Category Archives: Education

How to use ChatGPT to model exoplanet interiors as a lab for high school students.

Like the introduction of hand calculators into the classroom in the 1970s, ChatGPT offers enormous promise but currently suffers from a variety of negative expectations. Some of the arguments against students using calculators in the 1970s classroom are being used today against ChatGPT. I think there are some applications that work very well if you consider ChatGPT to be an intelligent ‘hand calculator’ in the math and physical science classroom. Here is an example I came up with without much effort!

I. Student Pre-Requisite Knowledge

The following example requires students to work with Scientific Notation, calculating the volume of spheres and shells, and working with the mass:density:volume relationship.

II. Exoplanet Interior Modeling

Astronomers have discovered over 5000 exoplanets orbiting other stars. We call these ‘exoplanets’ so that they don’t get confused with the ‘planets’ in our solar system. From a careful study of these exoplanets, astronomers can figure out how long they take to orbit their stars, their distance from the star, their diameters and their masses. How do they use this information to figure out what the insides of these exoplanets look like? This activity will show how a simple knowledge of mass, volume and density provides the clues!

III. Mass, Density and Volume

Mass, volume and density are related to each other. If two things occupy the same volume but have different masses, the less-massive one will have the lower density.

Density = Mass / Volume.

Example 1: A Prospector had his sample weighed to be 20 grams, and its volume calculated by water displacement and found to be 4 cubic centimeters. If pure gold has a density of 19.3 gm/cc, is his sample actually gold or is it iron pyrite (density 5.0 gm/cc)?

Answer:   Density =  20 gm/4 cc =   5 gm/cc   so it’s iron pyrite or ‘Fools Gold’.

Example 2: A basic principle of physics is that light things of low density float on top of denser things. Why do you have to shake a bottle of salad dressing before you use it?

ChatGPT Query: There are five different liquids mixed together in a bottle. After 10 minutes they sort themselves out. The liquids are:   Olive oil ( 0.92 g/cc ), water ( 1.0  g/cc), molassis ( 1.4 g/cc) , vinegar ( 1.0006 g/cc), honey (1.43 g/cc). From bottom to top, how will the liquids separate themselves?

IV. Designing Mercury with a One-Component Interior Model

Mercury was formed close to the sun where only iron and nickel-rich compounds could condense into a planet. Let’s model Mercury and see what we discover. The actual mass of Mercury is 0.055 times Earth.

Step 1 – Use the formula  for the volume of a sphere V=4/3 pR3 and with a known radius for Mercury of Rm = 2.43×106 meters to get the volume of the planet of V = 6×1019 m3.

Step 2 – Calculate the mass of Mercury for various choices of density. Give the predicted mass for Mercury in multiples of Earth’s mass of 5.97×1024 kg.

Step 3:  Test your knowledge: For a density of 5000 kg/m3 and a radius of 2.43×106 meters, what is the mass of Mercury for these selected values? Give your answer to two significant figures.

Volume = 4/3p (2.43×106meters)3     =  6.0×1019 m3

Mass = 5000 x Volume =  3.0×1023 kg

Mase(Earth units) = 3.0×1023 kg /5.97×1024 kg  = 0.05 times Earth

Use ChatGPT to generate data for plotting. Enter this question into the window:

ChatGPT Query: A sphere has a radius of 2.43×10^6 meters. What is the mass of the sphere if its density is 5000 kg per cubic meter? Express your answer in units of Earth’s mass of 5.97 x 10^24 kg. Give your answer to two significant figures.

Repeat the ChatGPT query four times to generate a mass estimate for densities of 4000, 4500, 5000, 5500 and 6500 kg/m3. Plot these points on a graph of mass versus density and draw a line through the values. Which density gives the best match to the observed mass of Mercury of 0.055 Mearth?  (Answer: about 5500 kg/m3).

V. Designing Mars with Two-Component Models

Now we add two components together for planets that have a high density core and a lower density mantle. These would have formed farther out than the orbit of Mercury but with masses lower than than of Earth. The mathematical model consists of a spherical core with a radius of Rc, surrounded by a spherical shell with an inner radius of Rc and an outer radius of Rp, where Rp is the observed planetary radius.  Mathematically the model looks like this:

M = Dc x 4/3p Rc3 + Dm x 4/3p ( Rp3 – Rc3)

Draw a diagram of the planet’s interior showing Rc and Rp and confirm that this is the correct formula for the total mass of the planet where Dc is the core density, and Dm is the mantle density.

Test Case: An exoplanet is discovered with a mass of 5.97×10^24 kg and a radius of 6,378 kilometers. If the radius of its core is estimated to be Rc = 3,000 km and its core density is 7000 kg/m3, what is the average density of the mantle material?

Vc = 4/3p Rc3  =   4/3p (3000000m)3 = 1.1×1020 m3

Vmantle = 4/3 p Rp3 – Vc  =  1.1×1021 m3 – 1.1×1020 m3 = 9.8×1020 m3.

Solve equation for Dm:

Dm =  ( M – Dc x Vc ) /Vm

Dm = (5.97×1024 – 7000 x 1.1×1020)/9.8×1020 =  5300 kg/m3

Check your answer with ChatGPT using this query. A planet consists of a core region with a radius of Rc and a mantle region extending to the planet’s surface at a radius of Rp. If the planet is a perfect sphere with a radius Rp = 6378 km and Rc = 3000 km, with a total mass of 5.97×10^24 kg, for a core density of 7000 kg/cubic meters, what is the average mantle density? Give the answer to two significant figures.

