Teaching fresh(wo)men calculus

I have just finished, for the second time, the calculus fall term for engineering freshwomen (and freshmen) ;) at UC3M. The classes were in English, split into two for practical sessions, around 70 students in total. It was a nice group (yes, some of my students will read this and no, I do not say this because of that… the teacher evaluation polls are already over) ;) I have been thinking about what do we teach, what is its purpose and how we should do it… and I have reached a few conclusions.

  • There are two reasons to teach maths to non-mathematicians: (a) because they will need some tools which are standard in their trade or (b) because they should learn to think, they should learn real problem solving techniques. The contents of the calculus term (derivatives and integration in one variable, basically) is already covered in high school, only a few new things are taught here (Taylor, polar coords…) So, the real reason must be the second one.
  • That’s why I have introduced two novelties: first of all, problems in “real life format”. With this I mean that they’re formulated vaguely, with no data. For example: “I want to leave my can of beer on the ground, but it is irregular and I’m afraid it might fall down. I should drink a little bit of beer so that it becomes more stable. How much?”
  • Another point that was important for me was the ability to give numerical solutions, approximations… I mean: to obtain numbers even when an analytical solution is not available. We also introduced numerical calculation techniques via octave, but it had to be out of the class hours.

My only complaint: such a course, if it has to be taught correctly, requires a rather low number of students per class. When I teach linear algebra, it’s ok for me to talk about eigenvalues to 120 students. That’s because the idea is fully different. I don’t know how to teach problem solving from the blackboard. There are always a few lucky cases in which you have to teach nothing: they already get your point, almost before you’ve finished stating it. With the rest, we normal mortals, it has to be done one by one…

Another important point. I would like to change the “blocks”. There should be four of them

  1. Visualization: sketching functions, curves in polar coordinates or parametric, surfaces… And the reverse: see data and “guess” an analytical expression. Fitting experimental data.
  2. Computing: approximation schemes, estimation skills. Tayor, mean value theorem… And numerical programming skills.
  3. Optimization: all sorts of problems where some target function has to be maximized or minimized. There are few “real life” problems which can not be re-cast in this form…
  4. Cutting into pieces and pasting back: (for want of a better name) with this I mean all kinds of problems which “reduce” to integration: areas, volumes, lengths, work of a force, average of a function, etc.

Calculus at this level can be seen by the students as a bunch of tricks. And they’re right. All of us making a life as “applied mathematicians”, we have a bag of ideas that come to our mind when we see a new problem. Applied mathematics is just that: the ability to tackle a new problem, to make the “right metaphor” with another problem that you solved years back.

Just a finishing remark: why do we have so few girls??????? I want a convincing answer, or I’ll move to nursery school next year! And I’m serious about that!

Is this the reason?


10 thoughts on “Teaching fresh(wo)men calculus

  1. Hello javirl
    If you mean the question to why there are so few girls in engineering, is something I also ask myself, it seems that generally prefer biological branches of knowledge to get to engineering.
    And within a branch of engineering there is more to “telecommunications” than in industrials and no longer speak of software engineering, to give an example of careers in the UC3M
    Although our class, together 3 different degrees, we have enough and we can not complain and we are fortunate that they’re nice :)

  2. hahahahahaa Ddm what it stands for? DElegado de Mecan… what else? hahahahaha but he’s right…

    let’s also say that we can go anytime to Periodism to get some alcohol and girl and they can’t do the opposite!

    Javi the course has been fine, altough i would have enjoyed more beer problems(and also make them real, I offer myself to drink the neccessary beer of the problem so that it doesnt fall xD)

  3. Yes, Delegado de mecanica, only that, subdelegado xDD
    for me the course was very good and I liked the books get out and apply everything we learn into real life, besides you could experiment with a computer the numerical computation

  4. @Ddm: yes you’re right… quality is important, and the girls in science & engineering, even if scarce, they’re quite nice… ;) And I’m happy u enjoyed the course! :D

    @Ycs: don’t be so sure, girls studying journalism may always come over to get alcohol and boys… ;) About the beer problems, I’ll write a full book of them and dedicate it to u XD

