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I have previously asked the question teaching mathematics to young children about reliance on aids, but now I face this issue with the students I tutor. Seeing as many parents I know have also complained about this issue, I decided to ask it here.

According to Piaget, the formal operational stage of cognition should occur sometime after age 11. I have students who are 17 years old and still cannot abstract away from concrete subjects (like number of tissues, etc.) towards more abstract concepts. Should I, as a tutor, and the parents of those students worry about this apparent delay?

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Piaget's developmental stage theory, despite its wide-ranging impact on education, is highly questionable today in many respects. His research methods were erroneous, and there is plenty of evidence that his overly rigid stage concept is wrong. Milestones based on the theory are also questionable, and in any event general milestones cannot be used to predict individual results.

That said, I would be concerned about students who can't think abstractly at all by age 17. I would think that either some sort of cognitive problem is involved, or that they've simply been poorly taught. What is the educational background of the children involved?

Just as one would not expect creationism to result in a strong science foundation, in my opinion the excessive focus of some modern curricula (e.g. Everyday Math) on "discovery" of number bonds and concrete play in elementary years, with so little focus on abstract thought, hobbles the development of abstract intellectual ability. I would expect a child who had been taught correctly with, say, Singapore Math to have some decent abstract thinking skills in place by middle school (barring some cognitive deficit).

There are of course plenty of ways to develop logical and abstract thought, so I wouldn't give up on those 17-year-olds. I'd start by figuring out where things went wrong for them, then start giving them the foundation in prealgebra, critical thinking and problem solving that they're probably lacking at present.

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I think the problem with Piaget is mostly with people interpreting his findings without understanding statistics. One thing is saying that an average child does X at age of N years. A completely different thing is insisting that every child must do X at age of exactly N years -- not a year sooner and not a year later, otherwise something horrible happens. No, people are different. (Just like finding an average height of a person does not mean that everyone taller or shorter is sick.) On the other hand, if more recent science is available, use that. That's how science works. –  Viliam Búr Jan 11 '13 at 9:11
    
No, that's not most of the problem. –  Iucounu Jan 11 '13 at 14:54
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Abstract thinking requires practice. While Piaget's theories about child development are somewhat reliable for young children, the wheels tend to come off the Piaget bus when it comes to abstract thinking and when different people acquire abstract thinking skills. New research now tells us that the adult brain is not really fully developed until well into a person's twenties. We also know that rational and logical thought is primarily controlled by the left hemisphere of the brain, but teenagers are not generally rational or logical people. They are emotional and irrational and spend most of their time living in the emotional right hemisphere of their brains thanks to their hormones and a general lack of life-experience.

In recent years, the push in mathematics has been to focus on the procedural knowledge (knowing how to do something, like work a math problem) and less focus on the declarative knowledge (knowing why we work the math problem the way we do). I think this has really hurt the critical thinking skills of our students because teachers are so focused on simply teaching the how and forgoing the why. Students can work a problem, but they cannot take those skills and transfer them over to another, similar math problem.

I can safely tell you that at age 17 my abstract thinking skills were terrible. I wouldn't say they were non-existent, but I wouldn't call them good by any stretch of the imagination. Unsurprisingly, I wasn't very good at algebra or geometry in high school and struggled with both. It wasn't until I enrolled in college and began taking music theory classes that I was REALLY forced to start thinking more abstractly and more analytically (sitting and analyzing music for hours at a time will do that). Even though I ultimately graduated with a degree in biology, those music theory classes were instrumental in my personal transition from concrete to formal operational cognition. By this time, though, I was 18/19 years old.

Honestly, I wouldn't be OVERLY concerned about a student who, at 17, isn't exhibiting formal operational thinking yet--unless there are other red-flags being thrown up which might indicate an undiagnosed learning disability such as dyscalculia or even a visual processing disorder which might make it difficult for a student to interpret symbols and patterns or understand math equations.

If you really want to work with your students on their logical and analytical thinking skills, you might try incorporating some simple logic puzzles like the ones you can find here or here. Or you might introduce them to Suddoku. You can use these as a warm-up in the first 5 minutes of your tutoring session. It will be frustrating for them at first because they are working with a part of their brain that they are unaccustomed to using on a regular basis, but with some practice it will get easier for them. And, perhaps most importantly, you need to sit there with them and guide them through it at first until they become more comfortable doing it on their own.

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