Problem Posing in Mathematics Education

Summary
The authors, Brown and Walter, highlight the importance of problem posing in math education to improve student learning. They begin by defining what appears to be a math problem is not really a problem. Their first example shows that finding Pythagorean triples in x^2 + y^2 = z^2 is not a problem per se because it is not contextualized in any way. In contrast, their next example is a simple triangle which lends itself to problem-posing more easily when there is no other information given. Students may pose problems about this triangle like finding its total area or dividing the triangle into 3 equal parts. Later, they suggest a strategy for posing a problem as in the famous Konigsberg bridge problem where one can keep all the original information of the problem and change only the question. In fact, historically, this original problem when re-posed repeatedly led to the invention of graph theory.

Next, the authors talk about the benefits of posing problems. By engaging with problem-posing, students can see the interrelationship between posing and solving. This activity gives students an opportunity to restructure the situation described in the original problem and allows them to find new connections that they don't expect to see by problem-solving. Finally, other benefits include promoting creativity, incorporating student learning into social settings, and reducing fears and anxieties caused by problem-solving.

Reflections
I do recognize problem-posing as an essential ingredient of both teaching and learning math. It seems to me that problem-solving (like an object) and problem-posing (like a mirror image) are complementary activities that act as "opposites". I associate posing/solving with "opposites" like  multiplication/division, squaring/square rooting, differentiation/integration, and so on. I think it is easier to do the former (like walking forward) in each pair than it is to do the latter (like walking backwards). When I tutor students privately in math, they may be struggling with a math concept which prevents them from solving the problem with this concept. I have to admit that in the past, I used to be afraid to take risks when posing problems for students to highlight certain aspects of a math concept. The reason is that I was not sure whether my posed problems would be solvable, yet some of these unsolvable problems offered good teaching moments. In addition, these moments also turned out to be good learning experiences for my students. Now, although I am not afraid to take risks anymore, I intend to give my students the opportunity to pose their own problems by modifying the original problem. Through posing, as they explore ideas, they reflect on every action they take and generate new questions to improve the flexibility of their thinking. On the other hand, most of my
low-achieving students struggle with posing more badly than they do with solving because they think posing a problem is like walking blind-folded.

On the Dual Nature of Mathematical Conceptions

Summary
The article talks about the relationship between operational (a process) and structural (a product) conceptions of how abstract math concepts are formed. A structural approach focuses on higher-level thinking, whereas a conceptual approach is grounded on algorithmic skills. Although a structural conception is more abstract and more integrative than operational thinking which is sequential and computational, they are complementary. In addition, Sfard states that abstract concepts are formed in the following three phases.

(1) interiorization which helps you learn a new concept through a process (for example, "counting leads to natural numbers (p18).")

(2) condensation which allows you to combine multiple processes to make generalization

(3) reification which helps you "convert abstract concepts into compact ... new ... self-contained (p14)" ideas

Next, Sfard talks about why historical mathematicians found it difficult to reify the concepts of numbers and functions. The reason is that mathematicians considered and defined these concepts operationally before they did them structurally. Whenever a mathematician defined properties of a number or a function through some operational manipulations, someone else found a new abstract object from these processes and re-defined the properties. As a result, the development of these concepts became cyclic before they were fully reified in structural form. In modern times, mathematicians put a greater emphasis on structural development when studying math concepts.

Lastly, Sfard emphasizes the beneficial effects of having both structure (related to reasons) and operation (related to rules) on student learning. Without one or the other, students will have difficulty in developing good procedural and conceptual understandings of math.

Reflections
In my tutoring practice, I frequently notice that some of my students have a good conceptual knowledge of math concepts shown by their abilities to reason. But, they cannot demonstrate their knowledge on my practice tests and on their school tests. In contrast, some of my other students who achieve high scores on their school math tests do not seem to have a sound understanding of math principles. In the first case, the students whose goal is to understand the structural aspects of math concepts seem to suffer badly in the current education system which favorbly assesses learners' operational proficiency. Memorizing rules promotes operational skills but hinders reification of abstract concepts. That said, my students who learn math procedurally without reasoning skills really enjoy the rewards of their high grades on their school math tests. They always want me to tell them which methods work best for which math problems. Unfortunatly, the reality is that my students, especially in senior grades, come to me with lots of drills (basically, 10 - 15 questions on  each concept) from their math teachers. These questions are no more than fillers preventing students from interiorizing, condensing, and reifying abstract concepts.

