Response to A Mathematician’s Lament by Paul Lockhart

Lockhart raises an important question for educators: if math is an art, should it not be taught in its authentic form—as an expression of creativity, curiosity, and authenticity? This reminds me of a discussion we had in my LLED 360 class (Teaching ELL Students), where teacher candidates from different subject areas sat together to discuss how we would develop lesson plans for our respective disciplines. I jokingly remarked that, as a math teacher, I had it easy because the textbook essentially created the lesson plan for me. In contrast, subjects like the humanities require more complex planning and effort from teachers.

Though it was meant as a lighthearted comment, was I entirely wrong? I recall my elementary school math teacher structuring lessons this way—acting more as a homework monitor while the textbook did most of the teaching. I vividly remember memorizing everything in the textbook. As Lockhart suggests, the modern math curriculum fails students by stripping away the essence of math—the inquiry that drives true learning. I strongly agree with this sentiment. If the curriculum had allowed for more creativity and critical thinking, learning math could have unlocked countless opportunities for students.

This connects with Skemp’s distinction between relational understanding and instrumental understanding. Math, when taught in isolation, often feels purposeless. Only when taught in an inquiry-based manner can students apply math skills to other aspects of their lives, thus fostering a more relational understanding.

However, I don’t fully agree with the idea of completely abandoning the current approach, which includes assessments and exercises. Young learners still need some structure and accountability to progress. While Lockhart’s vision of a more open-ended approach to math learning is inspiring, it may lack the practical framework necessary for teachers—at least in the near future.

My solution to the locker problem

 

My rendering of the locker problem is quite simple. Let us first discuss what happens to the first 500 lockers. All prime numbers (numbers that cannot be divided by any other number except for 1 and itself) would be closed because no student will change their state after the first student. Other composite numbers, unless they are the square of a certain number, will change their state an even number of times. Whereas those that are perfect squares of a number will change state an odd number of times, so they will be closed. Take the example of the 300th locker, since 300 is a composite number but not a perfect square, it changes state from open 14 times, so, in the end, it will still be open. After the 500th locker, every locker changes its state once. 

During the problem, I used visualization (through drawing), and analyzing by cases. I listed all the scenarios that are possible and tried to detect the pattern.

My favourite and least favourite math teacher

My favourite math teacher is Khan academy.  Khan academy is a nonprofit learning website. It has helped a lot of students worldwide. I found the videos useful and thorough with explanations. Two most important appeals about it is it’s free and easily accessible. I got me with my Calculus courses, especially when I was in dire need of help with a certain concept. Plus, no payment was needed.It alleviates a lot of financial pressure on students. Other than these features, I think the AP Calculus lecturer has a bed-side manner that calmed me down. I was able to follow his explanations very closely once I stopped being so tense about the difficulty of the materials. My most important takeaway from Khan academy is that learning could be as convenient as you want it to be, and I don’t have to psych myself up every time to do the practice questions. 

My least favourite is my kindergarten teacher. I believe my first math class with her is when my math phobia started. She didn’t ease me into the subject, which I think was a big mistake when teaching early math learners, one that I will try to avoid in my teaching career. Building relational understanding is an essential part in learning. I wish she had at least tried to use tools or interesting stories about basic operations before throwing questions at me. Additionally, my teacher cut my play time short to make me do additional algebra. I understand it was her job to do so, however I believe having empathy towards students is an important qualification of a good teacher. I learned from my experience with her that every student who comes to my class will have their own challenges, and being patient with their struggles is just as crucial. 

Discussion about relational vs instrumental teaching

Our group had a heated discussion and we all shared our experiences related to the topic. There are two most important questions.  

First question regarding whether the two kinds of understanding are distinct or separable. Our answer is they complete each other. Learning math is a fluid process where different parts of the brains are engaged. A student will need the logical side to do computations, whereas areas like geometry require visual imagination, and that is when the artistic part of the brain gets fired up. According to the article, learning that involves using a “mental map” could be considered as relational understanding, and that is equivalent to picturing different dimensions of an object in our example. In UBC course MATH 200, students extensively studied taking derivatives in multidimensional spaces. I found it amazing that calculus and geometry, when used together, is the perfect marriage for solving special problems. 

