In the particular case of 52, solving it more or less by your method, I think “pack of cards”, which makes me want to start somewhat non-optimally with factors of 2 and 13:
52²
(2² × 13)²
2⁴ × 13²
16 × 169
16 × (16 × 10 + 9)
16² × 10 + 16 × 9
2560 + 16 × 9
2560 + 16 × (10 − 1)
2560 + 16 × 10 − 16
2560 + 160 − 16
2660 + 60 − 16
2700 + 20 − 16
2700 + 4
2704
2×5 → 1 0
5×5 → 2 5|0 4 ← 2×2
1 0 ← 5×2
2 7 0 4..to people who learn about Common Core from the rants of the parents who don't get it, quite sadly.
The math shown in this article as well as much common core does not provide the user/student with understanding. Instead, they are taught incantations and rituals to get to an answer. This is a recipe for a society of math illiterates. Who is pushing this and why do they want such a society?
Common Core is not a better system than the one taught for a hundred years in this country. If you disagree, I would love to see the evidence. I'm sure before such a huge change was made to our educational system, there must be MOUNTAINS of such evidence that was presented. No? Not really?
My early school math involved memorising multiplication tables. Later on, there was rote memorisation of other aspects of math that I just never understood - you just "had to", and none of the teachers could explain why.
It was for this reason that I hated math, and it wasn't until I started understanding some of the more fundamental relationships many many years after school that I finally got it.
While this is but one anecdote, and having heard similar stories from others is just more anecdotes - there does seem to be research backing up that a lot of people don't understand the fundamental relationships between numbers and how math works.
I see Common Core math trying to teach relationships between numbers - showing that by being able to break down large unwieldy numbers you can get something that while it might have a bunch more steps, they're at least feasable without counting on fingers/toes or trying to remember times tables.
Why are we multiplying these big matrices of numbers, and what is this complicated determinant thing? Why would anyone ever care what a kernel is, and what are these subspaces that keep getting referenced?
Now every other somewhat mathy problem I see can be broken down into linear algebra and I wish I had learned it better (and that it was taught with more intuition).
It was sufficient to allow a few people to build atomic bombs and go to the moon. It wasn't sufficient to impart the mathematical intuition and literacy needed by the rest of the citizens in a modern democratic society.
"How to teach math to people who are way into that kind of thing" may be a solved problem, but "How to teach math to everybody else" isn't.
Bonfiglio, C. M., Daly, E. J., Persampieri, M., & Andersen, M. (2006). An experimental analysis of the effects of reading interventions in a small group reading instruction context. Journal of Behavioral Education, 15, 93- 109.
Burns, M. K. (2007). Reading at the instructional level with children identified as learning disabled: Potential implications for Response-to-Intervention. School Psychology Quarterly, 22, 297-313.
Burns, M. K., Dean, V. J., & Foley, S. (2004). Preteaching unknown key words with incremental rehearsal to improve reading fluency and comprehension with children identified as reading disabled. Journal of School Psychology, 42, 303–314.
Carney, J.J., Anderson, D., Blackburn, C., & Blessing, D. (1984). Preteaching vocabulary and the comprehension of social studies materials by elementary school children. Social Education, 48(3), 195-196.
Daly, E., & Martens, B. (1994). A comparison of three interventions for increasing oral reading performance: Application of the instructional hierarchy. Journal of Applied Behavior Analysis, 27,459-469.
Eckert, T. L., Ardoin, S. P., Daisey, D. M., & Scarola, M. D. (2000). Empirically evaluating the effectiveness of reading interventions: The use of brief experimental analysis and single-case designs. Psychology in the Schools, 37, 463-474.
Faulkner, H. J., & Levy, B. A. (1999). Fluent and nonfluent forms of transfer in reading: Words and their message. Psychonomic Bulletin and Review, 6, 111-116.
Gickling, E. E., & Armstrong, D. L. (1978). Levels of instructional difficulty as related to on-task behavior, task completion, and comprehension. Journal of Learning Disabilities, 11,559-566.
Hall, K. M., Sabey, B. L., & McClellan, M. (2005). Expository text comprehension: Helping primary-grade teachers use expository texts to full advantage. Reading Psychology, 26,211-234.
Levy, B. A., Nicholls, A., & Kohen, D. (1993). Repeated readings: Process benefits for good and poor readers. Journal of Experimental Child Psychology, 56, 303-327.
