from – How Not To Be Wrong: The Power of Mathematical Thinking, Jordan Ellenberg, Penguin, 2014:

The methods of calculus are a lot like linear regression: they’re purely mechanical. Your calculator can carry them out, and it’s very dangerous to use them inattentively. On a calculus exam you might be asked to compute the weight of water left in the jug after you punch some kind of hole and let some kind of flow take place for some amount of time, blah, blah, blah. It’s easy to make arithmetic mistakes when doing a problem like this under time pressure. And sometimes that leads to a student arriving at a ridiculous result, like a jug of water whose weight is -4g.

If the student arrives at -4g and writes, in a desperate, hurried hand, “I screwed up somewhere, but I can’t find my mistake,” I give them half credit.

If they just write  –4g  at the bottom of the page and circle it, they get zero even if the entire derivation was correct apart from a single misplaced digit somewhere halfway down the page.

Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense – or deciding whether the method is the right one to use in the first place – requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.

And let’s be frank: that really is what many of our math courses are doing….

The danger of over emphasizing algorithms and precise computations is that algorithms and precise computations are easy to assess. If we settle on a vision of mathematics that consists of “getting the answer right” and no more, and test for that, we run the risk of creating students can test very well but know no mathematics at all.

from When Kids Can’t Read: What Teachers Can Do, by Kylene Beers, Heinneman, 2003:

We talk about inferences. We make inferences all the time. We tell kids to make inferences. When pushed, we can even define inferences: an inference is the ability to connect what is in the text with what is in the mind to create an educated guess.

Right.

I once thought that if my students could make an inference, any inference, then my teaching woes and their comprehension worries would end…. The problem with comprehension, it appeared, was the kids couldn’t make an inference.

I shared this frustration with Anne one day…. On this particular day, I stood leaning against her office door, complaining that the kids she had given me that year could not make an inference. She quickly replied, “Well, teach them.”

”Teach them what?”

“Inferencing. Teach them how to make an inference.”

“You can’t teach someone how to make an inference. It’s inferential. It’s just something you can or can’t do, I said, beginning to mumble.

“Tell me you don’t really believe that,” she said.

“Well, it’s just really, really hard,” I said, now definitely mumbling…. She sent me back to class, and I began to wonder just how I’d teach inferencing. It took years for me to get a handle on that one.

from How We Think, John Dewey, Heath & Co., 1933

Ideas in the primitive and spontaneous sense are suggestions. In this primary sense, then, having of ideas is not so much something we do, as it is something that happens to us. So far as thoughts in this particular meaning are concerned, It is true to say “it thinks” rather than I think.

Only when a person tries to get control of the conditions that determine the occurrence of the suggestion, and only when he accepts responsibility for using the suggestion to see what follows from it, is it significant to introduce the “I” as the agent and source of thought.

Every inference… involves a jump from the known into the unknown. It involves a leap beyond what is given and already established. The inevitableness of mere suggestion, the lively force with which it springs before the mind, the naturally tendency to accept it if it is plausible or not obviously contradicted by facts, indicate the necessity of controlling the suggestion which is made the basis of an inference that is to be believed.

In every case of reflective activity, a person finds himself confronted with a present situation from which he has to arrive at something else that is not present. This process of arriving at an idea of what is absent on the basis of what is at hand is inference.

This control of inference prior to, and on behalf of, belief constitutes proof. To prove a thing means primarily to test it… Not until a thing has been tried do we know its true worth. So it is with inferences. The mere fact that inference in general is an invaluable function does not guarantee, nor does it even help out, the correctness of any particular inference. Any inference may go astray; as we have seen there are standing influences ever ready to instigate it to go wrong. What is important is that every inference be a tested inference…

There is no better way to decide whether genuine inference has taken place than to ask whether it terminated in the substitution of a clear, orderly, and satisfactory situation for a perplexed, confused, and discordant one. Ineffectual thinking ends in conclusions that make no difference in what is personally and immediately experienced. Vital inference always leaves one who thinks with a world that is experienced as different in some respect, for some object in it has gained in clarity and orderly arrangement.

Probably the most frequent cause of failure in school to secure or genuine thinking from students is the failure to ensure the existence of a situation of such a nature as to call out thinking. A teacher was troubled by the failure of pupils, when dealing with arithmetic problems in multiplication involving decimals, to place the decimal point correctly. The numerical figures would be correct but the values all wrong. One student might, for example, say $320.16; another $32.016; and a third, $3201.60.

The results showed that, while the pupils could manipulate figures correctly, they did not think. For if they had used thought, they would not very so arbitrarily in grasping the values involved. Accordingly he sent pupils to a lumberyard to purchase boards for use in the manual-training shop, having arranged with the dealer to let them figure the cost of their purchases. The same numerical operations were involved as in the textbook problems. No mistakes at all were made in placing the decimal. The situation itself induced them to think and controlled their grasp of the values involved.

The function of reflective thought is, therefore, to transform a situation in which there is experienced obscurity, doubt, conflict, disturbance of some sort, into a situation that is clear, coherent, settled.