Be afraid. Be very afraid. For hiding out in a classroom near you there is a maths zombie! You will be able to identify this supernatural algorithm-cruncher by the fact that she can answer complex mathematics questions yet has no understanding whatsoever of what she is doing.

You might however object that ‘understanding’ is latent and cannot be measured. You may in fact suggest that our best guide to mathematical understanding is the ability to answer maths questions. And our zombie has this ability. So what should we do? How can we know?

Well, it turns out that there is a way to make the distinction. Maths zombies will not be very good at the sorts of tasks beloved of constructivists and progressive educators. You know the sort of thing; explaining a mathematical method in words, solving a mundane problem several different ways, making a poster, composing an interpretive dance.

But wait; you may not yet be convinced. Perhaps you might wonder why these other things display a superior understanding of maths than the ability to do complex maths? Surely, a student could be as easily trained to deliver rote explanations or multiple methods without understanding as he could be drilled in the mindless application of a formula? And can we really conclude that a student who struggles to explain his thinking does not ‘understand’? He might just struggle with communication? What about English language learners?

And hang on a minute; exactly how would a traditional, instructivist teacher demonstrate that her students had understanding using her usual methods? She couldn’t. And yet by simply getting students to do the stuff that a constructivist teacher would prioritise, we will demonstrate that those students do have understanding; at least the ones that participate. Ergo, a priori and without any further need for investigation, constructivism wins. It’s about understanding, dude. Deep.

Hmmm… I suppose this is all predicated on the assumption that zombies exist…

Everybody knows that Math Zombies don’t exist. It is merely a tale told to naughty superintendents and policy makers to get them to pay for magic beans rather then invest in real PD or more money for veteran teachers.

The real threat, of course, is math Vampires! There are several ways of telling if you are dealing with a math Vampire:

1) They are very charismatic.
2) Quite often, they have an accent (who can resist an accent).
3) They can’t enter until they are invited. Usually by someone with the key to the public education door who has been seduced by the charisma and accent.
4) They constantly tell you that you have to see what they do “in action” to truly understand although the images of them doing this never seems to translate onto film.
5) When presented with questions, they pull their cape over their face and hiss “engagement” or “understanding” before disappearing into a cloud of accusations of racism/sexism/being right wing/wanting to live in the past.
6) You will notice that, as the money your district spends on education goes through the roof, you will see your child’s teacher pale from lack of sleep begging for photocopier paper.
7) You will notice that your child’s hunger for knowledge never seems to be satiated at school.
8) The math Vampire has a complete inability to take a cold, hard look at themselves in the mirror.
9) They seem to have a weakness to standardized tests.
10) Silver only seems to make them stronger.

However, like regular Vampires, the math vampire does seem avoid the light. Bringing one out into the open does seem to disperse it. Although, like any good horror movie, they always seem to come back. If you hear phrases like “I have always believed that ‘math facts’ are important” then you know that you have a resurgent Math Vampire.

Great comment, Nick! Constructivism as a main mode of teaching (not as a theory of learning) is detrimental — unless the pupils have parents who repair or even anticipate this mode of teaching (i.e., who provide tons of instruction and of “scaffolding”, to use a constructivist term). The expression “a priori and without any further need for investigation” in the original blog post is in conflict with lots of research, for instance on cognitive load.

Reminds me of the lead character out of the Music Man with his think policy. I wish I could just do my job without everyone else telling me how I should do it.

Reading posts like this leave me wondering what you think mathematics is. I think the vast majority of mathematicians would suggest that communicating rigorous arguments in a persuasive and explanatory way is absolutely fundamental to their subject.

Luke’s point is both fair and unfair. For those following the discussion Greg’s posting is clearly about how to test mathematics learning in grades 1 to 12. The issue there is what is an effective test of the math bit. If a student can correctly factor every quadratic you throw at them then it is likely they know how to factor quadratics.

But of course it would be so much better if they could draw a picture of what completing the square looks like and provide a well explained algebraic proof that it gives you the factors and all the factors and then trotting out the derivation of the quadratic formula to cap it off.

