At the start of August, Barbara Oakley, a Professor of Engineering, wrote an op-ed for the New York Times about maths teaching. It is a largely sensible piece about the fact that improvement in maths requires practice, not all of which is particularly pleasant.

Oakley’s article is still creating a stir in the maths blogosphere. Dan Meyer, a education software designer*, has now devoted two blog posts to attacking it. This is notable because Meyer has recently appeared to withdraw from the maths wars, having once been a constructivist/reform/inquiry/fuzzy maths partisan (see his 2010 TED talk and his blog motto, ‘less helpful’).

Meyer’s latest blog post drags up ideas about ‘conceptual understanding’ and ‘maths zombies’ that will be familiar to those of us who have been around the block a few times. I have written a lot of posts about conceptual understanding and it’s actually quite difficult to pin down. If you have the time, try reading some of the ‘productive failure’ literature. This often assesses ‘conceptual knowledge’ and yet the questions that are used to assess it will probably surprise you – they often look for all the world like recall of relatively simple declarative knowledge. This is why I am more impressed with evidence of transfer – but that’s a different blog post.

Understanding is perhaps a useful idea. We have a folk perception of what it means that I have written about before. I don’t want to be in a position of denying the obvious. However, I don’t think understanding is qualitatively different to knowledge. Otherwise, what is it? I also think that it provides a bit of a crutch for some maths teachers. The teacher down the hall may have her students acing all of the standardised tests but you can always console yourself by claiming that her students don’t *understand* the maths as well as yours. This cannot be easily disproved, particularly since the only objective measure we have has already been dismissed. So you can tell yourself this good night story and sleep easy.

Understanding is also necessary to the constructivist narrative. It has been comprehensively shown, from a range of different types of research, that explicit teaching is more effective than inquiry methods. Constructivists therefore need conceptual understanding as a fallback option. “OK,” they say, “a little bit of explicit instruction may be effective for learning mere procedures, but an approach based upon problem solving is more effective for developing conceptual understanding.”

However, there is very little evidence to support such a position. If it were true, there should be loads of it.

I think Oakley walks into something of a trap here; one that has also ensnared me in the past. She talks of there being too much of a focus on conceptual understanding in maths lessons. I now wouldn’t describe it like this. I think that there is too much of a focus on flawed teaching methods that are justified on the basis that they deliver conceptual understanding when they do not.

When Meyer talks of conceptual understanding versus procedural understanding, he is actually talking about the difference between experts and novices. Consider this claim of Meyer’s:

“A student who has procedural fluency but lacks conceptual understanding …

- Can accurately subtract 2018-1999 using a standard algorithm, but doesn’t recognize that counting
upwould be more efficient.- Can accurately compute the area of a triangle, but doesn’t recognize how its formula was derived or how it can be extended to other shapes. (eg. trapezoids, parallelograms, etc.)
- Can accurately calculate the discriminant of y = x
^{2}+ 2 to determine that it doesn’t have any real roots, but couldn’t draw a quick sketch of the parabola to figure that out more efficiently.”

These students are the mythical ‘maths zombies’ who are great at maths but don’t understand it. Yet if you examine all of these examples, they are just examples of students who are earlier along the novice to expert continuum. If you have been practising column subtraction and you are presented with 2018-1999 then, even if you could work it out by counting up, you will probably draw on the most familiar strategy because you lack the bandwidth to analyse it at the meta level. If I really felt it was important for students to be able to determine when counting up is more efficient than column subtraction then I would teach this to students explicitly. I would use contrasting cases where I presented two examples such as 3239 – 2747 and 2018 – 1999 and I would work both problems, both ways, drawing a distinction between the two.

Framed in Meyer’s terms, the idea that there are teaching methods that are better for conceptual understanding than explicit teaching is equivalent to the idea that there are teaching methods that can short circuit the novice-expert continuum and teach expert performance from the outset. This is the progressivist dream and one that, after a couple of hundred years, is yet to be realised.

Oakley comments that many of the engineers who work with her come from education systems that stress procedural fluency and a certain amount of rote learning. Is it really feasible that these people are all maths zombies who don’t understand what they are doing and who have been harmed by the way they were taught maths?

No. It’s not feasible at all. They started out as novices just like everybody else, they worked really hard and now they’re experts. Go figure.

**Meyer makes a point that Oakley and Paul Morgan, a critic of conceptual understanding, are from ‘outside the field’ of maths teaching, so I thought I would return the favour.*

Exactly.

