Monday, January 02, 2012

The Difference Between Curriculum and Pedagogy

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There's a difference between curriculum and pedagogy. Curriculum is all about what we teach. Pedagogy is about how we teach it.


There's also a difference between knowing how to do something and understanding what you're doing. In mathematics there are all kinds of "how-to", or computation skills, that kids learn and promptly forget right after the test; sometimes they forget before the test. The thing is though, it's difficult to forget something once you understand it.



Seven Principles of Learning by dkuropatwa, on Flickr

Creative Commons Attribution-Noncommercial-Share Alike 2.0 Generic License  by  dkuropatwa 


A few weeks ago I was part of a panel on the Richard Cloutier Reports show on CJOB radio here in Winnipeg. There were four of us: myself, Paul Olson (President of the Manitoba Teacher's Society), Robert Craigen (Associate Professor of Mathematics, University of Manitoba) and Anna Stokke (Associate Professor of Mathematics, University of Winnipeg). Robert and Anna are one-half of the group behind the wisemath blog.


There are some things we agree on:
  • All kids can and should learn basic computation skills (how to add, subtract, multiply and divide).
  • It's important for kids to understand what they're doing, not just to be able to perform by rote.
  • Manitoba's recent poor performance on the Pan-Canadian Assessment Programme test is not good news and we have some work to do in mathematics in Manitoba.
  • We'd like to see Manitoba place at the top of future national and international tests of this sort.



Understanding the concept by dkuropatwa, on Flickr

Creative Commons Attribution-Noncommercial-Share Alike 2.0 Generic License  by  dkuropatwa 


Some things we disagree on. I believe:
  • Learning with understanding should precede the learning of rote algorithms in mathematics.
  • To say Manitoba has placed 10th out of 11 provinces and territories in the 2010 PCAP test is a gross oversimplification of the the data represented on page 24 of the report (pdf). (Those confidence intervals are important. A repeat of the same test would likely have Manitoba place somewhere between 6th and 11th place. This isn't good news, but it's a little more nuanced than "10 out of 11". People knowledgeable about mathematics should be helping the public understand these nuances and promote informed discussion.)
So the crux of our differences are two-fold:


(1) I believe Robert and Anna conflate curriculum and pedagogy and are reading the Manitoba Curriculum documents as pedagogical texts when they were never intended to be read that way. Curriculum tells us "what" to teach, not "how" to teach.


(2) Robert and Anna believe the teaching of algorithms should be student's entry point to learning the basic operations (+, -, x, ÷). I believe the algorithms should be closer to the end-game of learning the basic operations.


John Scammel blogged about his take on the views expressed on Robert and Anna's blog. John points out in the comments the clear distinction the wisemath blog draws between Mathematicians and Mathematics Educators and the populations we teach. In K-12 classrooms we teach all students. The student body in University is different. Students taking math at University want to be there. That's not true of many students in the K-12 sector; the challenges are quite different.


On further reflection, there's a third difference: public (and private) debate should be open and sidestep insult.


The wisemath site seems to reject any comments that debate the blogger's views.


What I've read in the comments on John's blog and on Anna's blog (The last sentence of the last paragraph was recently edited; it used to say all future mathematics education research has no merit as a result of the issues Anna took with the article she blogged about. I regard this edit as a positive evolution in her thinking.) seems to hold K-12 teachers in a disdainful light.


Here's the audio from the CJOB panel we sat on together. It was a 2 hour broadcast, without commercials it's about 58 min. I took out the commercials. We talked about much more than was broadcast in the moments we were "off air". That was also an interesting conversation; unfortunately we didn't capture it. Next time I'll bring along my mp3 recorder. ;-)



Download (53.2 MB)




22 comments:

Mike Nantais said...
2/1/12 11:59  

Hi Darren,
you have made some excellent points here. I will listen to the broadcast when I get a little time. I agree we need to examine the meaning of the results, and should always want to improve instruction (in Math or any area). After an admittedly quick look at the math wise site and hearing about this group, I have a few observations. The mission statement of WISE contains the phrase "all students", I can't disagree with the overall intent of their mission statement. Some of their suggestions have some merit, however, they demean teachers and teacher education,it would seem some respect and dialogue would go further. I do wonder if they have looked at their own role. Students (at least at BU, and I would imagine at UW and UM too)are required to take university math credits to enter the Faculty if Ed. Perhaps they should start by looking at those courses, perhaps adding a lab component, and beefing them up - they also have control over the pre-requisites for these courses! Finally, I respect their deep knowledge of Mathematics, but that does not mean they understand learning & pedagogy.
My blog post on the test results is here: http://mnantais.blogspot.com/2011/12/do-manitoba-kids-suck-at-math.html

