Consider a solid ball of radius . Cut a cylindrical hole, through the center of the ball, such that the remaining body has height . Call this the donut . Use Cavalieri’s principle to calculate the volume of . Calculate the volumes of and .
Several students have asked what looks like. Here are some pictures that I found to illustrate the concept. The donut is the region between the red sphere and blue cylinder. The golden balls below show various views of the donut. The donut should fit between the two planes and , so that it has total height .
]]>There has been a minor change to the syllabus: Homework 6 will now be assigned in Week 11 and due in Week 12.
Additional resources:
Geogebra Demonstrations:
Suggested Exercises:
Past Finals:
I was looking through the Geogebra site and found this lovely applet Orthographic Projection by Malin Christersson.
This is a lovely tool for investigating one of my favourite facts about hexagons:
The area maximizing orthogonal projection of a cube is the regular hexagon.
It turns out that Malin has tonnes of awesome geometry stuff online!
Awesome math art!
]]>Homework 4 (tex) is now available.
Additional resources:
Suggested Exercises:
Past Finals:
Notes:
]]>The notes for the talk are available here.
The notes for the talk are available here.
]]>Additional resources:
Suggested Exercises:
Past Finals:
Notes:
]]>The Daily Reference Round-Up:
A couple books that jumped out at me!
Some quick ideas:
The workshop ended today. We had so much fun together. On the very last day, we did a full class video session. We watched a full class taught in IBL style. It was amazing to see all the pieces come together.
At the end of the day we had a graduation ceremony. We did it! We are now IBL practitioners. There was lots of cheering, and celebration. Everyone crowded around the door chatting. I think we’ll be seeing each soon.
]]>We started off the day with Kittens inspired by Kittens!
A hilarious (and deep) video about “productive” failure: amazing commitment and persistence in skateboarding.
I also bumped in to an essay A Message to Garcia by written by Elbert Hubbard, about the importance of individual committed to a task.
A math circle talk about Sperner’s Lemma by James Tanton, a very nifty highschool teacher.
Dana Ernst has a really nice post about Setting the Stage for an IBL course.
Today at the IBL workshop we did a fantastic bit of playful mathematics. Gulden Karokok introduced us to a mathematical game and encouraged us to play it together. As soon as we had played a couple matches, everyone started to conjecture about winning strategies. One way of teaching induction or formalizing using notation is to bore students with examples until they want to formalize things. The questions emerged naturally from the play and the whole room erupted in discussion. It was a perfect example of the mathematics generating the inquiry.
The game is quite simple: Two players have some number . They have the following moves: they can subtract one or they can “half it”. If is even, you half it by dividing by two. If is odd, you half it by subtracting one and dividing by two. A player loses if the number is zero. Players alternate turns until they reach zero.
The game is a nice little exercise in division. It could be good exercise for people learning arithmetic. However, it gets boring quickly. The natural questions are much more interesting: How wins? What’s the optimal strategy? These questions raise all sorts of fascinating problems both notational and mathematical. Lots of discussion groups puzzled over the question “What is a winning number? Should it be when the current player can force a win or when the next player can force a win?”
After chasing around examples on a page, I decided to take an algorithmic approach. Once I convinced myself that I could write up and algorithm for the problem, I tuned out of the workshop and wrote some code to generate all winning. The code is available here: http://pgadey.ca/teaching/ibl-game.py.
We were assigned the reading: Boaler, Jo. “How a detracked mathematics approach promoted respect, responsibility, and high achievement.” Theory into Practice 45.1 (2006): 40-46. I wrote up a bit of a response to it.
Jo Boaler describes a complex and interconnected set of practices which promote community development among students in a math class. The article discusses research conducted at the Railside School, an urban highschool in California. The purpose of the discussion is to highlight practices which help to foster equity in schools. It is well known, although not often discussed by math teachers, that test scores are strongly correlated with things like ethnicity, gender, and socio-economic status. This is seen, by many, as a fact of life despite being inequitable.
Boaler puts forward the idea of an “equitable test” in which student achievement is less strongly correlated with these factors. It is a noble goal. Boaler then describes several practices which were used at Railside to promote equitable behavior and relational equity, which is when people interact and relate to each other equitably. Boaler defines “relational equity” for students to be the ability to “appreciate the contributions of students from different cultural groups, social classes, genders, and attainment levels.”
I think that people learn best to relate to each other in equitable ways by interacting with each other towards a common goal. We get to know our neighbours, as people in themselves, when we sing together, when we work together, when we get caught up in something bigger than ourselves together. Often, attempts to foster relational equity come in the form of sermons about equality or inter-cultural understanding. It is my sense that these are much less effective than pursuing a common goal together.
Boaler goes on to describe some practices and concepts surrounding mathematics which might help for equity. I have summarized heavily, and included some reflections on how these things manifest in my own teaching practice.
Multidimensionality: the multifaceted aspect of mathematics. Recognizing that mathematics has more aspects than “solving problems quickly”. This needs to be promoted more by our pedagogy. It is present when students answer a question like “What does it take to be successful in math class?” with such things as “asking good questions, re-phrasing problems, explaining well, being logical, justifying work, considering answers, using manipulatives.” These are the The more ways that there are to succeed, the more people will be successful.
Roles: Giving people roles structures things and lowers the pressure to innovate. This did not happen in my encouragement to “group discussion”. By assigning roles like scribe, moderator, or presenter, people can have a more focused experience of group work.
Assigning Competence: Publicly promoting the importance of particular people’s work. We often assign marks which are viewed as punitive or measure an amount of error. We do not take time to “assign competence” or acknowledge competence in our students. In the workshop, we assigned names to particular strategies. For example, “working through concrete examples” became Jim’s Method.
