textbooks and how they position their reader

In the article, Wagner & Herbel-Eisennman critically examine the language of textbooks and how they “position” students relative to their teacher, their peers, and mathematics in general. They describe a “range of possibilities” that may result from the usage of certain pronouns and “modalities” within various contexts. As a former high school student and textbook user, I believe these implications would have definitely flown right over my head back in the day. As a slightly older slightly wiser teacher candidate, I think it’s a fascinating way to analyze a textbook. These subtle and incidental implications of language are likely undetected by the average student. That being said, I believe language has a way of washing over us, subliminally implanting and reinforcing ideas and perspectives. Even though the article looks at textbooks specifically, I feel as though much of the discussion around language could easily be applied to classroom teachers and how they communicate to their students. For instance, I like the idea of textbook authors having some vision of the “model student” which serve as their audience. And while a teacher may interact with their audience to a greater extent than a textbook's author, the teacher still has a lesson plan based on a conception of their students. I would argue then, that the use of certain pronouns and modalities is commonplace in a classroom for the same reasons it exists in textbooks. A lot of the way people speak about math is informed by the nature of the discipline. Generalizations and abstractions are likely to de-emphasize personal experience, and reverence of logical rigor is likely to result in extremes in modality. The tendency of textbooks to take this authoritative and objective position on teaching mathematics definitely adds to effect of learning math as this impersonal experience. But in the case of teachers speaking to their students, I believe the ways in which language is implemented is much more intuitive, and a good teacher will naturally strike a balance without needing to over analyze things like "modality" and "pronoun usage." All the same, I'm sure we have all been somewhat influenced by our childhood textbooks and whatever linguistical conventions, so it's good to be mindful of how specific language usage can subtly impact our students' learning experiences.

final reflection on the course

This was probably my favorite class this semester (don't tell 450). It's actually difficult to compare 342 and 450 as they kind of blend together in my mind. Especially looking through my previous blog posts in both courses, I realized that I couldn't remember which posts/readings/activities were associated with which of the two courses. I will say however that I did really enjoy the slightly heavier math slant of this course. Looking through my previous posts, I remember feeling very engaged, and really wanting to take my time in writing thoughtful reflections. That's true of 450 as well, and I don't want to repeat too much of what I already said in my 450 final reflection, but I will reiterate how much I enjoyed the writing aspect of these courses. It had been so long, that I had forgotten how much I enjoy writing. It really is a great exercise in developing and organizing one's thoughts, and it very much appeals to the perfectionist in me. I also want to mention here that I feel quite fortunate in being a part of such a truly excellent cohort of teacher candidates. That being said, I must must give credit to our esteemed professor here, because I think there's a lot about how you teach, and what you teach, that has cultivated this very supportive and creative learning environment. So thank you so much for a wonderful semester!

Unit Plan and Lesson Plans

Unit Plan:

https://docs.google.com/document/d/1t-DsbKYF_RbU1AgXlhgoLHg0u8XufAHU/edit?usp=drive_link&ouid=109961138128838636219&rtpof=true&sd=true

Lesson Plan 1:

https://docs.google.com/document/d/1SzeyukJh4PqeZ5jkeo3bJvXNt2l8_liR/edit?usp=drive_link&ouid=109961138128838636219&rtpof=true&sd=true 

Lesson Plan 2:

https://docs.google.com/document/d/1MiVOo2_EsM4Sq3tcmgJqcRIPn2fAt0Us/edit?usp=drive_link&ouid=109961138128838636219&rtpof=true&sd=true

Lesson Plan 3:

https://docs.google.com/document/d/1j1EPpmzLziZdQxmosbTxdMbVuMMyn8PX/edit?usp=drive_link&ouid=109961138128838636219&rtpof=true&sd=true

Unit Plan Draft

EDCP 342A Unit planning: Rationale and overview for planning a unit of work in secondary school mathematics

Your name: Mark LeBlanc
School, grade & course: Panorama Ridge Secondary School, Grade 10, Foundations and Pre-calculus

Topic of unit (NOTE: This should be a unit you will actually be teaching on practicum!):

Relations and Functions

Preplanning questions:

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words)   Relations and functions are the basis of analytical math whose applications are vast, especially within STEM. These ideas are fundamental to any kind of mathematical modelling and are therefore applicable to any field of research where mathematical modelling might be employed. Practically, in terms of academics, it serves as an entry point to upper year math which in turn, is the entry point of STEM fields for post-secondary school. More generally, the study of relations and functions provides a powerful framework for conceptualizing various processes and quantifiable relationships. At this level of study, the topic can easily be framed in terms of its applicability – a mathematical lens through which to view the world. However, more abstract representations of this topic’s ideas serve as a glimpse into set theory which may inspire interest in pure mathematics.