Ross Stokke, Math Professor said...
2/1/12 17:54  

Can someone please explain why asking for well-qualified, competent teachers of mathematics for our children is demeaning to teachers and teacher education? Is the fact that we expect a high level of competency and strength of firefighters demeaning to their profession? Is the fact that we expect a high level of competency and intelligence from doctors demeaning to those in the medical profession? Surely it is better for teachers and the teaching profession to have individuals in the classroom who are comfortable and competent with the subjects that they teach.

There is a difference between insulting teachers and working to improve a system. In fact, WISE Math has received a lot of support from teachers. It is difficult to see your comments as anything more than an attempt to create a divide between teachers and end-users of mathematics when in fact the two groups should be working together.

It is, incidentally, false to suggest that it is difficult to forget something after you have understood it. I have over the course of my career as a mathematician (I do research in math, and I teach math) understood a lot of mathematics, some of it at a rather deep level. Some of this math, I've now forgotten. I have discovered and proved original theorems whose proofs I cannot at the moment recall. This is the nature of the human brain. Furthermore, much of my understanding of the mathematics that I have learned came after I achieved "computational fluency" as you might say.

WISE Math simply asks for a balance between understanding and basic skills.

As you believe that the standard algorithms should be taught, you will surely agree with WISE Math's assertion that they need to be explicitly added as outcomes in the curriculum. Currently, they are not. In fact, we are aware that standard algorithms are banned from being taught in some schools. It is my impression from the many letters that Anna has received from parents and teachers that children are actively discouraged from ever using the standard algorithms at many schools.

Since you feel that a line should not be drawn between mathematicians and math educators, you presumably will also support WISE Math's advocacy for strong representation from mathematicians who teach content courses in math at the post-secondary level, and other end-users of math, on committees that design math curricula.

While I fully acknowledge that not all children will go to university, I and the founders / supporters of WISE Math want to ensure that children are not unfairly shut out of good careers in science and engineering, early on in their schooling, due to a weak elementary school math curriculum and other factors.

Surely, you will want the same thing.

Given that the vast majority of students who wish to obtain a degree in science, engineering, pharmacy, economics (et cetera, et cetera) are required to take a course in introductory calculus, does it not make sense that you should welcome input from the individuals who teach math content courses at universities?

This issue is not about the egos of mathematicians and/or math educators. It is about a generation of children and a subject area that is crucial to the success of our society.

R. Craigen (WISE Math co-founder) said...
3/1/12 01:38  

2. "To say Manitoba has placed 10th out of 11 provinces and territories in the 2010 PCAP test is a gross oversimplification of the the data represented on page 24 of the report (pdf). (Those confidence intervals are important. A repeat of the same test would likely have Manitoba place somewhere between 6th and 11th place. This isn't good news, but it's a little more nuanced than "10 out of 11". People knowledgeable about mathematics should be helping the public understand these nuances and promote informed discussion.)"

It was not us, but you and your colleague from the Teacher's union who obsessed over this particular report, Darren. As you'll recall from our converstation and the interview itself I mentioned both the TIMSS and the PISA assessments, which I had on hand, and drew out a well-defined trend in all three assessments across the western provinces over three PISA assessment periods starting in 2003. When Manitoba moves from almost dead-center among Canadian provinces to about two standard deviations below the mean in the international assessment over a 6 year period, that is something to pay attention to. That a similar pattern is observed in the 2010 PCAP is only the most recently publicized indicator.

Regarding your subsequent point (1) We do not, as you say, conflate curriculum and pedagogy. But I am glad you accept that curriculum is supposed to delineate the content of what is taught, which is what I've said all along.

