Heather struggled with reconciling her explanation and the new knowledge; the ability to try and see it from another perspective was blocked out, favouring instead her own understanding of how the seasons happened. Despite the fact that she struggled to explain it, her persistence in perpetuating knowledge that she believed to be true was staggering, or as Shapiro writes, “relat[ing] it to already existing ideas or to language which [she] already possesses” (Shapiro, 1988, p.99). As a result and as information for my own practice, I take this as a cry to have kids explain their thinking as much as possible, and to rely on tools like the Thinking Routines to help clarify and to get students to talk about their understandings at every opportunity, in order to expel the myths that may come up as a result of digging deeper.
I chose to explore equivalence and how the equals sign can be taken to mean “resulting in” or “computes to”. Vermeulen and Meyer describe this as students having an operational view of the equals sign, despite its relational meaning. Essentially, seeing the equals sign as a function of computation, and as such, students being likely to “reject equations such as 8=8 as false, because there is no obvious action” (Vermeulen & Meyer 2017). Many factors perpetuate this misconception, but they “…attribute an operational view of the equal sign to the use of calculators and direct verbal-to-written translation of mathematical sentences” (Vermeulen & Meyer 2017). The article by Vermeulen and Meyer had me thinking quite a bit about my own practice, not only with respect to technology, but also in the language I choose to use.
In thinking about how this affects educators, Vermeulen and Meyer write, “we are of the opinion that the results obtained from both teachers and students do suggest that, owing to these teachers’ limited MKfT of the equal sign, they were not aware that their teaching could, and possibly did, promote students’ misconceptions of the equal sign, nor were they able to identify students’ misconceptions or suggest how to prevent, reduce or rectify these misconceptions” (Vermeulen & Meyer 2017). As a teacher of young children, this is particularly striking, because it brings to light the fact that in something as simple as the way I chose to vocalize and draw attention to equivalence can help promote or expel the myths of math as solely arithmetic and computation.
Technology creates opportunities for students to be able to visualize the problem (and it’s potential solutions) in a different way, making room for them to be critical of their own misconceptions. In the case of algebraic equations and equivalence, the idea of a balance or see-saw helps the students envisage this concept. Sakow and Ruveyda write, “Modern tools like tablet apps may help middle school teachers end the thirty-year stagnation and put these algebraic misconceptions to rest at last” (Sakow & Ruveyda, 2015). In fact, there is an app that the article outlines as particularly effective to this end. Though I’m sure there are many such apps that use the same see-saw analogy, I particularly appreciate that “MathScaled’s weights change in value from problem to problem, erasing student notions of specific values for variables. Furthermore, the app allows students to save screenshots of their work to assist the teacher in efficiently assessing understanding and providing individualized support” (Sakow & Ruveyda, 2015). It is the constantly changing factors and reframing of the problems that allows the concept to solidify, and for the students to hone their skills in this respect.
References
Matthew Sakow, & Ruveyda Karaman. (2015). Exploring Algebraic Misconceptions with Technology. Mathematics Teaching in the Middle School, 21(4), 222-229. doi:10.5951/mathteacmiddscho.21.4.0222
Shapiro, B. L. (1988). What children bring to light: Towards understanding what the primary school science learner is trying to do. Developments and dilemmas in science education, 96-120.
Cornelis Vermeulen & Bronwin Meyer (2017). The Equal Sign: Teachers’ Knowledge and Students’ Misconceptions. African Journal of Research in Mathematics, Science and Technology Education Vol. 21 , Iss. 2.
You raise really interesting points here about how seemingly innocent teacher talk about numbers leads to misconceptions. I often fear that because primary school teachers are generalists without a solid foundation in STEM subjects that we lack deep understanding of math concepts. Having learned algorithms at school and having been successful in math classes, I didn’t realize that I didn’t understand some of the underlying concepts, like that of an equal sign, until I had already spent several years teaching using “__+__ makes ___”, likely leading to some of my students later developing misconceptions themselves. One of the ways we need to counter this is by providing teachers with solid foundations in subject areas through professional development not only in the teaching of subject areas but also in the concepts related to those subject areas. I understand why in middle-school and high school teachers are required (in Alberta) to have a bachelors degree in their subject area and wonder if this would be an appropriate requirement of elementary generalists, too.
