Some posters

For people with children, these high-resolution posters are intended for printing on A3 paper, and can be freely downloaded:

Mathematics posters: Flat shapes, Solid Shapes, Prime Factors (colour-coded)

Biology posters: Ladybirds of Australia, Plants and Fungi, Some Flowering Plants (monocots marked with a dot)

Astronomy/geography posters: Southern Cross (showing colours and magnitudes of stars), Orion (ditto), Geographical Features

Technology posters: Vehicles, Milestones in Materials

Solar racing teams: the US and Dutch models

Stanford at the finish of the World Solar Challenge in 2015

Everybody knows that I’m a big solar racing fan. Today I wanted to talk about solar car team models, comparing what I call the “US model” (although most other countries also use it) with what I call the “Dutch model” (also used by the Belgian team). In the “US model,” students work part-time on a solar car team, and new members are added each year. As an example of this, I will look at the Stanford Solar Car Project, and specifically at one team member: Rachel Abril, who is forever famous for her May 2014 TEDx talk.

Rachel Abril did a 4-year Bachelor degree in Mechanical Engineering (the blue blocks in the chart below show Stanford’s academic years) followed by a Masters degree. The hashed region on the chart shows her extensive involvement with the Stanford Solar Car Project, first as a junior Mechanical Team and Aerodynamics Team member, and later as Suspension Lead and Aerodynamics Lead. She did not, I believe, attend the 2013 World Solar Challenge, but she did attend the 2015 and 2017 races (Stanford was improving during this period, but so were the other top-twelve teams!).

Rachel Abril’s story highlights one great advantage of the “US model,” namely that long-serving team members develop enormous experience in the design, construction, and racing of solar cars. They can take the lessons of one race, and apply them to the next one (and Rachel’s TEDx talk mentions some lessons that Stanford learned).

There are a number of disadvantages to the “US model,” however. New recruits often have limited knowledge of relevant physics (especially in the US, where high school graduates are educationally about a year behind their European or Australian counterparts). What work can new recruits be given that is both interesting to them and useful to the team? How can they be properly integrated into the team, and feel that they are genuinely part of the group? How can the team stop new recruits from feeling “cheesed-off” and dropping out? Answering these questions well is the key to success for US teams. One of the answers lies in running internal training courses for new recruits (there is also the IEF Solar Car Conference), but teams do not always include “Education Lead” or “New Member Coordinator” as one of the key team roles.

Another disadvantage of the “US model” is that the mix of people with varying lengths of experience creates a power structure. It can be difficult for a new recruit to disagree with someone that has been on the team for many years (even if, objectively, the new recruit is right). This can be a trap.

A final difficulty with the “US model” lies in balancing solar car construction, academic study, and personal life. Conventional wisdom is that you can hope for at most two out of three. Privately, team alumni sometimes suggest that one out of three might be more realistic. I don’t know what support mechanisms might help with this.

Solar Team Twente at the finish of the World Solar Challenge in 2019

In contrast, in the “Dutch model,” a smaller group of people gives up a little over a year of their life to work full-time on a solar car. This is quite a sacrifice. The Belgian team’s recruitment page explains the return on investment for the year like this (my translation):

  1. A project filled with experiences that you won’t find in your regular studies;
  2. Discovering a genuine engineering project and its various phases: concept, design, production,
    and test;
  3. Connecting and collaborating with the largest companies in relevant industries;
  4. A close-knit group and a racing adventure never to be forgotten;
  5. The experience of a lifetime and so much more!

Essentially, the year on the solar car team functions as an unpaid internship (speaking as someone who has helped arrange engineering internships in the past, I can’t think of an internship where you would learn more). One positive feature of industry internships is normally industry networking; this is also worked into the Dutch/Belgian solar car experience (as #3 on that list indicates). Of course, the need to set up those industry connections is one more reason to have a really professional sponsorship team.