In your recent email to me you say "You will likely never see the sort of description you've written above [my one-line description of the standard algorithms - RC] included in any curriculum document", which I take to be your way of excusing the total absence of the four standard algorithms in the WNCP CCF, the implication being that they will end up being taught anyway.

Now either these algorithms MUST be taught or not.

In the latter case, why not just come right out and say "we don't care if students learn the four standard algorithms" (in which case we will necessarily have "haves" and "have nots" in a two--or more--tiered system, as only those taught the standard algorithms are adequately prepared for the more advanced algorithms of algebra and the required math for science, engineering and management courses).

In the former case, then I guess you have to admit that the curriculum must, after all, state this, which was my point in the first place. Why do you feel you must dance around this issue and use ad hominem and straw man arguments? I have taught professionally for 20 years; I married a junior high school math and science teacher; these things have been table-talk in our home for many years -- I assure you I have no difficulty distinguishing between pedagogy and curriculum.

But I wonder, since you also assert in the email I cite, that there is no such thing as "standard algorithms", whether you have difficulty understanding the "standard" part or the "algorithm" part. Perhaps you'd like to expand on this point here.

There are many algorithms for the four basic operations. However, among these are a unique set of clear-cut standard procedures taught world-wide with minor variations on how information is arranged (which you should know has nothing to do with the "algorithm"). This makes them "standard algorithms". This phrase is understand throughout the world of mathematical education and always refers to these procedures -- I am astonished that as a math teacher and curriculum consultant you are either unfamiliar with them or feign ignorance as some sort of diversion.

R. Craigen (WISE Math co-founder) said...
3/1/12 01:41  

Your additional point (2): "Robert and Anna believe the teaching of algorithms should be student's entry point to learning the basic operations (+, -, x, ÷). I believe the algorithms should be closer to the end-game of learning the basic operations."
We never said anything of the sort. I advocate the traditional approach to developing the standard algorithms, which develop aspects of the algorithms over a long period in small, well-ordered stages. I also advocate clearly motivating each stage as it is taught, so that understanding accompanies mastery of each still element. As far as I know Anna advocates the same. You'll find much discussion of such things spread throughout the NMAP report. What you're writing here is a simple fabrication.

But you're quite right that we differ about placing them at the end of the game. The standard algorithms are not an afterthought. Nevertheless, if you will concede that they are essential content I will concede that it is acceptable for the full algorithms to be held in reserve until near the end of the learning process (though sufficiently early to allow sufficient practise to permit mastery/fluency).

It is helpful that your word "should" hints that you might believe these algorithms are essential, but I would prefer you to say so outright, because I had the opposite impression after the CJOB interview, and I'm getting accustomed to a sort of doublespeak on this issue from WNCP proponents: When we pinned down Carole Bilyk of the Curriculum Branch on this point the best we got was that the standard algorithms are not forbidden. Well, thank goodness for small blessings, but this is by no means sufficient. One of your fellow curriculum consultants, a contributor to the framework documents, told me that they needn't be stated in the curriculum because students will learn them from their parents and older siblings. Is this your view?

You seem to think we are concerned about a tiny group of math majors and want schools to tailor math instruction only for the needs of this group. This also mischaracterizes our position and our expertise. This term one of my classes consists of 140 Engineering students, NONE of whom are math majors. They ALL need the established foundations we are speaking about. Read the comments on our JOIN page, and you'll see many representatives of various post secondary disciplines share these same concerns.

That you so badly misunderstand the nature and breadth of the concerns of the postsecondary community is noteworthy and a bit disturbing, because it is the educator's job to prepare students for what lies ahead. It is certainly not the business public schools to teach under assumption likely to harm the prospects of students in science, engineering, business and medicine.

R. Craigen (WISE Math co-founder) said...
3/1/12 01:42  

I agree with your point that debate should sidestep insult. To what are you referring? Is this a mea culpa for the simplification and fabrication of our views or the jab about conflating curriculum and pedagogy?

You point out that we do not take comments debating our position on the WISE Math website. Quite right, Darren, and (sigh!) one more time, please read the note at the top of our JOIN page: that is NOT THE PURPOSE of the site. It is our position that such debate should be public. Our site is for organizing a community of citizens sharing our concerns. There are plenty of public fora and alternative blogs (such as your own) where debate is welcomed and part of the purpose. Get used to it -- not every site is your personal soap box. WISE Math's web site has a focussed and relatively singular purpose, which we don't want buried under a free-for-all. We are quite happy to debate publicly, as we did on CJOB with you last month. As you'll see, Anna engages in two-way conversation on her blog.