Greg’s main point is that the explanatory stuff could also just be rote taught and learned. Being able to write it out in a test is not a sure test of understanding.

I think, and am ready to be corrected, that a second point is that learning the explanatory stuff is more effective if separated from the procedural stuff. That is it makes sense to teach and verify the learning of the procedures for factoring quadratics and then teach the explanations and proofs and verify the learning of that. In some cases the separation could be quite some time. This fits with any mathematician’s book on problem solving where playing around with the problem comes before discovering the solution. To play around with the problem or even follow it having fluency with the procedural stuff is a requirement. So this order of teaching is well aligned with what mathematicians proscribe.

The fairness of the post is that Greg has not addressed the issue of learning how to explain math and how to describe proofs or problem solutions. I think this can be taught and tested. The issue to how to avoid students learning by rote is the same as any. You don’t test them on deriving the quadratic formula or other well know problems. There is no shortage of questions that can test explanatory ability. See any higher level math contest. These manage to test exactly this without risk of rote learners getting high marks. It is worth noting that long answer math competitions tend to start in the very last grades of high school.
(See http://www.cemc.uwaterloo.ca/contests/contests.html all developed by mathematicians.)

There is also the question of why people keep suggesting the point of say grade 7 math is to train mathematicians. Even if this was the point in grade 12 it would be a massive waste of effort as few become mathematicians. Even the idea that it is to teach all those who need to take a university level math course would make it a waste of time for a large proportion of students. I suspect there are good reasons for teaching grade 7 math to every student and it would be worth knowing them.

I don’t think the point of grade 7 mathematics is to train mathematicians. However, I do think that if you want to remove a fundamental part of a discipline from the syllabus taught in schools, then you have to have a good reason for doing so. I can’t think of a good reason for removing argumentation from mathematics. Can you?

I think the vast majority of mathematicians would suggest that communicating rigorous arguments in a persuasive and explanatory way is absolutely fundamental to their subject.

Well it might be for Mathematicians, but we aren’t teaching kids Maths to become Mathematicians. We are teaching them a different way of thinking about the world. We teach particular skills in doing that because the skills cross over into Science, Computing and even Social Sciences (mostly statistics), so we kill two birds with one stone.

A physics teacher needs his students to understand how to add forces as vectors. A biology teacher wants her students to understand exponential growth. Social Studies teachers want their students to know what a mean is. Proof is not required.

We don’t teach argumentation at Grade 7 because the students don’t need it , and aren’t ready for it. This is not “dumbing down” because I’m all for them doing what they do rigorously. But at those Grades we have much better things to be teaching them than rarely used skills that they don’t yet have the intellectual maturity to grasp.

Luke,
I think the issue is always what balance? That has to change a lot from grade 1 to 12. A very superficial view would be that if Sudoku can be a pastime then not much understanding and explaining work is needed to keep students motivated and working through to some level, perhaps grade 7, 8 or 9 depending on a student’s interest.

So a good reason for keeping argumentation and explaining out of the assessment part of grades 1 to 7 might be it is not necessary and it takes up a lot of time that is needed for teaching all students procedures. It would be a particularly good reason if this meant more students were then ready to develop the understanding and explaining skills possible by grade 10 and 12. If math ability can be tested in early grades with multiple choice questions then adding a lot of work may have only a downside.

But less than 3% of students even try that so too much effort on assessing explaining understanding might be wasting time for 97% of the students.

I’m not convinced the person shown is a “Maths Zombie” at all. Although without seeing the kid in action it is hard to tell, it has the classic signs of someone who just isn’t very good at Maths, so compensates for that by getting extremely rule based.

Trying to teach understanding of abstract concepts to that person is likely to go awry. She isn’t interested in it because she can’t actually hold the concepts in her head for any length of time. The dependence on rules is not wilfulness.

(Note I regard the line “anyone can do maths” as contradicted by the evidence of every classroom I have ever been in, current wishful thinking about how we are all equal notwithstanding.)

