Zurab Janelidze's Blog: education

These are notes in progress for a talk given at the online user group conference of the advanced programme mathematics organized by ieb (19 February 2022)

1. Introduction

In my experience, what makes transition from school mathematics to university mathematics hard is the depth of engagement with mathematics that university mathematics requires of students, compared to the depth of engagement that school mathematics requires. Do you agree or disagree with the following thesis:

A school learner must understand school mathematics at the same depth that a university student is expected to understands university mathematics.

If you do not agree and think that a university student should understand university mathematics more deeply than a school learner understands school mathematics, this means that you expect a learner transitioning to university not only having to learn more advanced mathematics, but to understand it more deeply than they understand its foundation, the school mathematics. Is that possible?!

2. Misleading Questions

What hinders a school learner to understand school mathematics deeply? Learning is driven by assessment: eventually, the task of a school teachers is to bring a pupil to the point of passing the final exam paper. Learning happens through textbooks which engages learners in a learning process that is based on answering questions that resemble those in the final exam. So then, it is natural to conclude that the mathematical questions students have to work on, whether from the textbook or the exam, paint the image of the mathematics they are learning. What if these questions mislead the learner and provide them with a wrong image of mathematics?

Here we give some examples of questions that have a great potential of misleading the learners.

What is the next number in the sequence 0,1,2,3,4,5,6,7,...?

The school expects the answer to be 8. In actual mathematical reality, it can be any other number as well, since we have not given any restriction on the sequence apart from its first eight digits. There are 1669 entries on these eight terms in the Encyclopedia of Integer Sequences.

This question misleads a pupil to thinking that the first idea that comes to mind (which would be indeed 8 for most humans) is the right answer to a mathematical question, and hence, if no idea comes at first, then there is no way to answer a question.

What is the domain of the function f(x)=x^2?

The school expects the answer to be entire real line. In actual mathematical reality, a formula does not define a function uniquely. To have a function, you need to first name a domain (and a "codomain", which is not necessarily the same as the "range") and then, state the formula which is to be interpreted for the named domain and codomain. Otherwise, why the entire real line -- the domain could have been the complex plane just as well!

This question misleads a pupil to thinking that mathematical equations contain complete information about mathematical objects. This eventually results in them dismissing words in a complex logical statement, focusing on a flattened picture comprising of just the equations in the statement. For example, they read "if 1=2 then 3=4" as "1=2 and 3=4". The latter is a false statement. The former is not: from 1=2 we can indeed derive 3=4 (simply add 2 to both sides).

Solve the equation (x+3)/2=5.

The school expects the students to perform a sequence of manipulations eventually leading to the "answer" x=7. In actual mathematical reality, each step needs to be confirmed to be reversible to be certain that 7 is indeed a solution of the equation (that is it the only solution, is confirmed by the derivation).

This question misleads a pupil to thinking that solving an equation means applying some procedures in one direction. With this, not only the pupils do not understand the concept of a "solution", but they tend to think that any mathematical problem can be solved by a sequence of manipulations which need not be logically justified. This also results in them not understanding the difference between "implies" and "is equivalent to".

These are just some examples of misleading questions. There are many more!

3. Factual Teaching vs Insightful Teaching

Note:

"sketch" instead of "graph" (or "sketch of the graph").
Wilson says "-3 is not included" (it is rather the paint (-3,-1) that is not included in the graph) but "4 is included" (similarly, 4 is merely the x-coordinate of the point included in the graph).
Wilson says that the domain is "where your graph is on the x-axis", and "range is where the graph is on the y-axis".
Wilson says "if it is not defined, we put a round bracket, if it is defined, we put a square bracket".

In each of these examples we see a simplification of the language for pointing out something to the pupil. These simplifications may cost knowledge: a pupil who is not well familiar with the material receives logically incomplete information and hence enters into a mode of memorization where certain phrases are paired up with certain settings in a mathematical question. Instead of learning to actually comprehend the question, the learners become accustomed to break up the question in several keywords which in their memory relate (without any insight as to why) to some procedures that the learning needs to perform.

