So here they are then, this time for real!
1. Trigonometry in Grade 3?
This monograph, part of the LNS “Research into Practice” series, was published in August of this year. The authors suggest that we may be relying too heavily on Piaget’s stages of cognitive development, and thereby holding students back from what they are in fact ready and excited to learn about, were it presented in an engaging, age-appropriate fashion!
This suggestion seems to support the argument put forth by Will Richardson (“A Web of Connections”) – Why are we holding students back when they are ready to move ahead?! In a video segment I watched earlier this week, Richardson speaks about a 14-year-old boy in the US, who basically hosted his own TV show online during the 2008 election. What happens, muses Richardson, when this student returns to the classroom the next day and is given a textbook and paper to write on?
Similarly, the Trig monograph suggests that by following a strictly developmental approach with our students, we may be prohibiting them from “achieving the fulfilment and enjoyment of their intellectual interest” in mathematics (and surely in other subject areas, too!)
I am reminded of a lesson I recently did on growing patterns. The idea was for students to simply identify that when the numbers got bigger, that was called a growing pattern. But a few students noticed that there were predictable patterns in each of the tens and ones digits in the numbers, too, and they wanted to speculate about why that was. Several other students got excited about this once their peers pointed it out, and my lesson went out the window as I allowed them to explore this discovery.
Perhaps we *are* holding students hostage by clinging too closely to the curriculum. Of course we need a roadmap of our intended mathematical journey throughout any given school year, but if we focus on mathematical processes, and encourage student-generated dialogue in math class—even when it diverges from our intended plan – then we can foster the sort of rich, authentic mathematical thinking this monograph supports.
Even for students in Grade 3!
2. Paying Attention to Proportional Reasoning
This 16-page document is a support document to “Paying Attention to Mathematics Education”, and provides some specific comments and practical support for proportional reasoning, one critical factor in students’ understanding of mathematics.
Don’t be alarmed by the size of the document; many of the pages are taken up by mathematical illustrations and/or samples of student work. It is a relatively easy and engaging read.
The document begins with an overview of proportional reasoning and why it is important.
The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms. Students are using proportional reasoning when they decide that a group of 3 children growing to 9 children is a more significant change than a group of 100 children growing to 150, since the number tripled in the first case; but only grew by 50%, not even doubling, in the second case. (page 3)
It appears that developing a strong sense of proportional reasoning can help students with “reasoning” in general:
The ability to think and reason proportionally is one essential factor in the development of an individual’s ability to understand and apply mathematics. (page 4)
Several “sub-concepts” of proportional reasoning are explored in some detail, with illustrations for teachers who may be quite new to dissecting mathematics in this sort of detail. Scaling up or down, unitizing, and developing multiplicative reasoning, for example, are considered. For example, the monograph notes,
It is critical to spend time developing unitizing since “the ability to use composite units is one of the most obvious differences between students who reason well with proportions and those who do not” (National Research Council, 2001, p. 243). Unitizing is the basis for multiplication and our place value system which requires us to see ten units as one ten and one hundred units as ten tens.
As Cathy Fosnot emphasizes, this is complex since unitizing ten things as one thing almost negates children’s original understanding of number (Fosnot & Dolk, 2001, p. 11). It is therefore not surprising that a robust understanding of place value has been found by some researchers to not fully mature until fifth grade (Brickwedde, 2011, p.13).
One of the tips for getting started is to offer students problems that are both qualitative and quantitative in nature:
Qualitative problems (e.g.,Which shape is more blue?) encourage students to engage in proportional reasoning without having to manipulate numbers. (page 8)
For more information about Ministry resources related to math, including links to all recent monographs and several videos, too, click here.