Thoughs on the blogging Journey

Taking this class gave me my first experience with running a personal blog. I used to browse others’ blogs for lesson plans and recipes but never shared my feedback, being much more a borrower than a contributor. It took few sessions before I enjoyed the experience of logging in to share my thoughts with classmates, since I originally considered blogging to be just a digital form of traditional writing assessment. After reading other classmates’ posts, I realized that our blogs were much more than assignments.

I noticed that many posts in our class were immediate impressions to ideas mentioned in textbooks or other posts. Some are about experiences that resonate with research; others relate reading material to teaching experiences. Even though posts are published in an asynchronous manner, many suggest ideas that trend for a while. I view blogging in our course as a mixture of group discussion and group assessment. First, I am always aware that I have a larger audience when blogging compared to writing essays that are read by the marker’s eyes only. I feel very comfortable in this new sort of learning culture formed by this regular blogging and weekly meetings. Secondly, with knowing that the professor is a reader instead of a dominant blogger, I feel more comfortable expressing my ideas when compared to classroom discussions. My explanation would be that blogging offers more flexibility in space and time and helps thinking be translated into text more freely. Finally, I believe it helps to improve self-regulation. According to Ormrad, "Self-regulated" describes a process of taking control of and evaluating one's own learning and behaviours. I see opportunities to practice learning efforts that will give rise to academic success.

I really enjoyed the readings regarding problem posing. These readings opened a window for me and significantly impacted my understanding of my research questions.  My research will try to show that Problem-based-learning (PBL) allows students to actively explore various approaches to questions with more than one solution in real-world contexts, hence stimulating and constructing their own understanding of subject matter. In light of the reading, I realized that educational robotics has the potential to provide less time-consuming feedback to students’ questions so students need not wait their turn to receive feedback from teachers. Although I am still working on new models, I believe student posing questions will play a large role in their design.

Another greatest learning has occurred due to the incredible international setting of the classroom. Imagine math educators from west to east, from elementary level to university level, meeting weekly of refreshed minds? It is indeed a valuable eye-opening journey for any educator/researcher.  Thank you all for the excitements, intriguing questions, encouragement to the journey.

 

Well, my questions remain,

1. How do students view educational robots as a mathematics teaching agent? 

 

2. In which ways do elementary students’ dispositions towards mathematics carry over to their further studies?

3. In which ways do students pose math problems outside of school?

Building on community knowledge (Hispanic community)

This paper presents the author’s personal reflection on work in mathematics education in low-income, mostly Latino communities in Tucson. Over more than a decade, the author had been involved in the development of apprenticeship-style approaches to the teaching and learning of mathematics. It is claimed that “the research was driven by an equity agenda that capitalizes on building on the students’ and their families’ knowledge and experiences as resources for schooling” and was driven by the implications for the mathematical education of these children if their experiences and backgrounds were used as resources for learning in the classroom.

One of the key characteristics of the work was the involvement of the community; parents, students and teachers all collaborated to develop curricula. I see the benefits of offering teachers opportunities to learn first-hand about the experiences of the community and reconnect with their students’ families; however, this approach raises more questions about the role of teacher in this approach especially when Leslie (one of the teacher who worked with the author) wrote in reflecting on the impact of household visits: “it provides a real look at the whole child”. I agree that the approach helps the family to know teachers better while allowing teachers to learn ways to respect other cultures and therefore to think about the classroom in the community, but I am concerned with what elements are included in part of this “real” look of the child, and what options are available for the teacher to adjust classroom instruction accordingly.

The proposed approach requires parents (and other adults who are important in the life of their students) to provide resources towards the development of the mathematics learning modules. For example, in the construction module (asking students how to build an extra room in the yard), the teacher allowed parents to answer students’ questions. It was successful in this context, but would it work in a math-intensive situation? Even though the author intended for math to play a role in the construction module, would the students to view the activity as a math activity or just an activity which happened to use math in it? I am lacking confidence that both sample modules can be validated to be mathematical activities in terms of reasoning, abstracting, and generalizing math concepts. The author argues that there are many challenges to be overcome in the pedagogical transformation of household knowledge into mathematical knowledge for the classroom. The author further states that these challenges are related in part to teachers’, students’ and researchers beliefs about what counts as mathematics. Indeed, the activities introduced in the paper were more social than mathematical.

My question is: will apprenticeship in mathematics help a majority of students gain mathematics competence outlined in new curricula?  

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.

Response to " Problem Solving as a Basis for Reform in Curriculum..."

The article argues that reform in math curriculum and instruction should be based on providing opportunities to problematize the subject. The principles of the proposed approach are different from many historic and current views of problem solving.  In contrast of frustrating students with overly difficult tasks,  the article use “problematic” in a sense  to problematize what they study and “ allowing students to wonder why the things are, to inquiry, to search for solutions and resolve incongruities.” A specific example is presented from a 2^nd grade classroom where students were asked to find the difference between 62 and 37 and they were able to solve a seemingly routine math question by using alternative methods. The article concludes that the achievement is due to the choice of problematizing task over conventional teaching. The article argues that widely endorsed application-based approaches cannot fully resolve the misconception caused by a distinction between knowledge acquisition and application. The article draws on John Dewey’s notion of reflective inquiry to establish an alternative view of problem solving approach which advocates practices of problematizing the subject. The article further examines how the approach fits both views of mathematics understanding, functional and structural as well.  