With this method of learning, which is present at university too, the learner/student comprehends mathematics as a set of memorized principles whose only application that they can experience is writing an exam. Mathematics for them is not something in which they can look for a story, an insight, or meaning.

Factual teaching vs insightful teaching compares well with learning music based on reading music notation vs based on listening to the instrument:

4. Contribution from Dr. Cerene Rathilal

5. Challenge, Passion and Hard Work

It is customary to think that pupils lose passion for mathematics when they encounter challenges in it. I do not believe this to be the case. To the contrary, I believe that people get inspired by challenge, as long as it stimulates intellect. Nobody has passion for something that is easy to get! My thesis is:

What kills mathematical passion in learners is realization that to overcome a mathematical challenge all it takes is routine hard word (e.g., memorization), instead of intellectual enquiry.

Sure, it is important for a learner to develop the skill of working hard. However, mathematics is not the right subject for that. Mathematics is supposed to be the subject that awakens a genius in a human being. There is even scientific evidence to back something that true mathematicians know very well: to be smart, you need to be lazy. The reason for this seemingly paradoxical relation is that if you are not lazy, you are likely to involve yourself with various activities that makes use of the rest of your body more than your brain. As a mathematician, the more mathematical research I do, the less I want to do anything else, including going to a shop to buy bread -- I become too lazy to even do that, not to mention doing admin at the university! On the other hand, when I force myself to do hard work with the admin (or physical exercise, which for the sake of winning time involves bicycling to the bread shop), I am no longer able to do mathematics effectively. My brain sort of flattens out -- I am not able to think sufficiently deeply any more.

Doing mathematics properly and working hard are in direct contradiction. Part of mathematical ingenuity is about finding a solution to a problem that saves your time. Laziness actually drives mathematical enquiry. But as soon as you do mathematics the right way, your brain releases chemicals which enable you to work for long hours, without having a feeling that you worked hard. You get exhausted, and yet you feel you did not work hard enough. That is the truth about mathematics.

When mathematics is taught through hard work, it kills passion simply because hard work kills mathematical creativity in a professional mathematics researcher too. If you are doing mathematics and you feel you are working hard, as far as I can tell, you are not doing actual mathematics.

A mathematical challenge is supposed to ignite passion and not kill it. If it kills passion, it may not be a challenge worth pursuing. In fact, research mathematicians use this principle to navigate their way in research: selection of which problem to pursue and which not to pursue is very much determined whether the problem ignites or kills passion.

Thus, when mathematics is done the right way, the diagram is:

challenge => passion ignites => hard work

When it is done the wrong way, the diagram becomes:

hard work => passion dies => challenge

6. Final Note ♪

In general, school mathematics is much about learning procedures to solve computational problems, without understanding why do those procedures work, not to mention a chance to self-discover the procedures. Intellectual effort of the student is reduced to writing in one's memory bank these procedures and practicing their application as far as answering exam questions is concerned.

Instead, university mathematics is more about understanding concepts intuitively, allowing a student to apply the understanding to solve a problem by self-discovering a procedure. Not only students are expected to explain why a procedure works, but they are also expected to come up with a procedure (a proof) which would confirm validity of a mathematical statement.

Those who have a talent for memorizing procedures usually lack the talent of creativity in mathematics, and vice versa. In other words, school mathematics favors learners with a certain intellectual profile, which is likely to exclude those who are capable of taking their mathematics studies at an advanced university level.

The issues discussed here do not only apply to transition from school mathematics to university mathematics. Similar issues arise in transition from undergraduate mathematics to postgraduate mathematics, and from postgraduate mathematics to research mathematics. In all cases, incorrect approach to mathematics leads to lack of sufficient foundation to advance to the next step.

