Tony Gardiner, on mathematics education
This assessment by an unnamed expert is quoted in the short on-line version of the report A degree of uncertainty: an investigation into grade inflation in universities from Reform, a UK think-tank. A fuller quote:
There is considerable evidence to suggest that ‘degree algorithms’ (which translate the marks achieved by students during their degree into a final classification) are contributing to grade inflation. Approximately half of universities have changed their degree algorithms in the last five years “to ensure that they do not disadvantage students in comparison with those in similar institutions”. Research has also identified serious concerns about how these algorithms treat ‘borderline’ cases where a student’s overall mark is close to the boundary of a better degree classification. One expert concluded that “universities are essentially massaging the figures, they are changing the algorithms and putting borderline candidates north of the border”.
A brilliant blogpost from Double-entry bookkeeping and Galileo: abstraction vs idealization. He gives a definition of abstraction, as used in computer science:
abstraction is multiple realisability.
I am currently designing a first year, first semester course, something like Introduction to Algebra. One part of it will be about integers and polynomials:
- start with division with remainder of integers and polynomials
- develop, in parallel, divisibility theory and uniqueness of prime factorisation – at every step stating two theorems, one for integers, another for polynomials, but proving only one of them and leaving writing up a proof of the other as an exercise for students;
- conclude this part of the course by proving the Chinese Reminder Theorem for integers and Lagrange’s Interpolation Formula for polynomials and explaining why this is one and the same theorem;
- perhaps only then introduce rings, fields, homomorphisms, and Euclidean rings;
- and give one more exercise to students: uniqueness of prime factorisation of Gaussian integers.
I think that, at start of undergraduate mathematics, an abstract concept should be introduced only after students have well familiarised themselves with its several realisations.
This short dialogue started with someone posting on LinkedIn group Math, Math Education, Math Culture a link to the following blogpost:
Matrices are a common tool used in algebra. They provide a way to deal with equations that have commonly held variables. In this post, we learn some of the basics of developing matrices.
From Equation to Matrix
Using a matrix involves making sure that the same variables and constants are all in the same column in the matrix. This will allow you to do any elimination or substitution you may want to do in the future. Below is an example
Above we have a system of equations to the left and an augmented matrix to the right. If you look at the first column in the matrix it has the same values as the x variables in the system of equations (2 & 3). This is repeated for the y variable (-1 & 3) and the constant (-3 & 6).
The number of variables that can be included in a matrix is unlimited. Generally, when learning algebra, you will commonly see 2 & 3 variable matrices. The example above is a 2 variable matrix below is a three-variable matrix.
If you look closely you can see there is nothing here new except the z variable with its own column in the matrix.
When a system of equations is in an augmented matrix we can perform calculations on the rows to achieve an answer. You can switch the order of rows as in the following.
You can multiply a row by a constant of your choice. Below we multiple all values in row 2 by 2. Notice the notation in the middle as it indicates the action performed.
You can also add rows together. In the example below row 1 and row 2, are summed to create a new row 1.
You can even multiply a row by a constant and then sum it with another row to make a new row. Below we multiply row 2 by 2 and then sum it with row 1 to make a new row 1.
The purpose of row operations is to provide a way to solve a system of equations in a matrix. In addition, writing out the matrices provides a way to track the work that was done. It is easy to get confused even the actual math is simple
System of equations can be difficult to solve. However, the use of matrices can reduce the computational load needed to solve them. You do need to be careful with how you modify the rows and columns and this is where the use of row operations can be beneficial
What follows is a short exchange of comments:
AB: What is missed in this discussion of augmented [matrix] is the key issue: why do row operations lead to a solution of the original system of simultaneous linear equation? This crucial fact is stated, but not explained in any way.
Response: Well i wasn’t trying to be that deep. My goal is to support students who are learning algebra. Perhaps you can explain here why row operations work.
AB: It is a really simple but fundamental fact: if you have a system of simultaneous linear equations, then the following operations:
* swapping two equations
* adding a multiple of one equation to another equation
* multiplying an equation by a non-zero number
do not change (the set of) solutions; it is useful to start discussion of system of linear equations by proving this statement and emphasising that these operations are reversible; the proof crucially depends on this observation. Row operations on the augmented matrix are operations on the system of equations – we simply skip symbols for unknowns and the equation signs. If we do some row operations on the augmented matrix and then write a system of equations corresponding to the resulting matrix, the new system of equations is equivalent to the original one, that is, the two systems have exactly the same solutions.