As an example of the “Dutch model,” I will focus specifically on the 2018–19 “edition” of Solar Team Twente. Behind this team sits a part-time organisation (mostly of alumni) which handles recruitment and provides technical advice. This organisation began recruiting in February 2018, and a new team was announced on 9 June 2018. All these people were complete solar car novices, of course. The new team began work at the start of the 2018–19 academic year (with the aerodynamic and management subteams starting a little earlier). In the chart below, coloured blocks show academic years, and the hashed region shows the typical duration of full-time team involvement:

One of the first activities of the novice Twente team was to race the previous car, Red Shift, at the European Solar Challenge (iESC) on 21–23 September 2018. Team alumni raced the even older Red One, so that this was not only a training activity for the novice team, but an opportunity for knowledge transfer from alumni. Building on their iESC experience, the novice team then began designing and building their new car, RED E. The new car was revealed on 21 June 2019. After a test race on 17–18 August, the car was shipped to Australia on 30 August (a tragic crash due to wind gusts put RED E out of the race, but it was in the lead when that happened).

Engineering education in the Netherlands is traditionally a 5-year Ingenieur degree. Because of EU regulations, this is nowadays packaged as a 3-year Bachelor degree plus a 2-year Masters, but local students generally take the full package (because of the superior Dutch high school system, the 3-year Bachelor degree reaches at least the same standard as the 4-year US equivalent). As a result, the novice Twente team would have had substantially more formal education under their belts than new solar car recruits in the US. Dutch engineering schools also benefit from a close connection to industry, which drives a practical focus. The Eindhoven University of Technology, for example, is traditionally a feeder school for Philips, DAF Trucks, and other engineering companies in the Eindhoven area.

Of course, not every university teaches every skill needed for solar car design and construction. Dutch engineering schools typically teach agile project management, for example, but this does not seem to be the case in Belgium. The Belgian team therefore arranged industry training on the subject from their sponsor Delaware Consulting. Dutch teams also often benefit from industry-based “team building” activities (this video shows such an activity for Top Dutch). Practice races (including the European Solar Challenge) compensate for the fact that team members have never attended the World Solar Challenge before.

Because of team-building, educational initiatives, and good knowledge management, the “Dutch model” consistently produces top solar cars (Vattenfall/Delft has won the World Solar Challenge repeatedly, the Belgians won in 2019, Twente was on the podium in 2013 and 2015, Top Dutch came 4th in their first race, and Eindhoven has won the Cruiser Class every time). While the “Dutch model” relies partly on specific features of engineering education in the Netherlands and Belgium, I think there are several Dutch/Belgian practices that teams in other countries can learn from.

Nuon (now Vattenfall) at the finish of the World Solar Challenge in 2017

I should finish with a note on Vattenfall (Delft) Solar Team, which runs a variation of the “Dutch model.” Vattenfall (Delft) alternates what I call “big build” teams with “small build” teams. The “big build” teams design and construct new cars for the World Solar Challenge, while the “small build” teams modify existing cars for other events. For example, Nuna9 was a “big build” for the 2017 World Solar Challenge, while Nuna9S was a “small build” modification of the same car for the 2018 South African race (it included a clever radar system). Likewise, Nuna Phoenix was the same car modified again for the 2020 American Solar Challenge (that event was sadly cancelled, but Nuna Phoenix did set a world record). As part of providing a return on investment for the “small build” teams, Vattenfall (Delft) is careful to give these modified cars their own identity.

Answering Gracie Cunningham

A 16-year-old TikTok user called Gracie Cunningham recently went viral with two short videos (second video here) asking questions about mathematics. Like a few other people, I thought that they were sufficiently interesting to answer.

1. How did people know what they were looking for when they started theorising about formulas? Because I wouldn’t know what to look for if I’m making up math.

Well, first, contrary to your comment “I don’t think math is real,” mathematics is indeed real. Even if the universe was completely different from the way it is, mathematics would still be true. Edward Everett, whose dedication speech at Gettysburg was so famously upstaged by Abraham Lincoln, put it like this: “In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven.” (OK, not everybody has this view of mathematics, which is called “Platonism,” but in my opinion, it’s the only view that explains why mathematics works).

Second, mathematics is discovered, not invented. The great mathematician G. H. Hardy pointed out “that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.” When you embark on a journey of discovery, you don’t know where you’ll end up. That’s what makes it exciting. When the early Polynesians set off in canoes across the Pacific, thousands of years ago, they didn’t know that they would discover Hawaiʻi, Samoa, and New Zealand. They just headed off into the wild blue yonder because that was the kind of people they were.