I am hopeless at music, literally tone deaf. When I learned an instrument it was entirely rule based. My teacher would say things like “can’t you hear that you are in the wrong key?” and I would answer “No”. I literally could not hear. I have a choice of being a “music zombie” or not playing an instrument at all. I was not being wilful, despite what my parents thought. I just don’t have the mental equipment required. It was really very unhelpful when people suggested that “everyone has some musical talent”.

Everybody knows that Math Zombies don’t exist. It is merely a tale told to naughty superintendents and policy makers to get them to pay for magic beans rather then invest in real PD or more money for veteran teachers.

The real threat, of course, is math Vampires! There are several ways of telling if you are dealing with a math Vampire:

1) They are very charismatic.

2) Quite often, they have an accent (who can resist an accent).

3) They can’t enter until they are invited. Usually by someone with the key to the public education door who has been seduced by the charisma and accent.

4) They constantly tell you that you have to see what they do “in action” to truly understand although the images of them doing this never seems to translate onto film.

5) When presented with questions, they pull their cape over their face and hiss “engagement” or “understanding” before disappearing into a cloud of accusations of racism/sexism/being right wing/wanting to live in the past.

6) You will notice that, as the money your district spends on education goes through the roof, you will see your child’s teacher pale from lack of sleep begging for photocopier paper.

7) You will notice that your child’s hunger for knowledge never seems to be satiated at school.

8) The math Vampire has a complete inability to take a cold, hard look at themselves in the mirror.

9) They seem to have a weakness to standardized tests.

10) Silver only seems to make them stronger.

However, like regular Vampires, the math vampire does seem avoid the light. Bringing one out into the open does seem to disperse it. Although, like any good horror movie, they always seem to come back. If you hear phrases like “I have always believed that ‘math facts’ are important” then you know that you have a resurgent Math Vampire.

-Van Helsig

Great comment, Nick! Constructivism as a main mode of teaching (not as a theory of learning) is detrimental — unless the pupils have parents who repair or even anticipate this mode of teaching (i.e., who provide tons of instruction and of “scaffolding”, to use a constructivist term). The expression “a priori and without any further need for investigation” in the original blog post is in conflict with lots of research, for instance on cognitive load.

Reminds me of the lead character out of the Music Man with his think policy. I wish I could just do my job without everyone else telling me how I should do it.

This post & Nick’s comment certainly resonate with me right now. Thanks for a good laugh.

Definitely sounds like your dealing with a load of warlocks.

Reading posts like this leave me wondering what you think mathematics is. I think the vast majority of mathematicians would suggest that communicating rigorous arguments in a persuasive and explanatory way is absolutely fundamental to their subject.

You seem to be adopting the point of view of the beginning undergraduates Kevin Houston discusses on page 3 here:

https://www1.maths.leeds.ac.uk/~khouston/pdf/htwm.pdf

Luke’s point is both fair and unfair. For those following the discussion Greg’s posting is clearly about how to test mathematics learning in grades 1 to 12. The issue there is what is an effective test of the math bit. If a student can correctly factor every quadratic you throw at them then it is likely they know how to factor quadratics.

But of course it would be so much better if they could draw a picture of what completing the square looks like and provide a well explained algebraic proof that it gives you the factors and all the factors and then trotting out the derivation of the quadratic formula to cap it off.

Greg’s main point is that the explanatory stuff could also just be rote taught and learned. Being able to write it out in a test is not a sure test of understanding.

I think, and am ready to be corrected, that a second point is that learning the explanatory stuff is more effective if separated from the procedural stuff. That is it makes sense to teach and verify the learning of the procedures for factoring quadratics and then teach the explanations and proofs and verify the learning of that. In some cases the separation could be quite some time. This fits with any mathematician’s book on problem solving where playing around with the problem comes before discovering the solution. To play around with the problem or even follow it having fluency with the procedural stuff is a requirement. So this order of teaching is well aligned with what mathematicians proscribe.

