Comments Regarding Chapter 6

Posted at Aug 30/2018 12:33PM by Stu 4:
The book's discussion of individual differences (pg. 94-95) intrigued me because it implied that differentiated instruction is a byproduct of American culture. To be clear, I am referring to differentiated instruction in the vein that each student should be given instruction, classwork, and projects that are uniquely tailored to their abilities. Examples include simplifying assessments for low level students, providing enrichment activities for high level students, and the practice of tracking students based on academic ability (e.g. honors algebra vs. traditional algebra).

It appears that the Japanese philosophy (as described by the book) is that all children can learn something from the same activities, and they assume that the depth of understanding will be different for all students.

The Japanese approach seems to better embody one of the National Council for Teachers of Mathematic's (NCTM) "Effective Mathematics Teaching Practices" given below.

"Implement tasks that promote reasoning and problem solving.

"Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies." (National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.)

The key phrase in this to me is "multiple entry points." This gives the indication that mathematical ideas do not (and should not) have strictly one method or procedure used to solve them.

I've probably said enough for now... this post is getting pretty long.

Posted at Aug 30/2018 12:40PM by Stu 4:
Another thought--I'm wondering if the Japanese would prefer an integrated math curriculum as opposed to a traditional American sequence of Algebra I, Geometry/Algebra II, Trigonometry/Precalculus...

I have no idea. Is this even applicable?

Posted at Aug 30/2018 03:14PM by Stu 9:
The authors of The Teaching Gap claim that minimizing student frustration and confusion while working complex math problems is a cultural belief. It is then a widely accepted norm in education, for American teachers, that if a student is frustrated or confused this implies that the teacher has not effectively taught the concept. When in fact, if one has taken an upper division math course, they know this is not the case at all. The only way to solve a complex analytical problem is to bang your head against it. What I mean by this is to try different solution methods and refine what seems to be the best approach. Creating and testing hypotheses is fundamental to the scientific method. Why then is it culturally accepted in the US to "give" students the answer. What I mean by this is not giving them the direct answer to that specific problem, but rather immediately giving them the solution. I am personally pondering when exactly I made this jump in my own academic career.

Posted at Aug 30/2018 03:16PM by Stu 9:
Stu 4, I wonder if a Japanese math teacher would take the opportunity to extend a student's question in a lesson. If they have the learning goal to extend student's knowledge to engage in developing new ways to solve problems, I do not think that Japanese teachers would parse subject matter in this way into the different silos of algebra, geometry, and trig.

Posted at Aug 30/2018 03:53PM by Stu 9:
What is an example of a script you have experienced while teaching?

Posted at Aug 31/2018 07:00AM:
Pat: I have English translations of Japanese textbooks for grades 7-9 from the 1990's if anyone is interested. And, yes, @Stu 4, they employ an integrated curriculum rather than our approach of cutting up secondary mathematics into algebra, geometry, algebra, precalculus, and calculus.

Posted at Aug 31/2018 10:14AM by Stu 1:
Prior to answering Stu 9's question about an example of a script I have experienced while teaching, I wanted to point out something I found interesting. On page 87, the book discusses how teachers and students share the same script, and therefore know what roles to play in class. This is very similar to the hidden contract concept I mentioned in class. An example of a script I experienced during my k-12 math education involved the teacher always providing an example of every type of problem that we students would encounter on a homework assignment or assessment. It seemed as though students held an expectation that teachers would refrain from assigning a problem that was not discussed in class. This script is very different from the Japanese script in which teachers believe students learn best by first struggling through a problem.

Posted at Aug 31/2018 09:44PM by Stu 6:
Stu 9, I considered a similar thought. When does the ideas of learning procedures become learning about solving difficult problems without already knowing how. My thought was this occurs at the transition from high school to college. But it may be more on the idea that if you reach a certain level of mathematics, such as Calculus. However, I have contradictory evidence for both. Meaning I have seen a Calculus class taught in high school and college be very procedural. However, I have only seen a conceptual Calculus course taught in a college setting. Doesn’t mean it isn’t done this way somewhere just not common. I’m not sure if it is connected to the idea that students become self responsible at the college level and it is now okay to let them struggle. Thus, the script changes. But if it changes in this way then it is done without many students’ knowledge or understanding? Therefore a new script is written when mathematics courses are learned in a college setting AND at a certain level.

Posted at Aug 31/2018 09:51PM by Stu 6:
Teaching as a cultural activity allows us to examine how and why we make choices about our teaching practices. We are able to analyze the Japanese script of teaching and see how ours differs. But that doesn’t mean all people want to change the system of teaching or its culture. There are teachers that I have met that became teachers because they like the U.S. script. They enjoy teaching and learning mathematics as a set of procedures. They believe that this is a good way of knowing mathematics. Teaching as a cultural activity means we have a way to look at it and discuss it. However, we have to believe that the different system is an improvement if we want to change the script. The authors point out that changing one attribute does not change the system. Rather the system merely mutates the change to fit the current script. This prevents systemic change. Yes, and those who do not want it changed also prevent change. It is a desire to maintain the script.

Posted at Sep 01/2018 11:56AM by Stu 3:
Stu 4: I wonder if Japenese teachers even think of them as distinct subjects like we tend to or if it is integrated for them because they see it all as a part of one whole?

Posted at Sep 01/2018 12:01PM by Stu 3:
Stu 9: I think the desire to swoop in and save students who have any level of frustration is rooted in the cultural idea that it is the teachers responsibility to give the students the how. Struggle then may mean they didn’t give the ideas well enough. I think the Japanese teachers see their jobs very differently (much more of a guide then a dispenser of knowledge). I think the different cultural views of what they are supposed to do explains (at least in part) the different reactions to struggle.