Relational understanding requires a lot of time and effort, whereas it is much easier and faster to get the right answer through procedural thinking. How can we find a balance between mastery of computations and relational understanding given the large amount of content to be covered across the high school math curricula?

Strong is the Silence: Challenging the Interlocking Systems

This article addresses issues on the interlocking systems of privilege and oppression in math education. These systems benefit the privileged people and hurt those oppressed by injustices. The authors, prospective teachers, and practicing teachers open conversations about their identities and experience in working with students of diverse backgrounds and how these systems operate against the students. Through these discussions, the authors intend to develop educational programs to better prepare future math teachers and math teacher educators to handle these issues in a diverse classroom.

It seems to me that white children or even some white teachers may be ignorant about the math skills and knowledge that minority groups bring with them to the classroom. The authors state that math teachers set lower expectations for students in the minority groups. I believe that the lower expectations may make these students feel inferior to their white peers. The minority students need an equitable opportunity to demonstrate their unique math skills to others. So, everyone in class can reflect on and evaluate the diverse perspectives on math problems. Such communication may help students learn from each other's viewpoints and respect cultural differences. Is it possible that curriculum developers make math content more engaging to children of different cultures on an equitable basis if more educators from minority groups are involved in curriculum development?

No one is born with a racist or classist attitude. In an attempt to eliminate the interlocking systems, the authors address issues where privilege benefits white students and oppression hurts their non-white peers in the context of racism and classism. However, the authors seem to overlook the possiblity that racism and classism can exist within minority groups. If this happens, are there any resources available that math educators can use to resolve these cultural conflicts within the ethnic minorities?

A Research Programme for Mathematics Education

After reading the title, I predict that the article will focus on students' mathematical thinking. My

reading of the first paragraph makes me realize that this article covers a range of research ideas related to learning and teaching math. The project coordinator, Wheeler, publishes the proposals of Nesher, Bell, and Gattegno, who are the co-authors of the article.

Among the three educators, Nesher seems to be most concerned about interdisciplinary aspects

of math acquisition. He points out that the general public is strongly against “New Math” and demands “back to basics” because students are struggling with New Math. She wonders what makes math so difficult for students to learn compared to how easily they learn and master natural languages. She believes that current research totally overlooks the cognitive processes involved in learning math. To improve the ways students learn math, she suggests that a new research program examine the types of cognitive processes involved in learning math “as a language system (p.27)” in relation to its real-life interpretation and practicality as well as artificial intelligence, psychology, and sociology.


On the other hand, Bell suggests a program that examines methods of teaching which may affect
students' understanding of math. The reason he chooses to look at the teaching aspect is that many
students cannot understand fully the math concepts taught in their lessons. He points out that many
students can add two positive numbers or two negative ones or subtract a larger negative number from a smaller negative one accurately. But, some fail to subtract a smaller negative number from a larger negative one (for example, -5 – -12) logically without using a number line. Lastly, she believes that research should look at ways of teaching which help students understand the meanings of the math concepts taught and eliminate their math misconceptions.

From a mathematician's perpective, Gattegno states that the three “mother structures - order, algebra and topological (p29)” are fundamental to the development of classical math. He relates them to the structures and activities of the human mind on which math education needs to be based. He thinks that students generally learn math through seeing, hearing and other senses and that computer

technology can improve their learning of math. Therefore, he believes that a research program
examining the effect of technology on students' ways of learning math can help develop a better math
curriculum.

I think all of the authors' perspectives are valid and interesting. On the one hand, Nesher and Bell have different focal points on math educational research. While Nesher talks about the importance

of learning math effectively, Bell emphasizes the instructional aspects of math education. I believe that effective learning cannot occur without effective teaching and vice versa. To me, both are equally significant. Since there is no single teaching method best for teaching all math concepts, it is important to examine a variety of teaching techniques to improve student learning. In the same way, there is no one method most effective in learning math. Since different students have different learning styles, they can develop multiple ways of learning math concepts. Their preferred learning methods may shape lesson planning and class activities. On the other hand, Nesher and Gattegno consider math education from interdisciplinary views on their research interests which focus on cognitive processes and computer technology. These aspects are certainly important. Today's technology is more advanced than when this article was published. The use of computers can certainly increase students' interest and understanding of math. In terms cognition, if we think about how children think mathematically through task-based interviews, we can come up with different ways of teaching the subject in a fun-filled manner.