The general public is not invited to make disparaging remarks about the WNCP curriculum over at wncp.ca -- could it be that this simply isn't the purpose of that site? I certainly have no criticism for WNCP on this basis; that is the prerogative of those who manage web sites.

Speaking of Anna's blog you write, "...seems to hold K-12 teachers in a disdainful light." I don't know how you could honestly hold this view after talking with Anna during our time at CJOB. I believe her exact words on that day were something like "teachers are heros", which she expanded on at length. It is the conditions under which they are being expected to teach, the inadequate accreditation standards and preparation, and the materials and accompanying training that is the problem. We admire teachers, who carry the brunt of all this, many of whom do an excellent job in spite of it. And K-12 teachers seem to agree, judging by the number who have JOINed our initiative. Perhaps you should read some of their comments -- do these teachers also "hold teachers in a disdainful light"?

I would be interested in where your first commenter Mike Nantais regards what we've written at WISE Math as "demeaning teachers". I don't think we do, and if something comes across that way we would want to hear about it.

He also says we "demean" teacher education. I wouldn't use that word, but we certainly criticize the level of math accepted under Manitoba legislation currently governing accreditation, and echo the concerns of many that teachers simply aren't being provided adequate backgrounds for what they are expected to teach. If Mike disagrees with these positions, I suppose he might see our views on these matters as "demeaning". Once again, many teachers share our views, as you can verify from our JOIN page.

I agree with your statement that our off-mike conversation was quite interesting; I would add enlightening. However, you seem to have forgotten much of it, or you would not have mischaracterized our position so in this piece.

WISE Math said...
3/1/12 09:39  

I would like to reiterate what I’ve said many times before: I think that teachers are saints. They are some of the most hard-working, important professionals in society. Darren knows that I hold teachers in high regard – we’ve talked about this before – I therefore feel, as was stated earlier, that his comment about demeaning teachers is meant to be divisive. My mother and aunt were teachers, my mother-in-law was a teacher and my sister-in-law is a teacher. I know how hard they work and what a huge impact they have on society.

I would like to address some excellent and reasonable questions that were raised by the first commenter, Mike Nantais.

Mike is quite right that universities should be examining their own role in teacher preparation in math. All K-8 pre-service teachers are required to take one 3-credit hour university math course and, technically speaking, math depts set the prerequisites on their courses. For this reason, our math department members have had some intense conversations at our university (U of W), with the faculties that would be affected, about raising the prerequisite on the course that is taken by K-8 pre-service teachers. We know that Consumer/Essential Math from high school is simply not enough (even our students who come in with Consumer/Essential Math tell us that this course does not prepare them adequately to take any university math course). I do not wish to discuss internal interactions at our university in detail here but it is not possible for us to raise the prerequisite at this time. (If you would like more details about this, please feel free to phone me at my office number.) Furthermore, we are only one university in one province.

Some of our dept members have had several discussions with a member of the Faculty of Education, Jerry Ameis, who shares our concerns (Jerry is even more experienced with this situation than I am since he has been a faculty member for many more years than I have and has taught many more K-8 pre-service teachers than I have). Our plans are still in the early stages, but we are going to offer a math concentration program for pre-service K-8 teachers at our university. Students in the program will take 3 math content courses from the math dept that will be designed specifically for K-8 teachers (not calculus or linear algebra, but courses that focus on the math the K-8 teachers need to know and understand to deliver the material with deep understanding to their students). They will also take 2-3 math pedagogy courses from the Faculty of Education. Only some students will enroll in the program, but we hope that it will help. We also hope that other universities will follow suit.

Jerry wrote an article about this that appeared in the CMESG newsletter. It can be read here (on pg. 6, and titled “Towards improving teacher preparation for teaching K-8 mathematics”):

http://publish.edu.uwo.ca/cmesg/pdf/CMESGGCEDM%20-%2028(1).pdf

Anna Stokke

R. Craigen (WISE Math co-founder) said...
3/1/12 10:27  

The first part of my comment seems to have disappeared again (it appeared in the comments last night for a while) so I'm re-reposting it; if this is a duplicate please trash all but one copy:

Sorry Darren -- you mischaracterize our positions yet again. As we are in beginning-of-term mode for a while I will not be able to address all points, but I'll do a quick once-over.