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words)   The unit project will involve students researching a topic of interest involving the relationship between two continuous, quantitative variables. This can be a well-documented research topic, or something revolving observations from their own lives and experiences, or some experiment which they conduct themselves. Examples will be provided for students to serve as rough templates. The aim is to have students develop an understanding of the applications relations and functions, and how mathematical modelling is a useful tool with regards to the research of a given topic. Students will be asked to produce a brief power point presentation which introduces the topic, briefly discussing its relevance. In the presentation, they will be required to describe the two variables involved in the relation, the domain and range, and whether the relation is a function. They will be required to represent the relation in terms of a table, graph. If the relation is a function, they will be asked to represent the function as an equation as well, identifying the type of function. If the relation is not a function, students will be required to represent the relation in another way of their choosing. They will present their project to the class and submit the PowerPoint to the teacher. The project will be introduced about halfway through the unit, with class time being dedicated to working on the project each day, with the introduction of new concepts. Assessment will be based on a rubric TBD.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words) General, informal formative assessments will take place throughout the course in the form of teacher interactions with/observations of students as they work individually and within groups. More formal formative assessment will take place in the form of quizzes throughout the unit. Summative assessment will involve a unit project and unit test.

Elements of your unit plan:

a)  Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them.

Lesson	Topic
1	Sets and Set Notation
2	Relations and Mapping Diagrams
3	Domain, Range, Functions
4	Quiz & Function Notation
5	Sequential Patterns as Functions
6	Discrete and Continuous Variables (Unit Project Intro)
7	Quiz & Modelling pt. 1
8	Tables, Equations, and Graphs
9	Modelling
10	Quiz & Types of Functions
(11)	Unit Project Presentations
(12)	Unit Test

b) Write a detailed lesson plan for three of the lessons which will not be in a traditional lecture/ exercise/ homework format.  These three lessons should include at least three of the following six elements related to your mathematical topic. (And of course, you could include more than three!)

 

These elements should be thoroughly integrated into the lessons (i.e. not an add-on that the teacher just tells!)

a) History of this mathematics

b) Arts and mathematics

c) Indigenous perspectives and cultures

d) Social/environmental justice

e) Open-ended problem solving in groups at vertical erasable surfaces (“thinking classroom”)

f) Telling only what is arbitrary, and having students work on what is logically ‘necessary’

 

Be sure to include your pedagogical goals, topic of the lesson, preparation and materials, approximate timings, an account of what the students and teacher will be doing throughout the lesson, and ways that you will assess students’ background knowledge, student learning and the overall effectiveness of the lesson. Please use a template that you find helpful, and that includes all these elements.

 

Lesson 1: Sets and Set Notation

-        mathematician history research?

 

Lesson 5: Sequential Patterns as Functions

-        Open-ended problem solving in groups at vertical erasable surfaces (“thinking classroom”)

 

