Week 5 Reading Response on Mental mathematics under the lens: Strategies, oral mathematics, enactments of meanings by Jérôme Proulx
Summary
By looking into the nature of the mathematical activity generated in mental mathematics, Proulux gave several inspiring examples from number calculation, reading graphs and equation solving etc. His journal focuses on the solvers’ strategic processes in mental mathematics contexts in order to develop a better understanding of what doing mental mathematics calls for. He pointed out the topics related to meaning-preservation, meaning-making, and optimal ways of understanding mental mathematics strategies and their unique realm of mathematical discovery.
Stop 1: Street Mathematics versus School Mathematics
I found this is such an interesting approach to look at street-related and school-related tasks, which are identified being oral or written. Compared with focusing on the symbolic representation in written mathematics, and its application of learning standard routines, street mathematics tend to preserve much of the meaning of the situations at hand, more invented on the spot by the solvers. It reminds me of the vendors in the open market, they are so efficient of doing mental math and seems very familiar with the calculation around certain number patterns (the different number combinations of prices of the goods.
Question 1:
I’ve read a study on Brazilian working youngsters who are in everyday use of business transactions, showed distinct analytical methods from those taught in schools. Output on mathematical problems embodied in real-life situations was superior to that related to school-type word problems and context-free computational issues involving the same numbers and operations. Does this case speak to you in incorporate oral and written mathematics in everyday classroom?
Stop 2: the Toolbox Metaphor
Conceptualizing mental mathematics strategies as a posing and solving process, which theoretically ground the focus on meaning making and meaning preservation, I was stopped by some studies that centered on the assumption of ‘toolbox metaphor’, (or the “selection-then-execution” hypothesis), which implies that solvers tend to choose a strategies from a group of pre-determined strategies they have learned in the past rather focus on the operation they come up on the spot. Recently, the ‘toolbox metaphor’ has been challenged and some researchers illustrate that there is more to solve mental computational tasks instead of simply reusing of predetermined strategies.
Question 2:
From observing your students’ problem-solving procedure, which side would you take and to what extent do you agree or disagree?
Stop 3: Are we losing content when doing mental mathematics?
From my point of view, mental mathematics techniques act as interpreted narratives, as examples of behaviors occurring in the making. Such techniques are constantly generated as they unfold, and linked to previous forms of solving or entirely relied on thoughts in the moment. That’s why mathematics educators put efforts to make and maintain the sense; and this is how they can be seen as distinguished from paper-and-pencil work. But due to the nature of mental mathematics, that it is hard to carry every detail and restriction during problem-solving, such as mentioning the domain, range, and limit etc. Please see example below.
Question 3:
I went on to SyMETRI Presentation Proulx did on Feb 6. He gave us several ways that students solving this equation (see below), but usually when they do mental calculation, the domain of the equation, 
were not considered. What’s your opinions of why this happened, and how do we avoid it in teaching mental mathematics?
were not considered. What’s your opinions of why this happened, and how do we avoid it in teaching mental mathematics?
Reference:
Proulx, J. (2019). Mental mathematics under the lens: Strategies, oral mathematics, enactments of meanings. The Journal of Mathematical Behavior, 56, 100725. doi: 10.1016/j.jmathb.2019.100725
This sounds like a really interesting topic.
回复删除1. This reminded me of our class discussion about the validity of mathematics. I find it interesting the context we give to story problems for students often is around money. It makes me wonder why is it natural for us to do so. When I asked my students what is 2 - 6 they have no idea! But when I say "you have $2 and you want to buy something that is $6, how much money will you have to borrow from your parents?" all of a sudden they can tell me the answer. I wonder if this has anything to do with the street math that you mentioned here.
2. I often used the toolbox metaphor. I had to do a quick google search to see how others have defined it in the past and am intrigued why people no longer liked the term. I often describe the toolbox to my students as skills and strategies that will hopefully help them later on in life. I found it interesting that people saw it as something that you either have or don't have. I think skills and strategies helps us be flexible thinkers and by being flexible thinkers, we can be better problem solvers.
3. I find that I say "does this make sense?" a lot to my students whenever we are talking about estimation or mental math. But of course when we're doing such estimation or mental math, we are not focusing too much on the details and therefore we loose content. I also believe that there needs to be understanding in the content in order for people to know if something makes sense or not.
The idea that street math tends to preserve much of the meaning at hand reminds me of the article I read. Nunes, Carraher & Schlimann (1985) report that children tended to reason by using what they call a ‘convenient group’ in their natural working situations (p. 246). This makes me think about the importance of using context to determine which strategies to use. I find that some students can have difficulty representing their ideas in written form. I think focusing on the oral math before the written math could help students develop a stronger foundation of the concept.
回复删除From observing my students’ problem-solving procedures, I would say that they usually use strategies that they have learned. However, I have been encouraging them to think about solving the same problem in different ways. I find that sharing strategies can help students to spark new ideas that build on another person’s strategy. I often think of more ways to solve a problem myself during discussions with students.
Perhaps students have been taught that the denominator cannot be 0 so they did not consider it as a possibility. I think it is important to create the opportunity for students to explore all possibilities. In this case, I would ask them to consider the possibility of X=0 and explain the reasoning behind why X cannot be 0. I think exploring possibilities rather than learning rules could help students make sense of the concept and develop a deeper understanding.