Many of our detractors suggest we promote rote over understanding in our “back” to basics movement (a term that the media has come up with unfortunately; not us). However what we have ALWAYS been advocating for is *effective* math instruction. The reason we lobby so hard to ensure memorization and mastery of math facts be enshrined in primary maths curricula, is because most of them are devoid of these crucial steps. Do we prefer rote over understanding? I don’t know anyone who has ever suggested that. What I do know, is that there’s been a serious erosion in primary maths curricula over the past 50 years, and this is what is most significant in addressing, if we ever want to improve our student’s math performance.

There has been a lack of confidence in understanding the social aspects of such as rote. Fifty plus years I can still sense the annoyance at Forbes-minor (teacher fave) landing his sticker on planet 13 as my fingers touched the ceiling whilst he stumbled as his baying colleagues rolled their eyes as he eventually reached 169. By the time my children came to the subject it was eyes down and calculators at the ready – “all the eights …”

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Love your stuff, but I think that math zombies do exist. I work with first-year university students, most of whom took calculus in high school. Every year, there are a fair number of students who can differentiate pretty much any function I throw at them, but when asked what a derivative is or what it means, say something like “the derivative of x^2 is 2x”. (This is not an exaggeration. I’ve heard that answer, word for word, on more than one occasion.) Can you say that what these students lack is knowledge? Absolutely. However, I think the reason people use the word “understanding” here is that some information, such the fact that a derivative an instantaneous rate of change and the slope of a curve, is particularly powerful in organizing knowledge and enabling transfer.

Incidentally, I’ve tested students’ understanding of instantaneous rates by describing a real-world (or close enough) situation and asking how they would compute the instantaneous rate of change of something in that scenario. Another method is asking how they would explain some concept to a younger student who doesn’t know the vocabulary that they do. These methods are not perfect, but they do go beyond simply recalling definitions (which also has a place).

I suspect that most people who use calculus on a daily basis couldn’t easily give you a good definition. I had to think about this when I first taught it. I also think you could easily train them to quote an acceptable definition if that’s what you wanted them to do. Do they lack understanding? Perhaps. But it’s relatively easily fixed.

Yes, it’s definitely fixable. I teach them to both quote an informal definition and describe how it applies in situations where neither the word “derivative” nor a formula appear. (Without knowing what a derivative gives you, a student wouldn’t know when one is called for without being told.) I only cited this example because you referred to “math zombies” as mythical, while I meet a new pack of the creatures every quarter.

“Maths Zombies” are fictional. Just because these students can’t give you the definition that is in your head, it doesn’t mean they are great at procedural maths but don’t understand what they are doing. In my experience, procedural fluency correlates very highly with conceptual understanding and this two way interaction is supported by research. Furthermore, if “maths zombies” did exist then this would imply the independence of procedural and conceptual understanding and so we should also see students high in the latter and low in the former. I’m not aware of any.

Jane you missed the point. Your students simply know one piece of knowledge but not the other. You could argue that this is an oversight (which may be true) but all curricula have to start somewhere and therefore have to prioritise one thing other another.

I have a degree in physics and teach basic math to 16-19 year old SEN students, and part of my pedagogical development has been the recognizing of subtle flaws in my own understanding. (Including such gems as the correct definition of a edge or denominator as well as how to preform the order of operations). Interestingly I still correctly calculated I just started to realise that I was missing out steps.

You are likely thinking that this proves your point but the solution I found was to simply break the problem down even further and explicitly teach and address these misconceptions. For example

I presented multiple answers for the number of sides of a circle after teaching the definition of a edge and practicing it with simpler shapes.

I made a point to emphasise that AS and DM are reversible (by saying BIMDSA or placing them on top of each other).

I teach improper fractions early and hold back mixed numbers. I also alternate pictorial fractions with numerical ones for longer and even mix in denominators as units/or compare to currency, for example 3 quarters + 3 quarters gives me 6 quarters not 6 eighths. (The last misconception catches out 25/30 of my colleagues when presented pictorially – most get it correct when represented numerically or if they are allowed to convert it into a mixed number).

When Greg talks about curriculum development in teams and learning from each others best practice I believe this is what he means. If they don’t know the definition of a derivative that you have a clear guide on what to teach next (which i assume you do), this is Dylan Williams Assessment for Learning not some damn whiteboad gimmick (though you can use the white boards to show the answers it is merely a delivery system).

The alternatives miss the point. Group discussions, creative activities or problem solving activities might solve the misconception but they will take up much more time and potentially confuse students or lay down new misconceptions. This approach is easier to learn and easier to share with colleagues. Imagine the improvement in m early teaching if someone had told me the seemingly obvious bits of information I shared above.