STOP:

The notion of classroom culture is very clever. Given the current trend of advocating problem solving over computation in the classroom, many obstacles need to be well considered in order to provide the promised outcomes. In general, problem solving activities seem to offer fertile ground for interaction between students and instructors in classrooms, but there is not much proof showing that it will teach core concepts effectively and efficiently across a variety of settings. As a teacher, I always find myself needing to make compromises: giving students a challenge produces chaos, but telling students exactly what to do at every step produces boredom and little learning.

Question:

According to the article, real-life problems provide a legitimate context for problematizing mathematics. Will word problems provide a similar environment for engagement?

 

Response to "The Effects of Stereotype Threat.." by Keena Arbuthnot

In this article, Arbuthnot presents the findings that involve two studies that attempt to explain how stereotype threat affects standardized test performance in mathematics. The study concerns black eighth-grade students who are considered influenced by stereotype threat defined as “a social-psychological threat that arises when one is in a situation or doing something for which a negative stereotype about their group applies. According to Steele (1997) (cited by Arbuthnot) “this predicament threatens the individual: he or she fears being negatively stereotyped, judged or treated stereotypically, or inadvertently conforming to the stereotype."

 In the first study, the author presents findings from an analysis of black students’ test performances on standardized mathematics exam questions that do and do not include differential item functioning (DIF) which the author claims to signify a gap that exists between particular groups. The second study explores how stereotype threat influences students’ strategy choices in standardized exams. Findings from the two studies suggest that stereotype threat may have a negative impact on black adolescent students’ test-taking strategies and achievement on standardized mathematics tests.

Stops: 

According to the author, stereotype threat can negatively affect the intellectual performance of African Americans taking the math test, due to the stereotype that African Americans are less intelligent than other groups. This made me wonder what would occur to groups who are under the effect of stereotype promise, the opposite of stereotype threat. Such a group should seemingly benefit from the situation. However, I recall a different study concluding that positive stereotypes form a considerable burden and therefore adversely affect performance in the stereotyped domain. In Arbuthnot’s study, the entire sample consisted of 257 eighth-grade students: 36 Asian American, 159 black non-Hispanic, 40 white, and other participants. According to the author, although all of these students participated in the experiment, only black students are featured in this paper. The study could be more comprehensive and convincing if further analysis was done for other ethnicity groups.  

Question:

There many conditions under which people are likely to apply stereotypes. As teachers, stereotypes cause more problems than benefits, if any. What are the common stereotypes that we experience in K-12 settings?

Davis and Simmt: Mathematics-for-teaching

Davis and Simmt propose a theoretical approach to teach mathematics using mathematics for
teaching. The article argues that the coarse separation of  "teachers' subject matter knowledge" and "formal disciplinary knowledge" is inherently problematic. Rather, they argue that mathematics-for-teaching is likely neither a matter of "more of" nor "to a greater depth than" of an in-service session, but rather a "branch of mathematics". They use classroom multiplication to illustrate four categories of elements involved in  teachers’ mathematics-for-teaching, namely, “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”.

1. In the discussion of teaching multiplication, the authors assert that the understanding multiplication  is not about how many figurative aspects of multiplication are taught but physical actions out of which multiplication arises. First of all, if this is the sudden conclusion of 24 experienced teachers after lengthy discussion of teaching multiplication in classroom, I would be disappointed at the quality of the mathematics courses which qualify those teachers. Strangely, I have to agree with the authors that we do need a special "branch of mathematics" for math teachers.

2. I found that the following illustration in the article was very intriguing especially when I realized that the stem splits to symmetrical halves: stable knowledge and dynamic knowledge.






























Question: Compared to mathematicians, math teachers' expertise lies in the fact that their math is done through interactions with others. Should we value more their knowledge or their ability to engage students in meaningful discussion?

Books exhibited by Susan

I took a photo of the books our professor recommended in class. Will check some out later.

Romeo and Mari

In “ASSESSING UNDERSTANDING IN MATHEMATICS: STEPS TOWARDS AN

OPERATIVE MODEL”, Romero and Mari use an interesting example of a multiplication algorithm to demonstrate the complexity of diagnosis and assessment of understanding in mathematics for middle to high school students. By gathering explanations and interviews on the students’ work, a “multi-faceted assessment strategy” is proposed featuring the use of different tasks with an intention to obtain indicators of students’ understanding of particular mathematics aspects.

The authors assert that “to achieve this, it is important to make the learner feel involved in the situation, so it is convenient for the situations to be specific, simple to understand, without distracting elements and with controlled answers.”  One challenge is always to get students “involved” in the assessment procedure when the assessment is too general or too specific. More studies need to be put into the dimensions and scope of such assessment.

The study provides three categories of assessment when studying student’s use of algorithms and I find the Analytical category to be very innovative. According to the authors, it includes situations that require a reflexive use of the algorithm to lead to a solution rather than simply applying the knowledge. It requires an intentional analysis of the “External structure” as explained in the example of “The product of two consecutive natural numbers is 132. Find them.” This question is used as an analytical assessment of students' understanding of the algorithm, which goes beyond memorization from the original procedure (it is more like a reversed or decomposed procedure that the student is required to reconstruct), and makes it possible to identify analytical understating. 


Question: Are there other ways to assess analytical understanding of math?