Maths in Science - That's All There Is

Well, that's all there is from the Maths in Science project. A bit brief, I'm sorry, but there were some real gems in there to consider.

A Few Reflections:

Of the 39 scientists I contacted, I was able to post 6 interviews. Many of the scientists that I contacted were obviously very busy and not available to participate. There were several other scientists who said they would help out but didn't get back to me in time to get it posted. 

I had promised 2 Nobel Prize winners but the second one has gone overseas and is not available any more to share his ideas with us.

I do note the absence of female scientists in my line up. I did contact several but was not successful in getting any to write for the project which is disappointing.

I did also contact several Fields Medal winners. The Fields Medal is like the Mathematics equivalent of a Nobel Prize (there is no Nobel Prize for Maths - it's a long story). Sadly, I could not convince any of them to write for the project.

Highlights

Here are a few comments that stood out to me from the interviews:

Mathematics is simply the language of science.
- Andrew Knoll

I believe we need to rethink how we teach math and not use it to separate students.
- Don Kinard

Learning more maths, even if you don't know exactly how you'll use it, will always turn out to be helpful.
- Jack Szostak

I, most scientists, and all engineers, use math all the time. The more you know, the more you can do.
- George Whitesides 

I hope that you too found some gems to think about.




Once again, thank you to the scientists who participated in the Maths in Science project. It has been an interesting opportunity for us all to hear how these two fields overlap and interact.



 



  



 

Maths in Science - Andrew Knoll

Professor Andrew Knoll is one of the world's leading paleontologists and works at Harvard University. He has numerous awards and achievements to his credit including Paleontological Society Medal, the Wollaston Medal of the Geological Society (London), the Moore Medal of the Society for Sedimentary Geology, the Oparin Medal of the International Society for the Study of the Origin of Life, and both the Walcott and Thompson medals of the US National Academy of Sciences.  

Professor Knoll has done extensive research on the early life on earth, specifically the first 3 billion years, which is the focus of one of his more significant books. 

He was also a member of the NASA Mars Rover project team, using his skills and knowledge of early evolutionary life to support this significant endeavour.

Professor Knoll kindly gave some time to answer a few questions about is use and application of maths in his work.

Here is what he had to say:

The Questions:   

1. Describe what maths lessons were like for you at school.

I grew up in US schools much influenced by Sputnik.  The “new math” taught in primary schools was heavy on Venn diagrams, number sets and the like.  Thankfully, we still learned basic arithmetic and, as we got older, algebra and geometry.

2.  Was the maths that you learnt at school useful to you later in life?

Beyond a basic understanding of numbers and what they mean, the maths I learned in school provided a framework for the calculus and statistics I learned later at university.  And THOSE I use every day.

3. How good do you need to be at mental arithmetic to do calculations in your head?

One of the great skills that many people (including scientists) need to learn is how to approximate an answer.  To make a quick evaluation of a problem, you commonly don’t need to calculate the precise answer, but rather need to know whether the answer is close to ten, close to one hundred, or close to one thousand.  Thus, quick, approximate mental arithmetic is truly useful.

4. Mathematics teaches us that you can put two things together to make a new thing. Is this important in what you do?

In a sense, yes.  Commonly, I  would like to know the relationship between two sets of data, say, body length and weight.   If I plot length on the X axis and weight on the Y axis of a graph, the spread of points on the graph really does make a “new thing.”  For example, in the case mentioned, it may tell us about how the organisms in question grow.

5. Mathematics is about finding patterns. Do you need to look for patterns, or exceptions to patterns, in your research?

I do research in geology and paleontology and I spend my life looking for patterns.   Mathematical pattern may show how the chemistry of sediments changes along a depth gradient, or how biological diversity increased through time.

6. Mathematics also teaches us about balance and equality. Is this idea useful in your research?

Balance and equality are important  -- I’m not sure how I would apply either to my research.

6. Mathematics helps us to represent quantities and measurements numerically. Do you do this in your work?

All the time.  We describe fossils, using numbers to describe length, width, and many other features.,  And we characterize the chemistry of ancient sedimentary rocks, again using numbers of indicate how much carbon, or calcium, or iron is present in a sample.