McComas, J. J., Wacker, D. P. & Cooper, L. J. (1996). Experimental analysis of academic performance in an academic setting. Journal of Behavioral Education, 6,191-201.
Neill, K. (1979). Turn kids on with repeated reading. Teaching Exceptional Children, 12, 63-64.
O’Shea, L. J., Sindelar, P. T., & O’Shea, D. J. (1985). The effects of repeated readings and attentional cues on reading fluency and comprehension. Journal of Reading Behavior, 17, 129-142.
Pany, D., & McCoy, K. M. (1988). Effects of corrective feedback on word accuracy and reading comprehension of readers with learning disabilities. Journal of Learning Disabilities, 21, 546-550.
Rasinski, T. V. (1990). Effects of repeated reading and listening-while-reading on reading fluency. Journal of Educational Research, 83, 147-150.
Reitsma, P. (1988). Reading practice for beginners: Effects of guided reading, reading-while-listening, and independent reading with computer-based speech feedback. Reading Research Quarterly, 23, 219-235.
Rose, T. L., & Beattie, J. R. (1986). Relative effects of teacher-directed and taped previewing on oral reading. Learning Disability Quarterly, 9, 193-199.
Sanford, A. K., & Horner, R. H. (2013). Effects of matching instruction difficulty to students with escapemaintained problem behavior. Journal of Positive Behavior Interventions, 15, 79-89.
Sindelar, P. T., Monda, L. E., & O’Shea, L. J. (1990). Effects of repeated readings on instructional- and masterylevel readers. Journal of Educational Research, 83, 220-226.
Smith, D. D. (1979). The improvement of children’s oral reading through the use of teacher modeling. Journal of Learning Disabilities, 12 (3), 39-42.
Stoddard, K., Valcante, G., Sindelar, P., O’Shea, L., & Algozzine, B. (1993). Increasing reading rate and comprehension: The effects of repeated readings, sentence segmentation, and intonation training. Reading Research and Instruction, 32, 53-65.
Taylor, N. E., Wade, M. R., & Yekovich, F. R. (1985). The effects of text manipulation and multiple reading strategies on the reading performance of good and poor readers. Reading Research Quarterly, 20, 566-574.
Turpie, J. J., & Paratore, J. R. (1995). Using repeated reading to promote success in a heterogeneously grouped first grade. In K. A. Hinchman, D.J. Leu, & C.K. Kinzer (Eds.), Perspectives on literacy research and practice: Forty-fourth yearbook of the National Reading Conference (pp. 255-263). Chicago: The National Reading Conference.
VanWagenen, M. A., Williams, R. L., & McLaughlin, T. F. (1994). Use of assisted reading to improve reading rate, word accuracy, and comprehension with ESL Spanish-speaking students. Perceptual and Motor Skills, 79, 227-230.
Weinstein, G., & Cooke, N. L. (1992). The effects of two repeated reading interventions on generalization of fluency. Learning Disability Quarterly, 15, 21–28.
Wixson, K.K. (1986). Vocabulary instruction and children’s comprehension of basal stories. Reading Research Quarterly, 21(3), 317-329.
If you want to do a kind of literature review, where you cite these papers as a way of building some kind of actual thesis, that might be interesting. But what is the point of the above?
I can't speak for the situation in the US - and it's been a while for me -, but I remember that we were learning this stuff in that way, and these tricks, as you call them, as well as basic algebra, always seemed as a natural part of the process.
I don't know, but I have a hard time believing that fundamental axioms of math were not part of the teaching.
Oh, and by the way, you should check the result of your example again :)
Some parents are making a fuss because 'that's not how they learned arithmetic' [1], but actually anyone who is good with mental math uses these sort of tricks, particularly taking advantage of commutativity and distributivity.
In fact this builds far stronger intuition for the properties of numbers and is a good way to prepare for algebra and proof-based approaches later on.
Same sort of mental shortcuts for two-digit squaring:
52² = (50 + 2)² = (5 ⋅ 10)² + 2 * (50 ⋅ 2) + 4² = 5² ⋅ 10² + 200 + 4 = 2500 + 200 + 4 = 2604
[1] https://www.salon.com/2015/11/28/youre_wrong_about_common_co...