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?

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.

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.

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

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.

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.

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.

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.

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.

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.

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.