EDCP 342 course reflection

EDCP 342 was an amazing course, and I had so much fun learning about mathematics curriculum and pedagogy. I learned about school institutions and the evolution of mathematics through the ages beginning with the progressivist movement led by John Dewey, to New Math reform of the 1960s and to current math practices beginning the 1990s.

One of my favourite readings was Eisner’s Three Curricula that all schools teach as it discusses the implicit curriculum that students learn in school and is not explicitly listed in the official teaching curriculum. These remind me of the “life lessons” or “life skills” that usually come up in my elective courses in home economics and automotive, but also appear in academics unintentionally.

Being exposed to the different aspects of the BC curriculum such as core competencies and big ideas are very helpful from Susan’s guidance and microteaching activities as it got us thinking about ways to incorporate different elements of these goals into teaching and developing lesson plans. This has changed my ideas that every unit has to include all aspects of big ideas/core competencies, when in fact different units may be easier to promote different big ideas/core competencies. For example, in my unit plan assignment on data analysis for Math 9, I found it to be a great unit for developing mathematical communication and reflection rather than specifically focusing on mathematical computation.

I also enjoyed the discussion on Csikszentmihalyi’s idea of flow. It got me thinking about ways to develop lessons that promote focus and concentration while also submersing students into a sense of timelessness.

A suggestion I have for improving the course next year, would be to consider reducing the number of assignments so that there can be more time for reflection and stronger development of ideas from multiple draft review opportunities. Towards the end of the course after the short practicum, there have been several instances where I felt rushed to complete assignments such as this unit plan. Nonetheless, this was an amazing course and I am very thankful to have such a caring and knowledgeable instructor like Susan to teach this course.

 

Thank you, Susan.

Completed Assignment 3

See the link to my google drive for my completed unit plans.

https://drive.google.com/drive/folders/103f3B4C9PUc2OgapEh_gF6G93jzfBbXz?usp=sharing

Math Party

My friend and colleague, Curtis Chou, has a great thought mathematical thought experiment regarding the Mobius Strip that I would like to share with everyone.

Curtis Chou Returns to Science Slam Vancouver - YouTube

I also just realized that in a non-covid year, December is typically a time where PE classes do square dancing and I just wanted to share a few resources of embodied math and square dancing. 

https://www.maa.org/news/math-news/square-dancing-takes-a-mathematical-spin

https://eric.ed.gov/?id=EJ972388

Assignment 3 Lesson Plans Draft

 

See the link to my google drive for my draft lesson plans.

https://drive.google.com/drive/folders/103f3B4C9PUc2OgapEh_gF6G93jzfBbXz?usp=sharing

Dave Hewitt on Arbitrary vs. Necessary

 

Hewitt refers to the ‘arbitrary’ if it is something that students need to be informed by someone else and is concerned with names and conventions that students have to memorize (1999). It is the role of the teacher to assist students with retrieving this information through memory. The ‘necessary’ is something students can become aware of without being informed by someone else (1999). The teacher should provide tasks to help students educate their awareness to the arbitrary.

Thinking about this distinction between arbitrary and necessary will help me plan my lessons and develop as a teacher with regards to how I teach. Hewitt refers to ‘received wisdom’ as the fact given to students by the teacher and perceived by the student to be true as a result (1999). For example, when learning about the order of operations, students are told about BEDMAS OR PEDMAS by the teacher, which is an acronym reminding students the order they should perform arithmetic operations. If a student does not memorize this acronym, then they will incorrectly solve order of operation algebra problems. Even myself, I do not know why this order is the standard, but it is the agreed upon convention used everywhere. One of the points Hewitt makes that stands out to me is to give students time to use their own intelligence to become aware of why certain procedures must give certain answers instead of constantly giving them teacher’s wisdom. For example, when determining the area of a rectangle, students first must be told the definition of the area, and then be given the opportunity to “discover” how they can determine the area before being told the equation for finding the area. This allows for self-discovery before immediately being told the equation and eliminates the big question of “why is this the equation the area of a rectangle?”

 

Reading:

Hewitt, D. (1999). Arbitrary and Necessary Part 1: A way of viewing the mathematics curriculum. For the Learning of Mathematics, 19:3, 2-9.

