg-Schrödinger model of the atom, in which each electron acts as a wave (sometimes referred to as a “cloud”) around the nucleus of an atom replaced the Rutherford-Bohr model. One stipulation of the new model was that the ends of the wave that forms an electron must meet. In “Quantum Mechanics in Chemistry, 3rd Ed.” (W.A. Benjamin, 1981), Melvin Hanna writes, “The imposition of the boundary conditions has restricted the energy to discrete values.” A consequence of this stipulation is that only whole numbers of crests and troughs are allowed, which explains why some properties are quantized. In the Heisenberg-Schrödinger model of the atom, electrons obey a “wave function” and occupy “orbitals” rather than orbits. Unlike the circular orbits of the Rutherford-Bohr model, atomic orbitals have a variety of shapes ranging from spheres to dumbbells to daisies.

In 1927, Walter Heitler and Fritz London further developed wave mechanics to show how atomic orbitals could combine to form molecular orbitals, effectively showing why atoms bond to one another to form molecules. This was yet another problem that had been unsolvable using the math of classical mechanics. These insights gave rise to the field of “quantum chemistry.”

Also in 1927, Heisenberg made another major contribution to quantum physics. He reasoned that since matter acts as waves, some properties, such as an electron’s position and speed, are “complementary,” meaning there’s a limit (related to Planck’s constant) to how well the precision of each property can be known. Under what would come to be called “Heisenberg’s uncertainty principle,” it was reasoned that the more precisely an electron’s position is known, the less precisely its speed can be known, and vice versa. This uncertainty principle applies to everyday-size objects as well, but is not noticeable because the lack of precision is extraordinarily tiny. According to Dave Slaven of Morningside College (Sioux City, IA), if a baseball’s speed is known to within a precision of 0.1 mph, the maximum precision to which it is possible to know the ball’s position is 0.000000000000000000000000000008 millimeters.

The principles of quantization, wave-particle duality and the uncertainty principle ushered in a new era for QM. In 1927, Paul Dirac applied a quantum understanding of electric and magnetic fields to give rise to the study of “quantum field theory” (QFT), which treated particles (such as photons and electrons) as excited states of an underlying physical field. Work in QFT continued for a decade until scientists hit a roadblock: Many equations in QFT stopped making physical sense because they produced results of infinity. After a decade of stagnation, Hans Bethe made a breakthrough in 1947 using a technique called “renormalization.” Here, Bethe realized that all infinite results related to two phenomena (specifically “electron self-energy” and “vacuum polarization”) such that the observed values of electron mass and electron charge could be used to make all the infinities disappear.

Since the breakthrough of renormalization, QFT has served as the foundation for developing quantum theories about the four fundamental forces of nature: 1) electromagnetism, 2) the weak nuclear force, 3) the strong nuclear force and 4) gravity. The first insight provided by QFT was a quantum description of electromagnetism through “quantum electrodynamics” (QED), which made strides in the late 1940s and early 1950s. Next was a quantum description of the weak nuclear force, which was unified with electromagnetism to build “electroweak theory” (EWT) throughout the 1960s. Finally came a quantum treatment of the strong nuclear force using “quantum chromodynamics” (QCD) in the 1960s and 1970s. The theories of QED, EWT and QCD together form the basis of the Standard Model of particle physics. Unfortunately, QFT has yet to produce a quantum theory of gravity. That quest continues today in the studies of string theory and loop quantum gravity.

Unusual flashes of gamma rays could reveal that what appear to be giant black holes are actually huge wormholes, a new study finds.

Wormholes are tunnels in space-time that can theoretically allow travel anywhere in space and time, or even into another universe. Einstein’s theory of general relativity suggests wormholes are possible, although whether they really exist is another matter.

In many ways, wormholes resemble black holes. Both kinds of objects are extremely dense and possess extraordinarily strong gravitational pulls for bodies their size. The main difference is that no object can theoretically come back out after crossing a black hole’s event horizon — the threshold where the speed needed to escape the black hole’s gravitational pull exceeds the speed of light — whereas any body entering a wormhole could theoretically reverse course.

Assuming wormholes might exist, researchers investigated ways that one might distinguish a wormhole from a black hole. They focused on supermassive black holes with masses millions to billions of times that of the sun, which are thought to dwell at the hearts of most, if not all, galaxies. For example, at the center of our Milky Way galaxy lies Sagittarius A*, a monster black hole that is about 4.5 million solar masses in size.

Anything entering one mouth of a wormhole would exit out its other mouth. The scientists reasoned that meant that matter entering one mouth of a wormhole could potentially slam into matter entering the other mouth of the wormhole at the same time, a kind of event that would never happen with a black hole.

Any matter falling into a mouth of a supermassive wormhole would likely travel at extraordinarily high speeds due to its powerful gravitational fields. The scientists modeled the consequences of matter flowing through both mouths of a wormhole to where these mouths meet, the wormhole’s “throat.” The result of such collisions are spheres of plasma expanding out both mouths of the wormhole at nearly the speed of light, the researchers said.

“What surprises me most of all is that no one has proposed this idea before, because it is rather simple,” study lead author Mikhail Piotrovich, an astrophysicist at the Central Astronomical Observatory in Saint Petersburg, Russia, told Space.com.

The researchers compared the outbursts from such wormholes with those from a kind of supermassive black hole known as an active galactic nucleus (AGN), which can spew out more radiation than our entire galaxy does as they devour matter around them, and do so from a patch of space no larger than our solar system. AGNs are typically surrounded by rings of plasma known as accretion disks and can emit powerful jets of radiation from their poles.

Browne’s work on classical groups demonstrated simple proofs of important topological properties of and relations between classical groups. Her work in general focused on linear and matrix algebra. Browne saw the importance of computer science early on, writing a $60,000 grant to IBM to bring a computer to NCCU in 1960 – one of the first computers in academic computing, and probably the first at a historically black school. Throughout her career, Browne worked to help gifted mathematics students, educating them and offering them financial support to pursue higher education. Notable students included Joseph Battle, William Fletcher, Asamoah Nkwanta, and Nathan Simms. She established summer institutes to provide continuing education in mathematics for high school teachers. In 1974 she was awarded the first W. W. Rankin Memorial Award from the North Carolina Council of Teachers of Mathematics for her work with mathematics education.