This page serves as a sort of journal for Geoff’s doctoral program, separate from the Emergentmath blog. Here, Geoff will share thoughts and reflections on classes, insights into literature and other PhD-related musings. While you’re here, feel free to explore Geoff’s main blog, various “miniseries,” and select blog posts. Here is Geoff’s About Me page.

Week 4

Journal: Writing Every Day
Research Reflection: A Lack of Problem-Posing in Math Textbooks (Cai & Jiang, 2017)

Tuesday 9/15/2020

Journal: Writing every day (How’s that for meta?)

This week in Intro to Doctoral studies, we had a guest speaker, Dr. Nycole Courtey, VP of Student Affairs. Her dissertation was tied with keeping a work-life balance while conducting doctoral work. She offered the class her wisdom based on her expertise as well as her life experience. 

Among her sage wisdom was an imploration to “write every day, even if all you do is write word ‘the.’” I’ve found this to be undeniably true when working on a large undertaking, whether it be for work or for other work. 

Personally, there’s a catch for me. My most productive writing occurs in one of two settings:

Setting #1: Several hours at a coffee shop where the wifi is pretty good, but not great.
Setting #2: Late at night when the house is quiet. 

In setting #1 (the coffee shop) it sometimes took hours of staring, typing, deleting, and caffeinating before I became productive. Unfortunately, this setting is more-or-less unavailable currently. Even if I were to have access to a multi-hour coffee shop session, it’d be hard to do while facilitating the kids’ online learning. As long as they’re being remotely schooled, it’s hard to imaging a ton of day writing.

Setting #2 is more of an option, but everything’s pushed back. In the past, the kids went to bed around 8pm, giving me plenty of evening hours to, once again, stare at the screen, type, and delete (no caffeinating in the evening). Now the kids are older and they are typically off to bed around 10pm. Unless I’m willing to stay up later than I’d like, it’s hard to allow for the same amount of staring, typing, and deleting. I’ve either got to get to the brass tax of writing or call it an evening. 

At some point, either I’ll adapt or the kids will be back in school. And I can go back to writing more productively, even if the first hour of it I can only crank out the word, “the.”

Thursday 9/17/2020

Research Reflection: A Lack of Problem-Posing in Math Textbooks (Cai & Jiang, 2017)

Cai and Jiang (2017) examine U.S. and Chinese textbooks to determine the frequency of “problem-posing” tasks. These are not to be confused with “problem-solving” tasks, in which students must decode and divine their way to solving a problem. Problem-posing tasks solicit the student to develop questions or alternative problems based on a mathematical concept. Motivating their study are some cited works that suggest that problem-posing can yield conceptual understanding and promote mathematical communication and reasoning. Problem-posing tasks are cognitively demanding. They provide four categories of problem-posing tasks:

  1. Posing a problem that matches the given arithmetic operations. Their example: “‘Make up a word problem orally for 14 + 8 = ?’”
  2. Posing variations of a question with similar mathematical relationship or structure. An example: Change one of the pieces of given information and repose the question.
  3. Posing additional questions based on the given information and a sample question. 
  4. Posing questions based on given information. For example: given some information, “‘what problem can you pose? Can you solve it?’’” (p. 1527).

They analyzed elementary textbooks from China as well as the U.S.’s Everyday Mathematics and Investigations. Here are a couple of their findings:

  • The U.S. textbooks had a decent amount of problem-posing of types 1 and 2, but very few of type 3, and virtually none of type four. Investigations contained zero problem-posing tasks of type 4. 
  • When compared with older versions of the textbook series, it appears as if publishers are incorporating more problem-posing tasks, but it’s still a quite small number.
  • The distribution of problem-posing tasks vary wildly by grade level. That is, some grade levels include a lot of problem-posing tasks, others very little. 

Just because problem-posing tasks do not appear much in curricula, doesn’t mean they don’t occur in class, but it probably is a decent proxy. The lack of problem-posing tasks in textbook reflects a lack of imagination from publishers with the typical problem sets at the end of a section: after all, problem sets are supposed to have singular correct answers! 

It’s worth noting that this publication appeared in 2017 (the analysis occurred for publications in prior years). While that seems (and is) relatively recent, there have been tremendous strides in math curricula over just the past few years. Problem-solving centric curricula such as Illustrative Mathematics was only in its infancy at the time. While I can’t say for certain, I’d guess an analysis would show that curricula that emphasized problem-solving would more often emphasize problem-posing. Moreover, thanks to the efforts of the highly online math community, routines such as “Notice and Wonder,” a problem-posing routine, would be generative to incorporate more of what Cai and Jiang (2017) recommend. Recently, I’ve seen teachers asking students to create their own “Which One Doesn’t Belong?”s. And while the online community represents only a scintilla of math educators, it does have an outsize voice. 

Cai, J., & Jiang, C. (2017). An Analysis of Problem-Posing Tasks in Chinese and US Elementary Mathematics Textbooks. International Journal of Science and Mathematics Education, 15(8), 1521–1540. https://doi.org/10.1007/s10763-016-9758-2