7. Is estimation good enough or do you need to measure things accurately?

As I noted earlier, estimation is a useful skill.  But sometimes I really need make precise measurements or do exact calculations.  In a sense, estimation is the quick and dirty, whereas precise measurements and calculations enable the final product.

9. How do you use statistics to analyse your results?

Again, nearly every day.  As a paleontologist, I commonly need to know whether two fossil populations are different or not, and statistics allows me to draw this kind of conclusion.

10. Do you have any other insights to offer into how you use maths in your work?

Scientists need maths to do research, whether that research involves measurement, comparison of different types of data, or modeling.  Mathematics is simply the language of science.

What a great line to end on - "Mathematics is simply the language of science."

Thanks you so much Professor Knoll for your support of the Maths in Science project. 

 

Maths in Science - Don Kinard

Dr Don Kinard is a Senior Technical Fellow with Lockheed-Martin, involved in the development and production of the F-35 fighter jet. His early studies were in chemistry and his doctorate looked at physical (polymer) chemistry. Much of his later work has involved him in engineering, production and manufacturing of aircraft for Lockheed-Martin, one of the world's leading Aerospace companies. 

I was very interested to hear from a scientist who works in one of the most advanced technical industries in the world, to hear how his understanding of mathematics relates to his work as a scientist.

Dr Kinard was kind enough to answer my 10 questions about maths in science.

Here is what he had to say.
 

The Questions:

1. Describe what maths lessons were like for you at school.

I was always pretty good at math but it was generally pretty boring at least until Graduate school.  The lessons taught the mathematical basics but were devoid of substance and application.  I believe that we need to change the way we teach mathematics in schools to illustrate that mathematic formulas and patterns were invented to solve problems or define relationships in the physical world.  Calculus in college is a good example, we learned the formulas and solutions for three years but it would have been much more enlightening and stimulating if we had also learned why Newton had to invent it in order to solve his motion equations.  The first math class that really made sense to me was Differential Equations (here in the US this is taken after 3 years of calculus) where it was taught with an Engineering approach and we solved real problems.  I also really liked Statistical Thermodynamics where I learned how what the equations really meant and how they were derived.

2. Was the maths that you learnt at school useful to you later in life?

Absolutely, although I didn’t really appreciate it until I applied it to my research in Graduate school.   In Grad School I was studying the motion of water molecules in tight cavities such as collagen and elastin along with synthetic polymers.  Water in these tight spaces doesn’t freeze, it turns behaves like a polymer.   I described the behavior using viscoelastic equations and Laplace Transform equations.   Polymer mechanical and solution behavior is also typically modeled using statistical mechanics.  In general I have found that a good knowledge of mathematics is one of the key turning points for young people.  Those that understand it have the entire world of opportunities, those that don’t tend to back away from any subject that utilizes math to any great degree.  I believe that we need to rethink how we teach math and not use it to separate students.  For the past 30 years I have been involved in the design and manufacture of advanced fighter aircraft like the F-22 and F-35.  Much of the work utilizes mathematics in some form or another and although I may not be an expert in any particular mathematical field I find that a solid knowledge of math can provide a great deal of confidence and background when tackling difficult topics and working with the true mathematical experts.

3. How good do you need to be at mental arithmetic to do calculations in your head?

I’ve never been very good at mental arithmetic so it’s hard for me to judge.  What I do well is apply mathematical relationships (a particular variable varies as the inverse square of another for example) to understand behavior of complex systems and to find solutions to engineering problems.  At some point you have to get down and do the math but the relationships are the key to being a scientist and understanding what’s physically going on with your data and observations.  Being a scientist or engineer is sometimes like being a detective, you have a problem and you need to find a solution, you investigate the facts and try to deduce the culprit.

4. Mathematics teaches us that you can put two things together to make a new thing. Is this important in what you do?

Yes, Aircraft are very complex systems where problems or solutions are rarely black and white, everything is an optimization problem.  Often a problem is a combination of issues involving temperature, pressures, chemical reactions, mechanics, vibration, etc.  Again, it’s about relationships.  Systems such as flight controls and hydraulics for example are integrated with the vehicle system software.