Teaching Perspective Self-Test Results

This is the result I got from my TPI graph. It appears my nurturing perspective is a dominant feature, while apprenticeship is just above the mean and social reform just below the mean. Transmission and developmental are both almost recessive traits according to this test.

I am not surprised that nurturing is a beyond exceeding trait for myself as I often find myself being a very caring person and focused on social and emotional wellbeing of others as well. I was surprised at how apprenticeship was a little above the mean. Apprenticeship refers to teachers being able to know what learners are capable of and where they may need guidance. As teachers mature and become more competent, the teacher role changes and may offer less direction and give more responsibility to students (Pratt & Collins, 2002, p.3). When educating students, I like to let them ‘struggle’ a little bit when solving problems rather than giving them the answer immediately. The development was also not a surprise. Pratt & Collins (2002) refer to developmental as effective teaching planned and conducted “from the learner’s point of view” (p. 4). During my practicum, I noticed that I was sometimes explaining concepts a little more advanced than my students could understand; likely due to my additional knowledge coming from post-secondary math experience. It is important that as an educator, I teach concepts that could be understood by my students at a level to their understanding.

Given this information provided from the TPI test, I am curious to see how my results change after a few years of teaching and whether or not I will still score the same results. I hope to still score high on nurturing aspect, as I believe that one of the first avenues towards making math fun, is for the students to see that through a caring and passionate individual.

 

Source:

Pratt, D., Collins, J. (2002). Summary of Five Perspectives on ‘Good Teaching.’ TPI – Teaching Perspectives Inventory.

Thinking about math textbooks

As a student, I personally enjoy having textbooks as alternate sources of information to supplement teacher instruction. I also find that I greatly benefit from questions in textbooks to practice my understanding and refer to sample worked out questions if I need clarification. Comparing my answer with the textbook answer helped solidify my understanding of the material. From the teacher-bird view during my two-week practicum experience, all of the teacher I met referred to the textbook for practice questions, but did not know what questions were off the top of the head; when students asked, “I don’t understand question 9” or “Can you help me with question 18” teachers needed students to read out the question or refer to their teacher solution handbook before attempting to help. This changed my perception of how teachers use textbooks. When first developing lessons, they refer to textbook as sources of information and find relevant questions to assign students; then over the years, they may not refer back to the textbook unless they need clarification or a quick review.

I believe that textbooks are extremely valuable resources, however, I do not agree with having students to spend hundreds of dollars for new texts. An example of this is with booklists in post-secondary courses; students are required to purchase specific texts as described by the instructor. The Math department at UBC is doing a fantastic job towards making textbooks electronic and free to access. This would be an excellent idea for secondary schools to try and implement; to make education more accessible, especially if the school is in an area where many students have financial hardships.

In Herbel-Eisenmann & Wagner (2007), the discussion on how the textbook authors use pronouns to address the reader is something I resonate with. I enjoy reading a book where I feel a more personalized connection with the author. Hence, when a math textbook uses first person pronouns such as “I” and “we”, I feel more drawn in and connected to the learning experience. When an author uses third person or phrases like, “one can see that…” I feel more disconnected and almost feel talked down upon. Therefore, although a textbook can be helpful to students to learn by providing supplementary content and practice problems, it may not effectively achieve this if the language within is not welcoming.

 

Reading: Herbel-Eisenmann, B., Wagner, D. (2007). A Framework for Uncovering the Way a Textbook May Position the Mathematics Learner, For the Learning of Mathematics, 27:2, 8-14.

Flow

Csikszentmihalyi mentions a study he conducted in 1956, where he found that 30% of US Americans were unhappy and increases in material well-being does not seem to affect how happy people are. This reminds me of the saying, “money can’t buy you happiness.” A deeper thought to contributions of one’s happiness more than materialistic things are emotional or spiritual; feeling welcomed and loved or feeling at home as a few examples.