Press-release from ESA (European Society of Anaesthesiology):
New research presented at this year’s Euroanaesthesia congress in Copenhagen, Denmark and due for publication in the journal Medical Education shows that teachers who reward their students with chocolate cookies can score significantly better in evaluation surveys. The study is by Dr Christina Massoth and Dr Manuel Wenk together with colleagues at Department of Anaesthesiology, Intensive Care and Pain Medicine, University Hospital of Muenster, Muenster, Germany.
End-of-course evaluation of teaching (SETs) surveys are widely used and taken seriously by faculties, forming part of the decision making process for the recruitment of academics, distribution of funds, and changes to educational curricula. There is some doubt, however, as to whether this type of evaluation method can accurately measure the quality of course material and the extent to which important knowledge is transferred.
This study investigated whether a simple intervention by the teacher in the form of the provision of chocolate cookies to their students, could influence SET results in a significant way.
The team conducted a randomised, controlled trial using a group of 118 undergraduate third year medical students who were randomly allocated into 20 groups. During the first of four sessions of a curricular emergency course, 10 of the groups were each given 500g of chocolate cookies to share, while the other 10 groups made up the control and received nothing. Afterwards, all the students completed a 38 question evaluation survey which asked them about the teacher, course contents, teaching materials, and self-assessment.
The authors found that those groups who had received cookies evaluated their teachers as being significantly better than those who received nothing. They also considered their teaching materials to be better, and their summation scores for the overall quality of the course were significantly higher than those of the control group. All the results were statistically significant.
These findings fit in with other research that has described factors associated with better evaluation results such as grading leniency, environmental factors, or the attractiveness of a teacher. The authors note that: “Consequently, a higher student satisfaction does not necessarily correlate with a higher quality of education”.
The team conclude that something as simple as giving out chocolate cookies had a significant effect on course evaluation. They suggest that: “These findings question the validity of SETs when used to make widespread decisions within the faculty”. The authors go on to point out that: “On the upside, our findings may stimulate new ideas for teachers who seek to improve and control their SETs by manipulating food-related interventions”.
Dr Wenk notes that while this research may appear light-hearted at first glance, he cautions that: “Students’ end-of-course feedback and evaluation of teaching and teachers (SET) has become a standard tool for measuring ‘quality’ of curricular high-grade education courses. The results of these evaluations often form the basis for far-reaching decisions by the academic faculty, such as changes to the curriculum, the promotion of teachers, the tenure of academic appointments, the distribution of funds and merit pay, and the choice of staff. So this is a totally inadequate tool to measure quality if you can mess with the system that easily!”
Motivation by anticipation: Expecting rapid feedback enhances performanceThere are a number of factors that influence how well we do in school, including the amount of time we study and our interest in a subject. Now, according to new findings in Psychological Science, a journal of the Association for Psychological Science, how quickly we expect to receive our grades may also influence how we perform.Psychological scientists Keri L. Kettle and Gerald Haubl of the University of Alberta in Canada wanted to investigate how the timing of expected feedback impacts individuals’ performance. For this experiment, they recruited students enrolled in a class that required each student to give a 4-minute oral presentation. The presentations were rated by classmates on a scale from 0 (poor) to 10 (excellent) and the average of these ratings formed the presenter’s grade for that part of the course. Students received an email 1 day, 8 days, or 15 days before their presentation and were invited to participate in this research study. Students agreeing to volunteer in the study were informed when they would receive feedback on their presentation and were asked to predict their grades. Participating students were randomly assigned to a specific amount of anticipated feedback delay, which ranged from 0 (same day) to 17 days.The results reveal a very interesting relationship between how soon the students expected to receive their grades and their performance: Students who were told they would receive feedback quickly on their performance earned higher grades than students who expected feedback at a later time. Furthermore, when students expected to receive their grades quickly, they predicted that their performance would be worse than students who were to receive feedback later. This pattern suggests that anticipating rapid feedback may improve performance because the threat of disappointment is more prominent. As the authors note, “People do best precisely when their predictions about their own performance are least optimistic.”Although this experiment took place in a classroom, the authors conclude that these findings “have important practical implications for all individuals who are responsible for mentoring and for evaluating the performance of others.”
My answer in Quora to What are examples of polar coordinates in nature?