Now lets use ChatGPT to generate some models and then we can select the best one. We will select a mantle density from three values, 2000, 3000 and 4000 kg/m3. The core density Dc will be fixed at Dc = 9000 kg/m3. We will use the measured radius for Mars of Rp = 3.4×106 meters, and its total mass of Mm = 6.4×1023 kg. We then vary the core radius Rc. We will plot three curves on a graph of Rc versus Mm one for each value of the assumed mantle density. Use this ChatGPT query to generate your data points.

ChatGPT Query: A planet is modeled as a sphere with  a radius of Rp=3.4×10^6 meters. It consists of a spherical core region with a radius of Rc surrounded by a spherical shell with an inner radius of Rc and an outer radius of Rp. The core of the planet has a density of 9000 kg/cubic meters. The radius of the core Rc = 30% of the planet’s radius. If the density of the mantle is 2000 kg/cubic meter, what is the total mass of the planet in multiples of the mass of Earth, which is 5.97×10^24 kg? Give your answer  to two significant figures?

Repeat this query by changing the mantle density and the core radius values and then plot enough points along each density curve to see the trend clearly. An example of an Excel spreadsheet version of this data is shown in this graph:

This graph shows solutions for a two-component mars model where the mantle has three different densities (2000, 3000 and 4000 kg/m3). The average density of mars is 3900 kg/m3. Which core radius and mantle density combinations seem to be a better match for Mar’s total mass of 0.11 Mearth for the given density of the mantle?

VI: Modeling Terrestrial Planets with a three-component interior.

The most general exoplanet model has three zones; a dense core, a mantle and a low-density crust. This is the expected case for Earth-like worlds. Using our Earth as an example, rocky exoplanets have interiors stratified into three layers: Core, mantle, crust. 

Core material is typically iron-nickel with a density of   9000 kg/m3

Mantle material is basaltic rock at a density of 4500 kg/m3

Crust is low-density silicate rich material with a density of 3300 kg/m3

The basic idea in modeling a planet interior is that with the three assumed densities, you vary the volume that they occupy inside the exoplanet until you match the actual mass (Mexo) in kilograms and radius (Rexo) in meters of the exoplanet that is observed. The three zones occupy the radii  Rc, Rm, Rp

We will adjust the core and mantle radii until we get a good match to the exoplanet observed total mass and radius. Let’s assume that the measured values for the Super-Earth exoplanet mass is Mp = 2.5xEarth = 1.5×1025 kg,  and its radius is Rp = 1.5xEarth = 9.6×106 meters.

Core Volume  Vcore = 4/3p Rc3

Mantle Volume  Vm = 4/3 p (Rm3 – Rc3)

Crust Volume   Vcrust =  4/3 p (Rp3 – Rm3)

So the total Mass = (9000 Vcore + 4500Vm + 3300Vcrust)/Mp

Rc ,Rm and Rp are the core, mantle and planet radii in meters, and the total mass of the model is given in multiples of the exoplanet’s mass Mp.

Let’s do a test case that we work by hand to make sure we understand what we are doing.

Choose Rc = 30% of Rp and Rm = 80% of Rp. What is the predicted total mass of the exoplanet?

Rc = 0.3 x 9.6×106 meters =  2.9×106 meters.

Rm = 0.8x 9.6×106 meters =  7.7×106 meters.

Then

Vcore =  4/3p (2.9×106)3 = 1.0×1020 m3

Vm =  4/3p ( (7.7×106)3 – (2.9×106)3) =  1.8×1021 m3

Vcrust =  4/3p ((9.6×106)3  – (7.7×106)3) =  1.8×1021 m3

Then  Mass = (9000 Vcore + 4500 Vm + 3300Vcrust)/Mp

Mass = (9×1023 kg + 8.1×1024 kg + 5.9×1024 kg)/1.5×1025 kg  =  1.0 Mp

Now lets use ChatGPT to generate some models from which we can make a choice.

Enter the following query into ChatGPT to check your answer to the above test problem.

ChatGPT Query: A spherical planet with a radius of Rp consists of three interior zones; a core with a radius of Rcore, a mantle with an inner radius of Rc and an outer radius of Rm,  and a crust with an inner radius of Rm and an outer radius of Rp=9.6×10^6 meters. If the density of the core is 9000 kg/m^3, the mantle is 4500 kg/m^3 and the crust is 3300 kg/m^3, What is the total mass of the planet if Rc = 30% of Rp and Rm = 80% of Rp? Give your answer for the planet’s total mass in multiples of the planet’s known mass of 1.5×10^25 kg, and to two significant figures.

Re-run this ChatGPT query but change the values for the mantle radius Rm and core radius Rc each time. Plot your models on a graph of   Rc versus the calculated mass Mp on curves for which Rm is constant. An example of this plot is shown in the excel spreadsheet plot below.

For example, along the black curve we are using Rm=0.8. At Rc = 0.5 we have a model where the core extends to 50% of the radius of the exoplanet .The mantle extends to 80% of the radius, and so the crust occupies the last 20% of the radius to the surface. With densities of 9000, 4500 and 3300 kg/m3 respectively, the Y-axis predicts a total mass of about 1.1 times the observed mass of the exoplanet (1.00 in these units). With a bit of fine-tuning we can get to the desired 1.00 of the mass.    But what about the solution at (0.3, 1.00) ? In fact, all of the solutions along the horizontal line along y = 1.00 are mathematically valid.

Question 1: The exoplanet is located close to its star where iron and nickel can remain in solid phase but the lower density silicates remain in a gaseous phase. Which of the models favors this location at formation?