Csikszentmihalyi describes flow as the effortless, spontaneous feeling you get when entering an ecstatic state (2004). He lists seven conditions for being in flow: 1) completely involved in what you are doing; 2) a sense of ecstasy; 3) great inner clarity; 4) knowing that the activity is doable; 5) a sense of serenity; 6) timelessness; 7) intrinsic motivation. One thing that makes me experience flow is when I skate on the ice playing hockey. Skating around feels so natural and calm; I become mesmerized by the puck and the sounds of skate blades cutting into the ice. All my worries seem to disappear for the time being as if I entered a different reality. Another moment I experience flow was when working at Science World and demonstrating science experiments on stage. I become so focused in the moment interacting with the audience and teaching science, I often lose track of time. I like to think of myself as being completely in “the zone.”

Csikszentmihalyi presents this diagram during his TEDtalk, where flow is in the area with the optimal mix of challenge and skill. Other areas like arousal, control, and relaxation could also be desirable. As a mathematics teacher, or any kind of educator, anxiety, boredom, worry, and apathy should be avoided. To promote this zone of flow in the classroom, I think teachers need to teach at a level that is not considered too difficult for students but challenging enough so they can use their previous knowledge to assist with learning. Some students might feel that music will help get into flow, so educators should consider allowing them to listen to music while working and “peeking” to make sure they are on task. Opportunities to discuss and share ideas students are interested in can also promote flow in the classroom as it is easy to have conversations about something one is passionate about. Math puzzles or relevant realistic problems could be a great activity to help students feel immersed in problem solving as it challenges their thinking outside of the box and forces them to ask questions.

 

Video:

Csikszentmihalyi, M. (2004). Flow, the secret to happiness. TED. Retrieved from https://www.ted.com/talks/mihaly_csikszentmihalyi_flow_the_secret_to_happiness?language=en

The new BC curriculum and secondary math course pathways structure

One thing that surprised me was how the BC curriculum was redesigned to provide more flexibility to educators to structure their lessons and teachings to reflect personalization of student learning. This is done through the creation of core competencies and big ideas educators should address when developing lessons; things students are expected to do and expected to understand. Core competencies are sets of intellectual, personal, and social competencies students develop to engage in deeper learning and support lifelong learning (BC curriculum, 2020). Big ideas do not require students to know specific pieces of factual information or isolated bits of information but are broad and abstract statements central to one’s understanding in an area of learning (BC curriculum, 2020). The second thing that surprised me was the inclusion of the definition for 21^st century skills, which was defined as, “a term used to describe the combination of specific skills, content knowledge, expertise, and literacies that are essential for today’s graduates” (BC curriculum glossary, 2020). I really liked this definition. I think it is crucial that secondary students be taught skills that are relevant to them and take advantage of how easily information can be acquired in this day and age.

This is my flowchart for the pathways structure of BC curriculum math courses. The one-way arrows shows where you could go to at the course given. Notice that courses like computer science 11 and history of mathematics 11 have many arrows going into it. I think that those courses could be taken at many stages of a secondary student's math career. The two-way arrows means you could take the course at the same time as another course. There might be a few arrows I missed for concurrent courses as it was hard to keep track of everything. 

Figure 1: My interpretation of BC secondary math course pathways

Readings:

BC Curriculum orientation guide: https://curriculum.gov.bc.ca/sites/curriculum.gov.bc.ca/files/Curriculum_Brochure.pdf

BC Curriculum glossary: https://curriculum.gov.bc.ca/sites/curriculum.gov.bc.ca/files/pdf/glossary.pdf

BC mathematics curriculum (K-12): https://curriculum.gov.bc.ca/curriculum/mathematics/core/goals-and-rationale

Geometric/numerical puzzle

Question: Thirty equally spaced points on the circumference of a circle are labelled in order with the numbers 1 to 30. Which number is diametrically opposite to 7?

To solve this geometric/numerical puzzle, I drew a circle and made diametric lines across from numbers 1 to 30. Using my image, I find that the number diametrically opposite to 7 was 22. After drawing my image, I noticed that there are 15 ‘spaces’ between the starting number and the number diametric from it. 

Figure 1: My drawing 

A puzzle related to this can be to rearrange the numbers so that they are not organized in ascending order around the circle and ask students to find numbers diametric and next to a number, etc. If this were a word problem, it would be lengthy just to set up the problem alone. An extension of this is to use numbers 1-24 to correspond with letters of the alphabet and students can use it to create coded messages.