Lesson 9: Modelling

-        social/environmental justice

Flow State

As a student with a decent amount of school work, I’m often desperate to achieve some sort of “flow state” in order to complete tasks in a timely fashion. Personally speaking, I find flow states to be few and far between these days. I do know the feeling described by Mihalyi Czikszentmihalyi, and I wish I could tap into that state of mind whenever I liked. Unfortunately, if I’m trying to do some kind of assigned work especially, I consider myself lucky if I can be flowing for more than a few minutes at a time. I’m sure the required conditions for a flow state are different from person-to-person, but for myself, it’s largely dependent on the type of task I’m working on, and whether or not there’s a deadline. Discussing with Andy and Jacob, I realized that I’m more likely to have those moments of dedicating all my brain power when I’m focusing on something creative, or if I’m physically engaged in something like a sport. In those cases, I can lose track of myself for an hour or two at a time, and it’s a wonderful feeling to be locked-in like that. That’s not to say it doesn’t happen if I’m writing, or working on a math or physics problem, say. But usually in those cases, I’m flowing if I have an idea of how to proceed, and it may just last a few minutes before I either finish, or get stuck, or get distracted and need to remind myself to focus up. For a student, those moments of flow are precious and addicting. As teachers, I think we want to unlock that state for students as frequently as possible. From my own experience, it’s also more difficult to achieve flow when you’re not interested in what you’re doing - I don’t think you can force it like that. To me, that’s a clear point for trying to understand students’ interests. If we can get them engaged on a level which speaks to their “soul” or what-have-you, we can maybe facilitate entry into a flow state. Like I said, that feeling is addicting, and maybe if we can harness the flow state of our students, we can improve their engagement with their education. So I do think it occurs more frequently when our actions are aligned with our interests. And we want to encourage students to pursue their own interests, but at the same time, I also think we are often interested in more than we realize. For instance, I can claim that I don't like to cook, but that's in full denial of times when I've become completely absorbed in the process of preparing a meal. So anyway, I'm super skeptical of anyone and everyone who claims they do not like math. Sure, they may have had some bad experiences in a math class, but maybe the conditions for them to unlock that interest or that flow state are simply more stringent than they are for me. I'm optimistic that it's possible for anyone, we just need to be sensitive to the fact that the conditions for flow are different from student-to-student, and then try our best to work within those bounds.

soup can puzzle

Assuming the water tank is cylindrical in shape, and assuming it has the same proportions as a can of Campbell's tomato soup...

*The second assumption allows us to approximate the volume of the tank without needing to approximately measure both its length and radius.

                                                                  V = pi*r^2*h             (where h is the height/length of a cylinder)

We now just need to find how many times larger the length of the water tank is relative to the length of a soup can.

Length of the water tank:

We can use the image provided to approximate the length of the water tank. This process will first involve approximating the length of the bike in the picture. We will then use this to find a value for the scale of each pixel in the image, which can then be used to measure the length of the water tank.

*Research can be done to try to find the exact length of the bike. for example, based on the logo and shape of the frame, the bike appears to be some version of a Norco XFR Stepthru model. However, short of tracking down the owner of the bike, it is impossible to be certain of the bike's sizing, as well as the specific details of the wheels/tires. For these reasons, we will instead take an approximate average bike length of 175cm, based on Carl Ellis' article, "What size shed do I need for my bikes?" from thebestbikelock.com. 

 

 

The image was opened using ms paint and the x-coordinates of pixels on opposite ends of the bike and water tank were used to approximate lengths. This gives the water tank an approximate length of 432 cm.

Height of a Campbell's tomato soup can:

Taking measurements from campbellsfoodservice.com we find the height of a "typical" can to be about 4-inches, or 10.16 cm. 

 

*Here, we're assuming that these advertised measurements would be the same as the vintage soup can depicted on the water tank. It likely that the dimensions of Campbell's soup cans have changed over time, and as this work is a reference to Andy Warhol's paintings, we might attempt to track down a can from the early 60's and measure it ourselves for the sake of "accuracy." Instead, we will use the 4-inches, which is itself an approximation as the 4.125 inches on the website likely includes the thickness of the cardboard box in which the cans are delivered.

Ratio of the length and the height:

Dividing the length of the water tank by the height of a soup can...

                                        432 cm  ÷  10.16 cm  =  42.5 times larger (approximately)

 

Volume of the water tank:

 

                                                          V = pi*(42.5r)^2*(42.5h) 

                                                          V = 76765.625*(pi*r^2*h)

Having used the length/height instead of the radius, and again working with the assumption that the dimensions of the soup can would be the same as those of the water tank (assumed to be cylindrical)... It follows, from the volume formula of a cylinder, that the water tank's volume will be about 76766 times greater than that of a "typical" Cambell's soup can. Again from, campbellsfoodservice.com we have a volume of 10.75 oz (assumed to be US food label fluid ounces) or 322.5 mL

 

                                                        322.5 mL * 76765.625  =  24757 L

Since there was so much approximating anyway, giving a final approximate answer of "about 25000 liters or so" feels appropriate.

 

Is this enough to put out an average house fire?

