Questions and Answers about Addition Fact Research

Everything you want to know about memorization and strategies.

I started writing this two months ago and it slipped out of my grasp. I picked it up but it slipped out again. I trapped it in a corner but there was a wet spot on the floor and I hilariously fell backwards while this post flew into the air, landed behind me, and scurried away, laughing as she did it.

I had been putting together these Q&As on research with the ambition of giving teachers readable summaries of research (and maybe writers a head start also). I also wanted all my citations in place so that I could go back and write about this stuff later, perhaps even for long-suffering Maybe Edu Book #2.

I first tackled homework, grading, and I figured math facts were my home turf. I could handle that in a few weeks. No biggie.

Yes, biggie. I’ve been overwhelmed with the complexity of this literature, mostly (I’ve finally realized) because it has the most to do with learning, and so most to do with how the brain actually works. Unlike homework, which is basically a black box surrounded by a bunch of statistics, we can actually know something about how kids learn addition and multiplication.

What that means is that there are multiple literatures to juggle here. There is the cognition stuff, which comes out of cognitive scientists interesting in mathematical thinking, learning, and problem solving. Then there is the intervention literature that mostly focuses on students with learning disabilities. And then there is a much smaller literature that looks at instructional interventions for typically developing students, and to be honest I didn’t find much there to get excited about.

I’ve decided it’s time to get part of this out into the world, so you’re getting what is hopefully the first half. (“Hopefully,” because if this turns out to be less than a half…) It is just about addition. The next entry will be all about multiplication, and I swear to god if it takes me more than a few weeks I will lose my mind.

In the comments or via email, I would very much appreciate hearing what questions you feel were unasked or unanswered in this Q&A and what questions you have about research in general about how children learn basic arithmetic. If you see mistakes or embarrassing omissions, please let me know. (Any errors will, in the future, be attributed to you, so I hope you’re taking this seriously.) I am trying to actually get this right, so please hold me to a high standard.

OK without further ado, here are some questions followed by some answers and then followed by some citations.

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Q: What does it mean to “know your addition facts”?

A: It’s when you can answer single-digit addition questions by heart. It’s when you remember the answer to the question (quickly and easily) rather than figuring it out.

Q: Is this how most adults answer single-digit addition questions?

A: Yes, mostly. Though most of us at times use other strategies to figure them out.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Fuchs, D., Hamlett, C. L., Cirino, P. T., & Fletcher, J. M. (2010). A framework for remediating number combination deficits. Exceptional Children, 76(2), 135-156.

Research (e.g., Ashcraft & Stazyk, 1981; Geary et al., 1987; Goldman et al., 1988; Groen & Parkman, 1972; Siegler, 1987) documents that competent NC (number combination, i.e. single-digit addition) performance involves a mix of strategies, with counting strategies and decomposition strategies serving as back-ups for primary reliance on memory-based retrieval. In fact, individuals, even adults, use varying strategies at different times to solve the same NC.

Geary, D. C. (1996). The problem-size effect in mental addition: Developmental and cross-national trends. Mathematical Cognition, 2(1), 63-94.

In keeping with other studies of the mix of strategies used by North American college students to solve simple addition problems, the American students in this study used a combination of counting and decomposition to solve 27% of the presented problems.

Q: How do we know this?

A: First, because adults tell us that they’re remembering and not calculating. Second, from measuring how long it takes them to answer addition questions. This is called measuring their “reaction times.”

Geary, D. C. (1996). The problem-size effect in mental addition: Developmental and cross-national trends. Mathematical Cognition, 2(1), 63-94.

After each trial, subjects were asked to describe how they arrived at the answer…Chinese adults retrieved answers to 100% of the simple addition problems. In contrast, the American adults retrieved answers to only 73% of the problems and had to rely on some form of back-up strategy to solve the remaining problems.

Q: What is this reaction time evidence?

A: Young kids take longer to respond to 6 + 7 than they do to 2 + 3. But they don’t take much longer when they respond to things such as 7 + 3 or 2 + 8. In general, as the smallest addend gets a bit larger, the reaction time gets a bit larger by a more-or-less consistent amount. This suggests that young kids are figuring out the problem using a counting strategy.

