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.