It is an extremely serious undertaking and requires substantial long-term commitment over a number of years, so it is absolutely imperative that there is a strong underlying motivation, otherwise it is unlikely that you will stick with self-study over the long term.
You might be an individual at the beginning of your educational career, deciding whether to take a formal university program in mathematics.
You might have worked in a technical industry for years, but seek a new role and wish to understand the necessary prerequisite material for the career change. You might also enjoy studying in your own time but lack a structured approach and want a reasonably linear path to follow.
One of the primary reasons for wanting to learn advanced mathematics is to become a "quant". However, if your sole reason for wanting to learn these topics is to get a job in the sector, particularly in an investment bank or quantitative hedge fund, I would strongly advise you to carry out mathematics in a formal setting i.
This is not because self-study will be any less valuable or teach you less than in a formal setting, but because the credential from a top university is, unfortunately, what often counts in getting interviews, at least for those early in their career. An alternative reason for learning mathematics is because you wish to gain a deeper understanding of how the universe works.
Mathematics is ultimately about formalising systems and understanding space, shape and structure. It is the "language of nature" and is utilised heavily in all of the quantitative sciences.
It is also fascinating in its own right. If you are heavily interested in learning more about deeper areas of mathematics, but lack the ability to carry it out in a formal setting, this article series will help you gain the necessary mathematical maturity, if you are willing to put in the effort.
I want to emphasise that studying mathematics from the level of a junior highschooler to postgraduate level if desired will require a huge commitment in time, likely on the order of years. Clearly this is a staggering commitment to undertake and, without a strong study-plan, will likely not be completed due to the simple fact that "life often gets in the way". However, chances are if you are considering studying advanced mathematics you will already have formal qualifications in the basics, particularly the mathematics learnt in junior and senior highschool GCSE and A-Level for those of us in the UK!
In this instance it is likely that you might be able to begin learning at the start of the undergraduate level, or possibly at the level of an advanced highschool student.
Even if you have the equivalent qualifications in A-Level Mathematics or A-Level Further Mathematics, you will still have a long road ahead of you. I estimate that it will take approximately years of full-time study or years of part-time study, in order to have an equivalent knowledge base gained by an individual who has carried out formal study in a UK undergraduate mathematics program to masters level. While I don't think it is necessary to have postgraduate qualifications to become a quant, it is useful and can certainly put you ahead of the competition.
However, do not be put off by the time commitment for postgraduate study. It isn't absolutely necessary and is likely to be carried out in a formal, full-time setting regardless. If you are happy with this overall level of commitment, then the broad path that you will follow should look something like this:. As you can see, a mathematics education to a high level can take anywhere from 3 years to approximately 15 years or more!
Hence this is not something to be undertaken lightly. You must give it serious consideration and make sure that the payoff financial or otherwise from study will be worth the serious effort required. These days it is possible to study from a mixture of freely available video lectures, lecture notes and textbooks. There are those who learn better from watching videos and making notes, while others enjoy working methodically through a textbook.
I've listed what I consider to be the most useful resources below. At the undergraduate level, I am a big fan of the Springer Undergraduate Mathematics Series of textbooks, which cover pretty much every major course you will find on a top-tier mathematics undergraduate degree in the UK.
I will go into detail regarding choices of books for specific modules below. I've also found the Schaum's Outlines series of books to be extremely helpful, particularly for those who like to learn by answering questions. While they don't go into the detail that others might particularly the SUMS books above , they do help consolidate the basics by working through a lot of questions.
I highly recommend them if you've not seen any of the material before. Many Universities provide publicly accessible course pages that contain freely available lecture notes, often in PDF format, typeset in LaTeX or similar. Where appropriate, I've listed freely available lecture notes for particular courses. However, I prefer to recommend textbooks as they tend to cover a wider set of material. They aren't "cherry picking" material in a way that a lecturer will have to do so in order to fit the material into semester-length courses.
Despite this issue, there are some extremely good lecture notes available online. The rise of Massive Open Online Courses MOOCs has fundamentally changed the way students now interact with lecturers, whether they are enrolled on a particular course or not.