You have a very shallow understanding of our proposals. Have you yet looked at the 120 page NMAP report to which we link on our web site, and which I commended to your attention after our CJOB interview? Most of what we are proposing is completely in sync with the 45 recommendations of this panel.

The report addresses, in light of their survey of some 16,000 published papers in educational research, everything from pedagogy AND curriculum to the makeup of panels designing curriculum and assessment instruments, teacher training, where further research is needed, different ways in which math specialists might be incorporated into elementary school math classrooms, child-centered and teacher-centered learning models, and balancing different aspects of learning (mechanical skills, problem solving skills, and developing cognitive understanding).

If you want to know where we're going, get a grip on this report and let's discuss what you disagree with there. As yet all that we've written on WISE Math are summaries in plain language for public consumption.

Concerning your claimed two points of disagreement:

1. "Learning with understanding should precede the learning of rote algorithms in mathematics."

We have not taken any narrow or dogmatic position on this; so I don't know what you're referring to if your statement is supposed to characterize us as somehow taking an "opposite position".

If you want something to comment on, here's my own position: Sometimes skills naturally come first, sometimes understanding comes first. Many times they are learned simultaneously -- and this is the ideal.

To dogmatically insist on a single model is misguided, imbalanced and harmful. The current NWCP is claimed to emphasize understanding-first learning above all, which appears to be where you're coming from in this statement. Simply put, that is an imbalanced approach and we disagree with it. Having spent 5 years now looking at the CCF in detail, my view is that what is described there is quite inadequate for developing understanding, in any case -- a great deal is left to textbook publishers in connecting dots; in places their task is impossible, absent standard algorithms.

The CCF succeeds mainly in removing some important structural elements of understanding by devaluing well-defined, efficient and standardized procedures, developed by many great minds over a period of centuries, in the fool's hope that students' "personal strategies" will be suitable replacements for them. And the other fool's hope that teachers without adequate grounding in the abstract side of mathematics can guide students' explorations toward an equivalent level of understanding without an efficient or uniform approach.

See recommendation 10 of the NMAP report.

But since you insist on this interesting doctrine that understanding must always precede rote algorithms, please explain to me how you propose to teach young children how to count.

dkuropatwa said...
3/1/12 11:31  

@Mike Thanks for the link to your blog post. You made a number of good points; there and here. ;-)

@Ross I don't think having good mathematics teachers "is demeaning to teachers and teacher education"; where did you read that?

It's not clear to me how making a clear distinction between curriculum and pedagogy, emphasizing teaching for understanding and polite debate amounts to "an attempt to create a divide between teachers and end-users of mathematics".

I didn't say you can't forget things you understand, I said it's more difficult to forget things you understand. I'm sure the effort required on your part to recall the mathematics you've forgotten now would be far less than was required when you first learned it. Perhaps your experience is different?

You say the wisemath site is asking for a balance between understanding and skills. I would support such an approach wholeheartedly however, in conversation with Robert & Anna, it seemed to me they were advocating for a greater emphasis on drilling skills or rote learning. In Roberts comments above he says:

"Sometimes skills naturally come first, sometimes understanding comes first. Many times they are learned simultaneously -- and this is the ideal."

So maybe I misunderstood them because I quite agree with that statement.

There are many algorithms for the basic operations. None of them are explicitly written into the Manitoba curriculum. The curriculum (what we teach) says students must be able to add, subtract, multiply and divide. Pedagogically (how we teach), the algorithms have an important and proper place in all math classrooms. I have no objection to including a description of the algorithms in the curriculum documents. I just don't think that's likely because of the difference between curriculum and pedagogy. In any event teachers will continue to teach the basic operations and related algorithms along the lines I quoted from Robert above. I never have and never would tell a teacher not to teach column-wise operations. I am not aware of any school anywhere where teachers are told they cannot teach algorithms. The curriculum doesn't outline the limits of what we teach, it outlines the minimum that must be taught; it's a floor not a ceiling.

