Number Sense
Subitizing, recognizing a small quantity at a glance without counting, is well established up to about four items (Clements, 1999). Past four, students shift into groupitizing: chunking a larger quantity into subgroups they already recognize, which typically emerges around grade 1 (Starkey & McCandliss, 2014). Grouped, structured arrangements are recognized faster and more accurately than scattered ones (Ciccione & Dehaene, 2020), which is why the dot patterns in Subitizing are always grouped rather than random. The broader learning trajectories research (Clements & Sarama) shapes the number ranges and progression used across the whole suite. Number line estimation accuracy both predicts and can train overall math achievement (Siegler & Booth), and the IES/What Works Clearinghouse practice guide for elementary math intervention (2021) recommends the systematic instruction, explicit math language, and visual representations, including number lines, that the Flash, Wait, Ask, Reveal routine is built around.
Fractions
Whole-number bias, treating a fraction's numerator and denominator as two separate whole numbers instead of one quantity, is one of the most persistent errors in elementary math (Ni & Zhou; Braithwaite & Siegler). Every fraction routine here is built against it: fraction magnitude on the number line with feedback (Siegler and colleagues), equal partitioning and equal sharing as the K–2 roots of fraction sense (Common Core; Empson), unit fraction iteration through learning trajectories (Clements & Sarama), and deliberately varied representations so no single picture, especially the circle, carries all the meaning (Mao, Sutherland & Fainstein, 2026).
Fluency
Fluency here means accurate, efficient, and flexible, not simply fast. For multi-digit subtraction with regrouping specifically, jump strategies, adding or subtracting in friendly chunks, outperform split-by-place-value strategies (Beishuizen, 1993; Klein, Beishuizen & Treffers, 1998; Selter, 2001), which is why Split by Place Value is offered for addition only in Multi-Digit Strategy Flash.
Questions
Think something here is out of date, or want a citation for a routine not listed yet? Email dotsensemath@gmail.com.