Student Responsibility: A good community has its members support each other. If we promote inter-student support explicitly, and emphasize students supporting each other, than we are working towards a good learning community. Two practices that Boaler cites as particularly effective in promoting equity and inter-student support in mathematics are justification and reasoning. By asking students to describe their group’s reasoning or justification, we put value on a group’s collective work. The reasoning or justification for a problem can be said in many different ways, and each member might have their own approach to the problem. This is in distinction to a final answer, which has one representation. Another strong method for building student responsibility is making assessment value group processes: people write a quiz in groups and each group member gets the same mark. This involves students in each other’s lives via assessment.
High Expectations: Overall, we should make it clear to our students that we demand a lot of them. Often my teaching can seem patronizing or like I am “dumbing things down”. This makes the subject feel like it is not worthwhile. By demanding intellectually stimulating answers, we promote a culture of complexity and richness. How do we find that way in to our subjects? I do not know.
Effort over Ability: Make a big deal out of effort and attempting. Lots of praise for practice, struggle, and effort. Make a culture where people value effort. Promote the idea what talent/ability is acquired and not innate.
Learning Practices: Emphasize concrete ways of working that are known to be effective. Asking precise questions, framing uncertainties clearly, checking over work, are all valuable skills. We can direct our students to be aware/reflective of their own learning process. We need to make these in to an acknowledged part of our teaching.
]]>The Link Round-Up of the Day:
The MAA IP Guide has a whole section of queries about course design. I am eager to answer them in writing. I think that it will make my course design much more solid. The coolest thing that I bumped in to while researching my project is that Dana Ernst keeps a point form journal about each of his sessions in class! So Cool!
On a completely different note: Today we did a bit of change ringing! Amy Ksir brought a handmade set of tubular bells and we rang an eight-bell plain hunt.
]]>Some Amazing Resources:
There are some books that I want to track down:
I think that next year I will draw heavily on Ted Mahavier’s Calculus I, II, & III : A Problem-Based Approach with Early Transcendentals.
We were assigned the reading: Yoshinobu, Stan, and Matthew G. Jones. “The coverage issue.” Primus 22.4 (2012): 303-316. I wrote up a bit of a response to it.
The coverage issue is well known to all teachers. It is, informally put, the problem of having more course material than time. We have to “cover” so much material per course, that we are constantly struggling against the clock. The worst manifestation of the coverage problem is when students’ curiosity exacerbates the problem. In my experience, tricky topics elicit many questions and provokes in-class discussion. I am always eager to promote discussion and answer questions. Time limits get in the way of these things, and force me to shut down valuable discussion. Or, if I allow the inquiry to continue until the end of lecture, I find myself struggling to make up for lost time later.
Yoshinobu and Jones define the “coverage problem” to be “the set of difficulties that arise in attempting to cover a lengthy list of topics”. It is assumed that the teaching here is happening in normal higher education context with fixed lecture times and semesters. Their work focuses on the failures of what they call the “standard model” of teaching in which an expert lectures to passive students.
The authors identify several issues with the standard model of lecture. They note that the standard model creates an environment which promotes beliefs which will eventually harm students’ ability to do mathematics. Foremost among these is the belief that math is a spectator sport. The standard model forces students to passively receive information, and puts all active engagement with mathematics out of sight. Lectures also promote the belief that experts are the sole authority in mathematics. In an IBL model “students actively participate in contributing their mathematical ideas to solve problems, rather than applying teacher-demonstrated techniques to similar exercises.”
One thing that pleased me about their article is their concrete description of how IBL could work in an early math course, something without many proofs:
In Precalculus, students may focus extensively on building a cohesive understanding of basic classes of functions (linear, quadratic, higher-degree polynomial, rational functions, exponential functions, logarithmic functions, and periodic or trigonometric functions). This can be achieved through tasks such as having students interpret a data table or a graph in context, determining a symbolic description of a function from a given data table, graph, or written context, and asking students to select the most appropriate class of function to model a given situation. Students may also be asked to read key definitions or theorems, and give examples and non-examples or restatements, a task that helps students learn to make sense of mathematical text. Students will often be asked to solve problems without similar solved problems being given as models.
Yoshinobu and Jones give a brief treatment of how to “fix” the coverage problem. Teachers are asked to identify the “essential” aspects of their course and then handle the less than essential topics in one of three ways: readings and assignments, mini-lectures, or removal. If a topic is valuable but not essential, it can be treated outside of class time by assigning the students readings and assignments on those topics. This assumes a certain amount of autonomy and independence from the students. It could work. The other approach to keeping non-essential material in the course is to offer mini-lectures which treat the topic as an aside from the main line of the course. The last resort is to simply drop the topic from the course.
The authors present a short summary of the evidence in favour of IBL. To simplify things considerable, let’s suppose that assessments generally have two parts: procedural questions and conceptual questions. Procedural questions usually pose some task which needs to be accomplished: finding some values, or solving some equation. Conceptual questions involve some cognitive processing and usually require an explanation. One might need to explain why a process works or demonstrate that a theorem is true. Yoshinobu and Jones state that there is a great deal of evidence, especially in early math courses and highschool, that IBL students perform no worse than standard model students on procedural assessments and perform better on conceptual assessments than their standard model peers.
Sometimes I think about teaching our students to do “mathematics like mathematicians”. Lots of my students are not math majors. Becoming mini-mathematicians does not seem like a desirable goal for many students who are not math majors. It would be desirable to math majors and those seeking to become mathematicians but those students are a minority. I would prefer to make my student in to self-confident problem solvers in a general sense.
I would also be very curious to know how students in IBL courses express their self-efficacy and condfidence compared to other strudents. It seems to me that the main advantage of the IBL paradigm is that it makes mathematics in to a participatory activity and empowers students to be more mathematically confident.
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