I feel like all problems can be logic problems, but a problem is considered “geometric” when there are aspects that allow for visualization of the problem geometrically. When I read this problem, I immediately thought of it as a geometric problem because I began visualizing a circle with the numbers arranged around it. I drew diametric lines to help me think of the solution and did not make the logical connection between half of 30 and properties of a circle.

I think there is more value to give students problems with no single correct answer (in an academia, these are considered wicked problems). This encourages discussion and conversation amongst students to share their reasoning and logic when arriving at their answer. 

Updated Microteaching Topic: Introduction to the Guitar

Some feedback I received from my guitar microteaching lesson was to shorten some content so that I could ensure that my teaching pace was slowed down to fit into the 10 minute window while also emphasizing vocabulary and reviewing the different parts of the guitar. If this were an in-person lesson, one way to address the issue of not everyone having a guitar would be to pass my guitar around the class so that students can hold it and make connections between my presentation and embodied learning.

For my updated microteaching guitar lesson plan, I have chosen to omit the section on teaching the C major scale. I had also intended to play a short song at the beginning and end of the lesson, however, I did not do this during the actual lesson because of the short time constraint. 

Following along on the power point , I will be omitting slide 4. 

Subject: Introduction to the Guitar	Grade:	Date: Oct 14/20	Duration: 10 mins
Lesson Overview (What this lesson is about)	This lesson will provide students an introduction to the guitar.
Class Profile   Construct a hypothetical class profile, in which you specify how many students are in your class, their learning challenges, and their levels of English proficiency	EDCP 342A: Curriculum and Instruction in Secondary Mathematics (~27 post-secondary students)
Content Objectives (What the students will know)	By the end of the lesson, students should be able to: ·        Identify different parts of the guitar. ·        Define pitch and correctly label note associated with each (open) string. ·        Read chord charts and tabs.
Language Objectives   (What new language the students will learn)	·        Pitch: how high or low a musical sound is. ·        Tab: Short for tablature, which is a form of writing down music for the guitar, which mainly uses numbers instead of music notation.   ·        Chord: Set of harmonic pitches/frequencies consisting of multiple notes heard simultaneously.

Materials and Equipment Needed for this Lesson

Guitar

	Lesson Stages	Learning Activities	Time Allotted
1.	Warm-up   Get students’ attention, connect to previous knowledge and explain why the topic is important to learn.	·        Ask if anyone has picked up and tried to play a guitar before.	1 minute
2.	Presentation   Teach the new content and language.	·        Show students the different parts of the guitar.         ·        Name the pitch associated with each string.         ·        Reading tabs.         ·        Reading chords.	5 minutes
3.	Practice and Production       Practice, reinforcement, and extension of the new content and language.	·        Practice reading tabs.         ·        Practice reading chords.	2 minutes
4.	Closure	·        Summary of vocabulary from Language Objectives.         ·        Open the floor to questions.	2 minutes

Three curricula all schools teach

I think it is interesting that students spend 480 weeks or 12,000 hours in school by the time a child graduates from secondary education. If they continue to post-secondary immediately after graduation, they will typically add on another 110-140 weeks or approximately 2600-3000 hours in school. That is a long time to spend in an academic institutional setting consecutively.

The big point from this reading was the implicit curriculum that schools teach, one of them being to set up students for the real-world outside of secondary education. Eisner mentions how “children will not have jobs in adult life that is interesting; most jobs do not provide for high degrees of intellectual flexibility; most jobs depend on routine” (n.d., p.90), which implies that secondary education is purposely designed to set up students for that lifestyle afterwards. Secondary education prepares students for one-way communication and hierarchical organizational structures in the workforce. It is also mentioned that the current design of the curriculum fosters competitiveness through grading of students. Something I have noticed while in schooling is that students are very competitive amongst each other when it comes to grading, with some students even going as far as “sabotaging” classmates by not helping or lying when asked questions about assignments. Eisner mentions that grading “fosters differentiation of classes into ability groups” (n.d., p.91). This is the idea that students who achieve higher grades get placed into honours classes and programs. Some honours programs offer different credit track systems within the school curriculum speeding up a student’s progress. This now becomes unfair as it does not give students an opportunity to be part of this program. In an example from my high school experience, I have seen some students who excelled in mathematics opt out of the honours mathematics program and take “normal” math to get a better grade as the content is not as challenging. Implicit curriculum also teaches students how to be cognitively flexible, ie. be able to shift problems and adapt to new demands on schedule; teach intellectual and social virtues such as punctuality and willingness to work hard on tasks; and defer immediate gratification in order to work for distant goals. Interestingly, I have noticed these implicit curricular teachings through my elective classes in high school through mechanics, automotives, and even home economics. The teachers I had for these courses talked to us like adults and taught life-lessons in the classroom and made me realize that my courses were training us to be able to manage busy homework and study schedules and conform to authority.