5. Mathematics is about finding patterns. Do you need to look for patterns, or exceptions to patterns, in your research?

Yes, patterns are the relationships between system behaviors.  When I was researching composite materials we were looking for relationships between the elasticity of the resins compared with the stiffness of the fibers and the adhesion of these resins to the fibers.  Turns out the resin modulus is key to toughness and damage tolerance but is detrimental to the compression strength.  If you look at some sophisticated avionics like radar the key is to recognize patterns in the data and suppress all of the electronic noise in order to find targets and separate real threats from background noise.  Mathematics is especially important in electronics and electrical engineering where you are concerned not only about the performance of the system but the analysis of the data from your sensors while also understanding the heat generated from the avionics and issues like vibration and stability.

6. Mathematics also teaches us about balance and equality. Is this idea useful in your research?

Mechanics of materials is a good example of balance and equality.  How much load a particular structure will take is a relationship between the basic material properties and the geometry of the structure.  We apply loads to the structure either manually or using finite element analysis and determine if the structure will deform or break.  Another example is applying fluid mechanics to the heat transfer and flow of fluids such as fuel or hydraulic fluid in an aircraft.

7. Mathematics helps us to represent quantities and measurements numerically. Do you do this in your work?

Whether you are calculating and resolving loads on a structure, determining how much fuel you need to get to the engine per second, or understanding the heat that will be generated by avionics it’s all about quantities and measurements.  You measure variables in order to understand the system behavior and many times variables from one system impact variables from another system thus requiring for example a design of experiments approach in order to understand the interactions.

8. Is estimation good enough or do you need to measure things accurately?

Estimation is best when you are trying to understand complex interactions and are trying to determine the most important variables that may be affecting the system.   This could be when you are engaged in advanced design activities trying to come up with new aircraft shapes and you need to use your experience and estimation to help define the solutions space.  When you get down to the actual designs/drawings you need to measure and calculate as accurately as possible.

9. How do you use statistics to analyse your results?

Yes, Most complex systems cannot be resolved directly using basic principles and formulas.  Systems behave statistically whether it’s a strength of materials problem or the span time for building a particular air craft of component.  We determine and calculate the probability of failures of systems in order to determine maintenance frequencies.  We test various batches of materials and do statistically valid numbers of specimens in order to calculate the design-to properties.  We do monte carlo statistics on our schedule positon for example to try and get insight into the variability of accomplishing our tasks on time.

10. Do you have any other insights to offer into how you use maths in your work?

Scientists use mathematics to understand the behaviors of systems and by determining the relationships mathematically determine how the systems will vary as the parameters vary.  Engineers utilize mathematics to take scientific principles and turn them into useful products and services.  My basic philosophy is that mathematics is the language of the Universe (the language of God so to speak) and that understanding math is the key to understanding the universe.  Math is a fundamental tool to scientific advancement and will be more and more important to societies that want to be technological leaders in the world.  In addition there is a basic confidence that comes from really getting the math, confident students will be better able to take on challenges and become productive in their careers.  The US (don’t know about Australia) cannot afford to abrogate scientific discovery and technology to countries like China where much greater percentages of their college students are majoring in STEM (science, technology, engineering, and math) than here in the US.  Without math there is no understanding, without understanding there is no science, without science and engineering there is no future.  Take Global Warming as an example.  Although a vast majority of the science makes it clear that human’s use of fossil fuels contributes to planetary warming (regardless of any planetary cycles), the general public is not sufficiently understand the science to understand the concepts.   Many people simply tune this out and just assume someone else is taking care of it.  This leads to the ability of those that want to diminish the importance of the human impact and support current industries to delay us from taking any real action thus, perhaps, putting everyone at risk.  Science and politics are intertwined, the better we understand science the better we can make informed decisions and insure that our government is taking appropriate action on important issues.

 In his e-mail to me, Dr Kinard had some other interesting comments to make:

Interesting questions Mr. Ferrington; they made me think and allowed me to philosophize a bit.  I am pleased that you are taking the time to collect information on how math is used in the real world to better impress upon your students that math is fundamental to all science and engineering advancement and is literally the key to the future for your students as well as for society as a whole. 


A huge thank you to Dr Kinard for his time and interest in participating in the Maths in Science project. I hope you have enjoyed this interview as much as I have.



 

Maths in Science - Jack Szostak

Professor Jack Szostak is a Nobel Prize winning geneticist. He is ranked as one of the top 50 most influential scientists in the world today. His work that led to the awarding of the Nobel Prize related to his research on the functioning of the ends of chromosomes. He has also produced the world's first artificial yeast chromosome and continues to use his research in genetics to explore the origins of life on earth.

