Multiplication fact mastery: from acquisition to automaticity!

Q: “What does it mean to be fluent with multiplication bonds?”

The answer, more often than not, is, “FAST and ACCURATE.”

BUT, PLOT TWIST! Building fluency should involve more than speed and accuracy. In fact, fluency refers to having efficient and accurate methods for working things out (computing). We say Learners are  fluent with numbers when they can demonstrate flexibility in their ‘working things out’ method they choose (computational methods), understand and can explain these methods, and produce accurate answers efficiently.

The 'WORKING THINGS OUT'  STRATEGIES that a learner uses should be based on mathematical ideas that the learner understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships. 

With repeated experiences working with number, learners can come to “just know” that 2 × 6 = 12. At this point, we say students have mastered their multiplication facts, as they have become so fluent at applying STRATEGIES that they do so automatically, without hesitation.

When learners achieve automaticity with these facts, they have attained a level of mastery that enables them to retrieve them from long-term memory without conscious effort or attention. Automaticity in maths is defined as:

 The Commitment of maths facts to long-term memory frees up working memory (Ashcraft, 1994; Hunt & Ellis, 1999). Learning higher-order skills requires more working memory. When working memory resources are overburdened, “performance deficts are likely to occur” (Eysenck & Calvo, 1992: Faust, Ashcraft, & Fleck, 1999). In committing these facts to long term memory they lay the foundational skills for more difficult math concepts including fractions, decimals, ratios, proportions, and much more (O’Donnell & SanGiovanni, 2011). Without fluency and indeed, automaticity in those simpler facts, higher order math skills are very difficult for learners. Those basic facts translate into almost every aspect of mathematics in later years.

Becoming fluent with math facts in primary school will help children feel successful and want to continue to participate fully in mathematics for the rest of their years in the school and beyond.  

BUILDING TIMES TABLE ACQUISITION…

Build conceptual understanding

Learners need to build a strong sense-based foundation and understand multiplicative relationships before starting to work towards developing automaticity with multiplication. This means:

 

 

 

Brendefur et al (2015) showed that when learners learn timestables through developing STRATEGY  (such as derived fact strategies: adding or subtracting a group/ doubling or halving/ using a square product/ decomposing a factor) they learn timestable facts several times more efficiently than students using drill methods (memorisation and rehearsal). 

Forming a network of connections and relationships is key but this takes takes considerable, DELIBERATE practice that is distributed across time. 

BUILDING TIMES TABLE AUTOMATICITY…

It is not enough that pupils simply “learn” their number facts—they must be committed to memory, just as letter sounds must be memorised in development of phonics automaticity (Willingham, 2009). Automaticity is a different type of knowledge; it is based on memory retrieval.  Learning starts with understanding of concepts, to be sure, but memory skills must develop simultaneously. 

 

To move a fact (or skill) from short-term to long-term memory requires “overlearning”—not just getting an item right, but getting it right repeatedly (Willingham, 2004). And retaining the memory for a long interval requires spacing out additional practice after initial mastery - emphasising the importance of regular review of learned material (Rohrer, Taylor, Pashler, Wixted, & Cepeda, 2005).

Brain research indicates that repetitions actually produce changes in the brain, thickening the neurons’ myelin sheath and creating more “bandwidth” for faster retrieval (Hill & Schneider, 2006).

Not all practice makes perfect. Learners  need a particular kind of practice – deliberate practice – to develop expertise. When most people practice, they focus on the things they already know how to do. DELIBERATE PRACTICE  is different. It entails considerable, specific, and sustained efforts to do something you can’t do well – or even at all.

Research  shows that it is only by working at what you can’t do that you turn into the expert you want to become.

 

How you practice matters far more than how much you practice. Not all forms of practice are equal. Deliberate, purposeful practice. This involves attention, rehearsal, repetition over time, precise feedback and moving out of your comfort zone. 

You can read more about it in this blog I wrote:

 

 

Final thoughts...

Decades of drill and timed testing have failed our learners, often leading to a lack of fluency and a negative disposition toward mathematics. Even in cases where learners are able to successfully complete tasks, such as timed tests, one might question the value of such assessments. Does a perfect score on a timed test really tell us anything about that student’s understanding?  Couldn’t we learn more by carefully observing and questioning students as they engage in meaningful practice playing games, in class discussions of strategies, or even through brief interviews with individual learners? (Kling and Bay-Williams 2014) Such questions are worthy of careful consideration as one reflects on possible paths toward multiplication fact mastery.

It is our hope that by following these three steps:

• Understanding fluency
• Thoughtful sequencing and development of strategies
• Meaningful practice

grown ups can better support their learners as they develop mathematically robust, flexible understandings of multiplication facts.

As a teacher, I believe that it is our responsibility to make sure learners leave our classrooms at the end of the school year proficient in the fluency requirements, that  we nurture flexible thinking and we aren’t just focused on answer-getting but on building number sense.

If we want to help our learners gain MULTIPLICATION FACT MASTERY we need to help them on a journey from acquisition to automaticity!

Want to help your learner achieve AUTOMATICITY? Read our BLOG: 7 Principles for achieving AUTOMATICITY.

 

I’d love to hear your thoughts. Which ideas resonate with you? What do you disagree with?  All views are welcome - post in the comments.  

Thank you for all you do to support your children's number journey. Thank you for watching and listening.

Love, Janey x

 

Here are some LINKS to other REFERENCES which may add value to your thinking : 

Ashcraft, M. H. (1992). Cognitive arithmetic: A review of data and theory. Cognition, 44, 75-106.

Bloom, B.S. (1986). Automaticity: the hands and feet of genius. Educational Leadership, 70-77.

Burns, M. (1995) In my opinion: Timed tests. Teaching Children Mathematics, 1 408-409.

Ericsson, K., Krampe, R., & Tesch-Romer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100(3), 363-406.

 

Fosnot C.T. Dolk M. (2001). Young Mathematicians at Work. Constructing Multiplication and Division. Westport CT: Heinemann. 

Parkhurst, J., Skinner, C. H., Yaw, J., Poncy, B., Adcock, W., & Luna, E. (2010). Efficient classwide remediation: using technology to identify idiosyncratic math facts for additional automaticity drills. International Journal of Behavioral Consultation & Therapy, 6(2), 111–123.

O'Connell, S., & SanGiovanni, J. (2011). Mastering the Basic Facts in Multiplication and Division. Portsmouth: Heinemann.