Older kids and adults also take longer to respond to 6 + 7 than they do 2 + 3, but in a different way. First, it doesn’t depend only on the smallest addend but on their sum. Second, the connection between the sum and the time taken was non-linear; larger problems took a bunch longer. Finally, they were overall faster. These together suggest that adults were remembering the answers from memory (but it took longer to remember larger answers).

Ashcraft, M. H., & Guillaume, M. M. (2009). Mathematical cognition and the problem size effect. Psychology of learning and motivation, 51, 121-151.

Groen and Parkman’s analysis of their first graders’ data revealed a linearly increasing reaction time (RT) function across problems…Groen and Parkman concluded that first graders performed simple addition by min counting, a process known in educational settings as ‘‘counting on’’; set the counter to the larger of the two values, then count on by ones for the value of the smaller number, the min, as in 3, 4, 5 for the problem 3 + 2.

Counting could be rejected as the basis for adults’ performance to simple addition problems. Instead, we proposed that adults were searching through a memory representation for the sums of the problems. Because of the longer RTs for large problems, it seemed clear that the search through memory was somehow lengthier or more difficult for those answer

Q: Is that settled then?

A:  There’s still a bit of debate about whether people are just remembering or whether there is some super-fast automatic counting that happens in your brain when you try to remember a fact. It doesn’t seem like this would be very important for educators though, since it manifests as something like remembering.

Baroody, A. J. (2018). A commentary on Chen and Campbell (2017): Is there a clear case for addition fact recall?. Psychonomic Bulletin & Review, 25(6), 2398-2405.

The conventional wisdom in cognitive psychology has long been that fact recall (a reproductive process) replaces counting and reasoning strategies (reconstructive processes), because the former is more efficient than the latter (National Mathematics Advisory Panel, 2008). That, is, “true” mental-addition experts store all basic combinations not involving zero as a specific association between the addends and their sum and directly recall such facts. Recently, proponents of a compacted (automatic and non-conscious) counting model have revived the debate by suggesting that fast reconstructive processes play a role in the retrieval of small sums.

Q: How do people memorize stuff?

A: By trying to remember it.

Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the public interest, 14(1), 4-58.

Attempting to retrieve target information involves a search of long-term memory that activates related information, and this activated information may then be encoded along with the retrieved target, forming an elaborated trace that affords multiple pathways to facilitate later access to that information.

Q: But how do you try to remember an addition fact if you don’t already know it?

A: You read it or hear it, and then try to remember it that the there’s a connection between the numbers in the problem and the answer.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Fuchs, D., Hamlett, C. L., Cirino, P. T., & Fletcher, J. M. (2010). A framework for remediating number combination deficits. Exceptional Children, 76(2), 135-156.</a>

For each computerized drill and practice trial, students saw a complete NC [number combination] “flash” briefly; stored the question stem with its answer in short-term memory; and then reproduced the complete NC from short-term memory. Our assumption was that repeated pairings of a question stem and its correct answer would help students commit the NC to long-term memory.

Miller, S. P., & Hudson, P. J. (2007). Using Evidence-Based Practices to Build Mathematics Competence Related to Conceptual, Procedural, and Declarative Knowledge. Learning Disabilities Research & Practice, 22(1), 47–57. doi:10.1111/j.1540-5826.2007.00230.x

Constant Time Delay is an evidence-based instructional procedure designed to build declarative knowledge. The procedure involves a controlling prompt that results in near-errorless student responses.

Q: When do typical students commit addition facts to memory?

A: Once they have efficient techniques for figuring them out. This process should be moving along after a few years in school, in the US by 3rd Grade.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., ... & Zumeta, R. O. (2009). Remediating number combination and word problem deficits among students with mathematics difficulties: A randomized control trial. Journal of educational psychology, 101(3), 561

To answer addition NCs, typical children gradually develop procedural efficiency with counting. First they count two sets (e.g., 2 + 3) in their entirety (i.e., 1, 2, 3, 4, 5); then they count from the first addend (i.e., 2, 3, 4, 5); and eventually they count from the larger addend (i.e., 3, 4, 5). As conceptual knowledge about number becomes more sophisticated, individuals also develop decomposition strategies for deriving answers (e.g., [2 + 2 = 4] + 1 = 5). As increasingly efficient counting and decomposition strategies help individuals consistently and quickly pair problems with correct answers in working memory, associations become established in long-term memory, and individuals gradually favor memory-based retrieval of answers

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Fuchs, D., Hamlett, C. L., Cirino, P. T., & Fletcher, J. M. (2010). A framework for remediating number combination deficits. Exceptional Children, 76(2), 135-156.

