A Mathematician's Lament, By Paul Lockhart
A few years back one of my colleagues introduced me to A Mathematician's Lament, by Paul Lockhart.
I was so impressed with his comments, and they way he communicated them, that I have made it a reading in MAT194 ever since. One of our goals in this course is to come to terms with our own histories and our own interests in mathematics. And so I invite you to read "The Lament", and find out for yourself if there are lessons for you in Lockhart's work.
Many of us have experience with formal education. Lockhart has some very strong opinions about the state of mathematics education. Whether you agree with him or not, you will hopefully enjoy the amusing way he makes his points! Please consider the following, as you read his article.
- Respond to this comment: "There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it."
- Lockhart undertakes an attack on high school geometry: what do you think of this? "High School Geometry: Instrument of the Devil": Never was a wolf in sheep's clothing as insidious, nor a false friend as treacherous, as High School Geometry.... All metaphor aside, geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. (Apparently I am incapable of putting all metaphor aside.)
- Please share your experiences or your thoughts on the following section of the text:
- SIMPLICIO: But then how can schools guarantee that their students will all have the same basic knowledge? How will we accurately measure their relative worth?
- SALVIATI: They can't, and we won't. Just like in real life. Ultimately you have to face the fact that people are all different, and that's just fine. In any case, there's no urgency. So a person graduates from high school not knowing the half-angle formulas (as if they do now!) So what? At least that person would come away with some sort of an idea of what the subject is really about, and would get to see something beautiful.
1. Respond to this comment: "There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it."
It seems to be a prominent theme across the board that students completely dislike when things are mandatory. Most explained that as soon as they are forced to do it, they don’t want to. It is a wonder if this is because of our age and still growing in maturity? Or if this is just part of the human condition? Whatever the reason is, mandatory learning of subjects in the school curriculum does in fact kill most motivation and interest in students. It has to be hard for the teachers too. Amber Manning made a great point when she said, “In the past, some teachers have been so worried about getting their students to pass these standardized tests that they don't have time to make the students more involved and do engaging activities that they will enjoy because they have so much to cover that they have to speed through these topics.” Jenna Henderson made the same observation from her mother who was a math teacher. Jenna shares, “My mother who is a math teacher always complains about having to follow the lessons plans and curriculum given provided by the school board as it limits how she can teach to the students more effectively and creatively.”
So, if the teachers are having a hard time competing with curriculum and how they want to teach, imagine how hard it is for students who are forced to learn this curriculum. Which if we get really honest, probably 7/10 students don’t want to even be at school anyways. So, what about the other 3 out of 10? A seemingly preferred alternative among the students in the class was to be able to choose which courses to take and when. This brings up a question though, would some avoid taking math, or another subject of disinterest to them entirely? But therein lies another quandary, would students have complete disinterest in these subjects if they were presented in a completely different way from the beginning? Perhaps without the curriculum and standardized tests. Looks like we will never know because we have been doomed from the beginning. An even more daunting fact is that even though most students disagree with the current structure of the education system and curriculum, we have no clue how it would even begin to look different or what could be changed as a solution. It seems as though it [standardized testing and curriculum] may be a necessary evil. But for whom?
2. Lockhart undertakes an attack on high school geometry: what do you think of this?
Lockhart mentions that geometry class is “by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum.”
It seems as though a majority of people have the same feeling towards geometry but for multiple reasons. Colton Williams summed up one of these reasons when he said, “Geometry while I was in the class wasn't all that hard. But the class was all about memorizing formulas to solve for each shape more than anything else,” while on the other hand, people felt as though that geometry was a much different breed than other math classes, or that the class itself simply wasn’t enjoyable because of the way the class was taught. While students like Rachel and John go so far as to not even remember much of the class. Overall, the general point of this group is that they agree that the class was just not in a form which was deemed stimulating or enjoyable.
Then there is the other side of the coin in which there are those that enjoyed it or at the very least there are those who didn’t think it was as bad as Lockhart made it out to be. Craig McGhee mentioned that it helped him in other classes in terms of visuals and also noted that if other classes were taught more visually like geometry was then maybe it wouldn’t have been as bad of an experience. Regardless, others still mentioned that a lot of geometry was about memorization of rules and formulas.
In the end, it feels as if the general consensus is that high school geometry was more or less about memorization and the general opinion/enjoyment was left to each individual's preferences.
3. Please share your experiences or your thoughts on the following section of the text:
SIMPLICIO: But then how can schools guarantee that their students will all have the same basic knowledge? How will we accurately measure their relative worth?
SALVIATI: They can't, and we won't. Just like in real life. Ultimately you have to face the fact that people are all different, and that's just fine. In any case, there's no urgency. So a person graduates from high school not knowing the half-angle formulas (as if they do now!) So what? At least that person would come away with some sort of an idea of what the subject is really about, and would get to see something beautiful.
Everyone had different opinions on the two comments from the text. Some understood Simplicio's concerns. Some agreed with Salviati, some disagreed with Salviati. Students that disagreed with Salviati think not everyone can see the beauty in math since everyone's built differently. For people who are not interested in the subject, learning it would be a waste of time. Majority of students agreed with Salviati's opinion that we cannot guarantee students will all have the same basic knowledge. Some students also shared their experiences with math classes.
Learning math is not about learning how to find the derivative of f(x). It's about how to think logically and cortically. "Beauty is everywhere. It is not that she is lacking to our eyes, but our eyes which fail to perceive her." Learning the beauty of a subject is not about whether the subject is beautiful or not. It's about developing a pair of eyes which are good at seeking beauty.
4. What is a favorite part of the Lament for you? What resonated with you?
It seems that most of the class enjoyed the part where the reading mentioned that math is art.
Below are some of the responses that stood out:
Liam Painter, Shawn Huesman, Octavia Dieng, John Nuestro, Amber Manning, Nevaeh Moore, and Austin Paolucci: All of these students said that the part that resonated with them was the idea of math being art as mentioned above. Although all of these students enjoyed the same part of the reading, the students had various approaches to the reading. One example that the student gave was about learning a new language. Other students also connected this topic to real life.
Justin Horn: This student enjoyed activity-based learning that was talked about in the reading.
Ganga Adhikari: This student enjoyed the course description of the mathematical concepts ranging from lower-level mathematics to calculus. This student also heavily criticized the education system as it tests students on what they can memorize instead of using project-based learning.
Jenna Henderson: This student said that they enjoyed the part about math being "an act of discovery and conjecture, intuition and inspiration".
Students had various ways of approaching the topic and enjoyed different topics. Lots of students were engaged in this question and agreeing and commenting on what other students said.