I’ve never seen the app you shared before but it looks like an incredible way to show equivalence without assigning numbers to either side and I was struck by the ability to use fractions in the equation!
Hi Tracy,
Thanks for your thoughts and insight. Like you, the article provided me with a lot of “Oohh, I didn’t think of that!” moments, where my perspective changed because the problem was explained from a new angle. I like your thought about potentially having elementary teachers complete a bachelor’s degree in a specific subject area, though they teach an array. Maybe it would increase the amount of maths’ specialists at any given school or district, and could then provide additional resources for professional development and learning communities?
Thanks for your response!
Amanda
Hey Amanda, I hadn’t read the “Equal Sign” paper you were referencing until after I read your post, and I found it fascinating. Thanks for sharing!
You make some really good points about how simple language can have wide-ranging and long-lasting effects on student conceptions and overall understanding (or misunderstanding) of a concept. I found that this became “real” for me when I left Canada to teach in the UK, and the slang I was using like “plug this number in to x” no longer made any sense. I thought that the language I chose would make the concept easier to understand, but it turns out I was making it more complicated for the students. It took me being observed by a colleague to tell me “hey, maybe try to use another word than ‘plug in’ – we don’t use that here”. Go figure! (does everyone know “go figure”?)
These days I am always cognizant of the language I use especially in calculus classes, and I try to be as short and concise as possible… unlike some of my posts here… but I have been dropping the metaphorical ball in terms of ironing out student misconceptions. Reading your post reminded me that students may be stuck on terms and concepts that I am taking for granted when I’m introducing new topics, and that I should focus on designing a few new “diagnostic tests” to see where my students may be struggling.
As for the paper you shared, I found a few quotes particularly poignant:
1) “Teachers’ views of mathematical content and how to teach it is often influenced by
the textbooks they use; more so in the absence of curricular guidance. This is particularly true if teachers did not receive proper, specialised training in mathematics, as is the case with the teachers in this study.”
In an age where I do SO MUCH GOOGLING to help me with designing resources, I forget that some schools have little to no Internet access and are forced to deliver what’s written in textbooks. Perhaps, in this case, textbooks are partly to blame, and we should be working together as a mathematics community to create a better (perhaps partially open-source) resources?
2) The teachers received “very little, if any, guidance on the topic of equality or the equal sign … in … curriculum documents, [so] it is hardly surprising that teachers themselves show a low level of MKfT (Mathematical Knowledge for Teaching) regarding equality and the equal sign”.
It’s no surprise that students struggle with these operations when teachers don’t even realize they themselves are struggling to understand a concept, or have misconceived notions about what they are teaching. It makes me reflect on what I think I know versus what I truly understand.
Woo. So much to think about.
Thanks so much for the great post!
Scott
Hi Scott,
Thanks so much for your lengthy post- you’ve got so many great ideas and points!
The example you use of being in a different country and having the language you use affect your instruction (despite speaking the same language!) is hilarious, but very real. It’s true that we have to constantly be reminded that the language we use matters, and that while being concise and direct is likely very appropriate for your calculus class, it may also be necessary to find ways to explain concepts with multiple analogies that students understand and have a good grasp of. I’m thinking here more in terms of science rather than maths 😉
You bring up some excellent points about the article itself- thanks for your valued perspective as a specialist maths teacher. It’s really interesting for me to get a read on your ideas, as my resources and concept-based approaches are all at the elementary level, and not as sophisticated (read: more resources for visualization are easily accessible for me to explain concepts of addition, subtraction, etc. without the internet) as yours.
Thanks again for your thoughts!