summing up the course

When I reflect on the course, I think there's some value to mentioning the first few things that come to mind without flipping back through readings and blog posts as these are likely the most profound learnings I have taken away from the course

1) I recall in one of our early courses the discussion about what it means to have a deep understanding about something and the relevance of being able to express something in multiple ways. I know I ( and my students) have been frustrated at times with textbook questions that dictate how work should be shown, or ask for multiple representations of the same work or answer. I think this discussion helped me appreciate the value in not stopping as soon as one understanding is attained.

2) I appreciated the challenge of taking a few textbook questions and trying to 'open them up' as per the readings on the week we focused on problem solving. While my initial reading of the chapters had me thinking that they were trite and stating the obious, the challenge of re-writing and the discussions that ensured were valuable and brought up some ways to 'open up the questions' that I had not previously thought of.

Some burning questions...
1) How can assessment be fair (standardized?) yet still evaluate the softer skills fairly, while being possible to do with a courseload of 200 students?
2)How can we bring together the importance of practice and rote learning with the importance of student driven learning and abstract thought?
3) Will the new curriculum bring any real change to my classroom?

Esmonde - Ideas and Identities: Supporting Identity in Cooperative Mathematics Learning

In this review, Esmonde discusses how cooperative learning (CL) might assist students to learn in mathematics classes. She acknowledges that CL is an often recommended technique to increase equity in classes, but also that the benefits (group harmony, learning academic content with social skills) can also have detrimental effects (incorrect math strategies, undesirable social interactions). I know that when I did my teacher education program (2005-2006), cooperative education was certainly highly encouraged, and we discussed to great length the challenges and possible successes of having students learn together and/or teach each other in various cooperative organizations (random groupings, jigsaw teachings etc.) In general, I apply cooperative learning strategies in my classroom regularly. There are content-based and social based outcomes: students enjoy working together; it breaks up monotony of math class; they receive multiple options of ways to understand concepts; the teacher can circulate and have discussions with small groups, thus getting to know students better. Furthermore, I think we are providing students with valuable skills and messages that it’s important to work together (and not just with friends) and that mathematical learning and understanding is a process, which is often effective to discuss conceptions and misconceptions in order to understand.

One of the questions Esmonde raises is how we might group students. Do we group high achievers together so that they don’t dictate other students’ learning practices, or do we create mixed groupings such that they can teach each other? True cooperative education might likely be more effective with groupings of similar ability so that students can explore the concepts at levels that challenge and are meaningful to them. On the flip side, it can be valuable for students to learn from their peers – to see and understand how others have made sense of the material, and valuable for highly able students to have to clearly communicate their processes.

I’ve often considered this problem, and my short-answer response is both: I think it’s important to change up groupings often (and sometimes let them choose their own) so that both these can happen. There is an undeniable possibility that students will copy off each other and let the more able (or more driven) students do most of the work and solve the problems, yet I have managed to come to terms with this by considering that this form of education allows for multiple entry points, and the weaker student may only be able to imitate the stronger student, but that there is hopefully some learning in this.

QUESTION: Should cooperative activities be evaluated in Math class? If so, how?

Does Everybody Count?

In her article discussing curricular reform, Noddings questions the assumption that mathematics needs to be an emphasis for all students and that math skills are something everyone needs. She suggests that math courses might contribute more effectively to general education.

Through much of her article, Noddings comes across somewhat like an educated but whiny student 

asking “why do we need to learn this?”, arguing that technology has replaced the need for things like

arithmetic skills. Later on in her article, she comes across a bit like a whiny teacher suggesting that it 

would be nice if teaching was a “real profession, one like law and medicine.”  (p. 101) She then 

suggests that teacher would not be willing to dedicate three or four years to graduate school for a 

career in which they have little autonomy, poor pay and low status.

I think one of her main arguments is that there is not enough specific teacher education, and that in 

addition to being knowledgeable about their subject, a teacher should be more highly trained in 

education. I think I disagree with this!  Even the one year I spent in my teacher education program 

seemed mostly time-filler. I think teachers need to learn from practicums – I mentorship with an 

experienced and able teacher is the most appropriate way to learn our profession due to the fact that 

there needs to be a strong connection between the theoretical goals of education and the practical 

application of teaching within the constraints of the system we work in (eg. 30 kids in a class for 75 

minutes every other day etc.)