Some MOOCs are free, while others charge. On the whole, I've found MOOCs to be a great mechanism for learning as they are similar to how students learn at University, in a lecture setting. They provide the added benefits of being able to pause videos, rewind them, interaction with lecturers on online portals as well as easy access to supplementary materials. Some have suggested that the quality of MOOCs is not as good as that which can be found in a University setting, but I disagree with this.
There are some extremely good MOOCs available in data science, machine learning and quantitative finance. However, I have found there to be a lack of more fundamental courses and as such you'll see me recommending textbooks for the majority of the courses listed here. Gender gap in science, technology, engineering, and mathematics STEM : Current knowledge, implications for practice, policy, and future directions. Not lack of ability but more choice: Individual and gender differences in choice of careers in science, technology, engineering, and mathematics.
Williams, C. Physics Education, 38 4 , — Wu, X. The effects of different versions of a gateway STEM course on student attitudes and beliefs. Zeldin, A. Against the odds: Self-efficacy beliefs of women in mathematical, scientific, and technological careers. American Educational Research Journal, 37 1 , — Download references. The authors thank the students, student counselors, teachers, and principals of Oulu upper secondary schools, also the staff of the Department of Education and culture of Oulu, for giving their time and support for this study.
We are grateful to the Matriculation Examination Board and the CSC for opening their student registers for this research. We also owe a debt of gratitude to every Finnish research university that kindly supported our research by giving us their student selection information for research purposes.
The authors are grateful for receiving funding for this research from the University of Oulu, the Adult Education Allowance of Finland, and from the Oulu University project regarding student admissions, funded by the Ministry of Education and Culture. You can also search for this author in PubMed Google Scholar.
Satu Kaleva collected the cohort sample, carried out the analysis and drafted the manuscript together with Hanni Muukkonen and Jouni Pursiainen. Jouni Pursiainen and Jarmo Rusanen collected the register datasets, Mirkka Hakola and Jouni Pursiainen analyzed the data and Jouni Pursiainen contributed on writing the data analysis of the quantitative data.
All writers have approved the final manuscript. He is interested in the study paths from upper secondary school to the universities, especially the effect of subject choices in the matriculation examination.
Her master theses research concentrated in finding the background factors that are influencing student acceptance process. She has participated in the development projects of education. His research has been focused always on register based data, like matriculation examination in this study. Her research areas include collaborative learning, knowledge creation, learning analytics and methodological development.
To study learning processes and pedagogical design, she has been involved in and lead several large scale international educational technology development projects carried out in multidisciplinary collaboration. Correspondence to Satu Kaleva. All students in this study provided consent prior to study participation.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Reprints and Permissions. Kaleva, S. Download citation. Received : 20 June Accepted : 14 November Published : 13 December Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article.
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Download PDF. Conclusion Advanced mathematics was highly valued in Finnish universities, and many students chose advanced mathematics believing in its usefulness for their future studies or careers.
Introduction Reasons for the gender gap in STEM science, technology, engineering, and mathematic fields have been sought in several studies e. Methods Setting of the study: education system in Finland In Finland, there are 5. The Finnish education system consists of: 1 Early childhood education and care before compulsory education begins. General upper-secondary education After the 9-year compulsory basic education, school-leavers opt for general or vocational upper-secondary education.
Data collection The original register data , including all the students admitted to Finnish universities during —, was collected from the Finnish universities by CSC, the IT Center for Science Ltd. These questions were presented in the questionnaire as follows: 1 Please continue the applicable sentences that concern your own choice of mathematics: open-ended questions I chose advanced mathematics, because….
Results The results showed that the student admission process of Finnish universities significantly appreciates advanced mathematics. Full size image. Table 2 Mathematics and gender distribution in different university degree programs — Full size table. Conclusions The current study investigated the connection between STEM subject choice, especially the choice of mathematics, conducted in upper-secondary school and their relation to university admissions.
Availability of data and materials The questionnaire data were collected mostly from minor aged 16—year-old students with consent of confidentiality and therefore cannot be shared. References Allen, C. Article Google Scholar Bandura, A. Google Scholar Bilbao-Osorio, B. Article Google Scholar Bottia, M. Article Google Scholar Britschgi, V. Google Scholar Cannady, M. Google Scholar Ceci, S. Google Scholar Chow, A. Article Google Scholar Dabney, K.
