Dale recently wrote a post on Constructivism, and what to do when the students won't construct.

I had a similar experience to his in my own classroom. At first, I wondered if we might better not just let the students use the traditional algorithm. After all, if it's working for them... But then I realised that it in fact was NOT working for them.

I had a similar experience to his in my own classroom. At first, I wondered if we might better not just let the students use the traditional algorithm. After all, if it's working for them... But then I realised that it in fact was NOT working for them.

This became particularly apparent when one student in my room today was trying to subtract 22 from 51. As he was not able to deduct 2 from one, he simply took 1 away from 2, showing me that he had no understanding of the placement of the numbers when they were stacked one on top of the other -- he was jsut doing what worked for him, even if it made no sense whatsoever. A peer interjected, and tried to show him about "borrowing", a concept he seemed vaguely to remember from last year's class, but soon both students (and their "audience" on the carpet) were completely confused!

When I later attempted to have this same student model 51 using base ten blocks, and then take away 22 (to help him construct an understanding of trading and hence "borrowing"), it became clear to me that he had a complete lack of number sense. His actions repeatedly showed me that he had no understanding that 22, for example, was comprised of 2 tens and 2 ones. Or that 12 was made up of 1 ten and 2 ones, for that matter, or that 10 was the same as two fives!

(I checked this with my own kids at home at dinner tonight, to compare -- we talked about these numbers at the dinner table, without paper to work it out on. Both boys managed to work it out in their heads; one noted that he knew that 52 - 22 would be 30, so he just did that and then took one away, since 51 was 1 less than 52. My other son, also in Grade 3 like his twin brother, was less articulate about how he sorted it out in his mind, but it was clear that he had constructed some sort of method that worked for him. When I wrote the numbers on paper, one on top of the other, neither knew what to do with this -- clearly their teachers from previous years had not imbued them with some sort of a standard method for subtraction. Or if they had, it hadn't stuck. This intrigued me.)

Although some students in my class at school seem to have already "learned" a traditional method for addition (or subtraction, or multiplication, or whatever), upon digging a little deeper, the teacher can soon discover that many of these students have no real understanding of what these algorithms represent. They may have memorised the method, but they cannot tell you if their answers to a math problem are reasonable or not.

It seems that -- unless one is inheriting students from a group of primary teachers who are quite skilled at and confident in teaching math the "new way", or from parents who resist the temptation to instill in their childrent the traditional methods from when they were in school -- one must first help the students

When I later attempted to have this same student model 51 using base ten blocks, and then take away 22 (to help him construct an understanding of trading and hence "borrowing"), it became clear to me that he had a complete lack of number sense. His actions repeatedly showed me that he had no understanding that 22, for example, was comprised of 2 tens and 2 ones. Or that 12 was made up of 1 ten and 2 ones, for that matter, or that 10 was the same as two fives!

(I checked this with my own kids at home at dinner tonight, to compare -- we talked about these numbers at the dinner table, without paper to work it out on. Both boys managed to work it out in their heads; one noted that he knew that 52 - 22 would be 30, so he just did that and then took one away, since 51 was 1 less than 52. My other son, also in Grade 3 like his twin brother, was less articulate about how he sorted it out in his mind, but it was clear that he had constructed some sort of method that worked for him. When I wrote the numbers on paper, one on top of the other, neither knew what to do with this -- clearly their teachers from previous years had not imbued them with some sort of a standard method for subtraction. Or if they had, it hadn't stuck. This intrigued me.)

Although some students in my class at school seem to have already "learned" a traditional method for addition (or subtraction, or multiplication, or whatever), upon digging a little deeper, the teacher can soon discover that many of these students have no real understanding of what these algorithms represent. They may have memorised the method, but they cannot tell you if their answers to a math problem are reasonable or not.

It seems that -- unless one is inheriting students from a group of primary teachers who are quite skilled at and confident in teaching math the "new way", or from parents who resist the temptation to instill in their childrent the traditional methods from when they were in school -- one must first help the students

*de-*construct what they have previously "learned", before one can help them construct an understanding of mathematical concepts.*(I should add, as a post script, that I am/was until recently one of "those" teachers: The truth is that there are many mathematical concepts that are just now beginning to make sense to me. As a student, I was very good at memorising algorithms, and although I spewed them forth as needed through most of elementary school, and managed to pull off A's and B's in math, my lack of deep understanding of what I was doing soon caught up with me, and by Grade 10 I was failing math class quite thoroughly! I soon dropped math altogether, and it was not until I began to work on my Masters Degree during my 4th year of teaching that I slowly started to revisit math, first with Rita Cohen's "Math for Math-Phobic Teachers" course at OISE, then later, through provincial strand-specific training as a program resource consultant with my board followed by daily absorption from Trevor Brown who was my colleague at Tyndale for a year, and now through self-directed learning with this project. It has taken me nearly two decades to realize what a lousy math teacher I was "back then", and to become aware of just how critially important teaching math the "right way" is in order to ensure strong mathematical literacy in students, especially our youngest learners. The work we are dumping on our intermediate school colleagues when we hand off primary and junior students who have not only major gaps in their mathematical abilities, but who have also developed a whole host of mathematical MIS-understandings over the five years they are with us amounts to a seemingly insurmountable challenge, I would imagine! I would argue that this is a testimony to the desperate need we have for teachers to get a whole heap more job-embedded training in the are of mathematics... but that is perhaps a topic for a )*