Now the Babylonians and others developed mathematics primarily because it was useful – for astronomy (which you need to decide when to plant crops) and engineering and business. But the Greek started to do mathematics just for fun. They discovered mathematical truths not because they were useful, but simply because, as Joel Spencer, put it, “Mathematics is there. It’s beautiful. It’s this jewel we uncover” (quoted in The Man Who Loved Only Numbers, p 27).

It is also worth pointing out a subtle kind of prejudice (almost a kind of racism) that is widespread – many people think that the ancients were primitive. They weren’t. They had plumbing, and architecture, and astronomy. They even knew about zero. Some of them were very smart people. There were Babylonian versions of Albert Einstein, whose names are now long forgotten. There were many Greek versions of Albert Einstein (including Archimedes, among others). I kind of wish that schools would teach young people more about what the past was actually like. If people learn anything about the past at all, it’s usually from popular literature:

2. Once they did find these formulas, how did they know that they were right? Because, how?

Short answer: Euclid. Around about his time, the Greeks started to ask themselves exactly that question, and developed the concept of rigorous mathematical proof as an answer. The fact that typical mathematics classes don’t introduce simple proof is yet another indication of how badly broken modern education is. Even this simple visual proof (uploaded by William B. Faulk; click to view animation) gets the idea across:

3. Why is everyone being really mean to me on Twitter? Why are the only people who are disagreeing with me the ones who are dumb, and the physicists and mathematicians are agreeing with me?

Well, that’s also an interesting question. First, for reasons that I don’t fully understand, Twitter just makes people mean.

Second, as Dunning and Kruger famously pointed out, it is the people who know the least that are the most confident.

Third, the original video was in teen-girl English, with multiple uses of the word “like” (I have a sneaking suspicion that this was deliberate). Using teen-girl English for a “serious” subject like mathematics makes people’s heads explode (this will be useful to know when you have your first job interview).

But thank you, Gracie, for asking some really good questions.

Some follow-up remarks on what mathematics is not are here.

Laws Guide to Nature Drawing and Journaling: a book review

Laws Guide to Nature Drawing and Journaling by John Muir Laws

Having written before about nature journals, a while ago I purchased the Laws Guide to Nature Drawing and Journaling by John Muir Laws of This is a wonderful guide to both the scientific and artistic aspects of keeping a nature journal. There are chapter on how to observe as well as chapters on how to draw flowers, trees, and other things. Laws provides three useful observation cues: “I notice,” “I wonder,” and “it reminds me of” (click page photographs to zoom):

This wonderful book is full of practical tips, both on the scientific side and the artistic side. I particularly liked this little curiosity kit:

I haven’t quite finished with the book, but I really love it so far. Other reviews online are also very positive: “I can’t find a thing lacking in this book” (; “informative and inspiring” (; “the best book for nature journaling in your homeschool” ( Goodreads rates the book 4.67.

* * * * *
Laws Guide to Nature Drawing and Journaling by John Muir Laws: 5 stars

The modern Trivium and the teaching of science

The “trivium” approach to education derives from “The Lost Tools of Learning,” a 1947 speech by scholar and detective story author Dorothy L. Sayers. This approach takes the seven liberal arts (illustrated above), drops the all-important quadrivium, and applies the remainder in a largely metaphorical way. It is an interesting approach, although it inevitably under-emphasises mathematics. The door to Plato’s Academy was marked “Let no one ignorant of geometry enter (Ἀγεωμέτρητος μηδεὶς εἰσίτω),” and this referred to the most advanced mathematic of his day. I’m not sure that the “trivium” approach to education delivers that level of mathematical knowledge. Then again, does the standard approach?


Science, on the other hand, can be fitted quite well into the “trivium” model. The three stages of this model (largely metaphorical, as noted) are “grammar,” “logic,” and “rhetoric.”