7. Some Feedback from Students

The content of the first-year math courses is a lot of new work which builds onto the foundation that high school laid, therefore it was ideal to enter the first year with my mind still fresh from the matric exam the previous year. I personally found the way of learning maths in high school - listening in class, practising and using textbooks - quite similar to learning 1st-year math, only the workload increased and the pace at which it was presented. One big difference from high school is that you are given the free will to attend lectures and use your time wisely for practising problems. You are given examples to work through weekly, but it is basically up to the student if they want to make it their "homework" and that is what will separate the cream of the crop when they get tested. Thus if one did not practice self-discipline in high school, University is where they learn to. Another difference from high school is different lecturers for different courses with their own unique teaching styles. Compared to one high-school teacher, students now face the reality that they must adapt to new lecturers for every course each promoting new environments of learning, but most important growth. The biggest difference for me from high school is the inquisitiveness and enthusiasm. I remember in high-school everyone always complained about this assignment and that test, and actually lost the marvel of learning. In University however you are surrounded by a sea of individuals with the same interests as you, further igniting your passion for math. I just started my second year, and I just keep discovering new benefits of studying math. It depends on what module one takes, but the math still continues to enlighten sides of myself that I forgot I had, for example thinking outside of the box, or getting excited if a math problem sounds crazy or makes absolutely no sense! My advice would be to any scholar coming to university to think about math creatively, observe it around you, talk about it, be brave, ask many questions and most importantly have fun! - Nina Smit

The transition from school Mathematics to University Mathematics is quite challenging. For one, you do not know what to expect and I feel the NSC system fails students who take a deep interest in the STEM field. The mathematics you learn in school is pretty straight forward and answers to questions often follow an obvious algorithm that you can use without even understanding the fundamentals of what you were taught. Mathematics forms an integral part of STEM and I feel schools (or schools in my town) did not emphasize this enough. I know some schools offer Ad Math but the schools in my town didn't which was sad. I feel all schools should offer students the opportunity to challenge themselves with "difficult Math". This would lay a good foundation for students transitioning into university. - Cole Tymothy Paulse

Transitioning from school mathematics to university mathematics can be tricky. The main difference is the way it is taught and examined. In a school mathematics test/exam, you will be expected to answer questions that are similar to the examples you have been shown in class or your homework questions. This is very different in a university setting. Many questions don't look the same as the given examples so you will have to apply the knowledge acquired from the "homework" to solve a problem that is presented in a format that you probably haven't seen. You can't only use the fact that you know the formula and format to solve a university mathematics problem, you need to develop your logical and critical thinking skills and get creative to understand and answer the question. - Refentse Makweya

School mathematics relies heavily on memorization while university mathematics relies on more problem-solving. At school, we are taught a method to solve a specific problem. At university, we are taught certain tools that can be applied to solve various problems in interesting and creative ways. University mathematics courses contain far more work than high school mathematics and learning happens fast. University mathematics assessments often feel more like mathematics olympiads in the sense that the questions are often testing problem-solving abilities over simple memorization. - Anonymous

For my transition...I was really blown out of my mind at how much math in university depended on trigonometric work as well as graphs. I wish the teacher in school could emphasize more on the importance of those topics. It also on the other hand showed me just how much i did not know about maths and to what extent it can be applied. I really do look forward to my journey in maths and all the components I will still learn in the years to come. - Ancois Huysamen

I enjoyed my primary School very much. It should not be surprising that I excelled at mathematics and I was given enough space to attempt other problem-solving methods and also try some harder problems in competitions. In high school, I was given similar freedom to do those things, even if teachers did not know much other mathematics outside the syllabus. I stayed in a traditional high school for two years before being homeschooled. I did the Cambridge syllabus as opposed to CAPS. Cambridge syllabus was significantly more robotic. Every past paper has very similar questions that you can learn how to solve. Also, the memos of these past papers gave marks for some unnecessary steps so memorising solutions on past paper memos was even more vital. In university, I feel a bit more freedom in the methods I can use and I can be confident that as long as I'm clear about what I'm doing, I'll get all the marks. I can also be confident that my lecturer can answer any math-related question I have. - Jayden Thomas Dickson

In short, it was and still is one of the toughest but most rewarding challenges I've ever faced. In my experience, there was a big shift in what we were trying to achieve with mathematics. In high school, I was almost always finding a single answer or a number. The goal was mostly just to find the value of something and use Mathematics as the tool to do so. When I first started University Mathematics I saw quite quickly that most of our time was spent doing something much more useful and amazing. We were using Mathematics to understand and describe things. I started to see and understand how Mathematics was not only a method of problem-solving but also an amazing language of description. A very clear example I could give is my completely different experience with Probability and Statistics. Throughout high school, Statistics was a section I really disliked. It felt boring, repetitive, and pointless. However, when I started with my first-semester module on Probability Theory and Statistics (which thankfully was a compulsory module or I never would have chosen it), it quickly become one of the subjects I most enjoyed. It really challenged how I perceived so many everyday events and interactions. The necessity of going from axioms to proof made the well-established formulas I was shown in high school come to life because I could see with more intuition every term, constant and co-efficient was meaningful and not just conveniently helpful.

