Is speed up of arithmetic calculations as a result of memorisation? In elementary school, it is typical for a student to learn "times tables" to increase their speed of mental multiplication. What is the mechanism for the increased speed of calculation resulting from the repetition of "times tables"? Is it memory based? That is, the pattern is recognized and the result is retrieved from memory. Is it optimisation based? Does the brain actually just become faster at multiplication? The same question could apply to the progression of addition as a result of memorised counting.

This is assuming a naive, limited approach without mental imagery optimisation, such Chisanbop, where a mental abacus is used.
I asked this question back when I was building a neural model of progression through counting strategies [1]. The quotes in this answer are taken from that publication.
Psychologically, there is some evidence of addition speedup being a result of memorisation.

Typically, children progress through various strategies before finally
memorizing the results of addition, as shown in Table 1 (Siegler,
1987). The Counting strategy involves choosing the larger number and
incrementing it a number of times equal to the smaller number. The
Recall strategy is where the two numbers form an association with a
previously memorized answer, which is then recalled from long-term
memory. Other strategies have been identified, although we focus here
on recall and counting [...]
[The] recall strategy [...] improves reaction time and accuracy,
consistent with the data from Tables 2 and 3, respectively.



There is also some evidence from the Cognitive Neuroscience point of view which ends up being supported by the neural model I built:

First, when doing mental calculation, age is correlated with the use
of parietal brain areas. However, it is inversely correlated with the
use of prefrontal and hippocampal brain areas, as well as the use of
the dorsal basal ganglia area (Rivera, Reiss, Eckert, &amp; Menon, 2005).
Within the model, this transition can be framed as older children
using fewer iterative processes (no use of working memory in
prefrontal, no iterative process control requiring the basal ganglia,
and no loading of instructions from the hippocampus) and more
memorization.
Second, those with dyscalculia show greater activation
of the prefrontal cortex compared to individuals with normal numeracy
(Kucian &amp; von Aster, 2015). Although this model make no claims about
why dyscalculia occurs, given that it is a complicated disability
usually accompanied by various comorbidities and no direct cause
(Rubinsten &amp; Henik, 2009), it does provide part of a possible
explanation as to why such compensation occurs. Specifically, those
with dyscalculia are unable to consolidate the functional role of the
prefrontal cortex during the counting task within the frontal-parietal
region and must instead rely on their working memory. Given an
excessively noisy input, inaccurate feedback, or inappropriate
modulation of the error signal, the Fast-Net could fail to learn the
mapping between addends and sum. Consequently, there would be limited
progression from counting to recall and a continued dependence on
working memory.



Improving With Practice: A Neural Model of
Mathematical Development