Article Google Scholar European Commission Article Google Scholar Jang, H. Article Google Scholar Linnansaari, J. Article Google Scholar Margot, K. Article Google Scholar Pajares, F. Google Scholar Palmer, T. Article Google Scholar Parker, P. Article Google Scholar Perez, T. Article Google Scholar Seyranian, V.
Article Google Scholar Slavit, D. Article Google Scholar Stage, F. It seems that in almost any industry, you can find an app for certain mathematical functions.
The reality is that someone has to develop the formulas and write the apps. Moreover, you as the end user have to be able to appreciate which app is right for which job, and why. It should be clear that in our technical society advanced mathematics is a vital part of your future. Hannah Whittenly is a freelance writer and mother of two from Sacramento, CA. She enjoys kayaking and reading books by the lake.
The farther to the left in Figure 1, the higher are unobserved costs relative to benefits and the higher earnings it takes to induce the individual to choose mathematics. The vertical axis displays the causal impact of advanced mathematics on earnings years after high school entry in percentage terms.
The dark red lines display the figures for women, while the light blue lines display the figures for men. The shaded bars highlight those who would be at the margin of choosing advanced mathematics if costs were reduced even further. For young men and women with identical abilities, the earnings gains from studying mathematics are equal. This can be seen by the overlapping red and blue lines in Figure 1, which illustrate the earnings gains caused by advanced mathematics across the latent ability distribution.
Lost mathematical talent can be retrieved by allowing for more flexible bundling of courses in high school curricula.
The fact that earnings gains are equal across genders for individuals with identical latent ability indicates that there is no gender discrimination in the labour market as to rewarding individuals with similar mathematical ability equally for their advanced mathematics qualifications.
This also indicates that the underlying mathematical ability distribution is equal, at least around the top decile of the distribution. What if costs were to be reduced even more? The shaded areas in Figure 1 show the students who would be at the margin of choosing mathematics if costs were to be reduced further.
This means that there is a considerable fraction of young women who would gain from choosing mathematics when only those of the highest ability choose mathematics. There is considerable unexploited mathematical talent among young women waiting to be realised.
Our results suggest that this mathematical talent can be retrieved by allowing for even more flexible bundling of courses in high school curricula. For example, one could allow advanced mathematics to be bundled with other advanced science courses that appeal more to young women. How and why mathematics boosts earnings. Our study also analyses how advanced mathematics qualifications affect earnings. We find that mathematics moves women to the top of the earnings distribution, while it prevents men from falling to the bottom of the earnings distribution.
Overall, the effects across the mathematical ability distribution and the earnings distribution suggest that advanced mathematics pushes high mathematical ability students to the top of the earnings distribution. At the same time, it pulls more mediocre mathematical ability students from the bottom and it does not have a significant impact on low mathematical ability students. Studying advanced mathematics provides a large earnings boost to high ability women.
Why does studying advanced mathematics provide such a large income boost to high ability women? But the fact that advanced mathematics makes women drift away from the humanities subjects where they have been traditionally focused towards highly paid education in health and technical sciences also indicates that qualifications, preferences, self-confidence or self-perception may be affected by succeeding in advanced mathematics in high school.
We even find that women become more likely to work in more competitive career tracks in the private sector and more likely to become chief executives. The high earnings gains thus reflect the fact that high ability women switch towards more advanced and more mathematics-intensive fields at university. They also embark on more competitive career tracks, and subsequently climb higher up the earnings distribution and also, to some extent, the corporate hierarchy.
How to identify, attract and foster talent. The reasons why someone chooses mathematics or not are complex, but we believe that our conclusions generalize to populations where substantially more men than women choose the most mathematics-intensive university majors and careers. This includes most West European countries and the United States. For example, the Netherlands and Sweden have high school tracking similar to the period we study in Denmark, while there is more freedom to choose and combine courses in US high schools and A-levels in the UK.
In , following the period we study, there was a major reform of the Danish high school that broke up the course bundling and gave students as good as free course choice. This implied a further increase in students choosing advanced mathematics. Fostering the most talented people to succeed in STEM careers could increase innovation and growth.
This suggests that restrictive course bundling matters for the gender gap, but it does not explain it fully.
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