P.s Ask both math and non-math colleagues to define the denominator (a seemingly trivial piece of maths). Note how many say the total number of parts rather than the number of parts of one whole. This lack of immediate recall of detailed definitions or even the loss of key relationships between ideas is not unique to math but affects all knowledge. We don’t need to conclude that we are all knowledge zombies it is simply an inherent property of human understanding that can be mitigated by good planning.

I totally agree that what they’re missing is knowledge and am not at all a fan of discovery methods. (The closest I come to that is highly guided, step-by-step derivations for some procedures.) However, the information missing is so critical that I think these students do qualify as mathematical zombies because they are highly successful at doing calculations but have little or no idea what these calculations mean. To some extent, like you say, we all do this, but I would argue that some knowledge is so critical at explaining and unifying that if you’re missing it, the “zombie” label does apply.

BTW, I can’t imagine a math colleague defining “denominator” as anything other than “the quantity being divided by”. The whole thing about parts and wholes goes away by the time a kid starts algebra.

The last bit was interesting. I don’t teach much algebra (technically I shouldn’t teach any) due to my cohort and I definitely show that a fraction is simply a division however how you answered it (and the same for me) was very context dependent. I assumed you were considering fractions as representations of physical objects but I failed to make that explicit. Knowledge recall is cue dependent and one person’s idea of profound understanding is another’s idea of a flawed one. It is trivially easy to identify misconceptions in any scheme of work that could be more aggressively addressed and no reason to believe someone else would have the same priorities. Again this applies to all knowledge. The easiest approach is not to claim maths zombies but to simply present an improved approach or scheme for others to judge.

As an example when being taught Jujitsu I was frequently taught moves poorly by instructors more skilled than me. However because I had been taught aikido I new a few techniques extremely well that were fringe in the Jujitsu style. My instructors were not zombies but rather lacked omnipotence.

To use your example most people don’t need to understand derivatives beyond being able to calculate a few common example problems. We should of course strive for more but there are lots of other pieces of knowledge we could transmit instead. What we choose to teach in this space is driven by our own knowledge and priorities rather than societies. (Though good teacher training and subject knowledge will make these closely correlate).

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Reblogged this on traditional math and commented:

Worth reading and commenting on

I teach high school math, Algebra and Geometry, and I long ago settled on a guided approach. The students come in with a variety of “holes” and a wide birth of ability levels as well as various degrees of intrinsic motivation. I only use discovery method as an interesting warm-up or anticipatory activity, after that its straight to the guided approach with plenty of questions along the way. I find that the research showing mastery before inquiry is spot on. Students are far more motivated to stay with a challenging application when they know they have been prepared with the skills they will need for the problem. I also find the greatest tool for learning is a side by side application of two previous and similar skills. For example, to show a student how to choose the best method for finding the roots simple quadratic such as was discussed above, I would have them find the roots by graphing and by quadratic formula for both an easy to graph problem and a harder to graph problem. The side-by-side comparisons of problems and methods are where most of the ah-ha moments happen. I have several homework assignments where my students have to choose between two methods and must use both methods at some point in the work. And my students do well, if I do say so myself.

Reblogged this on The Echo Chamber.

Can accurately subtract 2018-1999 using a standard algorithm, but doesn’t recognize that counting up would be more efficient.I’m late to the party, but I’m just going to point out that this isn’t true.

The problem is that it can be very easy to include the starting and ending point. 2018-1999 = 1999 — 2000 — 2001 — — 2017 — 2018 = 20 which is quite unfortunate. It takes longer to recall how one subtracts by counting, for fluent users of mathematics, than it does to simply do it the “hard” way.

Fluent users have one method that works

in all situations, and save time and energy going over possible other methods that might be useful in very peculiar situations. One perfectly reliable method is simplyI would point out that almost everyone with even the remotest idea of number subtracts 2018 – 1999 by going 2018 – 2000 = 18 and adding one.

And, furthermore, almost everyone else does it on a calculator. Now some people rail about how that is lazy, yadda, yadda, yadda. But the average person has a single 100% effective technique, and they stick to it. They don’t bother having alternative techniques that are sometimes marginally quicker, they use the one that they

knowwill work.Can accurately calculate the discriminant of y = x2 + 2 to determine that it doesn’t have any real roots, but couldn’t draw a quick sketch of the parabola to figure that out more efficiently.I have yet to meet a single person like this. If they can handle the discriminant, they know far too much to be worried by this question.