Typically developing students commit NCs to long-term memory through repeated pairings, but those repeated pairings occur naturally with development of efficient counting and back-up strategies.

Ashcraft, M. H., & Guillaume, M. M. (2009). Mathematical cognition and the problem size effect. Psychology of learning and motivation, 51, 121-151.

This suggested that the transition from counting to memory retrieval was well under way by third grade.

Q: Why would students typically only memorize a fact after they have efficient strategies?

A: Less efficient strategies make it harder for us to hold in mind the numbers of the problem along with the answer.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., ... & Zumeta, R. O. (2009). Remediating number combination and word problem deficits among students with mathematics difficulties: A randomized control trial. Journal of educational psychology, 101(3), 561.

As increasingly efficient counting and decomposition strategies help individuals consistently and quickly pair problems with correct answers in working memory, associations become established in long-term memory, and individuals gradually favor memory-based retrieval of answers.

Q: Does it actually matter if kids learn their addition facts?

A: Yeah.

Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: a 5-year longitudinal study. Developmental psychology, 47(6), 1539.

The current results confirm these findings and demonstrate that early arithmetic skills are important for later mathematics achievement, above and beyond the influence of domain general abilities and several other quantitative competencies. Skilled use of counting procedures to solve addition problems and the ability to decompose numbers to solve these problems appear to be particularly important, with the benefits of knowing basic facts in first grade increasing with each successive grade.

Powell, S. R., Gilbert, J. K., & Fuchs, L. S. (2019). Variables influencing algebra performance: Understanding rational numbers is essential. Learning and Individual Differences, 74, 101758.

In work related to algebraic reasoning in the elementary grades, Fuchs et al. (2016) determined second-grade fluency with mathematics facts predicts fourth-grade equation solving with an unknown and function tables processing. In both elementary and middle school, Lee, Ng, and Bull (2018) identified mathematics fact word problems as predictive of algebraic word-problem performance. Beyond the elementary grades, Britt and Irwin (2008) worked with middle-school students and learned a deep understanding of additive, multiplicative, and proportional relationships related to algebraic reasoning. Even at the college level, computation of whole numbers significantly predicts algebra performance (Tolar, Lederberg, & Fletcher, 2009), and mathematics fact fluency significantly predicts success in an algebra course (McGlaughlin, Knoop, & Holloday, 2005). This literature demonstrates a connection between arithmetic and algebra, a connection that likely needs greater emphasis in elementary and secondary school (Banerjee, 2011) and perhaps requires explicit instructional attention in college remedial courses.

Q: So what do you do for kids that are behind or don’t know them?

A: You either help kids memorize or you help them develop more efficient addition strategies. A roadblock for some students with learning disabilities seems to be learning to “count up” by the smaller addend. When compared, both an intervention focusing just on memorizing and one aimed at teaching these kids to “count up” seemed to do equally well at improving kids’ addition fact retrieval.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Fuchs, D., Hamlett, C. L., Cirino, P. T., & Fletcher, J. M. (2010). A framework for remediating number combination deficits. Exceptional Children, 76(2), 135-156.

We examined the effects of drill and practice to encourage automatic retrieval, conceptual lessons to promote decomposition strategies, and the teaching of efficient counting strategies. Regardless of intervention approach, effect sizes were of similar magnitude, suggesting the potential efficacy of all three approaches.

Q: But I should probably do both if I can though, right? Both efficient counting and memorization? Along with conceptual stuff?

A: Yeah, it all seems to be valuable.

Q: What are the implications for the classroom?

A: Just to actually try to teach this stuff. Don’t ignore memorization and efficiency. Don’t ignore strategies either. Teach all these things, but make sure kids are actually making progress in those early years, because it absolutely makes a difference.

Q: Terrific. And I assume all of this applies in a straightforward way to multiplication as well? And also to subtraction and division?

A: Uhhhhh.

TO BE CONTINUED.