My last blog post promised to summarize two new Ministry Monographs related to math, but alas, I went off on a tangent…

So here they are then, this time for real!

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!

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.

So here they are then, this time for real!

1. Trigonometry in Grade 3?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)

The resource then proceeds to consider some key concepts of proportional reasoning, as well as some examples and non-examples. Starting points are offered for teachers, including practical illustrations across grades and strands. Following this, some specific examples from EQAO are disseminated at the Grades 3, 6 and 9 levels.

It appears that developing a strong sense of proportional reasoning can help students with “reasoning” in general:

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)

Interestingly, Dale and I have noticed that this tends to be a weakness of our students, this inability or unwillingness to reason. All too often I have noticed that students are far more eager to write – as an explanation for a solution – things like “I thought it in my head”, or “I remember my teacher from last year told me”, rather than really think through a solution, asking themselves, “hey, does this make sense? How do I know? How can I prove it?” On some level, we both believe that the students just struggle with paying attention in general, and not reasoning specifically. On the other hand, perhaps if we exposed them to more activities that developed their abilities in proportional reasoning, then our students might be better able to reason, period.

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,

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).

We have certainly noticed this “thin” understanding of place value in our own students, and indeed, have planned our cross-class differentiated math groups accordingly. Interestingly, the note about this concept not fully developing until a certain age seems to contradict the point made about developmental learning in the previously mentioned monograph. So, are students in Grade 3 not understanding place value deeply because it has (in some cases) not been well introduced in the early primary years, or are their brains simply not ready for it yet?

One of the tips for getting started is to offer students problems that are both qualitative and quantitative in nature:

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)

Skimming and scanning this monograph reminds me of how very,

For more information about Ministry resources related to math, including links to all recent monographs and several videos, too, click here.

*very*limited my own mathematical understanding is. Best practices for teachers include solving problems oneself before sharing them with students, in order to anticipate what sorts of errors and solutions might arise. I know that. And I even have the luxury of living with a mathematician (who scoffs at my pathetic abilities to reason and think!) Alas, when does the average teacher have the time to sit down and just do math?!For more information about Ministry resources related to math, including links to all recent monographs and several videos, too, click here.