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Integration/separation

11/19/2017

5 Comments

 

Being asked to recall something previously encountered might improve the likelihood of recalling that something at a later time, a phenomenon known as the testing effect [although there is some non-laboratory evidence that it is not an effective pedagogical approach for all].  

The word testing is unfortunate, because it might imply that testing / quizzing is the best way to revisit previously encountered material. This has been recognised by various cognitive scientists, who advocate various forms of 'retrieval practice', of which testing/quizzing is only one. [I shall note here that these cognitive scientists describe retrieval practice as "a learning strategy, not an assessment tool" (*).]

There are a number of teachers I have spoken to recently who advocate regular quizzing, often as part of a teaching routine: teach a topic, do some practice on it, set a quiz (possibly a few weeks) after you have finished it, see if they have learned it, perhaps give feedback and act on it, in the hope that they remember it.  It is easy to see why regular quizzing is so popular: reasons that I have heard in support of regular quizzing include strengthening memory, finding out if something has been learned (see *), preparing learners for exams...

The recent emphasis on the role of memory in mathematics education has had positive and negative effects. On the positive side, it has re-stressed the usefulness of memorising certain facts and automating certain procedures, and has focused teachers on the importance of revisiting previously learned material. But I perceive a negative effect to be an over-emphasis on the usefulness of testing / quizzing. ​The following observations stem from my experience:

(1) Even with so-called 'low stakes', ungraded testing (I shall call this type of testing 'quizzing'), it is my experience that there are still negative effects on the self-image of some learners, particularly if the quiz is timed. Why revisit material via a quiz, and why time it?

(2) I have experimented with various degrees and frequency of quizzing, and have found that too-regular quizzing results in (some) learners not applying initiative to quizzes, rendering the process ineffective/inefficient.

(3) The class time spent on regular quizzing + feedback + acting on feedback adds up. For all of those minutes of quizzing, the teacher could have been interacting with learners and providing stimuli for learning, rather than sitting idle while learners complete only those questions they could already do. 

(4) This brings me back to assessment. It is my experience that reading what a learner has written, after the event (and providing feedback after the event), is not as informative for the teacher, or as useful for the learner, as speaking and listening to the learner as they work on a problem.

(5) The organisation of regular quizzing can be difficult, as described here. One solution to the mounting organisational complexity is to give regular short quizzes, on one topic only. This is tempting, as it also gives the teacher the illusion that they can then find out what learners 'can and can't do' (see *). 

And this brings us to the main issue, and an alternative.

The approach teach-practice-quiz-feedback seems to lead to separating / simplifying mathematics teaching and learning into bite-sized chunks. Sometimes it is necessary to separate / simplify, with the aim of re-integrating at some later stage. Gattegno (e.g. here) described the process of learning as integration and subordination: integration of new material with the old, after which the old is subordinated to the new.

I am choosing to stress the importance of integration here. In the mathematics classroom, it is possible to create tasks and activity that continuously integrate previously encountered material. It is necessary, and often desirable, to exploit the connectivity of mathematics, to work on discerning details from wholes, to identify relationships. Successful mathematicians must be able to identify which facts and properties are relevant in which context.  I am not talking here about creating professional mathematicians, I am talking about helping learners to be successful on any exam that requires them to solve problems.  
​

There is the danger that the quizzing teacher stresses topic-by-topic 'retrieval practice', whilst ignoring these other aspects of mathematics. Given the limited benefits of regular quizzing, my contention is that it is not necessary or desirable. The classroom is not bounded by the limitations of experiments into memory; different ways of creating memorable experiences in the mathematics classroom are possible. 

5 Comments
Mike Ollerton
11/19/2017 01:52:54 pm

For me the crux issue resides in your penultimate paragraph:

"I am choosing to stress the importance of integration here. In the mathematics classroom, it is possible to create tasks and activity that continuously integrate previously encountered material. It is necessary, and often desirable, to exploit the connectivity of mathematics, to work on discerning details from wholes, to identify relationships. Successful mathematicians must be able to identify which facts and properties are relevant in which context."

I offer an example of the understanding of Place Value and how it might be embedded in learners experiences from primary to secondary education. One sequence of development (I say "One" because I certainly do not believe in there being a single linear development) is to draw children's attention to the relationship between Th, H, T, U to the structure of 10x10x10, 10x10, 10, 1 and developing this to t, h, th to the right of the decimal point.

There exists the same structure in other bases, so in base 3 we have 3x3x3, 3x3, 3, 1 with 1/3, 1/9, 1/27 to the right of the tricimal point. (Made up word). Some children will only be a gnat's crotchet away from 10^3, 10^2, 10^1, 10^0, 10^-1 and therefore 3^3, 3^2, 3^1 etc yet exponents and certainly negative exponents is usually within the domain of secondary school mathematics.

