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.