In the Middle Ages knowledge of the complete table was considered a substantial feat of memory reserved for the truly expert.
Graham Flegg, Numbers: Their History and Meaning, p. 106
In another post, I suggested some questions as part of a thought experiment about what is needed to teach well. In this post, let’s consider what’s needed to make learning something very, very difficult. In this respect, I’ve been thinking about Graham Flegg’s description of multiplication in the Middle Ages. How, for instance, might we teach the multiplication table so that it might be “considered a substantial feat of memory reserved for the truly expert”?
Here are a few of the things I might do if I wanted to make learning the multiplication and related tables very, very difficult. (Hardly anyone does all of these, although nearly all of us have used at least a few. But we are thinking about making learning REALLY, REALLY DIFFICULT!)
Call them facts and treat each fact as a separate bit of information unrelated to any other fact.
Tell students that they need to memorize each one.
Avoid showing how any fact builds upon anything that students have learned before.
Present the facts for each operation (addition, subtraction, multiplication, and division) as totally discrete sets of information, over a period of years, so students can’t use prior knowledge to help them to make sense of what they’re learning.
As students start to learn, put all of the facts on separate flash cards and present and practice them in a random order, without any background information.
Have students work on them very hard once a week or every so often for a long period of time. (Mass practice is less effective than distributing practice more often over shorter amounts of time, within a school day or over several days each week.)
Have students only practice in one way, in one situation.
Have each student work on learning facts by themselves, instead of working or playing together to learn them.
Never use recognition of the correct answer out of possible choices, while students are learning. (Recognition is easier than recall and can provide a useful stepping stone on the way to recall.)
Never use sorting, card, bingo or other games tailored to educational content that directly involve students in learning through exploring, using, and practicing new information in enjoyable ways.
Suggest websites, apps, or worksheets that have children work on all math facts instead of working on a few related facts at a time, then adding more over time.
OR we might just expect children to pick them up without any practice activities. We might think that understanding them is enough.*
Any more ideas on how to make learning math facts more difficult?
* Sometimes people get caught up in an odd dichotomy about what it means to learn math facts and other detailed information well. They seem to focus on whether children can understand concepts (usually through lots of discovery activities) OR whether children can remember details (usually through rote memorization).
Understanding aids memory. A child who understands that doubling quantities is at the heart of the two times table can construct and reconstruct the table. However, what cognitive psychologists call automaticity, so that one can quickly remember the specific information that 2 x 12 = 24 without effortful thinking, depends on focused practice, in which kids use the details of the necessary information again and again, in different contexts, over time. Practice does not have to disregard meaning, but we don’t want to ask children to reconstruct how 2 X 12 = 24 each time they have to use that knowledge.
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