One size does not fit all. That is a big issue with CC. That, and it does not teach a formula for problem solving.

Myself, I would say the answer is E. 7 + 3 = 10, 15 - 10 = 5, 5 + 3 = 8. It's different for everyone. Ultimately there is some memorization involved in math and CC ignores and attempts to make up work arounds.

This, far from being a weakness, is the Common Core math curriculum’s greatest strength. In this very thread we have been discussing both the necessity and insufficiency of deductive reasoning in primary and secondary education. We bemoan the perpetuation of misinformation by news outlets and the lack of critical thinking by the populace, but you’re, essentially, asking that your child or children go back to a method of learning that emphasizes results and memorization instead of deductive reasoning and conceptual learning.

When I learned my times tables, I was told to memorize that 8x7 = 56. This fact was presented as incontrovertible and worthy of committing to memory so that it can be retrieved instantaneously for future reference. We did some exercises like arranging pennies in 8 by 7 arrays and counting them to verify that we had 56 pennies total and grouping to demonstrate that 8 groups of 7 pennies were equal to 56 total pennies, but at the end of the day we were only tested over the problem 8x7 = __. I still don’t have 8x7 = 56 memorized. If I need the answer to the problem I think, “Eight time five is forty plus eight times two is sixteen. The answer is fifty-six.”

Common Core is attempting to teach and test something similar to this second approach to the problem 8x7=__. It is attempting to give students deeper, more abstract understanding of numbers that will serve as a foundation for further mathematical study.

I agree that once a student has demonstrated both an understanding of the concepts behind and proficiency in performing arithmetic they should be able to choose the method that works best for them in actually computing sums and multiplication. However, as long as we educate students in classrooms with twenty children of different learning styles and abilities there will need to be a standard evaluation method. With the new emphasis on conceptual learning in math, the Common Core problems are a shortcut to evaluating a student’s conceptualization of numbers and arithmetic without necessitating one-on-one evaluation.

Looking at your second example, fifteen minus seven is equivalent to fifteen minus five minus two because seven can be decomposed into the sum of five and two. If a student has a thorough conceptual understanding of what seven is, then C should be the obvious answer to your second example since your solution E was not given as a possible solution. This is an example of deductive reasoning. Working from the general concept of what fifteen minus seven means, we come up with various “subtraction sentences” that are equivalent. Instead of testing whether a student can memorize the algorithm of how to take seven from fifteen, they are encouraged to conceptualize seven’s abstract meaning and work from that general definition to evaluate which of a group of possible correct answers is a correct implementation of this general definition of seven.

This is not the math you’re used to where there is only one possible solution. There are many possible solutions. That’s why the problem was multiple choice. Only one of the multiple choice answers fit the question being asked. If asked the question “Which of the following is a type of vegetable? A. Tomatoes B. Grapes C. Bread D. Green Beans” would you answer “E. Carrots”? No. You would answer D. Green Beans, because you understand what a vegetable is, and you understand that both green beans and carrots are vegetables. The Number Sense question you posted is exactly equivalent to my hypothetical question about vegetables, but without the necessity of knowing what a vegetable is. It is a distillation of deductive reasoning.

In college I became the de facto math tutor in my sorority. Invariably my students would complain about “college math”. Now, I can personally attest to the fact that the algebra, trigonometry, and calculus that they were learning is exactly the same as the algebra, trigonometry, and calculus that I learned in high school. The difference between “high school math” and “college math” was a difference in teaching methods. The high school teachers that these women had learned from were mainly concerned with teaching algorithms, or as you put it “a formula for problem solving”. However, “a formula for problem solving” is a contradiction in terms. If you are using an algorithm or formula you aren’t really solving problems in a purely mathematical sense. You are applying the algorithm, and you might be providing the answer to a word problem, but you have not participated in mathematical deductive reasoning which is what is generally meant by the word problem solving.

My sorority sisters constantly complained about the lack of examples in their math classes. Because they had not been provided with a simple algorithm applicable to all problems of a certain type, they did poorly on tests that asked questions even slightly different than what they had been assigned as homework. Sometimes all it took was a professor who assigned homework out of a book but wrote their own exams. The slight difference in wording between one problem set and the other was enough to stump some of my algorithm dependent friends. They resisted using the deductive reasoning necessary to draw comparisons between the homework problems and the test problems, because they had been coddled with exact formulae and algorithms in math classes all of their lives.

The real challenge of the information age and the deciding factor to whether our kids will succeed is how to isolate and evaluate which information is valid and important while ignoring the miscellany of misinformation that is widely available. Deductive reasoning is the difference between a gullible dolt who is convinced by confirmation bias and equates correlation with causation and a critical thinker who can evaluate their sources and question the knowledge presented by authority. While the implementation of Common Core can be flawed and the methodology is foreign to adults who learned arithmetic by rote memorization, the motivation behind the Common Core math curriculum is loftier than the motivation behind the curriculum that I learned as a primary scholar, and frankly, I’m jealous of the kids that get to learn in such a conceptual way.