This use of exponents can/is likely to be revisited when Standard Form appears on the menu. For me, this is all about exploiting the connectivity of mathematics; a far cry from "bit-sized chunks" of mathematics...

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Mike Ollerton
11/19/2017 01:55:24 pm

For me the crux issue resides in your penultimate paragraph:

"I am choosing to stress the importance of integration here. In the mathematics classroom, it is possible to create tasks and activity that continuously integrate previously encountered material. It is necessary, and often desirable, to exploit the connectivity of mathematics, to work on discerning details from wholes, to identify relationships. Successful mathematicians must be able to identify which facts and properties are relevant in which context."

I offer an example of the understanding of Place Value and how it might be embedded in learners experiences from primary to secondary education. One sequence of development (I say "One" because I certainly do not believe in there being a single linear development) is to draw children's attention to the relationship between Th, H, T, U to the structure of 10x10x10, 10x10, 10, 1 and developing this to t, h, th to the right of the decimal point.

There exists the same structure in other bases, so in base 3 we have 3x3x3, 3x3, 3, 1 with 1/3, 1/9, 1/27 to the right of the tricimal point. (Made up word). Some children will only be a gnat's crotchet away from 10^3, 10^2, 10^1, 10^0, 10^-1 and therefore 3^3, 3^2, 3^1 etc yet exponents and certainly negative exponents is usually within the domain of secondary school mathematics.

This use of exponents can/is likely to be revisited when Standard Form appears on the menu. For me, this is all about exploiting the connectivity of mathematics; a far cry from "bit-sized chunks" of mathematics...

Reply
Michael Pershan link
11/19/2017 02:48:58 pm

As far as I can tell, in this piece you're making two related claims.

(1) Quizzing doesn't really help you assess or teach math.
(2) Teachers overemphasize the bite-sized chunks of math at the expense of math as an integrated whole.

Now, I totally buy the second claim. The first claim I don't believe, and I also don't believe that quizzing needs to chunkify math. If you're curious what your kids can do on their own with a nice integrated problem and you give them a problem and they work on it by themselves -- that's a quiz.

You say that you don't find quizzes useful because you'd rather just listen to the kid talk through the problem. I also like listening to kids think, but there are many, many times in my teaching life when that's not possible.

So I think that your two claims don't imply each other. They're independent. Not that you assert otherwise in the piece, but I just wanted to make that clear for myself, as I think through your ideas.

As far as the benefits of quizzing go...look, if you don't need quizzes then you don't. I'm not in the business of telling other teachers what tools they need. (I'm more in the business of trying to deeply describe my own teaching.)

And so when I look at the list of points you make against quizzing, I'm not sure how to take them. I believe that there are classrooms where quizzing doesn't help very much. (In particular, with my 8 year old students I only rarely quiz for various reasons.) It takes too long, they aren't interested in revising their work, the material lends itself to integrated tasks more easily.

With my algebra students, things are different. There really are benefits to getting quite good at solving equations.

So I guess that's my feedback to your post. I don't agree, I don't disagree, I just want more details. I assume you're making smart teaching decisions given your context -- what are those decisions? why do they make sense in your context? what generalizations should other teachers, in other contexts, take from your experiences?

Reply
Danny
11/20/2017 05:21:27 am

Hi Michael

So, any work that is done independently is a quiz? Of course I would agree that this is useful, but this is not common usage, nor what I am referring to. Perhaps it would have been more useful to use the word testing? What I'm talking about here is setting learners (sets of) questions on certain topics that must be done independently (without teacher intervention), to be completed usually (but not always) in writing, perhaps (but not always) under timed conditions.

I am not in the business of telling others how to teach either, I'm sorry if you felt that this is what I was doing... I was trying to offer an alternative, and give reasons why I think this is beneficial... I was hoping to invite the reader to (re)consider the effects of testing in their situation, the mathematics classroom, beyond what the research seems to them to be suggesting. I am a fan of researching my own practice, I my experience of testing is that it was less than useful.

Re: material, my conjecture is that the material for 18 year olds lends itself to integration as much as that for 8 year olds, but of course I don't know the material you are working with.

Of course I agree that it is desirable for learners to "get quite good at solving equations".. but I am saying that testing is not necessary, or even a good way, of doing this... I am saying that learning about solving equations, including 'practice' on procedures, can be integrated and subordinated to solving problems that involve (forming and solving) equations, which has many benefits that testing does not have, and yes that the teach-test structure leads to separation of connections.

Regarding your questions at the bottom, I am working on something that might shed more light on how all of this looks like in practice, I'll send it to you when it's done.




Reply
Danny
11/20/2017 05:28:37 am

Oh, and yes, I am saying that quizzing is not as useful for assessment as it is assumed it is.

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