You said:
"Since you feel that a line should not be drawn between mathematicians and math educators, you presumably will also support WISE Math's advocacy for strong representation from mathematicians who teach content courses in math at the post-secondary level, and other end-users of math, on committees that design math curricula."

Yes, given they're also well versed in modern approaches to pedagogy.

And:
"While I fully acknowledge that not all children will go to university, I and the founders / supporters of WISE Math want to ensure that children are not unfairly shut out of good careers in science and engineering, early on in their schooling, due to a weak elementary school math curriculum and other factors."

Agreed.

@Robert Reply coming soon.

Ross Stokke said...
3/1/12 12:33  

Hi Darren,

I wish I had more time to reply to your comment but I am very busy preparing for classes which start tomorrow. However, I'd like to point out that you did write the following:

"What I've read in the comments on John's blog and on Anna's blog seems to hold K-12 teachers in a disdainful light."

This is not true and we all think very highly of teachers (my mother and my sister were/are both teachers in the public school system). Again, we advocate for an improved curriculum and improved K-8 teacher preparation. How does this hold teachers in a disdainful light?

I can, incidentally, give quite a few examples of schools and math educators who are under the impression that standard algorithms are banned and discourage kids from using them. I will not do this in this public forum, though.

I also completely agree with Rob's statement about learning math: "Sometimes skills naturally come first, sometimes understanding comes first. Many times they are learned simultaneously -- and this is the ideal."

Did you learn all of the intricacies of real analysis at the same time you learned introductory calculus? Should you have? Should one start by building the real numbers from the rationals using Dedekind cuts?

Why are you surprised that Anna and Rob advocate for a balanced approach? Read point 3. of the WISE Math Mission Statement:

"... Math curricula must support the development of understanding of math concepts AND the learning of basic skills and concepts through incremental practice. The two are NOT mutually exclusive. There must be a healthy balance between understanding and practice, to allow students to grow in both senses."

You hear them arguing for what's missing - this does not mean that they are opposed to a balanced approach. The key word here is balanced.

You wrote that you're in favour of mathematicians being included in curriculum design "given they're also well versed in modern approaches to pedagogy".

Perhaps, by a similar argument, those on curriculum committees should also be well-versed in higher level mathematics which is taught at the post-secondary level. In fact, one might advocate that they should have taught higher level math courses themselves so that they understand what they are preparing K-12 students for. (I'm not saying that I'm advocating for this but it is as extreme as your view.)

Furthermore, simply because someone doesn't agree with what's currently in vogue within the math education community, doesn't mean that they're not well-versed in pedagogy.

Again, given that the vast majority of students who wish to obtain a degree in science, engineering, pharmacy, economics (et cetera, et cetera) are required to take a course in introductory calculus, does it not make sense that you should welcome input from the individuals who teach math content courses at universities?

dkuropatwa said...
3/1/12 13:08  

@Robert Regarding PCAP, TIMMS and PISA I agree that Manitoba's poor performance on these tests is a cause for concern and we have some work to do in mathematics in Manitoba. See both my post above and Mike's post
.

Your insistence on the curriculum documents explicitly delineating how mathematics should be taught does confuse pedagogy with curriculum. You go on to misrepresent what I said in my email to you. You then accuse me of straw man and ad hominum arguments. It seems to me that's what you're doing here. I wrote in my email:

There is a difference between a curriculum document and a pedagogical reference or text. They are not the same. Your question conflates the two. You will likely never see the sort of description you've written above [Ed. Note: describing column-wise operations] included in any curriculum document. You might see it in a text or reference about how to teach mathematics; a pedagogical text.

Also, a curriculum does not provide a ceiling for what students should know or can be taught; it's a floor. There are very few teachers who stop teaching at the floor. I never did. I'm sure you don't either.

As for what the curriculum says about the basic operations, they're all in there described as curricular outcomes, not as pedagogical approaches. 

e.g. 5.N.5. Demonstrate an understanding of multiplication (2-digit numerals by 2-digit numerals) to solve problems.

e.g. 5.N.6. Demonstrate an understanding of division (3-digitnumeralsby 1-digit numerals) with and without concrete materials, and interpret remainders to solve problems.

