Adding up the price of fruit-what is the cost? Week 2 Post about Mathematics in the Streets and in Schools

Mathematics in the Streets and in Schools

Carraher, Carraher & Schielmann

British Journal of Developmental Psychology 3(1) 21-29 1985

The article starts with the premise that there may be a difference between solving mathematical problems using methods learned in school and solving them in more familiar contexts out of school. Carraher et. al. tested the notion that a child who might have difficulty with routines learned at school might at the same time be able solve mathematical problems for which these routines were devised in more effective ways in a more familiar context. Five street vendors (between the ages of 9 and 15) from Recife, Brazil completed an informal test in their natural setting and formal test with pencil and paper.

Carraher et. al. findings state:
-participants scored significantly better in natural settings compared to decontextualized math problems in a formal test; 

·      -context-embedded problems were much more easily solved than ones without a context

·      -that in natural settings problem-solvers reasoned using quantities and convenience groups (knowing the price of three items together) rather than school learned routines;

·      -in a natural setting problem-solvers were focused on quantities and in formal testing the participants were focused on manipulating symbols

The authors conclude that their findings conflict with the implicit pedagogical assumption of mathematical educators according to which children should first learn number operations and only later apply them to verbal and real-life problems. In conclusion, Carraher et. al. ask how do we introduce math systems in contexts in order to sustain human daily sense and then depart in order to teach mathematical systems? 

Carraher et al. claim that there is an implicit assumption that math educators should teach math in a decontextualized way and then apply their knowledge. I stopped and wondered if this is true. When I was in elementary school (in 1990-97) we had several worksheets with number operations (12 X 4 etc.) and then word problems on the bottom as extensions. The screenshot on the left is a comic by Charles Schulz makes a joke about the types of word problems often found in math classes; in my teaching, I include it as a part of an introduction handout for 10-12 year olds. 

I realize now, that this joke might be more funny to people my age than my students because the big push when I was in teacher education (for ages 5-12 at least) was starting with a 'student's daily human sense.' Students figure out problems using a method that makes sense to them and then write it down, draw pictures etc. in math journals or on vertical surfaces to share with other mathematical thinkers.

One example from this article is: How much does it cost to buy 10 lemons?

Vendors say: Lemons are 35 cents. Three lemons are 105 so 105...105...105 is 315 and 35 is 350.

If a student were to write this in a math journal, I might say “Oh, wonderful,” and then compare it to another student who uses 35 X 10 and have them explain. I could claim this is a mathematical proof. We came to the same conclusion two different ways. There is a quote I share by Polya, “It is better to solve one problem five different ways to than to solve five different problems.” As Callaher et. al. suggest, I might then introduce an example where it is too cumbersome to do repetitive addition in order to push the multiplication method.

The questions that arise for me relate to motivation and thinking processes: If I were to ask the participants in this study to draw their thinking would they do it or think I am crazy for asking them? After all, they already figured out the problem. What motivation is there to continually ‘explain one’s thinking’? I also wonder, how do the thought processes that are involved in explaining mathematical thinking verbally different than those involved with writing? The novelty of writing on a whiteboard only lasts for so long and drawing out a long answer to a problem is not something one would do in a natural setting. Is drawing on a whiteboard actually sustaining one’s daily mathematical thinking then? Is this teaching method effective and for whom? What implications does this have for people whose mother tongue is not English? The long term implications of these questions will affect a whole generation of students who will make jokes about writing their mathematical thinking on whiteboards.

Please feel free to touch on any of the questions I asked above. Also,

-Based on your math teaching, researching and learning do you think that there is an implicit pedagogical assumption that math educators should teach math in a decontextualized way and then apply their knowledge?

-Carraher et. al. end the article with the question that asks, how do we introduce math systems in contexts in order to sustain human daily sense and then depart in order to teach mathematical systems? Where is the point of departure?