Answer: The exoplanet should have a large iron/nickel core and not much of a mantle or crust. This favors solutions on the y=1.00 line to the right of x=0.5.

Question 2: The exoplanet is located far from its star where it is cool enough that silicates can condense out of their gas phase as the exoplanet forms. Which of the models favor this location?

Answer: The exoplanet will have a small iron/nickel core and a large mantle and crust. This favors models to the left of x= 0.5.

So here you have some examples for how ChatGPT can be used as an intelligent calculator once the students understands how to use the equations and is able to explain why they are being used for a given modeling scenario.

I would be delighted to get your responses and suggestions to this approach . Just include your comment in the Linkedin page where I have posted this idea.

The Last Total Solar Eclipse…Ever-Updated

One year ago, I posted a fun problem of predicting when we will have the very last total solar eclipse viewable from Earth. It was a fun calculation to do, and the answer seemed to be 700 million years from now, but I have decided to revisit it with an important new feature added: The slow but steady evolution of the sun’s diameter. For educators, you can visit the Desmos module that Luke Henke and I put together for his students.

The apparent lunar diameter during a total solar eclipse depends on whether the moon is at perigee or apogee, or at some intermediate distance from Earth. This is represented by the two red curved lines and the red area in between them. The upper red line is the angular diameter viewed from Earth when the moon is at perigee (closest to Earth) and will have the largest possible diameter. The lower red curve is the moon’s angular diameter at apogee (farthest from Earth) when its apparent diameter will be the smallest possible. As I mentioned in the previous posting, these two curves will slowly drift to smaller values because the Moon is moving away from Earth at about 3cm per year. Using the best current models for lunar orbit evolution, these curves will have the shapes shown in the above graph and can be approxmately modeled by the quadratic equations:

Perigee: Diameter = T2 – 27T +2010 arcseconds

Apogee: Diameter = T2 -23T +1765 arcseconds.

where T is the time since the present in multiples of 100 million years, so a time 300 million years ago is T=-3, and a time 500 million years in the future is T=+5.

The blue region in the graph shows the change in the diameter of the Sun and is bounded above by its apparent diameter at perihelion (Earth closest to Sun) and below by its farthest distance called aphelion. This is a rather narrow band of possible angular sizes, and the one of interest will depend on where Earth is in its orbit around the Sun AND the fact that the elliptical orbit of Earth is slowly rotating within the plane of its orbit so that at the equinoxes when eclipses can occur, the Sun will vary in distance between its perihelion and aphelion distances over the course of 100,000 years or so. We can’t really predict exactly where the Earth will be between these limits so our prediction will be uncertain by at least 100,000 years. With any luck, however, we can estimate the ‘date’ to within a few million years.

Now in previous calculations it was assumed that the physical diameter of the Sun remained constant and only the Earth-Sun distance affected the angular diameter of the Sun. In fact, our Sun is an evolving star whose physical diameter is slowly increasing due to its evolution ‘off the Main Sequence’. Stellar evolution models can determine how the Sun’s radius changes. The figure below comes from the Yonsei-Yale theoretical models by Kim et al. 2002; (Astrophysical Journal Supplement, v.143, p.499) and Yi et al. 2003 (Astrophysical Journal Supplement, v.144, p.259).

The blue line shows that between 1 billion years ago and today, the solar radius has increased by about 5%. We can approximate this angular diameter change using the two linear equations:

Perihelion: Diameter = 18T + 1973 arcseconds.

Aphelion: Diameter = 17T + 1908 arrcseconds.

where T is the time since the present in multiples of 100 million years, so a time 300 million years ago is T=-3, and a time 500 million years in the future is T=+5. When we plot these four equations we get

There are four intersection points of interest. They can be found by setting the lunar and solar equations equal to each other and using the Quadratic Formula to solve for T in each of the four possible cases.:

Case A : T= 456 million years ago. The angular diameter of the Sun and Moon are 1890 arcseconds. At apogee, this is the smallest angular diameter the Moon can have at the time when the Sun has its largest diameter at perihelion. Before this time, you could have total solar eclipses when the Moon is at apogee. After this time the Moon’s diameter is too small for it to block out the large perihelion Sun disk and from this time forward you only have annular eclipses at apogee.

Case B : T = 330 million years ago and the angular diameters are 1852 arcseconds. At this time, the apogee disk of the Moon when the Sun disk is smallest at aphelion just covers the solar disk. Before this time, you could have total solar eclipses even when the Moon was at apogee and the Sun was between its aphelion and perihelion distance. After this time, the lunar disk at apogee was too small to cover even the small aphelion solar disk and you only get annular eclipses from this time forward.

Case C : T = 86 million years from now and the angular diameters are both 1988 arcseconds. At this time the large disk of the perigee Moon covers the large disk of the perihelion Sun and we get a total solar eclipse. However before this time, the perigee lunar disk is much larger than the Sun and although this allows a total solar eclipese to occur, more and more of the corona is covered by the lunar disk until the brightest portions can no longer be seen. After this time, the lunar disk at perigee is smaller than the solar disk between perihelion and aphelion and we get a mixture of total solar eclipses and annular eclipses.

Case D : T = 246 million years from now and the angular diameters are 1950 arcseconds. The largest lunar disk size at perigee is now as big as the solar disk at aphelion, but after this time, the maximum perigee lunar disk becomes smaller than the solar disk and we only get annular eclipses. This is approximately the last epoc when we can get total solar eclipses regardless of whether the Sun is at aphelion or perihelion, or the Moon is at apogee or perigee. The sun has evolved so that its disk is always too large for the moon to ever cover it again even when the Sun is at its farthest distance from Earth.