You'll notice both of these outcomes also emphasize the importance of students understanding what they're doing, the development of which is laid out in the K-4 curriculum. 

A couple of other points of clarification. You and Anna both spoke of "the standard algorithms". I'm sure you know there is no such thing as a "standard" algorithm. There are several multiplication algorithms for example.

It seems to me the crux of our differences turn on this: You spoke passionately about introducing calculation algorithms as soon as possible in students mathematical careers. I also believe students should learn calculation algorithms, as many as possible, but after understanding has been developed.

I know you are concerned about mathematics education. I know some of the teaching methods you're learning about are concerning to you. I know you have a depth of knowledge in mathematics. I wonder if you're familiar with some of the modern literature on pedagogy or specifically the pedagogy of mathematics?

Have you read How People Learn? How Students Learn? or perhaps Adding It Up? All of these are available to be read for free online.


continued ...

dkuropatwa said...
3/1/12 13:08  

...

There are differences in how basic calculation algorithms are taught around the word. Watch how professor Roger Bowley in the UK carries out column subtraction. While we might understand what he's doing elementary school kids here would be confused by his method had they never seen it before. Showing this to kids here would foster some good discussion. Understanding what he's doing would perhaps deepen their own understanding of place value and how subtraction works. And it would give them another algorithm to use for subtraction; another tool in the toolbox so to speak.

The students you teach at University are successful students be they Math, Engineering or Arts students; they've graduated high school. Approximately 30% of high school students in Canada go directly to post-secondary schools after graduation; about 15% go to university. Those are the students you see in your classes. We see them too, and all the rest. As I said above, the demands of teaching math in the K-12 sector are quite different, as is the approach to teaching.

You agreed that debate should sidestep insult. Let's both try to do that, shall we?

@Anna Thanks for the link to the CMESG article. Looks like the U of Winnipeg is up to good things. I also quite liked the Free Trade game.

Rob Fisher said...
10/1/12 07:36  

I hesitated adding this comment because most of the comments are about math pedagogy not so much about our results but in conversations with people this might be being missed

I was reading the PCAP website and found this

"Some of the key findings about the performance of our students include
the following:

Over 90 per cent of Canadian students in Grade 8 are achieving at or above their expected level of performance in mathematics, that is to
say, at level 2 or above. Almost half are achieving above their expected level.
In math, there was no significant difference in the performance of girls and boys at the national level. However, more boys than girls were able to demonstrate high-level math knowledge and skill
proficiency.
For Canada as a whole, girls performed better than boys in both science and reading. More variation was seen at the provincial and territorial level.
In most provinces and territories, students attending
minority-language school systems outperformed students in
majority-language systems in mathematics. This was reversed, however, for reading, where students in majority-language school systems
outperformed students attending minority-language systems. There was no significant difference by language in science performance."

I don't want to get into a statistical analysis war with mathematicians but this summary makes me think that as Canadians, and Manitobans, we're doing well.

Anna Stokke, Math professor said...
10/1/12 22:38  

There are a few things to keep in mind about the PCAP test. To be honest, I'm quite uncomfortable interpreting the results of any test when I have not been able to see the test questions. For example, I could give my calculus class a test and write the questions for that test so that they are very simple and don't really involve much calculus. If I wasn't required to show the test to anyone and simply announced the results and claimed that my students were doing very well in calculus because they scored really well on the test, I would be misleading people.

I phoned the CMEC and asked for copies of the PCAP test questions but they do not release them. There are sample questions in the PCAP report, starting on Pg. 20. These questions raised some red flags for me and for some of my colleagues. First note that students were allowed to use calculators on the test. The sample question for Level 1 involves the addition of a column of numbers. Considering that students could use calculators, I'm not sure what they were trying to test with this type of question (perhaps whether students could punch the correct buttons).

The sample question for level 2 is trivial and simply requires that a student be aware that diameter is twice the radius. From this sample question, which is the only one we're given for level 2, I don't find it at all reasonable that this level was designated as the acceptable level of performance for Grade 8 Canadian students. Considering that 66% of MB Grade 8 students scored at or below level 2, I would say that there is nothing to be proud of here. I don't think that either the sample level 3 or 4 questions are especially difficult either but, again, I have not seen all of the questions that were given because the CMEC will not release the questions.