If the [explicit] curriculum of secondary education is to teach students to have English literacy, numeracy, and academic knowledge to prepare them for post-secondary education, then the mandated BC Provincial Curriculum achieves this. The mandated BC Provincial Curriculum also includes some aspects of the implicit curriculum that Eisner discusses in this paper. The BC curriculum tries to connect academic learning to life skills for students to succeed if they choose not to further their education after secondary. The BC curriculum also includes some of Eisner’s idea of the “null curriculum.” Eisner defines the “null curriculum” as information schools do not teach. For example, most students complain about not knowing how to do their taxes however, I think this is taught in a BC mandatory graduation class; it could use further emphasis so students can retain this information longer than from a single lesson on taxes and finances. There are some skills schools may not offer completely as they do not have the resources to teach such as an automatives program. BC curriculum should work with school districts to try and accommodate this by bringing in guest presentations of people who work in those fields to educate students and provide opportunities to learn more through apprenticeship or volunteer programs.

Reading:

Eisner, E.W. n.d, The Three Curricula That All Schools Teach. The Educational Imagination On the Design and Evaluation of School Programs. 3rd Edition. Stanford University. Pp. 87-107

Battleground Schools

I really liked how this article was like a brief history of mathematics education throughout the 20^th century. Starting with the Progressivist Movement led by John Dewey from 1910-1940, New Math reform movement of the 1960s, and present-day Math Wars. Dewey advocated for a shift in math education where students should form and test hypotheses and perceive patterns and relationships through experimentation, inquiry and interpretation. Dewey is a very influential person and strong advocate for educational reform. Dewey challenged the system of impractical solutions derived from complicated steps to solving math problems. Another moment that made me stop during reading was through the introduction of the Bourbaki math group. They were a secret-invite only group of French-mathematicians publishing work under the pseudonym Nicholas Bourbaki. They published mathematical textbooks and introduced new math notations. I thought it was funny that there existed a secret group of mathematicians in which entry was invite only; like a superhero organization. The last section that made me stop was on the discussion of New Math. New Math was introduced following the Soviet Union’s successful launch of Sputnik in the great space race. The goal of New Math was to introduce abstract concepts earlier from K-12 so that students can develop into rocket scientists and engineers to help US win the space race. Teachers and parents who learned math previously did not understand New Math and was not able to teach it effectively. New Math was highly conservative and reflects traditional university style teaching such as lecturing, presentations, favouring deductive methods and viewed math as infallible. This reminded me of the scene from the movie Incredibles 2, where Bob was trying to teach Dash “new math” and was getting frustrated questioning, “why would they change math? Math is math.”

 

Paper:

Gerofsky, S. 2008. Mathematics Education. Battleground Schools Volume 2 (L-Z). Greenwood Press. Westport, Connecticut, London. Pp. 391-400

Movie Scenes. 2018. Incredibles 2 – ‘Math is Math’ Scene. Retrieved September 29, 2020 from https://www.youtube.com/watch?v=3QtRK7Y2pPU.

Goldberg Polyhedrons

In geometry, a polyhedron is a 3D shape with polygon faces, straight edges, and sharp corners or vertices. It derives from the Greek word “many-base.” Polyhedrons are convex, meaning that their faces are curved outward. A Goldberg polyhedron is a convex polyhedron made of pentagon or hexagon faces, ie. five or six sided polygons and exactly three faces meet at each vertex.