The fairness of the post is that Greg has not addressed the issue of learning how to explain math and how to describe proofs or problem solutions. I think this can be taught and tested. The issue to how to avoid students learning by rote is the same as any. You don’t test them on deriving the quadratic formula or other well know problems. There is no shortage of questions that can test explanatory ability. See any higher level math contest. These manage to test exactly this without risk of rote learners getting high marks. It is worth noting that long answer math competitions tend to start in the very last grades of high school.

(See http://www.cemc.uwaterloo.ca/contests/contests.html all developed by mathematicians.)

There is also the question of why people keep suggesting the point of say grade 7 math is to train mathematicians. Even if this was the point in grade 12 it would be a massive waste of effort as few become mathematicians. Even the idea that it is to teach all those who need to take a university level math course would make it a waste of time for a large proportion of students. I suspect there are good reasons for teaching grade 7 math to every student and it would be worth knowing them.

I don’t think the point of grade 7 mathematics is to train mathematicians. However, I do think that if you want to remove a fundamental part of a discipline from the syllabus taught in schools, then you have to have a good reason for doing so. I can’t think of a good reason for removing argumentation from mathematics. Can you?

I think the vast majority of mathematicians would suggest that communicating rigorous arguments in a persuasive and explanatory way is absolutely fundamental to their subject.Well it might be for Mathematicians, but we aren’t teaching kids Maths to become Mathematicians. We are teaching them a different way of thinking about the world. We teach particular skills in doing that because the skills cross over into Science, Computing and even Social Sciences (mostly statistics), so we kill two birds with one stone.

A physics teacher needs his students to understand how to add forces as vectors. A biology teacher wants her students to understand exponential growth. Social Studies teachers want their students to know what a mean is. Proof is not required.

We don’t teach argumentation at Grade 7 because the students don’t need it , and aren’t ready for it. This is not “dumbing down” because I’m all for them doing what they do rigorously. But at those Grades we have much better things to be teaching them than rarely used skills that they don’t yet have the intellectual maturity to grasp.

Luke,

I think the issue is always what balance? That has to change a lot from grade 1 to 12. A very superficial view would be that if Sudoku can be a pastime then not much understanding and explaining work is needed to keep students motivated and working through to some level, perhaps grade 7, 8 or 9 depending on a student’s interest.

So a good reason for keeping argumentation and explaining out of the assessment part of grades 1 to 7 might be it is not necessary and it takes up a lot of time that is needed for teaching all students procedures. It would be a particularly good reason if this meant more students were then ready to develop the understanding and explaining skills possible by grade 10 and 12. If math ability can be tested in early grades with multiple choice questions then adding a lot of work may have only a downside.

(See the Canadian math competitions for the sort of multi choice test that I am pretty certain you have to understand the math to do well on http://cemc.uwaterloo.ca/contests/past_contests/2015/2015PascalContest.pdf)

These same people think by grade 10 you need a good explanation to score well on one of their competitions.

http://cemc.uwaterloo.ca/contests/past_contests/2014/2014CIMC.pdf

But less than 3% of students even try that so too much effort on assessing explaining understanding might be wasting time for 97% of the students.

I’m not convinced the person shown is a “Maths Zombie” at all. Although without seeing the kid in action it is hard to tell, it has the classic signs of someone who just isn’t very good at Maths, so compensates for that by getting extremely rule based.

Trying to teach understanding of abstract concepts to that person is likely to go awry. She isn’t interested in it because she can’t actually hold the concepts in her head for any length of time. The dependence on rules is not wilfulness.

(Note I regard the line “anyone can do maths” as contradicted by the evidence of every classroom I have ever been in, current wishful thinking about how we are all equal notwithstanding.)

I am hopeless at music, literally tone deaf. When I learned an instrument it was entirely rule based. My teacher would say things like “can’t you hear that you are in the wrong key?” and I would answer “No”. I literally could not hear. I have a choice of being a “music zombie” or not playing an instrument at all. I was not being wilful, despite what my parents thought. I just don’t have the mental equipment required. It was really very unhelpful when people suggested that “everyone has some musical talent”.