Posted at Sep 02/2018 01:33PM by Stu 9:
@Stu 3, after watching the videos Pat posted to Course Forum, I 100% agree with you. The comparison between the teaching styles of the Japanese and American teacher were easily recognizable once it was contextualized for me in that way.

Posted at Sep 02/2018 02:41PM by Stu 5:
I think viewing teaching as a cultural activity (TACA) is very beneficial to understanding the complexities involved in trying to change teaching. If we think about teaching the way we think about any other cultural norm, we begin to see how difficult it will be to change our behaviors. Changing how we greet people or the clothes we wear would be incredibly difficult to do nationwide. And, if just a few people tried to make such changes, they would be ostracized for their actions.

The way we view so many areas of education (teaching, learning, mathematics, school, intelligence) are seen throughout our culture. If someone makes a mistake, they will frequently say "I'm so dumb" implying an inherent characteristic about themselves. Verses, they could say "that action was dumb" which makes no assumptions about their personal abilities.

I think that making cultural shifts like this is necessary to make long lasting change in education. How can we try making these changes?

For now, my own answer to this question is to pay attention to how I discuss education and teaching with others. My language should reflect that teaching is something that can be improved upon with practice, not an inherit skill, and intelligence should be treated in the same way.

Posted at Sep 02/2018 04:56PM by Stu 8:
Viewing teaching as a cultural activity was very beneficial for my understanding as well. I recall becoming somewhat cynical in college as I watched teacher after teacher try to implement experimental ideas (usually with little success) because they were told these ideas would improve teaching. The issue can now be reframed as pertaining to the cultural values and beliefs rather than the practices involved with a lesson (although these are important, they seem to be more of a result than a cause of the problem).

Reading about the TACA view allowed me to think of teaching as part of a layered cultural system. Perhaps teaching and other cultural activities are influenced significantly by the larger culture in which they are performed. This larger culture provides the script for how to teach in a way that fits the culture. When TACA is influenced by this larger culture, the specific practices and methods used in a lesson are developed according to what would fit the design of the lesson as pert of teaching as a cultural activity. I like the idea the authors present that when reform of a few methods/practices is attempted, the result is more often a faulty method (the method cannot function without an overarching structure that supports it) and a dysfunctional teaching system (the irregular method causes holes in the mostly unchanged TACA system) than the desired improvement.

Perhaps many teachers have good intentions and want students to gain more from their lessons than they currently do. But it is unlikely very many teachers know how to do this or even what these gains are.

Posted at Sep 03/2018 09:22PM by Stu 7:
Commenting on Stu 1's post I have had a similar experience. I am even nervous about assigning problems as an instructor now. This may be connected to the teacher from the video in the first lecture, when she expressed that she was afraid that students would fail. From the book we can see it results in limiting students' experiences with math. Unfortunately, this is how the unproductive cycle persists.

The way Stu 5 described how students discuss their mistakes is evident in my actions as well as my previous classmates. If I or another classmate asked a question that seemed trivial to the rest of the class, they tend to think that their question may have been "dumb" or "stupid." Then this can lead to thinking that everyone around them knows more than they do. It's as Stu 5 mentioned, we see learning as an inherit skill, rather than a process that can always be revised, refined and improved.

Posted at Sep 03/2018 10:14PM by Stu 2:
It is easy to see as a student, who is experiencing the education system in the U.S., implements that the repetition to learn the skills of how to deal/work with procedures to solve for a solution is a common normality. To Stu 4: I have seen where some school systems try to have integrated mathematics (in k-8) to make sure that every student, no matter their success level with math are all on the same page. For example, here at ASU, Dr. Calra van de Sande, had a conversation with me about how in certain states at each grade level the same key topics are discussed every year, but as they progress “the more complex and in depth” of material is taught. Which in reality, does not successfully work. Most teachers focus on the procedure aspect than conceptual. The cultural activities can change (most likely not successful), but what are some examples on how you think we can improve the cultural scripts for teaching and express the possibility of effective reforms? The Teaching Gap mentions that if this is possible, then the improvement of the skills of individual teachers shall prevail.

Posted at Sep 03/2018 10:26PM by Stu 2:
Also, another thing. To what Stu 6 said about the script changing once students go to college is the common transition in cultural scripts. I had a very rare occasion in high school where my statistics teacher changed the script of a traditional problem solving & learning procedures type of class room to procedural. I can admit to myself that I first started falling in love with math because it was "easy" to catch on and solve for this or manipulate that. My statistics teacher instead introduced to use a formula sheet. In the eyes of a student in 10th grade that was very exciting. We KNEW we would pass the class. But when the first test came and we could use our formula sheets. The test seemed to be not so of a clear cut answer, but intended on us to use the formulas correctly & conceptually what each of the formulas meant. Understanding the difference between a sample mean and a population mean. The homework were something we had to write solutions to & we did not go over them in class, but we were given the solutions if we came after school to understand them. Made us have to think more. This lead me to wanting to teach in the same manner, & believed ALL classes in the k-12 could be done in a similar way. And not be influenced by the larger culture that performed in the structure of the classroom. Yes, it is a desire to maintain the script, but bravery to whom ever embraces change (correctly).

Posted at Sep 04/2018 11:18AM by Stu 7:
Stu 2: That's really interesting how your teacher made you use the formula sheet. I have only witnessed teachers using a formula sheet to help the students on the test because they thought memorizing everything "wasn't how the real world worked anymore." (basically they used it as a safety net. For example, in my lower level stats class, my professor said that in today's society and unnecessary. So she taught us only the procedural part using a calculator. This is one of the ways that the book mentions using technology in a way the limits student learning.