What concerns me most about the performance of MB students are the two extreme ends. I don't like that 16% performed at or below level 1 and I'm very concerned about the top end - why did only 1% of MB students perform at level 4 while 5% of Ontario students performed at this level? Certainly there are plenty of intelligent and hard-working kids in Manitoba and I hope that they are not being neglected.

Again, I think it's important to be careful about interpreting the results of a test when the questions on the test have not been made available to the public. (For instance, I am quite hesitant to say Canadian kids are doing really well in math based on scores from a test with invisible questions.) However, it is a math test of some sort (as is the math assessment portion of the PISA) and comparisons can be made in performances across provinces. In this aspect, MB students did not perform as well as they should have. Darren points out that if one considers the confidence intervals, several other provinces did just as poorly. However, this does not mean that MB students didn't do poorly and that we don't have a problem - it simply means that those other provinces (like Nova Scotia, for instance) also did poorly and need to improve as well. (Confidence intervals aside, It is a fact, though, that MB scored 10th out of 11 - the same is true of the 2009 PISA results.)

If my students in university were all coming in well-prepared and if I didn't see so many students who have problems with basic arithmetic (particularly arithmetic of fractions), I might not be so concerned about the results of these tests.

1. University professors are extremely concerned about declining math skills based on their experiences with incoming students.

2. The PCAP test scores show that MB students are lagging behind in math.

3. The 2009 PISA results give the same picture (in fact, one can see a decline in MB math scores if previous test years are considered).

We need to be honest and realistic about this and work hard to improve the situation.


OverlookTutorialAcademy said...
13/1/12 17:01  

Great article! There was a study of math teachers around the world as part of the Third International Mathematics and Science Study [TIMSS]and the way their point of view on these very questions influenced their lesson planning.

For quick look at that check my blog: http://overlooktutorialacademy.blogspot.com/2011/10/tutor-tips-turn-your-math-lesson-plans.html#more

Rita said...
31/1/12 19:49  

As a retired teacher ,I agree that it is ao important that students learn what they are doing as they learn new concepts.It is not enough to just memorize material but to understand the concepts and what the concepts mean.

Jamie said...
12/2/12 20:04  

Hello,
Teaching the curriculum is important, but I do feel that what is more important is they way we teach students. The strategies, the volume, and the presentation is key to teaching students of all ages. Adults and children need to have the opportunity to learn in a multitude of ways. Educators need to be open to new ways to teach so that the pedagogy of each educator continues to bloom.

Classof1 said...
15/3/12 08:00  

Interesting post Darren, as an educator I do want my students to understand the concepts and applications of it. So, they can know how to apply it when the need arises. And that is the true purpose of learning.

Jordyn said...
10/4/12 14:28  

Great post. As a current College student wanting to become a teacher I agree with what you have to say here. I love how you say, "there is a difference between knowing how to do something and understanding what you're doing." Many teachers need to understand this concept. Thanks for sharing.

mrgant said...
20/4/12 07:08  

M.gant- very interesting post. This is considered a major concern in our education system. I for example struggle with college algebra and was able to easily pass statistics based on a technique used by the instructor.

B. Goode said...
8/5/12 19:17  

Interesting post. In nursing education a great deal of emphasis is being placed on understanding the concept and not just memorization. Once an individual fully understands the concept it becomes easy to apply it in different and real world situations.

Jen B said...
27/5/12 15:32  

I have to agree here. I have seen in other countries how children learn by rote-- repeating after the teacher-- and I don't understand how they can call that "learning." I believe that when learning, the individual has to be able to understand the concept being taught and be able to apple it. Memorization means nothing if you don't know how to use what is being taught.

Dee Harp said...
9/12/12 16:52  

There are some great points made in this article. There is a big difference between curriculum and pedagogy! I loved math when I was in school but I was one of those people who looked for the answer to a problem instead of understanding the problem and the process to get the answer. I've noticed that now days most students look to find the answer instead trying to understand the subject being taught. If students were taught to understand a math problem and it's process on whatever subject being studied than the student would actually learn the subject and hold on to the information taught instead of loosing the info after a test was taken.

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