Woo!

Thanks for the question, Kelly. In this situation, it feels like students are (a) focusing on a narrow feature of a given equation, (b) manipulating it in a way they don’t really understand, and (c) getting a result that they also don’t really understand, all without really understanding how the algebraic representation of the function connects to the domain.

So one possible move here is to give them a rational function and several x-values they need to evaluate, some of which should be outside of the domain of the function. They need to know what it looks like to successfully evaluate a function for an x-value and then what it looks like to unsuccessfully evaluate the function for an x-value. In some case they’ll have wasted effort, only realizing deep into their evaluation that the x-value doesn’t work.

That leads to a useful question: “How do we know in advance which x-values work and which ones don’t?”

Thanks again for getting my brain moving a bit tonight. Really interesting question.

Here’s something scary, I was helping a Pre-calc student with some homework on solving rational adding and multiplying rational expressions and solving rational equations. Their teacher told them to they have to limit the domain when they multiply, but not when they add or solve. In fact, one of the examples the teacher did for them showed the solution to be 3, but the denominator of one of the terms was x-3. Talk about fluency without understanding!

Thanks to Alibali for sharing your 2014 article with Crooks about two potential definitions of conceptual understanding of math (https://alibalilab.wiscweb.wisc.edu/wp-content/uploads/sites/371/2018/09/CrooksAlibali2014.pdf).

I noticed that some of the “fluency without understanding” examples could be considered the lack of number sense (such as “Kids that know 7+7 by memory, but have to count 7+8 on their fingers, or know 8×5 is 40 but have no idea that 8×6 is just 8 more.” https://twitter.com/KristaPottier/status/1037890779606343681 and “Student can round 84,899 to the nearest thousand (thanks to silly rhymes), but can’t place 84,899 accurately on a number line to see that 85,00 really is the ‘nearest thousand.'” https://twitter.com/MsOlivaEIS/status/1037843325712248832) .

How do you see number sense fit in these definition?
Is it a foundational to conceptual understanding or one of the related math concepts to understand (see Tosto, et al. 2017 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5611774/ )?

Additionally, what is the point of being able to use procedures without understanding where they came from or why we are using them? Would it be so we can mindlessly calculate? I am with Conrad Wolfram on this one, the world does not need human calculators…

Thaslam –

Could you (or any readers) possibly provide the filename of the post with the typos? I don’t doubt you are seeing them, but the file I am downloading at http://bit.ly/2tMzJ08 does not have those errors. That said, with tech, I do expect disasters.

I believe readers looking at the sentences you cite as “without references” will see plenty of them.
On the need for memorizing facts as a necessary foundation for building conceptual frameworks, permit me one additional reference.

In the theory of learning described by Jean Piaget, moving facts and procedures into long-term memory (LTM) is termed “assimilation.” Resulting changes to the brain’s conceptual frameworks in LTM are “accommodation.” Herbert Simon and his colleagues write that a

“careful understanding of Piaget (shows) that assimilation of knowledge also plays a critical role in setting the stage for accommodation–that the accommodation cannot proceed without assimilation.”
— from Anderson, J.; Reder, L.; Simon, H. (2000). Applications and misapplications of cognitive psychology to mathematics education. Texas Educational Review, Summer (emphasis added)

The article, downloadable from https://files.eric.ed.gov/fulltext/ED439007.pdf , takes some hard shots at the 1989 NCTM standards which I suspect you will take issue with.

But how math can be applied to a variety of problems is research for which Dr. Simon was awarded a Nobel Prize. Readers will find the article to be especially interesting.

For the record, I think everyone was highly disappointed to hear from science, 10-20 years after the 1989 NCTM standards were written and adopted in most states, that the standards asked students to do what their brains could not. Absent clear science at the time, the NCTM made an optimistic guess about how the brain worked. It worked out the way optimistic guesses about science usually do.

Everyone hoped the CCMS would fix what was wrong. It was disappointing to find the CCMS, though better aligned with science than the NCTM’s standards, are not well aligned.

If we must have “state” standards, they must work for students both mathematically and cognitively. But for kids and the nation, it would be better if standards in education were set by front-line practitioners who learn from and respect scientific research, in the manner doctors set the standards of medicine.

— Eric (rick) Nelson

Then show this research by these “cognitive experts”, because as of right now no one who makes your argument on this page has followed up with anything.

(you provided research regarding direct instruction vs discovery learning, that is a completely different conversation. Your report by Geary et. al. actually supports Math ed’s stance: “Instruction focusing on conceptual knowledge of fractions is likely to have the broadest and largest impact on problem-solving performance. Procedural knowledge is also essential, however, and although it must be learned separately, it is likely to enhance conceptual knowledge and vice versa.” as does the Clarebout/Elen article: “The development of competence requires foundation knowledge, a conceptual framework, and the ability to organize and retrieve knowledge.”).

As for Nelson’s article, That is not a scholarly article–lots of spelling errors, poorly written, agenda driven. You make the claim then try to prove it, scholarly articles present the findings and then comment on them–you have no findings, just a slew of claims with citations jammed in for ‘support’. …Two times you make your claim: “The cognitive science of 2017 suggests that with ‘decreased atention’ (sic) to ‘memorizing rules and algorithms,’ skill in calculations would decline” and “The 2010 CCMS assumes the brain can work efectively (sic) with facts before they have been well-memorized. Science in 2017 says the student brain cannot.” still have no reference! Also, claiming that U.S. is last in 2012 doesn’t help support your claim at all, since many of these practices weren’t widely adopted by that time–I’d say based on empirical observation, they still aren’t widely adopted. Not to mention there are a lot of other reasons why the US struggles in educating its masses.

I’d be a fool to follow you and your (still unproduced) science without a better argument or SOME ACTUAL SOURCES!!!

Thaslam —

Much of what you have quoted above (not all, but most) involves people with no expertise in how the brain works claiming that the brain can do what cognitive experts say the brain of students cannot do. That does represent how U.S. math standards have been set since 1989.

And what has been the outcome? By the international standards, at or near the end of their schooling, how are U.S. students doing?

In 2012, the OECD tested skills in 22 developed-world nations including the United States. In “numeracy” (solving problems with mathematical content), U.S. 16-24 year olds ranked 22nd out of 22: Dead last. See http://bit.ly/2tMzJ08 .

In a world where a nation’s technical and engineering skills are essential for self-defense, that outcome devastating for our children and grandchildren. Your math standards that deny science have been tried for 30 years. It is time to ask cognitive scientists to join in making decisions on what our math standards should be.

— rick nelson