The answer to our initial question is that the last total solar eclipse is likely to occur about 246 million years from now when we include the slow increase in the solar diameter due to its evolution as a star.

Once again, if you want to use the Desmos interactive math module to exolore this problem, just visit the Solar Eclipses – The Last Total Eclipse? The graphical answers in Desmos will differ from the four above cases due to rounding errors in the Desmos lab, but the results are in close accord with the above analysis solved using quadratic roots.

Exploring the Heliosphere

This is my new book for the general public about our sun and its many influences across the solar system. I have already written several books about space weather but not that specifically deal with the sun itself, so this book fills that gap.

We start at the mysterious core of the sun, follow its energies to the surface, then explore how its magnetism creates the beautiful corona, the solar wind and of course all the details of space weather and their nasty effects on humans and our technology.

I have sections that highlight the biggest storms that have upset our technology, and a discussion of the formation and evolution of our sun based on Hubble and Webb images of stars as they are forming. I go into detail about the interior of our sun and how it creates its magnetic fields on the surface. This is the year of the April 2024 total solar eclipse so I cover the shape and origin of the beautful solar corona, too. You will be an expert among your friends when the 2024 eclipse happens.

Unlike all other books, I also have a chapter about how teachers can use this information as part of their standards-based curriculum using the NASA Framework for Heliospheric Education. I even have a section about why our textbooks are typically 10 – 50 years out of date when discussinbg the sun.

For the amateur scientists and hobbyists among you, there is an entire chapter on how to build your own magnetometers for under $50 that will let you monitor how our planet is responsing to solar storms, which will become very common during the next few years.

Basic book details: 239 pages; 115 ilustrations; 6 tables; 70,000 words;

There hasn’t been a book like this in over a decade, so it is crammed with many new discoveries about our sun during the 21st century. Most books for the general public about the sun have actually been written in a style appropriate to college or even graduate students.

My book is designed to be understandable by my grandmother!

Generally, books on science do not sell very well, so this book is definitely written without much expectation for financial return on the effort. Most authors of popular science books make less than $500 in royalties. For those of you that do decide to get a copy, I think it will be a pleasurable experience in learning some remarkable things about our very own star! Please do remember to give a review of the book on the Amazon page. That would be a big help.

Yep…I want to get the e-book version ($5): Link to Amazon.

Yep…I want to get the paperback version ($15): Link to Amazon.

Oh…by the way…. I am a professional astronomer who has been working at NASA doing research, but also education and public outreach for over 20 years. Although I have published a number of books through brick-and-morter publishing houses, I love the immediacy of self-publishing on topics I am excited about, and seeing the result presented to the public within a month or two from the time I get the topic idea. I don’t have to go through the lengthy (month-year) tedium of pitching an idea to several publishers who are generally looking for self-help and murder mysteries. Popular science is NOT a category that publishers want to support, so that leaves me with the self-publishing option.

Other books you might like:

Exploring Space Weather with DIY Magnetometers. ($7). Link to Amazon.

History of Space Weather: From Babylon to the 21st Century. (paperback, $30) (ebook, $5). Link to Amazon.

Solar Storms and their Human Impacts (e-book; $2) Link to Amazon.

The 23rd CycleL Learning to live with a stormy star. – Out of print.

Your Brain..On Math!

Credit: SciTechDaily https://scitechdaily.com/researchers-discover-how-the-human-brain-separates-stores-and-retrieves-memories/

In my earlier blogs, I talked about Math Anxiety, about how the brain creates a sense of Now, and various other fun issues in brain research too. Branching off of my long, professional interest in math education, I thought I would look into how ‘doing’ math actually changes your brain in many important ways, especially for children and adolescents. Brain research has come a long way in the last 15 years with the advent of fMRI and sensors that can listen-in to individual neutrons [1]. For a detailed glimpse of modern research have a look at my reference list at the end of this blog.

Here is what we know about how math affects brain structure and maturation. My previous blog on Math Anxiety covered this topic but here are some additional points.

The Basic Anatomy of Math

First of all, let’s put to rest a popular misconception. Its a complete fallacy that we only use 10% of our brain. The misconception probably arose because glial cells that support neurons account for 90% of the cellular matter in the brain, so neurons account for 10% [9,11,10]. The truth is, by the end of each day, your brain has used nearly all of its neurons to facilitate movement, sensory processing, advanced planning, and even day-dreaming!

The architecture of our brains is controlled by about 86 million neurons and the trillions of synaptic connections between them. At the lowest level, our brains are composed of numerous modules that are specialized for specific tasks. Each has its own local knowledge system and ‘data cache’ and can act much faster than the whole-brain, which is the way evolution designed this system to help us respond quickly and not get eaten. We benefit from this ancient architecture because craftsmen, musicians and dancers cannot tell you how they perform their tasks because it is largely unconscious and controlled by specific modules. [6:p45, 198].

Before the age of 2, children use a general knowledge ‘program’ that takes up all of their working memory [2:p151] to interact with the environment. Children require more working memory to do math than adults. Number facts and basic opeations are not yet in long-term memory so they use more of their prefronal cortex (PFC) to keep math in working memory so that they can solve problems [2:p155]. But through training they develope a growing multitude of specialized modules and automatic ‘subroutines’ for specific tasks and skills. [6:p56]. Consciousness occurs when these non-communicating modules begin to share their knowledge across many communities of modules spanning the entire cerebral network. Some of these global communication pathways are highlighted by the so-called brain connectome map. This sharing of multiple representations of similar knowledge leads to problem solving and creativity which now draw inspiration from the experiences of many different modules [6:p58] spanning the entire cortex.