  5. unless i dont have to solve them I would be really glad!! xD

    by the way…do the girls of periodism como to leganes?Havent seen em yet…may it be we only have eyes for studies(we’ve began doing jokes with scientific content) or you are hoarding all of them and we cannot find em by the campus?xDD

  6. Now that we’ve finished the course and that you’ve put our marks… XD,I would like to tell you one thing. I’ve been talking with the rest of my classmates and we all seem to think in a similar way when talking about series and sequences. We still don’t know how to solve them despite having done our final exam last Friday.I don’t know if the reason is that they are extremelly difficult, which i don’t think, or it’s that we don’t really understand what they mean. When you give us our final exams I’ll go to your office to see if you can explain me a bit about series and sequences;They can be very important in our future and i’d like to know all about them so i don’t have any problem when taking further steps in this world of calculus.
    I also have to thank you for all you’ve taught us. You made us learn by making us think and realise why things happen.
    You are a very good teacher.I want this words to make you go on in teaching, because with teachers like you, learning is easy.
    I’ll never forget you.
    Thank you Javi :)

  7. Thanks for the compliments… This year was specially good, things have worked out quite well, I’m very proud of the group…

    About sequences and series, I take my responsibility. I don’t think the topic is specially hard either, so I must have done something wrong about it… Of course, you’re welcome to come to discuss it whenever you want :)

  8. Ok, I don’t know if you see this a good idea, but as some professors have a acknowledgements section in their websites, I will start it for you by copying here the mail I sent to you.
    PD: Feel free to delete this post if you want…

    Sunday, February 7, 2010
    Dear Javier:
    Desde la primera semana de clase sabía que te tenía que escribir esta carta.
    Evidentemente, no la he podido enviar hasta ahora que ya has puesto las notas…

    Lo que tengo que decirte puede resumirse en esto: Estoy más que contento, orgulloso, de haberte tenido de profesor.

    Antes de entrar a la universidad, esperaba encontrarme con clases de 100 alumnos y profesores vestidos de camisa y corbata, que entraban a clase, llenaban cuatro pizarras de fórmulas, y salían de clase sin apenas despedirse. Personas con las que difícilmente ibas a poder hablar cara a cara.

    Cómo iba a imaginar yo que me iba a encontrar con un profesor que da clases
    haciendo “Magic tricks”, que recorre uno de los temarios más temidos por los
    estudiantes sin que apenas nos hayamos dado cuenta, que enseña cálculo hablando de física, socorristas y latas de cerveza, que pone examenes con viñetas de XKCD, que enseña, investiga y redacta usando software libre, que da conferencias sobre cosmología utilizando fotogramas de The big bang theory, que nos anima a que intentemos resolver matemáticamente cualquier problema que se nos pase por la cabeza (the evil goat, the CD cases stack, el potencial de dos átomos respecto a su distancia…), que participa en torneos de robótica (eso sí que no me lo esperaba), y que además, sorprendentemente, consigue que alcancemos un nivel de conocimientos bastante aceptable.

    ¿Y ésto a qué se puede deber? A la visión que tienes de la enseñanza. Tu post
    “Teaching fresh(wo)men calculus” lo dice todo. En vez de centrarte en aburridos teoremas matemáticos, nos has enseñado con ejemplos, con ejercicios. Te has centrado en los aspectos más útiles de las matemáticas, y es lo que has evaluado. Y eso es algo que pocos profesores son capaces de hacer. Has conseguido que cada problema que tuviésemos que resolver se nos hiciese un reto, un juego, y clase tras clase, me has ido enseñando sin yo darme cuenta, unas valiosas herramientas, pero sobre todo, una nueva filosofía de aprendizaje.

    Bueno, ahora sólo me queda desearte suerte en tu viaje por USA, y esperar verte por aquí algún que otro día, dando otra conferencia sobre astronomía (por cierto: te dejo unas imágenes de mi etapa de astrónomo aficionado: http://tinyurl.com/astrpctrs Mientras tanto, nos vemos en Physic Napkins !!!

    Yours Faithfully:

  9. How would I delete it? :) I feel so proud!! Really, the term went really smooth. But it was not just me, the class responded in a very nice way… I am very happy and proud of many of you, and I hope our paths will meet again in the future, maybe to undertake together some scientific project! :) In the meantime, we have the web, and you may drop by my office to discuss whatever you want…

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