Reflection on Curricular Change

The chapter discusses the key factors that form national curricula and the interplay among the nature of social values, goals, cultures, tradition and curriculum change. Though curricular change is commonly viewed as evolutionary to respond to the surrounding political and academic settings, there is no international trend evidently noticeable. Different nations aim in different directions due to different premises about the nature of thinking and learning. Focusing on curricular trends in the United States, the chapter briefly mentions changes in other 6 countries, including 4 European nations and two Asian countries.

The anecdote in the chapter cleverly catches differences in values and goals between Western and Eastern nations: a Korean student questioned an American teacher’s approach of class discussion by asking why she was listening to other students, even though “the teacher knows best”. To my understanding, national pedagogical styles differ more greatly than the capacity of students. For example, in China, it is the teachers’ sole responsibility to unpack mathematical concepts and to deliver lessons with clarity and depth, while it is the students’ responsibility to acquire the information and understand it in class and after class if necessary. Not surprisingly, we witness at the same time different outcomes and ideas about teaching and learning in TIMMS and PISA assessments.

Another interesting discovery made by the author showed that teaching trends in England are oscillatory and bounced between problem-solving and basic skills approaches. The current (I found online)curriculum adopted many surprising changes to mathematics teaching which East Asian countries have started to phase out:

• ·         Five-year-olds will be expected to learn to count up to 100 (compared to 20 under the current curriculum) and learn number bonds to 20 (currently up to 10)
• ·         Simple fractions (1/4 and 1/2) will be taught from KS1, and by the end of primary school, children should be able to convert decimal fractions to simple fractions (e.g. 0.375 = 3/8)
• ·         By the age of nine, children will be expected to know times tables up to 12x12 (currently 10x10 by the end of primary school)
• ·         Calculators will not be used at all in primary schools, to encourage mental arithmetic

Question: Many countries adopt other countries’ curricula. For example, California has promoted Singapore math; Japan is inspired by the problem-based curriculum from the United States. However, according to the author, curriculum is only a tool in teachers’ hands, and its success depends on its successful integration with the culture. How do we prepare ourselves to teach “foreign” curriculum borrowed from other countries?

Response to "The Answer Is Really 4.5" Beliefs About Word Problem

This study provides some insight into current school practices in relation to word problems, which authors claim to result in a "suspension of sense-making". It is suggested that the beliefs of students and teachers towards word problems significantly contribute to the development of tendencies among children to disregard realistic considerations in arithmetic word problem solving. The author suggests two main purposes of word problems in schools: The first main purpose is practising solving classes of problems that represent applications of mathematics to physical and social phenomena, and the second, to be vehicles for reasoning about mathematical structures, using imaginable manipulatives which often not realistic. The chapter focuses on the first purpose, studying word problems that describe practical situations arising in people’s lives. The authors claim that students develop their beliefs towards word problems during the classroom activities in which they are engaged:

“Students’ beliefs develop from their perceptions and interpretations of the socio-mathematical norms that determine –explicitly to some extent, but mainly implicitly – how to behave in a mathematics class, how to think, how to communicate with the teacher, and so on.”

The study further suggests that teachers are at fault for sharing the belief that realistic considerations should not interfere with the "real" mathematics principles that word problems are intended to relate to, which assumedly impact on their actual teaching and on their students' learning processes and outcomes.

STOP 

I find examples of beliefs specific to word problems quite convincing for the argument that our practice of word problems is built on shaky ground. Let’s look at two of those beliefs that students often hold:

There is a single, correct, and precise numerical answer.

Violations of your knowledge or intuitions about the everyday world should be ignored.

These findings and many others suggest, indeed, that many children in elementary school develop very specific beliefs that a word problem should be solved by simple operations of numbers embedded in the text, leading to the confusion of many teachers: “Tim did the same word question correctly yesterday but he made it wrong today”. Maybe Tim just tried adding two numbers yesterday and he decided to subtract them today, with no thought to the actual meaning of the problem.

The experimental class introduced in the study also provides many constructive insights. I would give it a try by replacing some “routine” or “classic” problems by some non-routine problems designed to provoke awareness about the complexities involved in taking into account realistic considerations such as “the fastest way running from classroom to the gym”.

  

Question:

According to the author, it is essential to establish a classroom culture by explicitly enhancing mathematical norms about what counted a good solution; what resources and support are available for school teachers? 

The What-If-Not strategy in action

This chapter uses several classroom examples to demonstrate the 'What-If-Not?' strategy with respect to problem posing and problem solving in mathematics teaching. The strategy is developed in 4 stages: 1. Attributes Listing, which brainstorms mathematical facts and previous knowledge to re-examine the given question and list the attributes of the question; 2. What-If-Noting, which encourages modification and adjustment of certain attributes to yield closely related questions; 3.Question Asking, generating new questions by modifying attributes of the original question; and then 4. Problem Analysing where students focus on interesting investigations. The strategy enables reducing rigid teaching while remaining focused on the topic.

Stop:

One of the obvious merits of the What-If-Not approach is that it makes students believe that there is more than one way to approach a question, rather than simply focusing on finding the “right answer”. We can expect that when the approach is actively implemented, students will have more opportunities to discuss their ideas and give more consideration to the meaning of the problem. However, some factors might affect the learning outcomes. Classroom culture and motivation are two of those factors. For example, how do teachers encourage less capable students to pose questions and maintain their focus on the activities? I have observed that some teachers do not prefer such students posing problems because they believe the approach is less helpful for these students than traditional learning. But if we want to our students to solve their problems, problem posing is a very important activity that encourages students to think creatively and critically. I will look into more research of the matter.

The author suggested that the value of the problem posing approach is not merely having the potential to engage students in creative activity, but for teachers to gain valuable knowledge during each stage of What-If-Not practice. Since transferring mathematical facts and computation is less stressed in the approach, teachers are able to pay attention to discoveries or errors made by the students, instead of giving the “right” answer. Students will make teachers aware of the significance of mathematical theorems and algorithms by witnessing the consequences of violating essential conditions of theorems and algorithms.

In the author's example below, we can see an instance of a student making a modification to the standard long division algorithm, yielding a negative number as a remainder. As teachers, we must understand the significance of having a negative remainder - what does it mean, and what does it teach us about the algorithm? In this case, some might argue that the remainder is by definition positive, but it ultimately depends on the lesson that the teacher is trying to convey.