The wiring diagram of a human brain revealing connections. Courtesy of the consortium of The Human Connectome Project

Development of the Brain

At birth, the average baby’s brain is about a quarter of the size of the average adult brain. Incredibly, it doubles in size in the first year. It keeps growing to about 80% of adult size by age 3 and 90% – nearly full grown – by age 5 [12]. Over 1 million new neural connections are created every second among the synapses of the growing population of neurons and dendrites [13]. What then ensues is a process of pruning as seldome-used connections wither and dissappear while others are strengthened [20].

The growing brain does not start out as a tabla rasa but through genetics and evolution there are already features in place that anticipate the growth of mathematical knowledge.

Number Line Maps

At the most elementary level, neurons already exist at birth that are active for specific numbers. These ‘number neurons’ have been found in both monkeys and in humans. In humans they are mostly found in the lateral prefrontal cortex (l-PFC) and the intraparietal sulcus (IPS). [2:p129], but also the mediotemporal lobe (MTL) [2:p98]

Our brain’s hippocampous has place and grid cells that form a direct map written on its cortex that represents the location of objects in space [7p219]. The posterior cingulate region has neurons tuned to the location of objects in the outside world, and is connected to the parahippocampal gyrus where “place cells’ are found. These neurons fire whenever an animal occupies a specific location in space like the northwest corner of your room. These place cells are so advanced that readout of individual nerve cell firings can be used to tell a researcher where the object is in the subjects visual field of view. This even works when the subject closes their eyes and imagines an animal located there. [4:p149].

A curious feature of how the young brain processes quantities is that it perceives quantities as being located on a mental number line. Called the SNARC Effect, even three-day-old infants will look-right for large quantities and look-left for smaller quantities.[2:236]. That calculation-related activity is being processed like mental movement on a number line was also tested in older subjects by studying neuron activation in the superior parietal lobule (SPL) where information is being manipulated in working memory. They found that eye motion alone predicted the answers to simple addition and subtraction problems [2:239]. So just as the brain uses an internal map in the hippocampus to locate objects in space, it also uses an internal map to locate numbers in space along a line! The number line however is not uniform.

Kindergarten students with no math knowledge see number intervals as quantities mapped out in logarithmic intervals just as many animals do, so that quantities are perceived almost the same way as light brightness or sound volume [2:87]. Large numbers with smaller intervals are crowded together in the right-hand of the mental number line while smaller numbers are more spread out in the left-side of the line.

Meanwhile, the concepts of addition and subtraction are already known to infants as young as nine months[2:196]. Thinking about quantity as symbolic numerals like 1,2,3 etc instead of dots like [.], [..], […] etc at first occupies children up to age 7 who have to use their working memory to keep track of this, but within a few years the relationship between number symbols and dots becomes automatic and unconscious [2:185]. By the way, although algebra looks like a language, algebra is not processed in the brain’s language centers [2:p222] You can think and reason logically without language. In fact, when professional mathematicians are studied and asked to solve advanced problems, their language centers are not activated. Instead, the bilateral frontal, intraparietal and ventrolateral temporal regions were active, which are connected to the regions associated with processing numbers [2:232].

Math Remodels the Brain.

For mathematicians, an interesting recycling of brain areas occurs in order to accommodate advanced mathematics. Afterall, the brain volume is fixed by the volume of the skull, so the only way that new skills are learned and mastered is by appropriating cerebral real estate from other adjacent functions. The inferior temporal gyrus (ITG) is an area where face recognition occurs. For mathematicians, part of this region is invaded by adjacent regions used in number processing [2:191], in some cases making it harder for mathematicians to recognize faces!

Admittedly, this is an extreme result of brain reorganization, but there are other examples that are more relevant to children and young adults and the answer to the question ‘Why do I need to know math?’

Researchers have proposed that math training not only makes us better at math, but also strengthens our ability to moderate our feelings and our social interactions because of the brains proclivity in  sharing brain regions for other purposes.

Example 1: In my previous blog on Math Anxiety, I mentioned that the sub-region called the dorsolateral prefrontal cortex helps us keep relevant  problem-solving information ‘fresh’ in our working memory. In math it is activated when the individual is keeping track of more than one concept at a time. As it also turns out, this region is also activated as we regulate our emotions. For example, most children learn how to tone-down their glee at winning a game when they see their friends are mortified at  having lost.  It is also important in suppressing selfish behavior, fostering commitment in relationships, and most importantly inferring the intentions of others, which is called a Theory of Mind.

Example 2: The long-term effect of not continuing math education and problem-solving in adolescents has also been documented. A recent study of adolescents in the UK shows that a lack of math education affects adolescent brain development. In the UK, students can elect to end their math education at age 16.  The neurotransmitter called gamma-Aminobutyric acid (GABA) is present in the middle front gyrus (MFG), which is a region involved in reasoning and cognitive learning. GABA levels are a predictor of changes in mathematical reasoning as much as 19 months later.  What was found among the older adolescents was that GABA showed a marked reduction[14]. This neurotransmitter is also correlated with brain plasticity and its ability to reconfigure itself by growing new synapses as it learns new skills or knowledge having npothing to do with math [16].