The “grammar” stage (intended for ages 6 to 10 or so) covers basic facts. Science at this level logically includes what used to be called natural history – the close observation of the natural world. Maintaining a nature journal is an important part of this, as are simple experiments, the use of a telescope, collections of objects (rocks, shells, etc.), and simple measurements (such as recording measurements from a home weather station).

Mother and child nature journaling examples from Nature Study Australia Instagram and website

Dorothy L. Sayers has nothing to say about science in the “logic” stage (apart from fitting algebra and geometry here), but the “logic” stage would reasonably include taxonomies, empirical laws, and an exploration of how and why things work the way they do – that is, the internal logic connecting scientific observations and measurements. A degree of integration with history education would provide some context regarding where these taxonomies and laws came from, and why they were seen as important when they were formulated.

Exploring Boyle’s law with a simple apparatus

In the “rhetoric” stage, the “how” and “why” of science would be explored in more detail, along with practical applications and project work (such as entering a science competition, or possibly even collaborating with local academics on a scientific conference paper).

A US Army engineer helps judge high school science projects (photo: Michael J. Nevins / US Army)

I suspect that quite a decent science education programme could be worked out on such a basis. If any reader knows of it having been done, please add a comment.

Nature journals

Having said something about phenology wheels, I thought that I should mention nature journals too. Some years ago, I blogged about the professional aspects of this, but nature journals are a powerful educational tool, because of the way that they focus observational attention. John Muir Laws has good advice on getting started, including “Do not focus on trying to make pretty pictures. That just leads to journal block. Open your journal with the intention of discovering something new. Use the process to help you slow down and look more carefully.

Mother and child nature journaling examples from Nature Study Australia Instagram and website

The very useful Nature Study Australia website also has good advice and several examples, as well as other nature study resources for Australians. Artist Paula Peeters, aiming more at adults, runs nature journaling workshops around Australia, and offers an introductory book for sale or free download.

Nature journaling example from Paula Peeters, who runs workshops around Australia

Nature journals need not only contain pictures and text: a spiral-bound sketchbook will easily accomodate flat objects such as leaves, pressed flowers, feathers, and sun prints. Drawings are an essential aspect, however.

The CNPS curriculum

The California Native Plant Society offers a superb nature journaling curriculum for free download. It includes the observational prompts “I notice… I wonder… It reminds me of…” It advises parents and teachers not to say things like “that is really pretty” or “what a good drawing,” but instead to say things like “Oh, you found a spider on top of the flower! Great observation.” It also provides excellent practical advice on drawing, poetry, and other activities.

With so many excellent guides to nature journalling, why not get started on your own?

A drawing of mine (from quite some time ago)

Phenology wheels

Recently, somebody pointed me at phenology wheels, which are a popular tool for nature study among teachers and homeschoolers. Nature study is all about careful observation and finding patterns, and phenology wheels help with both. Every month, students draw a picture of what they see in the garden or on a nature walk, and the completed phenology wheel then shows an annual pattern. Other activities are possible – see this University of Wisconsin-Madison Arboretum document.

The picture below shows a pair of partially complete mother/daughter phenology wheels from the very useful Nature Study Australia website (they are using the central circle to show indigenous seasons). It is helpful to outline each month’s section in felt-tip pen:

Mother and daughter phenology wheels from

I’ve generated blank wheels for the Northern Hemisphere and for the Southern Hemisphere, and produced a partially complete wheel of my own (from a European perspective):

Like nature journals, this is an activity both fun and educational!

Credits: lavender watercolour painting by Karen Arnold, sunflowers by Vincent van Gogh, butterfly from here, font is Jenna Sue, wheel constructed using R (with DescTools::DrawCircle, rasterImage, and the showtext package).

Reflections on school performance in the US

The US has just had a release of the 2017 National Assessment of Educational Progress (NAEP) results. They are not good. Of grade 8 pupils in public schools, 65% failed to meet proficiency standards in reading, and 67% failed to meet proficiency standards in mathematics. This is a serious problem, and it is worth getting to the bottom of it.

Doing a multiple regression on average state grade 8 reading scores, the politics of the state governor has no effect (p = 0.67). States vary enormously in the money they spend on education, ranging from $6,575 per pupil in Utah to $21,206 per pupil in New York. This makes no difference either (p = 0.93). What does make a difference is the state poverty rate (R2 = 0.49, p = 0.000000014).