Question: What do you think are key lessons people in a teacher education program need to learn? How long should such a program be?

Response to Palme

In this article, Palme notes that teachers and textbook companies are struggling with the development of school tasks that resemble out of school situations. He criticizes traditional math word problems in lacking realism and not being authentic. For example, a question involving an elevator and a number of people does not take into account that people arrive at different times, and the elevator would not always leave full. Palme attempts to build a framework that will allow people to judge whether problems are real and authentic.

In particular, he focuses on the inherent requirement in problem solving to accept different interpretations of the problem and methods of coming to solutions. He states, “The students must believe that their solutions are going to be judged according to the requirements of the real life situation and not have to think about what different requirements the teacher might have.”  In theory, this seems logical to me, yet I question how it would work in practice for two reasons: 1) if students over-simplify a problem, or always focus on simpler strategies such as guess and check, they may miss learning important math concepts; 2) Assessment becomes very complicated – it would be difficult to fairy assess the validity of different solutions. 

For my question, I will focus on a dilemma I recently had:

Recently, we had a question on a pre-calc 11 midterm where students were given the perimeter and area of a rectangle and asked to find the dimensions. The intent was that students would solve the perimeter equation for one variable, substitute the expression into the area equation, then expand and solve a quadratic equation. The question was worth 3 marks, and many students who had partially correct students got part-marks on the question.

A few students used guess and check methods, and since this is a valid strategy, I gave them full marks if they provided the correct solution. One student tried a guess and check strategy, and came up with an answer that was close but not exactly the solution – I gave him zero marks. He argued that he had partially completed a legitimate strategy that other students received full marks for, and thus should receive part marks.

Thoughts?

Response to New Questions for Old

In this chapter, the authors encourage teachers to optimize the use of their textbooks by opening questions up for students. They identify four ways to do so:

1) Change the existing question - in particular, adding or omitting a bit to the question to make it more open ended to encourage students to explore the material further. They give many examples such as instead of asking students to plot 3 points on a coordinate grid and decide if it is an isoceles triangle, ask students to plot 6 points, and determine all the isoceles triangles they can.

2) Give the answer rather than the question - for example, tell students the hypotenuse of a right angle triangle is 17, and ask them to list as many triangles as they can.

3) Change the resources - Allowing, or not allowing technology changes the scope of the question. For example, coming up with fractions equal to 1/3 with or without a calculator.

4) Change the format - ask questions in different ways. The authors provide a whole page of different ways to express an equation like 4x+2=10.

These methods do not seem particularly innovative to me. Just about every textbook I use applies all of these quite regularly, so I'm not sure what new questions I can offer by analyzing three questions, but I will try...

I choose to work with the Grade 10 McGraw Hill Ryerson text I currently use.

Question 1: Earth has a diameter of approximately 8000 mi. Land forms 29% of the surface of the Earth. Assume Earth is a sphere. Estimate the area of the land on earth. 

This question could be opened up by giving different aspects of the information - for example  stating the area of the land on earth and the diameter and asking for the percent covered. Alternately, students could be asked to compare their result (assuming Earth is spherical) and research how far off they are from the reality, given that Earth is not an exact sphere.

Question 2: Evaluate without a calculator: (3^(1/6))(3^(5/6))

Students could be asked to follow up by listing a few other rational powers of 3 they could multiply (3^(1/6) by that they would be able to evaluate without a calculator.

Question 3:  A circus recently had a sold-out performance. There were varying admission prices. The admission for premium seating was $250 for adults and $175 for students. The total revenue for premium seating was $29 125. The receipts showed that 130 premium seats were sold. Determine how many adults and how many students were in premium seats.

Students could be asked if they could find any other combination of adult and student tickets that had the same revenue, or they could be asked to model the situation using graphing technology, or they could be asked to come up with another scenario that could also be modelled with the equations they came up with for this question.