Professor Szostak very kindly agreed to answer 10 questions for me about his experience with maths and how he uses it in his work..

Here is what he had to say.

The Questions:

1. Describe what maths lessons were like for you at school.

I don’t remember much about the lessons as such, but I do remember learning about fractions when I was quite young, and a bit later about quadratic equations, and getting very excited about these things!  I also liked learning about the beginnings of math, for example, how much the ancient Greeks were able to figure out (like the size of the earth) using simple math.

2. Was the maths that you learnt at school useful to you later in life?

Basic maths has been very important, it comes into almost everything in the lab. Its not that I use advanced math very often, but its necessary for understanding how a lot of our lab methods work.  Some people in my lab know and use much more math, for example in doing computer simulations of how molecules move and react.

3. How good do you need to be at mental arithmetic to do calculations in your head?

I think its really useful to be able to do rough calculations in your head, just so you have a quick sense of whether an answer is reasonable or not.  This can save a lot of time and effort.

4. Mathematics teaches us that you can put two things together to make a new thing. Is this important in what you do?

I’m not exactly sure what you mean by this, but we’re always on the lookout for new and surprising things, and often a bit of math is helpful in figuring out if a surprising result is likely to be real or not. 

5. Mathematics is about finding patterns. Do you need to look for patterns, or exceptions to patterns, in your research?

Sometimes, for example when we’re studying the copying of DNA or RNA, we look at a lot of sequences (long strings of the letters A,G, C and T).  It takes some math, mainly statistics, to look at how accurate the copying has been, and to tell if different copying conditions really change the accuracy or not. 

6. Mathematics also teaches us about balance and equality. Is this idea useful in your research?

Perhaps in the sense that we may for example measure something in different ways, and use simple math to see if these different measurements are consistent.

7. Mathematics helps us to represent quantities and measurements numerically. Do you do this in your work?

Yes, we are always measuring things, for example, how fast a reaction goes, or how accurate a DNA or RNA copying reaction is.

8. Is estimation good enough or do you need to measure things accurately?

That depends a lot on the experiment.  Its nice to be as accurate as possible, but sometimes a rough estimate is all you need to understand whats going on in an experiment.

9. How do you use statistics to analyse your results?

Yes, we use simple statistics all the time.  For example, we might repeat an experiment several times to get error bars on a measurement, so we can tell if one experimental condition really gives a different result from another condition.

10. Do you have any other insights to offer into how you use maths in your work?

I would say that the study of biology is changing in ways that involve more maths than before.  There is a whole field of ‘bioinformatics’ which is the mathematical and computer-aided analysis of DNA sequences.  Also, studying how the brain works is becoming quite mathematical in surprising ways.  So learning more maths, even if you don’t know exactly how you’ll use it, will always turn out to be helpful.

Thank you Professor Szostak for your generosity and support 

of the "Maths in Science" project!

 

Maths in Science - George Whitesides

Professor George Whitesides of Harvard University is one of the most significant chemical scientists in the world today. He is well known for advances in NMR spectroscopy, nanotechnology and physical and organic chemistry. He has been the recipient of many prestigious awards for his work, including the AIC Gold Medal, the Priestly Medal, the National Medal of Science, the Benjamin Franklin Medal, the IRI medal, the Othmer Gold Medal and many other national and international awards.

Professor Whitesides is also a prolific writer and has produced hundreds of articles for scientific journals and magazines. He took time out of his very busy schedule to answer some questions for me about his experience with mathematics.

The Questions:

1. Describe what maths lessons were like for you at school. 

   I loved algebra, geometry, and calculus; I was not so fond of arithmetic and trigonometry. I prefer abstraction to numbers

2. Was the maths that you learnt at school useful to you later in life?

Absolutely.  I’m a scientist, and I use mathematics all the time.  Also geometry (and drafting) were very useful in developing the ability to visualize things in 3D.

3.     How good do you need to be at mental arithmetic to do calculations in your head?

I can’t answer. I don’t think of myself as being good at mental arithmetic, but I estimate numbers all the time (in my head, on paper, wherever).

4. Mathematics teaches us that you can put two things together to make a new thing. Is this important in what you do?

Again, I’m not sure of the question.  A combination of two things—in math, in physics, in art, in music—is often a new thing.