For grade 8 mathematics scores, the story is similar. Politics of the state governor (p = 0.76) and money spent on education (p = 0.51) have no effect, but the state poverty rate does (R2 = 0.55, p = 0.0000000008).

Clearly, poor children do much less well in school, and spending money on schools does not address the problem. Why do poor children do less well in school? Research shows that on day one, poor children have a cognitive and behavioural disadvantage. Poor children eat less well. Poor children are starved of words, because their parents, on average, spend less time talking, singing, and reading to them.

The problems lie at home; the solutions must also lie at home. Rather than spending more money in schools, the US seems to need more assistance to parents at home. For example, the State Library of Queensland has started a wonderful Dads Read programme in Australia. Bookstart in the UK offers a free pack of books to children at 0–12 months and at 3–4 years. Also helpful would be guides to teaching number skills, guides to nature walks, discounts for families at museums, and other assistance in STEM areas (I’m start to feel like it’s time to write another children’s book). Surely this problem with reading and mathematics needs to be addressed with urgency!

Story time at the Dover Air Force Base library, Delaware (USAF photo by Roland Balik)

Measuring the Earth this (Southern) Christmas

In around 240 BC, Eratosthenes calculated the circumference of the Earth. The diagram above (from NOAA) shows how he did it. This Christmas, people in the Southern Hemisphere can repeat his work!

Eratosthenes knew that, at the summer solstice, the sun would be directly overhead at Syene (on the Tropic of Cancer) and would shine vertically down a well there. He also knew the distance to Syene.

On 21 December, the sun will be directly overhead on the Tropic of Capricorn at local noon. This table show the time of local noon on 21 December 2017, and the distance to the Tropic of Capricorn, for some Southern Hemisphere cities:

City Local Noon Distance to Tropic (km)
Adelaide 13:14 1270
Auckland 13:19 1490
Brisbane 11:46 450
Buenos Aires 12:52 1240
Darwin 12:45 1220
Hobart 13:09 2160
Johannesburg 12:06 310
Melbourne 13:18 1590
Perth 12:15 940
Santiago 13:41 1110
Sydney 12:53 1160

At exactly local noon, Eratosthenes measured the length (s) of the shadow of a tall column in his home town of Alexandria. He knew the height (h) of the column. He could then calculate the angle between the column and the sun’s rays using (in modern terms) the formula θ = arctan(s / h).

You can repeat Eratosthenes’ calculation by measuring the length of the shadow of a vertical stick (or anything else you know the height of), and using the arctan button on a calculator. Alternatively, the table below show the angles for various shadow lengths of a 1-metre stick. You could also attach a protractor to the top of the stick, run a thread from the to of the stick to the end of the shadow, and measure the angle directly.

The angle (θ) between the stick and the sun’s rays will also be the angle at the centre of the Earth (see the diagram at top). You can then calculate the circumference of the Earth using the distance to the Tropic of Capricorn and the fact that a full circle is 360° (the circumference of the Earth will be d × 360 / θ, where d is the distance to the Tropic of Capricorn).

Height (h) Shadow (s) Angle (θ)
1 0.02
1 0.03
1 0.05
1 0.07
1 0.09
1 0.11
1 0.12
1 0.14
1 0.16
1 0.18 10°
1 0.19 11°
1 0.21 12°
1 0.23 13°
1 0.25 14°
1 0.27 15°
1 0.29 16°
1 0.31 17°
1 0.32 18°
1 0.34 19°
1 0.36 20°
1 0.38 21°
1 0.4 22°
1 0.42 23°
1 0.45 24°
1 0.47 25°
1 0.49 26°
1 0.51 27°
1 0.53 28°
1 0.55 29°
1 0.58 30°
1 0.6 31°
1 0.62 32°
1 0.65 33°
1 0.67 34°
1 0.7 35°
1 0.73 36°
1 0.75 37°
1 0.78 38°
1 0.81 39°
1 0.84 40°
1 0.87 41°
1 0.9 42°
1 0.93 43°
1 0.97 44°
1 1 45°