All this did come with a much higher requirement for practice, repetition, and the need for failure which was a new experience for me in mathematics. Getting things wrong for the process of learning became a common theme in anything I encountered at university. I realized extremely quickly that battling for hours on a topic and having assessments go terribly on a somewhat frequent basis was a reality I need to accept when it happened and learn from those mistakes because feeling disheartened and frustrated was going to only hurt the process of doing better in future. What made this process less demotivating is also a change in the culture around getting things wrong. In High School getting something wrong felt like a worse fate than trying. Success often seemed the only acceptable outcome. What changed dramatically was that failure became something celebrated as a learning opportunity and was met with countless resources to capitalize on it and the focus from my lecturers wasn't getting me to pass the module but to understand the content and become a better mathematician.

To summarize the transition was a massive one. It has challenged me more than I ever thought it would, but it has given me a viewpoint and appreciation for mathematics that I have never had before. I do wonder how many more of my friends in high school would have a changed perception of mathematics if they could get a taste of what the subject can be in a different environment. Looking back I am also amazed by how much I have progressed throughout the last year and hope that in the future more high school students will be inspired to consider a career in Science and Mathematics. - Chad Robert Davies (see also https://foabma.blogspot.com/2022/02/school-vs-university-mathematics.html)

The process of creation of mathematics has the following hierarchically dependent components:

Coming up with a concept.
Coming up with a question dealing with a relationship between concepts (this includes formulating a hypothesis, as well as finding an example or a counterexample of a concept/phenomenon).
Answering a question dealing with a relationship between concepts (this includes proving theorems as well as solving problems without being given the recipe for solution).
Applying the answer to a question dealing with a relationship between concepts to answer another such question (this includes solving problems by applying a given recipe for solution).

Modern mathematics education (both at the school and at the university levels) focuses mainly on the last two points. What is regarded as a low quality mathematics education would focus only on the last point. For a more whole mathematics education, the first two points must receive as much attention as the last two points do.

It is not difficult to implement the first two points in the practice of mathematics teaching. Here is an example of the structure of a class that focuses on the second and the third points:

The teacher proposes one or two concepts that the pupils are familiar with (perhaps, by taking suggestions from the class).
The teacher then asks the pupils to explain the concepts, helping the pupils in the explanation, when necessary.
Then, the teachers asks the pupils to think of a question that would combine the named concepts. The teacher helps in this process.
After this, the teacher and the pupils engage in answering the question together.
If the question is too hard to answer, it should be concretized to a simpler question. If the question is too easy to answer, it should be abstracted to a more difficult question.

Concepts arise in mathematics as a necessity to help one express a general phenomenon. Incorporation of the first point in a classroom can be achieved by explaining this necessity for the concepts that the pupils are already familiar with, or by taking pupils on a journey that would help them identify such a necessity and will result in (re)discovering a mathematical concept. Teaching concepts by first showing examples and then asking the pupil to develop a concept that fits those examples is another, perhaps simpler, way. The activities which ask a pupil to identify a pattern in a sequence of numbers or figures is in some sense of this type. However, these activities are sold as activities that fall under the third point, as the pupil is being convinced that the question must have one definite answer.

Zurab Janelidze's Blog

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Mathematical Structures Course 2022

Lecture 1: Magmas

Lecture 2: Join Semi-Lattices

Lecture 3: Relations

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Lecture 5: Posets

Lecture 6: Groups

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Lecture 8: Posets II

Lecture 9: Posets III

The Transition from High School Mathematics to University Mathematics