Response to Teaching Mathematics for Understanding: An analysis of Lessons Submitted by Teachers Seeking NBPTS Certification

In this study, the researchers analyze portfolios submitted by teacher candidates. They conclude that the lessons included many tasks involving hands on activities or real world contexts and technology, multiperson collaboration and hands-on material, but rarely required students to provide explanations or demonstrate mathematical reasoning.

I wondered as I read this article how representative the study would be. When I am in a job interview and asked to describe a lesson, I generally describe a very hands-on activity such as building clinometers and using them to measure heights as a way to make trigonometry meaningful.  While I’m not misleading anyone, as I do run this activity almost every year, it is the exception and not the rule in my class, as I generally follow a fairly traditional lesson structure. Indeed, if asked to provide a portfolio of my lessons, I would these ‘special’ lessons which are not what the students in my classes experience most days. I would suggest many other teachers might follow similar patterns.

The authors criticize that that tasks, while hopefully engaging and meaningful, tend to have a ‘low frequency of high demand tasks’ in exchange for a ‘higher incidence of innovative pedagogical features.’ This made me think of two things: First, I often feel pressure to be ‘performing’ and ‘entertaining’ my classes, which I think does not need to be a teachers’ role. Secondly, particularly with new curriculums coming into place, I hope math class remains challenging as it is one of the last bastions of challenge (some) students have in schools. Some of my students tell me they are used to getting near-perfect marks in most other subjects simply for completing their work to an acceptable degree, with little regard for quality.  While I don’t mean to torture students, I think part of what schools need to teach students is how to work hard to achieve something that is difficult, and when we settle by catering to a lowest common denominator, we may be robbing students of the opportunity to have to work hard for something.

Finally, in their conclusion, the authors bring up the concern of many studies including theirs focusing on classroom lessons and not on assessment. For example, if teachers use methods of instruction that include group work, hands on activities and technology, but their assessment focuses on pencil and paper knowledge and problem solving, we are not being fair to students. Indeed, lessons should prepare students for, and resemble, assessment. Students get (rightfully) frustrated if they have done and understood the coursework, yet are not able to be successful in assessments.

Question: How should policymakers determine appropriate levels of difficulty for math classes?

Response to 'When Learning no longer matters: Standardized testing and the creation of inequality' (Boaler)

In this article, Boaler tells the story of a school in an underprivileged area which has a forward thinking math department. The students demonstrate strong increases in achievement and strong results on independently developed forms of assessment, but still perform poorly on Standardized assessments. Boaler outlines the damage that is done to students’ self esteem as they work hard to achieve and feel like they have learned and understand, yet are still told they are ‘below average’ on these high stakes assessments. If I had to criticize the article, it would be that Boaler does not adequately describe the practices the math department undertakes in order to increase students’ mathematical understanding. She non-specifically states that they observe each other’s classes, meet regularly, and take part in professional development opportunities, but I would have been interested in more specifics about their teaching practices.

A stop for me in the article occurred when Boaler that students might perform poorly on standardized assessments as they are set in contexts that are confusing to linguistic-minority and low-income students. She gives an example of the following question:

A cable crew had 120 feet of cable left on a 1,000 foot spool after wiring 4 identical new homes. If the spool was full before the homes were wired, which equation could be used to find the length of cable (x) used in each home?

F. 4x + 120 = 1000

G. 4x - 120 = 1000

H. 4x = 1000

J. 4x - 1 000 = 120

This brought to mind the only high stakes provincial testing we still have in BC – the Math 10 Provincial exam. All Grade 10 students write a provincial math exam, but they are divided into one of two groups based on course they are enrolled in – the Foundations/Pre-Calculus (FPC) class exam focuses more on algebraic skills, and the exam reflects this. Alternately, the Apprenticeship/Workplace (AW) exam, which usually attracts students who are more challenged in math, features mostly extremely ‘wordy’ problems similar to the one above. In my experience, students in the FPC course generally score close to their term marks, but the majority of students fail the AW exam. I believe the intention of the writers of the exam is that the math will make more sense to students if it is put into context, but there is a disconnect and students find reading the long problems challenging, and often don’t know how to apply the math to the problem, even if they understand the mathematical concept. 