Example 3: The mediotemporal lobe (MTL)  includes the hippocampus, amygdala and parahippocampal regions, and is crucial for episodic and spatial memory. The MTL memory function consists of distinct processes such as encoding, consolidation and retrieval, and supports many functions including emotion, affect, motivation and long-term memory. The MTL also has numerous number neurons [2:p98] and is involved in processing mathematical concepts. Activity in this region represents a short-term memory of the arithmetic rule, whereas the hippocampus may ‘do the math’ and process numbers according to the arithmetic rule at hand.”[15].

Example 4: Memory-based math problems stimulate a region of the brain called the dorsolateral prefrontal cortex, which has already been linked to depression and anxiety. Studies have found, for example, that higher activity in this area is associated with fewer symptoms of anxiety and depression. A well-established psychological treatment called cognitive behavioral therapy, which teaches individuals how to re-think negative situations, has also been seen to boost activity in the dorsolateral prefrontal cortex. The ability to do more complex math problems might allow you to more readily learn how to think about complex emotional situations in different ways. Greater activity in the dorsolateral prefrontal cortex was also associated with fewer depression and anxiety symptoms. The difference was especially obvious in people who had been through recent life stressors, such as failing a class. Participants with higher dorsolateral prefrontal activity were also less likely to have a mental illness diagnosis.[17]

The bottom line for much of the research on how the brain functions with and without mathematics stimulation is that low numeracy is a bigger problem for the brain than low literacy [2:p307] It affects your economic opportunities in life, handeling personal finances, operating as a savvy consumer, and it even connects with your ability to logically process complex social situations and predict what your best course of action might be in many different circumstances.

Many of the brain regions needed for math performance are still under development between ages of 16 and 26 including most importantly the frontal cortex essential for judgment and anticipating future consequances of actions.

So when a student asks what is math good for, take a step back and walk them through the Big Picture!

Books that are definitely worth the time to read!

[1] The Tell-Tale Brain, V.S. Ramachandran, 2011, W.W. Norton and Co.

[2] A Brain for Numbers, Andreas Nieder, 2019, MIT Press

[3] The Consciousness Instinct, Michael Gazzangia, 2018, Farrar, Straus and Giroux

[4] Consciousness and the Brain, Stanislaus Dehaene, 2014, Penguin Books.

[5] Being You: A new science of consciousness, Anil Seth, 2021, Dutton Press

[6] The Prehistory of the Mind, Stevem Mithen, 1996, Thames and Hudson Publishers.

[7] The Idea of the Brain, Matthew Cobb, 2020, Basic Books

[8] The River of Consciousness, Oliver Sacks, 2017, Vintage Books

[9] Myth: We only use 10% of our brains. Stephen Chew ,2018, https://www.psychologicalscience.org/uncategorized/myth-we-only-use-10-of-our-brains.html

[10] Neurological glial cells – https://www.ncbi.nlm.nih.gov/books/NBK10869/

[11] Unsung brain cells play key role in neurons’ development, 2009, Bruce Goldman, https://med.stanford.edu/news/all-news/2009/09/unsung-brain-cells-play-key-role-in-neurons-development.html#:~:text=Ben%20Barres’%20research%20has%20led,90%20percent%20of%20the%20brain.

[12] https://www.firstthingsfirst.org/early-childhood-matters/brain-development/

[13] https://developingchild.harvard.edu/science/key-concepts/brain-architecture/

[14] www.sciencedaily.com/releases/2021/06/210607161149.htm and DOI:10.1073/pnas.2013155118

[15] Math Neurons” Fire Differently Depending On Whether You Add Or Subtract, 2022, https://www.iflscience.com/math-neurons-fire-differently-depending-on-whether-you-add-or-subtract-62658

[16] https://www.theguardian.com/education/2021/jun/07/studying-maths-beyond-gcses-helps-brain-development-say-scientists

[17] https://today.duke.edu/2016/10/could-mental-math-boost-emotional-health

[20] https://coverthree.com/blogs/research/kids-brain-development

The Big Bang: Explained at the reading level of Genesis.

The universe began with a bang. Can we explain it all as simply as in many religious stories. (Image credit: Shutterstock)

I have often wondered how the modern description of the Big Bang could be written as a story that people at different reading levels would be able to understand, so here are some progressively more complete descriptions beginning with Genesis and their reading level determined by Reliability Formulas.

Genesis (from MIT Bible Gateway)

In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. And God said, “Let there be light,” and there was light. God saw that the light was good, and he separated the light from the darkness. God called the light “day,” and the darkness he called “night.” And there was evening, and there was morning–the first day. And God said, “Let there be an expanse between the waters to separate water from water.” So God made the expanse and separated the water under the expanse from the water above it. And it was so. God called the expanse “sky.” And there was evening, and there was morning–the second day. And God said, “Let the water under the sky be gathered to one place, and let dry ground appear.” And it was so. God called the dry ground “land,” and the gathered waters he called “seas.” And God saw that it was good. Then God said, “Let the land produce vegetation: seed-bearing plants and trees on the land that bear fruit with seed in it, according to their various kinds.” And it was so. The land produced vegetation: plants bearing seed according to their kinds and trees bearing fruit with seed in it according to their kinds. And God saw that it was good. And there was evening, and there was morning–the third day. And God said, “Let there be lights in the expanse of the sky to separate the day from the night, and let them serve as signs to mark seasons and days and years, and let them be lights in the expanse of the sky to give light on the earth.” And it was so. God made two great lights–the greater light to govern the day and the lesser light to govern the night. He also made the stars. God set them in the expanse of the sky to give light on the earth, to govern the day and the night, and to separate light from darkness. And God saw that it was good.”

