The Dishes Problem

The problem: 

"How many guests are there?" said the official.

"I don't know," said the cook, "but every 2 used a dish of rice, every 3 used a dish of broth, and every 4 used a dish of meat between them". There were 65 dishes in all. How many guests were there?

Classroom talk on Skemp's Article

 On September 13th's class, our group responded to the following questions:

1) What 'should' come first in learning math: instrumental or relational understanding? Why? In what contexts?

2) What is our aim in teaching math: to teach fluency in procedures? To get kids excited about the possibilities in math? To help them have a deep understanding of some (or all) topics? How can we address all the goals that we (and our society in general) have for math learning? How to make principled choices about how we teach what, and when?

3) How can we, as mathematics teachers, work to improve the negative attitudes of math anxiety or math avoidance in our communities? Are these attitudes the same in all cultures, or could Canadian society learn more positive approaches to mathematics from other cultures?

Our responses were summarized on the blackboard: 

Letters from my future students!

Letter from a student who loved my class:

Dear Ms. Mohamed, 

Hope you have been taking care! 

I know it's been 10 years, but I really wanted to check in and thank you for my experience in your math class. Every so often I'll be minding my business throughout the day, and then I think, "oh look! So much math might have gone in to that" or "hmm, I wonder if that set of ideas came from a different origin place." I'm really grateful for the emphasis you placed on shifting our perspectives of math in the real world, and how much you passionately talked about different cultures, times, and people. You used to encourage us to share our math experience and our identity, and had so many thoughtful activities that facilitated that growth. I felt my confidence grow in your class, because I got to appreciate the subject I once hated and felt the satisfaction of all the mathematicians of the past cheering me on as I learned and mastered their topic of studies! 

Thoughts on Lockhart's Lament

I think that Paul Lockhart's, "A Mathematician's Lament", hit the nail on what mathematics education looks like from K-12. The examples of the musician and painter felt eerily familiar and I completely agree with him that students are generally taught to analyze the qualities and properties of concepts, without getting a chance to experience it in its full glory. The Fibonacci sequence, for example, is often blown over or briefly introduced as a series of numbers, but rarely is there a dedicated lesson highlighting the presence and applications for it in real life. The same thing can be said about prime numbers, where students understand that primes are defined as having factors of (1, P) - but they may not understand why that is important or what primes can do for us! Students learn about calculating the area of a standard parallelogram, but have they ever seen the application rhombs have to tilings? There is a significant disconnect between the mathematics that is taught in schools and the mathematics that many mathematicians interact with. This distance seems to deter students from the beauty of this study. 

My favourite and least favourite math teachers!

In terms of formal mathematics education, due to being a math major, I've had 20 MATH instructors during university, and another 8 during secondary school. I've also received assistance from my parents, some family friends, TAs and formal supports. So many have had a really positive influence on my journey, shaping the inner math teacher I hope to one day bring to the world! Others have, unfortunately, been examples of what not to do. 

Here were some of my favourites: 

The Locker Problem (1000 lockers edition)

Before attempting the problem, I thought the solution would have been a messy array of numbers that needed to be tracked via extensive diagrams. However, I began the problem by setting up 10 lockers, perhaps due to our connection to the base 10 system. Very quickly, it seemed that base 60 would be great for a more obvious illustration of the pattern since it has significantly more factors than 10. That level of effort may be commended, but would end up redundant. 

Reflection: “Relational Understanding and Instrumental Understanding” by Richard R. Skemp

Over the years I have had many opportunities to assist students struggling with mathematical concepts. Before and after the shift in curriculum for math assessments (2016-2018), I have found that teachers that communicate and assess mathematics according to the instrumental understanding are generally considered “better,” and teachers that opt for structuring their classroom around the relational understanding are deemed as more difficult. I agree with Skemp that this is a partial consequence of “over-burdened syllabi”, “backwash effect of examinations”, and “difficulty of assessment” (page 11).

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