 Goldberg Polyhedrons have notation of the form GP(x,y), where x is the number of steps in one direction, and y is the number of steps taken after turned 60^o. We can use x, y to calculate the triangulation number, T, where T = x² + xy + y² = (x+y)² – xy. Multiplying the triangulation number and the number of original faces the polyhedron has will determine the number of triangles the new polyhedron will have. This process is a type of subdivision known as principal polyhedral triangle (PPT), which breaks down the structure of a polyhedron. By subdividing the Goldberg polyhedron and slicing the spherical shape in half, we create a geodesic dome. The triangular shapes are structurally rigid and create a sturdy and durable structure. Any pressure applied to one triangle is evenly distributed to other triangles.  

Figure 1:  A GP(5,5) Goldberg Polyhedron (left).

Figure 2: A GP(5,5) subdivided by triangulation number (left). 

For this project, I tried to recreate creating several different kinds of Goldberg polyhedrons by colouring golf balls. The divots in the golf balls are all hexagon shaped. A problem I ran into while working on this project, was that divots were not entirely uniform, ie. same size. Nonetheless, it was an interesting opportunity to try and draw my own Goldberg polyhedron. An extension of this activity in the classroom could be to give students toothpicks and marshmallows and challenge them to first create a Goldberg polyhedron, then subdivide the faces to create a geodesic dome and compare strengths between their creations. This activity teaches students teamwork and cooperation, but also lets them explore and experiment on creating integral structures resistant to falling. There is another interesting resource online that is found on https://levskaya.github.io/polyhedronisme/, which generates different polyhedrons. It would be useful to integrate this resource in the digital classroom as it is a great visualization tool that demonstrates easily how polyhedrons change when different operators are applied such as truncating, orthogonality, duality, reflecting and snubbing. 

Figure 3: A GP(5,0) created from a golf ball. 

Figure 4: Virtual recreation of Fathauer's Goldberg Polyhedron using polyHedronisme. 

This was my first group project since starting the teacher education program at UBC, and I had a blast. I’ve learned so much through creating Goldberg polyhedral patterns on golf balls and researching polyhedrons. My favourite part of this project would be the interactive session using polyHedronisme. As a teacher, I hope to integrate many different digital tools in the classroom for learning. We are living in such a technologically advanced generation, I feel that incorporating these technologies into the classroom can be opportunities for students to engage with mathematics on a deeper level than equations seen in a textbook. 

Sources:

Fathauer, R. 2019. Robert Fathauer. Retrieved September 27, 2020 from http://gallery.bridgesmathart.org/exhibitions/2020-bridges-conference/fathauer. 

Hart, G.W. n.d. A Twelve-Part Puzzle Based on the (4,2) Goldberg Polyhedron “Tectonic Plates.”  Retrieved September 27, 2020 from https://georgehart.com/puzzles/GBpuzzle-4-2/goldberg-puzzle-2.html. 

Levskaya, A. 2019. polyHedronisme. Retrieved September 27, 2020 from https://levskaya.github.io/polyhedronisme/.

Mathematical understanding and multiple representations

Pape and Tchnoshanov argue that helping students utilize internal and external representation of mathematical concepts are important at interpreting physical, social and mathematical phenomena and can be beneficial to student learning experience. I agree that multiple representations facilitate student development of math concepts. When students are taught different forms of representational mathematics, they may be able to use internal representations to organize ideas and solve problems. Then they can transform those cognitive schemas to external representations such as tables, graphs and diagrams as tools to draw ideas and inferences to facilitate argument and support conclusions. I also agree that teaching students in different instructional styles to be beneficial to encourage sharing of ideas and communication of math concepts. For example, some students may prefer pure-analytic approach to solving problems while others take on a visual style. Students can be introduced to explore alternative ways of reasoning and inquiry when learning to solve problems, which encourages creativity to problem solving. They are also more likely to share ideas to students who solved problems differently.

One form of mathematical representation that interested me while learning was on slope fields, which is a visual representation of a differential equation solution. Slope fields can also tell information on stability of an equation. This is could be a useful tool when teaching students AP Calculus as it could help students visualize how differential equations behave. I can teach the students how to use this tool analytically at first by creating tables of values and sketching them out, but I can also integrate web tools like Desmos to show slope fields quicker.  