The Flesch Reading ease Score gives this an 87.9 ‘easy to read‘ score. Flesch-Kincaid gives this a grade level of 4.5. The Automated Readability Index gives it an index of 4 which is 8-9 year olds in grades 4-5. Amazingly, the scientific content in this story is completely absent and in fact promotes many known misconceptions appropriate to what children under age-5 know about the world.

Can we do at least as well as this story in a 365-word summary that describes the origin of the universe, the origin of the sun, moon and earth, and the appearance of life? Because the reading level of Genesis is only at most Grade-5, can we describe a scientific treatment using only concepts known by the average Fifth-Grader? According to the Next Generation Science Standards, students know about gravity, and scales of time but ideas about atoms and other forces are for Grade 6 and above. The average adult reader can fully comprehend a text with a reading grade level of eight. So if the text has an eighth grade Flesch Kincaid level, its text should be easy to read and accessible by the average US adult. But according to Wylie Communications, half of all US adults read at or below 8th-grade level. The American Academy of Arts and Sciences survey also shows that US adults know about atoms (51%), that the universe began with a Big Bang (41%) and that Earth orbits the sun (76%) so that US adults rank between 5th and 9th internationally in our basic scientific knowledge.

The genesis story splits itself into three distinct parts: The origin of the universe;The origin of stars and planets; and The origin of life and humanity. Only the middle story has detailed observational evidence at every stage. The first and last stories were one-of events for which exact replication and experimentation is impossible.

Because we are 3000 years beyond the writing of Genesis, let’s allow a 400-word limit for each of these three parts and aim at a reading level and science concept level not higher than 7th grade.

First try (497 words):

Origin of the Universe. Our universe emerged from a timeless and spaceless void. We don’t know what this Void is, only that it had none of the properties we can easily imagine. It had no dimension, or space or time; energy or mass; color or absence of color. Scientists use their mathematics to imagine it as a Pure Nothingness. Not even the known laws of nature existed.

Part of this Void exploded in a burst of light and energy that expanded and created both time and space as it evolved in time. This event also locked into existence what we call the Laws of Nature that describe how many dimensions exist in space, the existence of four fundamental forces, and how these forces operate through space and time.

At first this energy was purely in the form of gravity, but as the universe cooled, some of this energy crystalized into particles of matter. Eventually, the familiar elementary particles such as electrons and quarks emerged and this matter became cold enough that basic elements like hydrogen and helium could form.

But the speed at which the universe was expanding wasn’t steady in time. Instead this expansion doubled in speed so quickly that within a fraction of a second, the space in our universe inflated from a size smaller than a baseball to something many billions of miles across. Today, after 14 billion years of further expansion we see only a small fraction of this expanded space today, and we call it the Observable Universe. But compared to all the space that came out of the Big Bang, our entire Observable Universe is as big as a grain of sand compared to the size of our Earth. The Universe is truly an enormous collection of matter, radiation and energy in its many forms.

Meanwhile, the brilliant ‘fireball’ light from the Big Bang also cooled as the universe expanded so that by one million years after the Big Bang, it was cooler than the light we get from the surface of our own sun. Once this light became this cool, familiar atoms could start to form. As the universe continued to expand and cool, eventually the light from the Big Bang became so cool that it could only be seen as a dull glow of infrared light every where in space. The atoms no longer felt the buffeting forces of this fireball light and had started to congregate under the force of gravity into emmence clouds throughout space. It is from these dark clouds that the first stars would begin to form.

Mixed in with the ordinary matter of hydrogen and helium atoms was a mysterious new kind of matter. Scientists call this dark matter because it is invisible but it still affects normal matter by its gravity. Dark matter in the universe is five times more common than ordinary matter. It prevents galaxies like the Milky Way from flying apart, and clusters of galaxies from dissolving into individual galaxies.

Flesch Reading Ease 60. (Average difficulty); Flesch-Kincaid Grade: 10.2; Automated Readability Index: 11

Second Try (538 words):

Origin of the Universe. Our universe emerged from a timeless and spaceless void. We don’t know what this Void was. We think it had none of the properties we can easily imagine. It had no dimension, or space or time. It had no energy or mass. There was no color to it either blackness or pure white. Scientists use their mathematics to imagine it as a Pure Nothingness. They are pretty sure that not even the known laws of nature existed within this Void.

Part of this Void exploded in a burst of light and energy. Astronomers call this the Big Bang. It  expanded and created both time and space as it evolved in time. This event also locked into existence what we call the Laws of Nature. These Laws describe how many dimensions exist in space. The Laws define the four fundamental forces, and how they operate through space and time.

At first the energy in the Big bang was purely in the form of gravity. But as the universe expanded and cooled, some of this energy crystalized into particles of matter. Eventually, the familiar elementary particles such as electrons and quarks emerged. This matter became cold enough that basic elements like hydrogen and helium could form.

But the speed at which the universe was expanding wasn’t steady in time. Instead this expansion doubled in speed very quickly. Within a fraction of a second, the space in our universe grew from a size smaller than a baseball to something many billions of miles across. After 14 billion years of further expansion we see only a small fraction of this expanded space today. We call it the Observable Universe. But compared to all the space that came out of the Big Bang, our entire Observable Universe is as big as a grain of sand compared to the size of our Earth. The Universe is truly an enormous collection of matter, radiation and energy in its many forms.

