New Zealand's New Curriculum

In her article "More complex than skills: Rethinking the relationship between competencies and curriculum content," Rosemary Hipkins outlines the 2007 overhaul of the New Zealand Curriculum (NZC). The previous NZC of the 1990s was a detailed-oriented curriculum with prescribed outcomes for each subject and grade. With a desire to move away from skill based learning, the curriculum was transformed to focus on five "key competencies" including (1) managing self, (2) relating to others, (3) participating and contributing, (4) thinking, and (5) using language, symbols, and texts. These key competencies act as a framework for individual schools and teachers to build a more detailed curriculum of their own. Hipkins presents an argument for why this transformed curriculum could be beneficial to the modern student, emphasizing that learning should not be the passive acquisition of skills and facts, but should be active, engaging, and participatory. Finally, Hipkins presents an example of how the curriculum is transformed when a key competency is added. She argues that such a change emphasizes the "big picture" and that learning becomes the process of meaning-making for students, rather than acquiring knowledge from the teacher.

The new NZC certainly means well. The intent of building lifelong learners is a novel goal, but I am somewhat worried about the lack of empirical evidence supporting that this sort of change would be beneficial to students. In theory, the new curriculum appears as though there could only be benefits to students' learning. Unfortunately, it seems to be assumed that we live in a world where all teachers and schools are willing and highly capable of engaging with content, students, life-experiences, and the "big picture." How can one assume that all teachers will be able to engage with content at such a deep level? The new NZC puts a great deal of responsibility on individual schools and teachers, and although the author recognizes that schools and teachers should not be expected to work out what is meant by "key competencies" themselves, she does not provide any examples of pathways for schools to do so. Teachers who have been in the classroom for 20+ years may claim that what they have been doing works, and might be unwilling to make such an overhaul. Similarly, new teachers, who have the additional weight of being in a unfamiliar environment, might be overwhelmed with trying to juggle the new experience of working with a classroom full of kids, trying to understand the content they need to teach, as well as how to include these key competencies. Furthermore, if national testing and assessment still aligns with the old, skills-based curriculum, how are schools, teachers, parents, and students supposed to believe that this is actually beneficial to student learning? We've come full circle to the issue of empirical evidence. The goal of the new curriculum foster lifelong learners is of immense value, and is a goal that educators should be striving for in their classrooms. But, it should be noted that transforming theory into practice (particularly at a large scale) does not always turn out the way we hope it might.

Word Problems

This week, I read "Beliefs about word problems" by Greer, Verschaffel, and De Corte. In this article, the authors argue that word problems should not have a bijective relationship with arithmetic computations, but be thought of as exercises in mathematical modeling. The article presents a number of empirical examples of students and student teachers engaging with mathematical word problems that are firmly based in the real world. That is, the problems require the participant to think realistically, rather than simply mapping the problem to an arithmetic computation. They found that only a very small percentage of students and about half of student teachers placed the problems within a real world context. Why it is "ok" to disconnect mathematical word problems from reality, the authors suggest, is due to the beliefs surrounding word problems. They state that although beliefs about word problems are most prevalent in the classroom setting, a broader perspective that includes the school, the education system, and society in general is necessary to understand how beliefs shape mathematical practice. The authors provide the following illustration:

As you can see, I was wondering how and where parents fit into this picture. Just the other day, I came across a friend's post on Facebook which was a video of a mother complaining about the common core standards and advocating for a "just teach the algorithm approach". My friend said something along the lines of "this is stupid and makes me contemplate homeschooling." Of course, I felt that I should at least chime in from a neutral standpoint addressing the pros, cons, and overall intent of the standards. Then, someone replied to me saying "You say the intention is to create independent math thinkers. .... but by frustrating children to the point if tears.... how about we take that money we are spending on this foolish frustrating system.... and we reduce class sizes so teachers can effectively. ... oh I don't know.. TEACH." I gave this woman a well-informed response, but she hasn't said anything back yet. :P

This dialogue was running through my head the entire time that I was reading this article. If parents are so convinced of what constitutes mathematical reasoning and the doing of mathematics, in what ways does this affect the classroom? Moreover, where does the unwillingness to accept that mathematics might be more than plug and chug stem from? How can teachers (who understand and appreciate the new standards) convince parents otherwise?

New SAT Math Questions

Hi all,

Found this article/quiz on the NY Times website and figured I might share it.

http://nyti.ms/23Tqbtg

For a more in-depth article, see http://www.nytimes.com/2016/02/09/us/sat-test-changes.html

New Questions for Old

This week, I read "New Questions for Old" by Prestage and Perks (2001). The authors start out the chapter with stating "take an old question, change it a bit, and hey presto, a new question appears." This was an instant red flag for me and may have influenced my opinion of the chapter. I'll summarize before I state my opinion of the chapter.

The authors recommend four strategies to alter an existing questions: (1) change a bit of an existing question, (2) give the answer, rather than the question, (3) change the resources, and (4) change the format. The authors provide explicit examples of altered questions in each of the categories, why the changes are significant, and how they may have a positive influence on student learning.

Generally, I did not like the examples that the authors provided. I found that many of their examples were arbitrary and would not necessarily help students understand a particular concept better, since they didn't necessarily provide anything meaningful for students to hold onto in the future. For example, the problem "What right-angled triangles can you find with an hypotenuse of 17cm" is an extremely general question. What are students supposed to get out of this exercise? The authors claim that since students have to apply the algorithm many times and make decisions about the number and types of solutions. But do they have a more rich understanding about right triangles or do they just know how to compute better?

Ok....enough of a rant for now. Onto my altered problems! I've chosen a problem from differential calculus, integral calculus, and


For integral calculus:

 It is a very common problem for students to find the volume of a cone using the disk method for solids of rotation. As a change, have students extend this to a tetrahedral of side length a, and then to a cone with an arbitrary polygonal cross-section. This could lead students to thinking about some topology, if we want to take it that far. :P This would be changing the question a bit, according to Prestage and Parks.

For differential calculus: 

Original problem: Show that 2x^2 + x -2 has a zero on the interval (-1,1) using the intermediate value theorem. As an extension, we could ask students to consider any odd degree polynomial and have them prove that the function has at least one zero using the intermediate value theorem.This changes the format, since students cannot do the normal trick of finding explicit x values to evaluate the curve at.

For linear algebra:

Original problem: Given u = [5,2] and v = [-2,4], and the linear transformation T(x) = [.5 0 | 0 .5] [x_1, x_2] (that's supposed to be a 2x2 matrix and a vector in R^2). Find the images of u and v under the transformation T.

As an extension, have students plot this into a computer program and discuss geometrically what T does to each vector x in R^2. As a further extension, transformations in R^3 and do the same in a computer program. This could allow students to better visualize linear transformations in three dimensions. This change would be an example of changing the resources.