5. Mathematics is about finding patterns. Do you need to look for patterns, or exceptions to patterns, in your research?

Both.  One tests for a pattern, and if that does not work, one tests for another pattern.  Sometimes you don’t find one.

6. Mathematics also teaches us about balance and equality. Is this idea useful in your research?

If you mean here that “=” is equality and balance, of course. All the time.

7. Mathematics helps us to represent quantities and measurements numerically. Do you do this in your work?

Constantly.

8. Is estimation good enough or do you need to measure things accurately?

Both. Estimation helps to follow the course of experiments, to design them, and to look for trends.  Accurate measurement and analysis is critical  to quantitation of prediction, and for both confirming compatibility of result and prediction, and maybe more importantly for showing incompatibility of result and prediction.

9. How do you use statistics to analyse your results?

Constantly in some programs.  Understanding what a number or a result means is absolutely crucial to any experimental program; statistical analysis is the best chance we have to answer that question.  Statistics is a hard subject to get excited about until you actually care (for whatever reason) about the meaning of data.  Simple questions like “Are these two numbers—with different numerical values—distinguishable or indistinguishable?” are core questions in science.  If I had one subject that I wish were better taught (I don’t know how) to students who would go on to work with data in any form, it would be statistics.   

10. Do you have any other insights to offer into how you use maths in your work?

There are areas of science that are purely qualitative, and for which math is not necessary, but not many.  I, most scientists, and all engineers use math all the time.  The more you know, the more you can do. 

----------------------------

Thank you so much Professor Whitesides for your participation in the "Maths in Science" project. I certainly value your contribution.

Maths in Science - Professor Ian Frazer

Professor Ian Frazer is best known, particularly by Year 7 students around Australia and probably other parts of the world, for his work in developing the Human papilloma virus (HPV) vaccine against cervical cancer. 

Professor Frazer grew up in Scotland and moved to Australia in the early 1980's. His dedication to his work has resulted in many significant awards, including the Prime Minister's Prize for Science, The Howard Florey Medal, the Balzan Prize, the Australian Medical Association Gold Medal, the CSIRO Eureka Prize and the William B. Coley Award. Professor Frazer was elected as a Fellow of the Royal Society in 2011.

It is a great honour to have been able to ask Professor Frazer 10 questions about how he uses maths in his work and I have chosen him to be the first scientist to introduce the "Maths in Science" project.

The Questions:

1. Describe what maths lessons were like for you at school. 

We were the guinea pigs for a new curriculum focussed on sets and matrices - it was exciting to see the power of these tools to solve problems.

2. Was the maths that you learnt at school useful to you later in life?

Mostly I use the statistics and probability theory I learnt – sometimes calculus.

3. How good do you need to be at mental arithmetic to do calculations in your head?

I was lucky enough to find a book that had a series of “tricks” to do quite complex arithmetic calculations in your head – however they did require good mental arithmetic.

4. Mathematics teaches us that you can put two things together to make a new thing. Is this important in what you do?

Inferences from independent observations are at the core of research hypothesis formation.

5. Mathematics is about finding patterns. Do you need to look for patterns, or exceptions to patterns, in your research?

Pattern recognition, and outlier recognition, give rise to new theories for testing.

6. Mathematics also teaches us about balance and equality. Is this idea useful in your research?

Systems analysis requires understanding of homeostasis maintenance – which is all about balance and equality.

7. Mathematics helps us to represent quantities and measurements numerically. Do you do this in your work?

We measure and compare numbers routinely as part of bioinformatics.

8. Is estimation good enough or do you need to measure things accurately?

In biological systems significant changes generally occur in half log steps so estimation is often good enough – more than single digit precision is hard to achieve in my field of science.  

9. How do you use statistics to analyse your results?

Virtually everything in biological sciences is about demonstrating that differences observed between control and test measurements are (a) statistically and, if so,  (b) biologically significant. 

10. Do you have any other insights to offer into how you use maths in your work?

It would be fair to say that 98% of the interpretation of our work relies on mathematical tools. 

   ---------------------------------------------------------------

Thank you so much Professor Frazer! I am so grateful for your participation and generosity with your time. Your answers were thoughtful and will give us all something to contemplate.