 

From a linkedin post by DryTech Water Damage Restoration Services titled "How Many Gallons Does It Take To Put A Building Fire Out?" we see that it is "fairly normal" for fire departments in rural areas employ water tanks that carry between 1500 - 3000 gallons of water. Converting this to liters, we have a rough estimate of 5678 - 11356 liters water being a "fairly normal" size of water tank with which rural fire fighters would arrive on the scene and expect to extinguish a typical fire.  

*There is a lot of guess work in this problem. As a student, I can see it being both interesting and frustrating to play detective and wrestle with how much research are you willing to do vs. how many assumptions/approximations are you willing to make. One could obviously take it a step further in either direction, and/or work from different assumptions. One might also choose to work with small and large estimates in order to get a range of values for the answer. As a teacher, I think whether or not the student is able to accurately estimate the volume of the tank is one thing, but it's probably more interesting just to note how a given student approaches a problem like this. I think you might get a student who enjoys the challenge of being as accurate as possible, and consequently, does lots of research and justification of their method. On the other hand, you might get a student who is much more willing to estimate without putting too much time into it, and that is also valid. Math wise, I would especially want students to be aware of how the scale factor of a single dimension would affect the scaling of the volume, and being able to connect that to volume formulas. In other words, I would want to see students aware of how scaling volume will essentially have you cubing your scale factor.

 

Because so much of this puzzle involves research/assumptions/approximations, I think the natural extension would be to make it explicitly clear to students that you want them to do research, citing their sources, and justifying their usage of them.

 

Hewitt response

Hewitt does an excellent job of putting into words certain ideas that I've only ever thought of on a surface level. I appreciate the distinctions he makes between aspects of math curriculum and the ways in which students interact with those aspects. My understanding is that "arbitrary" refers to terminology, notations, conventions, and other choices about how math is actually physically done. These need not be the way they are, but are mostly propagated due to convenience and perhaps some long forgotten historical basis. I really like the idea of addressing this aspect of math with students. For starters, I think math is more interesting when it seems less like a set of seemingly pointless rules and viewed more as a creative endeavor. In practice, the arbitrary must be taught by teachers for students to memorize, so they're able to actually communicate and interact with the standard mathematical landscape. In contrast, what is “necessary” is more so the mechanics, processes and results of mathematical reasoning. These ideas can also be taught in such a way as to be simplified to seemingly arbitrary methods and algorithms, but they are more so necessary in the sense that they represent a set of idealistic learning goals which focus on students' relational understanding of the material. On that note, there is definitely some similarities between the dichotomies of arbitrary vs. necessary and instrumental vs. relational. If I were to try and combine these ideas (as I understand them), I would start by saying that necessary content can be taught either relationally or instrumentally. However, the ideal situation, at least in my mind, is that necessary content is taught relationally. When it is taught instrumentally, the necessary is reduced to the arbitrary in the eyes of the students, making demands on their memory instead of engaging any kind of deeper awareness. Again, I think the perfect-world image of a math class involves students deriving necessary ideas from their own relational understanding of the curricular content, perhaps making use of their own personal set of arbitrary conventions. In practice, I think it’s understandable to sometimes focus more on instrumental teaching, and blur the lines between what Hewitt might consider either arbitrary or necessary. He raises a good point about how students may or may not be willing to engage with the more creative aspects of mathematics. For example, they may have no interest in distinguishing between the necessary and the arbitrary and may just consider the subject as entirely arbitrary, only valuing an instrumental understanding. To be honest, I think that’s also probably the prevailing student opinion, which is a failing of math education. As it stands, for practical reasons such as covering a course-load’s worth of content and maintaining student engagement, teachers definitely need to balance their lessons in terms of the degree to which the necessary and arbitrary are separated, oscillating between relational and instrumental teaching. It also occurs to me that what is considered necessary or arbitrary is at least slightly subjective and may differ slightly from one teacher to the next. I would also add that even the arbitrary can be taught somewhat relationally, providing some rationale and historical context for the choices which have been made about math over the years. I think part of an excellent math course involves equipping students with the tools to do math, but also giving them ideas about how to develop their own mathematical tools. I would argue then, that the idea of the arbitrary existing within any mathematical framework, is in itself a necessary idea, and worth not only pointing out to students, but also celebrating.