Trihexaflexagons

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I’m digging into abstract thinking this week and building connections between abstraction, mathematics, art and epistemic cognition.

Here are some responses I’ve posted to discussion forum topics, questions, and ideas.

  1. I personally think that abstraction promotes interactions and builds relationships where there were none before. Willingham (2009) suggests that having shallow knowledge is better than having no knowledge, and that acquiring deep knowledge is dependent on the attention paid to the details. Vi Hart shares abstractions and relationships developed between mathematical principles, art, music, sound, and all kinds of other fields of interest. Her work is based on playfully paying attention to the details, as well as acquiring deeper knowledge from previous researchers e.g. see this series of videos on trihexaflexagons. Abstraction, in Vi Hart style, is remixing an 1884 romance novella, adapted to geometric shapes and a mobius strip setting [https://www.brainpickings.org/2011/01/20/vi-hart-flatland-on-a-mobius-strip/}. By seeing how Vi Hart creates these abstractions, building connections where there weren’t any before, others can learn beyond our previous shallow knowledge. Just some thoughts on how art, formula, and observation are tied together in abstract thinking.
  2. Have you seen Dan Meyer’s TED Talk on math and problem solving. It’s not one I use directly, but reference, since it connects to media production and digital literacy connections to mathematics instruction. His blog site is a rich repository of math teaching based on the ‘be less helpful’ mantra he mentions in the video. I can’t help but wonder how this perspective fits into the ‘back to basics’ push that will happen in Ontario education, particularly if the use of mobile technologies is also curbed at the same time. Dan Meyer, in 2010, suggests it’s an interesting time to be a math teacher. In 2018, it’s an interesting time to be a teacher. Period. And an even more interesting time to be a teacher of teachers! Exclamation!!
  3. I see a connection between your thoughts and the use of manipulatives when introducing algebraic thinking. Since I was not taught with this technique, it took some time for me to get my cognitive processes around using algebra tiles when teaching math concepts to grade 7 & 8 students, just as Gibbons & Bush (2015) reveal. Willingham (2009) suggests that “understanding new ideas is mostly a matter of getting the right old ideas into working memory” (p. 91). Since students are now familiar with the use of manipulatives well into their middle and senior elementary school years, it’s logical that these old ideas of manipulating mathematic concepts from concrete to abstract, for example using base ten blocks to manipulate multiplication equations, will ultimately help students rearrange them, make comparisons, and think about features that were previously ignored, when they begin constructing algebraic equations with algebra tiles.What the political agenda, from their perspective of how mathematics is taught, needs to grasp, is that the basics are embedded into this learning. Just because a grade six student can’t rhyme off the multiplication facts to 12 doesn’t mean they don’t have deeper understanding and are primed for abstraction when math facts are required in new or meaningful  contexts. This political ‘hot-button’ issue of going back to the basics may ultimately impact the deeper knowledge required to make abstraction easier in future learning, in my opinion.
  4. This is an interesting connection to abstract thinking! I had to search and read a bit about epistemic cognition to understand what it is referencing and will continue to read more from the references you’ve provided. In the context of teacher education, I agree that there is a real need to rethink how teachers are being prepared for the complex and abstract world of teaching since it’s not just about art or algebra. Willingham (2009) talks about the notion of deep knowledge – this caused my mind to abstractly build a relationship to my prior knowledge of the work done by Fullan, Quinn & McEarchern (2017). This is exactly the type of connected thinking, beyond the surface or rote knowledge teacher educators have about topics in the field, to the whole interconnected complexity of teaching.One challenge is the barrier Faculties of Education, and academia in general, often impose on student assignments and learning tasks, that silo their learning, e.g. in math class we only learn about teaching math. I push students to make these cross subject, cross course connections, encouraging them to make connections and think about topics from multiple perspectives. In a sense, this primes my 20 students to “know more about the subject, and the pieces of knowledge are more richly interconnected” (Willingham, 2009, p. 95). They may not have deep understanding, but their minds are better prepared to make the abstract leap described by Goldwater & Getner (2015). I’d like to think that by making connections between math, media, and digital literacies, I’m explicitly teaching teacher candidates to practice abstraction and building deeper knowledge. I have lots more thinking to do with your connection to epistemic cognition – thanks for building this connection for me.
  5. I wonder if it’s more than just a negative attitude that interferes with our ability to persevere with abstract thinking. Abstraction is a key goal in school (Willingham, 2009) which is more than just being able to understand art or solve algebraic equations. It’s being able to see relationships, as Willingham (2009) suggests, between measuring the area of a table top and applying that learning, transferring that skill, to measuring the area of a soccer field. So the practice of transferring a problem solving strategy from one location to another is important.The typical teacher response when students say they’ll never need to know algebra, for example, is ‘you’ll never know when you’ll need this’. This gets to the core underlying thinking skills that teaching algebra brings. My teacher candidates are crafting lessons using a lengthy lesson plan template and complain that they’ll never need this when they’re teachers. I agree, but it’s the foundational skills they are building to be able to transfer this lesson planning skill to a variety of lessons, classroom contexts, and grade level situations that is a necessary outcome of this preliminary work. Just as teaching basic algebra is in Gr. 7 & 8 – they’ll never know when the foundational problem solving and thinking skills will transfer to new and unexpected scenarios. So the strategy that Willingham (2009) suggests is an important one – let students in on the secret of why it’s important to learn the  specific skills, but also the underlying metacognitive skills that give it purpose.

References

Fullan, M., Quinn, J., & McEachern, J. (2017). Deep learning: Engage the world change the world. Corwin Press. Retrieved from http://npdl.global/deep-learning-book/

Goldwater, M. B., & Gentner, D. (2015). On the acquisition of abstract knowledge: Structural alignment and explication in learning causal system categories. Cognition, 137, 137-153. doi:10.1016/j.cognition.2014.12.001

Gibbons, K. & Bush, S. (2015, May 11). Advocating for algebra tiles. [weblog post]. National Council for Teaching of Mathematics Publications. Retrieved from https://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/Blog/Advocating-for-Algebra-Tiles/

Goldwater, M. B., & Gentner, D. (2015). On the acquisition of abstract knowledge: Structural alignment and explication in learning causal system categories. Cognition, 137, 137-153. doi:10.1016/j.cognition.2014.12.001

Willingham, D. T. (2009). Why don’t students like school? San Francisco, CA: Jossey Bass.

Image Attribution: Photo by Luis Alfonso Orellana on Unsplash