Slope field for dy/dx = -x/y

 

Paper: Pape S.J., Tchoshanov M.A. 2001. The Role of Representation(s) in Developing Mathematical Understanding. Theory into Practice. Vol. 40, No. 2. Pp. 118-127

The kind of teacher I want to be

My favourite math teacher is a teacher I had for several years from grade 9 to 11. My favourite thing about him was his teaching style; he taught instrumental learning very well through detailed examples and clear notes. He had a very friendly personality and had a very good humour which made his classroom a safe and welcoming environment. He made himself readily available for help afterschool most days or during tutorial hours, but also uploaded class notes to his website which made it easy to review missed material or a resource to go to for clarification. My least favourite math teacher was a prof I had for introductory Number Theory because it was difficult to follow along with his lectures as proofs skipped lots of steps and weren’t very intuitive. Often, it was difficult to read and felt rushed. Assessment of learning did not feel fair, as only unspecified questions in the problem set would be marked; I could have understood all questions in the problem set except for one and if that one question is selected for grading, it would be an unfair representation of my understanding of the material.

 

Letter #1, the student who liked me:

Hi Mr. Y,

Do you remember me? I was the student you had a few years ago who started off the year struggling but with your patience, persistence and help, I managed to finish off the year with 86%. You made every class enjoyable to attend because your humour and relevance to teaching made learning fun. I saw real-world application of what we were learning in class in other subjects like economics and art. My favourite part of coming to class was how you greeted everyone individually at the door as the 9am bell rang. You made tests fair by testing us on things we learned and no “surprises.” Anyways, I had so much fun being in your class and hope to visit you soon,

Student A.

Letter #2, the student who disliked me: 

Mr. Y,

I hated your class. Your math class was the hardest class to sit through every day. I’m not sure if you had a lot on your plate, but it seemed like you had to review your notes often before or even while you’re teaching, so sometimes you make class confusing. You emphasized how there are multiple ways to solve problems, which confuses me. I just want to know how to solve problems one way and always use one way. I also think you assign too many practice problems.

From,

Student B.

Skemp's two approaches to teaching and learning math

Skemp’s paper on comparing relational and instrumental understanding in mathematics was philosophical and interesting, however there were a few moments that made me stop and think when reading. Skemp defines instrumental understanding as learning “rules without reasons” while relational understanding is more knowing what to do and why.

The first moment that made me stop and reflect was when Skemp mentions one of the mismatches in mathematical learning, where a student’s goal is to understand instrumentally when the teacher wants them to learn relationally. This is extremely evident throughout my undergrad studies of mathematics, where the prof would spend a long time going over the proof and theory behind an equation, but it would be completely overlooked as the proof is more complicated than the equation. When I taught students math, they also only cared more about the equation as it would lead to the right answer quicker than understanding why the equation works; an advantage of instrumental understanding as mentioned by Skemp. The second moment which made me stop and think was when Skemp listed the advantages of relational mathematics, several of which were it being more adaptive to new tasks and being easier to remember. I noticed that with relational understanding, students draw conclusions on new (but similar) questions based on their previous learning as they see similarities in the questions. For example, I had students use the area of a triangular prism to solve for the area of a pyramid. Both shapes are similar but are different enough to require a different equation to solve for the area. The third moment I stopped while reading was when Skemp discussed the “over-burdened syllabi” where a single line of math contains many concentrated ideas. For example, Pythagorean Theorem is the sum of two small squares equals the big one. A student would have to draw out two squares, square their lengths, add them, and take the square root of that to find that diagonal line. In instrumental understanding, that would be the equation a²+b²=c²_.

Skemp argues that relational understanding would take too long to achieve and may be too difficult depending on the topic. I agree, however believe the advantages of relational understanding outweigh the disadvantages. Teaching students the theory behind an equation and why it works can get them to ask questions and think outside the box; which are important aspects to learning and critical thinking.

Paper: Skemp, R. Relational understanding and instrumental learning. Mathematics Teaching, 77, 20-26, (1976)