Question: How can we put math concepts into meaningful concepts without overwhelming students with language and concepts they might not be familiar with (like spools and cable in the question above)

Response to Hill, Ball and Schilling

In this article, the authors attempt to develop measures of teachers’ combined knowledge of content and students by writing, piloting, and analyzing results from multiple choice items.

To be honest, this articles’ attempts to conceptualize and measure teacher knowledge did not resonate strongly with me. A 30 page paper and detailed questionnaires allowed the authors to come up with conclusions I consider trite such as “teachers have skills and insights and wisdom beyond that of other mathematically educated adults” (pg. 395) and “Teachers know that students often make certain areas in particular areas or that some topics are likely to be difficult… but teachers often reason about students’ mathematics:  They see student work, hear student statements, and see students solving problems. Teachers must puzzle about what students are doing or thinking, using their own knowledge of the topic and their insights about students.” (p. 396) While these conclusions are valid, they seem quite obvious, and the task of trying to measure and quantify something that so clearly would be extremely different for every teacher seems disingenuous. Certainly, I’m a better Math teacher now than I was in my first year – while my mathematical knowledge has not greatly increased, my knowledge of how students learn, understand, and make errors has increased, allowing me to teach more effectively.

The article also challenges the use of Multiple Choice testing. This has been a regular struggle for me – when teaching in Ontario, my colleagues and I almost never used multiple choice testing. When I wrote my first test for my first job teaching in BC, I showed it to the department head for input, and he said “it looks good, but where’s the multiple choice?”  Having now taught and subbed at multiple schools in Vancouver, I notice that multiple choice testing is the norm, and often can even make up students’ entire grades in some classes. As per the authors, I think it’s important that we “think carefully about the multiple-choice format” (p. 396).  This form of testing can lead or mislead students to correct or incorrect answers, and does not test skills in the same way that a full solution allows us to evaluate process and thinking. I believe the long tradition of provincial exams, along with bigger classes and less prep time encourage multiple choice testing in BC classrooms, but that the Ministry should be more involved with standardizing evaluation so that students are not being primarily evaluated in a multiple choice format.

Question: Do you think that Multiple Choice testing accurately evaluates students?  Do you think the Ministry of Education has a role in dictating the types of assessments in BC classrooms?

Response to Thurston

Response to On Proof and Progress by William P Thurston

Thurston wrote this article as a response to another (by Jaffe and Quinn). In it, he discusses some of the positive aspects of the article and criticizes some of the assumptions the authors of the article made. In particular, he focuses on the questions of: 1) What is it that mathematicians accomplish? 2) How do people understand mathematics? 3) How is mathematical understanding communicated? 4) What is a proof? And 5) What motivates people to do mathematics. After he considers different aspects of how mathematicians understand these important questions, he discusses some personal experiences that highlight how he gained his perspectives.

In particular, I was interested in Thurston’s discussions about how the social interactions mathematicians partake in through their research motivate them to continue researching particular aspects of the discipline. He states: “most mathematicians don’t like to be lonely, and they have trouble staying excited about a subject, even if they are personally making progress, unless they have colleagues who share their excitement.” (Thurston, 48)

This made me consider how we teach math in schools, particularly secondary schools. As featured in the video we watched in class last week, math classes are often one of the last vestiges of the classroom where students sit in rows in individual desks, and spend a great deal of time expected to be silent either listening to the teacher explain concepts or trying concepts. Group work and discussion regarding approaches to solving problems are often discouraged in math classes, and are rarely evaluated, and students often feel that if they share answers or approaches, they are cheating. Most, if not all of students evaluations come from work which is done individually, and often under pressure. This is in contrast to how mathematicians work in collaborative environments, recognizing that colleagues can play a key role in seeing problems in different ways and helping to maintain motivation and gain understanding.

I try to address this in my class by encouraging group and partner work while working on concepts, and providing assessments in class where students are encouraged to work together and yet still receive marks for their work. Yet, I have been criticized by colleagues for this approach, as they feel it does not accurately reflect student abilities, and some students will simply copy the answers of their classmates.

Question: Do you think it’s appropriate for student evaluation in secondary school to be partly made up of work that is done in collaboration? 

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