With and Across The Grain
Anne Watson (2000) came across some learners asked to copy and complete a table based on the following structure
7 x 1 = 7 1 x 7 = 7 7 ÷ 1 = 7 7 ÷ 7 = 1
7 x 2 = 14 2 x 7 = 14
14 ÷ 2 = 7 14 ÷ 7 = 2
… … … …
She noted that when learners follow a simple number pattern to
anticipate the next and future terms, they are acting in a manner which
is similar to going with the grain of a piece of wood: fresh wood
splits relatively easily along the grain. This matches my
experience of offering people sequences of terms in which everyone
(mathematicians and non-mathematicians alike) quickly work out the
pattern and can predict the next and future terms.
Going with the grain on sequences and grids means following simple
patterns such as writing all the 7s in the first column, then all the
multiplication signs, then the numbers 1, 2, 3, … and so on.
Cutting across the grain reveals the structure of wood, so going across
the grain can be used to refer to the act of making mathematical sense
of relationships, here, between the different entries in a row of the
table, which is presumably what the authors intended learners to
The phrase with and across the grain
can shift from description to action when it reminds teachers to prompt
learners to make mathematical sense and so turn copy-and-complete from
a clerical exercise into a significant and relevant mathematical
experience. Another way of saying this is that in order for doing
a task to influence learning, it is necessary to prompt learners to see
the general through (each of) the particulars, and then to see each of
the particulars in (as instances of) the general. This two way
process was summarised by Alfred Whitehead (1932):
To see what is general in what is particular
I prefer to rephrase it more expansively:
and what is permanent in what is transitory
is the aim of scientific thought. (p4)
‘to see the general through the particular and the particular in the general’
With and Across the Grain, when internalised as a description of
actions which a teacher can take to direct learner attention, has
become a teaching framework which enhances or structures
learning. When taken up by learners, it acts as a framework for
learning. As with other teaching-learning frameworks, it serves to
bring to mind actions which might enrich learning, but which might
otherwise have slipped by unnoticed.
and ‘to be aware of what is invariant in the midst of change’
is how human beings cope with the
sense-impressions which form their experience, often implicitly. The
aim of scientific thought is to do this explicitly.
Watson, A. (2000). Going across the grain: mathematical generalisation in a group of low attainers. Nordisk Matematikk Didaktikk (Nordic Studies in Mathematics Education). 8 (1) p7–22.
Whitehead, A.(1932). The Aims of Education and Other Essays. London: Williams & Norgate.