Meanwhile, the brilliant ‘fireball’ light from the Big Bang also cooled as
the universe expanded. By one million years after the Big Bang, it was cooler than the light we get from the surface of our own sun. Once this light became this cool, familiar atoms could start to form. As the universe continued to expand and cool, eventually the blinding light from the Big Bang faded into a dull glow of infrared light. At this time, a human would see the universe as completely dark. The atoms no longer felt the buffeting forces of this fireball light. They began to congregate under the force of gravity. Within millions of years, immense clouds began to form throughout space. It is from these dark clouds that the first stars would begin to form.

Mixed in with the ordinary matter of hydrogen and helium atoms was a mysterious new kind of matter. Scientists call this dark matter.  It is invisible to the most powerful telescopes, but it still affects normal matter by its gravity. Dark matter in the universe is five times more common than the ordinary matter we see in stars. It prevents galaxies like the Milky Way from flying apart. It also prevents clusters of galaxies from dissolving into individual galaxies.

Flesch Reading Ease 66.7. (Average difficulty); Flesch-Kincaid Grade: 7.2; Automated Readability Index: 6.6 (11-13 year olds).

Third Try ( 410 words )

Origin of the Universe. Our universe appeared out of a timeless and spaceless void. We don’t know what this Void was. We can’t describe it by its size, its mass or its color.  It wasn’t even ‘dark’  because dark (black) is a color.  Scientists think of it as a Pure Nothing.

Part of this Void exploded in a burst of light and energy. We don’t know why.  Astronomers call this event the rather funny name of the ‘Big Bang’. It  was the birth of our universe. But it wasn’t like a fireworks explosion. Fireworks expand into the sky, which is space that already exists. The Big Bang created space as it went along.  There was nothing for it to expand into. The Big Bang also created  what we call the Laws of Nature. These Laws describe how forces like gravity and matter affect each other.

As the universe expanded and cooled, some of its energy became particles of matter. This is like raindrops condensing from a cloud when the cloud gets cool enough. Over time, these basic particles  formed  elements like hydrogen and helium.

The universe continued to expand. Within the blink of an eye, it grew from a size smaller than a baseball to something many billions of miles across. Today, after 14 billion years  we see only a small piece of this expanded space today. Compared to all the space that came out of the Big Bang, what we see around us is as big as a grain of sand compared to the size of our Earth. The Universe is truly enormous!

After about one million years  the fireball light from the Big Bang became very dim. At this time, a human would see the universe as completely dark. There were, as yet, no stars to light up the sky and the darkness of space. Atoms  began to congregate under the force of gravity. Within millions of years, huge clouds the size of  our entire Milky Way galaxy began to form throughout space. From these dark clouds, the first stars started to appear.

Mixed in with  ordinary matter  was a mysterious new kind of matter. Scientists call this dark matter.  It is invisible to the most powerful telescopes. But it still affects normal matter by its gravity, and that’s a very good thing! Without dark matter,  galaxies like our Milky Way and its billions of stars would fly apart, sending their stars into the dark depths of intergalactic space.

Flesch Reading Ease 73.3. (Fairly easy to read); Flesch-Kincaid Grade: 5.9 (Sixth grade) ; Automated Readability Index: 5.1 (8 – 9 year olds) Fourth to Fifth grade.

Summary.

The Third Try is about as simple and readable a story as I can conjure up, and it comes in at a reading level close to Fourth grade. Scientifically, it works with terms like energy, space, expansion, matter  and gravity, and scales like millions and billions of years. All in all, it is not a bad attempt that reads pretty well, scientifically, and does not mangle some basic ideas. It also has a few ‘gee whiz’ ideas like Nothing, space expansion and dark matter. 

So, what do you think? Leave me a note at my Facebook page!

Next time I will tackle the middle essay about the formation of  stars and planets!

Things to do with your Smartphone!

We all have smartphones, but did you know that they are chock-full of sensors that you can access and use to make some amazing measurements?

Here is an example of a few kinds of data provided by an app called Physics Toolbox, which you can get at the Apple or GOOGLE stores.

Sensor data accessed by Physics Toolbox app

Each of these functions leads to a separate screen where the data values are displayed in graphical form. You can even download the data as a .csv file and analyze it yourself. This provides lots of opportunities for teachers to ask their students to collect data and analyze it themselves, rather than using textbook tables with largely made-up numbers!

There are also many different separate apps that specialize in specific kinds of data such as magnetic field strength, sound volume, temperature, acceleration to name just a few.

I have written a guide to smartphone sensors and how to use them, along with dozens of experiments, and a whole section on how to mathematically analyze the data. The guide was written for a program at the Goddard Space Flight Center called the NASA Space Science Education Consortium, so if you know any teachers, students or science-curious tinkerer’s that might be interested in smartphone sensors, send them to this blog page so that I can count the traffic flow to the Guide.

Here is an interview with the folks at ISTE where I talk about smartphone sensors in a bit more detail:

https://www.iste.org/explore/classroom/students-can-do-citizen-science-their-smartphones

OK…here’s the URL for the actual Guide!

http://spacemath.gsfc.nasa.gov/Sensor/SensorsBook.pdf

If you really want to look into using smartphone sensors in a serious way, here are some articles I have written about them that you might also enjoy:

Odenwald, S., 2019. Smartphone Sensors for Citizen Science Applications: Radioactivity and Magnetism. Citizen Science: Theory and Practice, 4(1), p.18. DOI: http://doi.org/10.5334/cstp.158

Odenwald, S., 2018, The Feasibility of Detecting Magnetic Storms With Smartphone Technology, IEEE Access, https://ieeexplore.ieee.org/document/8425973