I quoteabout bees communicating to fellow bees direction and distance to the food source, that is, polar coordinates of the food source in relation to the hive. It is hard to make an example closer to nature than that one.
It has long been known that successfully foragingperform a upon their return to the . The laden forager dances on the comb in a circular pattern, occasionally crossing the circle in a zig-zag or waggle pattern. described this behaviour in his Historia Animalium. This waggle pattern of movement was thought to attract the attention of other bees. In 1947, correlated the runs and turns of the dance to the distance and direction of the food source from the hive. He reported that the orientation of the dance is correlated with the relative position of the sun to the food source, and the length of the waggle portion of the run is correlated to the distance of the food from the hive. Von Frisch also reported that the more vigorous the display is, the better the food. Von Frish published these and many other observations in his 1967 book The Dance Language and Orientation of Bees and in 1973 he was awarded the for his discoveries.
Imagine a mental experiment: what would happen if the atmospheric conditions on Earth were, in the last 5 or 10 thousand years, slightly different: a light haze obscured the stars without limiting solar radiation and thus not affecting development of agriculture etc. Would mathematics, as we know it, develop without the principal source and paradigm of precision: the movement of stars in the sky? Can anyone point to studies of history of precision as intellectual and technological concept?
I ses this example in my lectures when I explain the difference between arithmetic and harmonic means:
A car traveled from A to B with speed 40 miles per hour, and back from B to A with speed 60 miles per hour. What was the average speed of the car on the round trip?
Anatoly Vorobey and Vladimir Kramchatkin made a useful comment on Facebook on this quite standard and well-known problem:
“The answer is obviously 48 [miles per hour]. 95% can not solve this problem the first time. But if they are told in advance that the distance between A and B is 120 [miles], 95% of schoolchildren will easily solve this problem.”
A concrete number, 120 km, serves as a strong hint that students are expected to do something with this number. But, for majority of students, if a magnitude or a quantity is not assigned a concrete numerical value, it does not exist. This is one of the flaws of mathematics education at schools: no-one tells students that they have to be able to see hidden parameters in arithmetic problems.
But this is not the only flaw: students are also not told how to check solutions. Checking answers frequently benefits from seeing a problem in a wider context and varying the data. The standard answer that students give to the problem with the car is 50 miles per hour, the arithmetic mean of the two speeds. But this solution immediately collapses if we slightly change the problem: what would happen if the speed of the car on its way back from B to A was 0 miles per hour?
Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.
I heard that from Israel Gelfand in a private conversation. Because of Gelfand’s peculiar style of work, I was often present during his conversations with his biologist co-authors about structure of proteins. The quote was included by me in an earlier version of the book ‘Mathematics Under the Microscope‘, but is not present in the published version (originally I planned to insert it in Section 11.6).
Besides being one of the most influential mathematicians (and mathematical physicists) of 20th century, Gelfand also had 50 years of experience of research in molecular biology and biomathematics, and his remark deserves some attention.
Indeed biology, and especially molecular biology, is not a natural science in the same sense as physics. Indeed, it does not study the relatively simple laws of the world. Instead, it has to deal with molecular algorithms (such as, say, the transcription of RNA and synthesis of proteins which ensures the correct spatial shape of the protein molecule) which were developed in the course of evolution as a way of adapting living organisms to the changing world. If they solve a particular problem in an optimal way, they should allow some external description in terms of the structure of the problem. Indeed, this is the principal paradigm of physics; it is an experimental fact that the behavior of physical systems is governed by various minimality / maximality principles, and the optimal points have, as a rule, especially nice mathematical properties.
But why should a biological system to be globally optimal? Evolution is blind, and there is no reason to assume that the optimal solution is reached. The implemented solution could be one of myriads of local optima, sufficiently good to ensure survival. Lucky strikes could be so rare that the huge search space and billions of years of evolution produced just one survivable algorithm, which, as a result, dominates the living world, and is perceived by us as something special. But it might happen that there is absolutely no external characterization which allows us to distinguish it from other possible solutions, and that its evolutionary phylogeny is its only explanation.
However, I am not a philosopher and cannot claim that my solution of Gelfand’s paradox is correct. What I claim is that philosophers ask wrong questions. The classical conundrum of relations between mathematics and physical world starts to look very different — and much more exciting — as soon as we include biology into consideration. I will try to continue this discussion.