Our group also talked about the kinds of activities that promote each kind of understanding. Academic institutions such as Kumon, Math drills and Time drills are examples. Their goal is to help students succeed in math by practicing a lot of questions, focusing on quantity over quality. Students get exceptionally skilled at doing math by memory instead of actually understanding the concepts. On the other hand, relational understanding happens when a teacher explains a deeper connection, when the students seek out to understand a math topic independently before learning it in-depth, and many other examples.  

Response to Eisner’s Three curricula all schools teach

The first time the article stopped me was when Eisner mentions that the differentiation of classes based on students’ level of learning will create competitiveness. I found his point similar to what I have experienced in secondary school. Students in my school were segregated to take different versions of calculus classes. Those who did additional math after school tended to have better grades and got “bumped” into calculus or AP courses. Students in the regular classes were more often likely to struggle with the subject for various reasons; however, some have felt labeled as incompetent for a lack of math proficiency because they were behind their peers. My friend who felt this way once told me that it wasn’t because of the difficulty of math that held her back, but the implicit judgement coming from others. This kind of implicit favor of math competence is not helpful for creating an encouraging learning environment for students, and might have a damaging effect on their academic performance. 

The discussion of the null curriculum, specifically the question of what the school doesn't teach, also stopped me. After completing my math degree in undergraduate studies, I took courses in Diploma in Professional Accounting at UBC, because I aspired to be an accountant at the time. What got me thinking the most was the importance of financial literacy. When I was learning tax laws, I felt severely lacking in understanding policies and applying to real life situations. Yet, this is such an important skill to have as an adult. Having knowledge in tax and accounting can help one’s financial life enormously, and even open up to professional opportunities. However, the current school system hasn’t yet fully incorporated these subjects. I personally advocate implementing such topics as part of core curriculum courses in secondary schools. 

The article changed my opinion of the curriculum. I used to perceive the curriculum as an entity consisting of only courses and standardized testing. After reading, I realized how connected everything is-- the course curriculum, students’ (adolescents’) developmental stages, and society, etc. The mandated BC Provincial Curriculum has a framework that encompasses all three curricula Eisner mentions in the article. However, each school district is shaped uniquely by its cultural environment and demographic components, studying each school case by case might be a better choice for the evaluation of the three curricula.  

Response to Skemp's article on instrumental and relational ways of understanding mathematics

I resonate with Skemp’s standpoint that relational mathematics learning has longer beneficial influences on students’ learning. Specifically, he claims that when the pupils have the relational mindset but the teacher has a more instrumental teaching approach, the mis-match will cause more damage. I relate to this point because it reminds me of my past experiences as a student. 

Growing up in my home country, where math learning is heavily practice-focused, I have always believed that success in math is dependent on one’s ability to apply equations from a textbook and score a full mark for every question. However when I saw the movie The Imitation Game in grade 8, I was deeply fascinated by Alan Turing and how computation could be performed in machines. Impulsively, I brought these questions to my teacher. She gave me a puzzled look, and replied, “Why do you care? Your responsibility is to learn the material in the textbook to get good grades.” Dishearteningly, this didn’t address my original question, but it revealed to me that I should continue my inquiry regardless. Years later, I decided to pursue math in university. Perhaps part of me wanted to heal my (wounded) inner child, and continue to seek more answers. 

As educators, teachers are expected to understand that every student comes to the classroom with their very own challenges and questions. Instrumental teaching, as Skemp states, may contribute to students’ negative attitude to math learning. It is essential to create a learning scheme that sustains students’ curiosity and passion for the subject, by incorporating relational philosophy into teaching. He expands on this concept by comparing the two ways of learning music, once that is strictly written based, and the other that associates music with different sensories. I agree strongly with his stance because math learning involves not only the logical part of the brain, but one’s emotions, feelings and perception around them. Just like when he is exploring a new city, the adventures don’t consist solely of the destinations, but the conscious effort to remember how everything is connected. 

I agree with Skemp’s concern regarding the possibility of successfully implementing a relational understanding model in the current school systems. As mentioned in the article, teaching relational mathematics would require revision of lesson plans for teachers. Students, on the other hand, often find themselves juggling a heavy academic course load and extracurricular activities. Time is of the essence, and so are resources. Other challenges will also arise when the plans are put into execution. Nevertheless, after reading the article, I decided to intentionally practice the instrumental mindset in my future teaching days!

 

Introduction

Hello! This is the first post of Caris EDCP342 blog.