As a starter, on page 53 of

Metcalf, R. C. (2007). The Nature of Students' Understanding of Quadratic Functions. ProQuest.

found at

https://books.google.fr/books?hl=fr&lr=&id=vXWJ6cDuddcC&oi=fnd&pg=PR10&dq=conceptual+obstacles+in+the+learning+of+quadratic+functions&ots=exXUJtyyW2&sig=KPBboueH3fS8fD865LWMc5yoQx8#v=onepage&q=conceptual%20obstacles%20in%20the%20learning%20of%20quadratic%20functions&f=false

there is a reminder of Zalavsky's (1997) Five main conceptual obstacles.

Zaslavsky, O. (1997). Conceptual Obstacles in the Learning of Quadratic Functions. Focus on Learning Problems in Mathematics, 19(1), 20-44.

As well as the papers noted above by Christian Boissinotte, we have a section entitled "Typical problems in reasoning about functions" in our book: Key Ideas in Teaching Mathematics: Research-based guidance for ages 9-19

http://ukcatalogue.oup.com/product/9780199665518.do

Let’s begin with this classical high school question:

What , if anything, is wrong with

 1= square root of 1= square root of (-1)(-1)= square root of -1 x square root of -1=-1

 Proof. Theorem. Every element  (a,b) de C, (a,b) different of (0,0)  has two square roots in C.  Then, unlike R, C cannot be made into an ordered field.

 And so on, we can find many other obstacles in learning of quadratic functions.

Carlos, at what level do these teachers work? do they deal with complex solutions of equations? if so, as Eugenio pointed out, there is a whole lot of new obstacles related to the field change.

For a quadratic function that has the real numbers as its domain and range, one can ask if one or more points in the domain map onto zero.  Further exploration of this function can lead to a larger domain and range--the complex numbers. The sam question about solving quadratic equations can be asked.

At some stage, one asks the question of whether there are procedures one can use to find solutions to quadratic equations, either in the domain of real numbers or the domain of complex numbers. There are a variety of such procedures, the quadratic formula being one.  My point is that I want to deal with your question at a conceptual level that is different than the conceptual level you are observe in some of your students.

Moreover, I want to deal with the quote at a quite general level. Functions (by definition) are mappings that have a domain and a range. (For example, the domain might be 2 by 2 matrices of real numbers; it might be a differential function). One can then ask for a particular function what are the specific domain values.  So, as one teaches the topic of function, one also can teach the topic of "solving" equations. Here is a question for thought: We emphasize graphing of functions. In what ways is this useful, and in what ways does it limit a learner's insights into functions?

I dont know conceptual obstacles in learning of quadratic functions.

Good idea planning to make a diagnostic test for in-service math teachers about quadratic functions. I know misconceptions in experimental sciences.

There is a bunch of research investigating difficulties of students in calculus. I list here some reading that might be useful.

Ainsworth, S. E., Bibby, P. A., & Wood, D. J. (1997). Information technology and multiple representations: New opportunities—new problems. Journal of Information Technology for Teacher Education, 6(1), 93-104.

Anderson, M., Bloom, L., Mueller, U., & Pedler, P. (1999). The impact of the graphics calculator on the assessment of calculus and modeling.International Journal of Mathematical Education in Science and Technology, 30(4), 489-498.

Coulombe, W. N., & Berenson, S.B. (2001). Representations of patterns and functions: Tools for learning. In A.A. Cuoco (Ed.), The roles of representations in school mathematics (pp. 166-172). Reston, VA: National Council of Teachers of mathematics.

Dubinsky, E., & Harel, G. (Eds.) (1992). The concept of function: Aspects of epistemology and pedagogy (Mathematical Association of America (MAA) Notes No. 25). Washington: MAA.

Dunham, P. H., & Osborne, A. (1991). Learning how to see: Students’ graphing difficulties. Focus on Learning Problems in Mathematics, 13(4), 35-49.

Eisenberg, T., & Dreyfus, T. (1994). On understanding how students learn to visualize function transformations. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in Collegiate Mathematics Education I (Vol. 4, pp. 45-68). Rhode Island, RI: American Mathematical Society.

Ellis, A. B. (2007b). Unexpected connections across function families: students' generalizations about quadratic data. Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, University of Nevada, Reno, Nevada. Retrieved January 15, 2010 from http://www.allacademic.com/meta/ p190328_index.html

Ellis, A. B., & Grinstead, P. (2008). Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. Journal of Mathematical Behavior, 27(4), 277-296.

Movshovitz-Hadar, N. (1993). A constructive transition from linear to quadratic function. School Science and Mathematics, 93(6), 288-298.

Wilson, M. R., & Krapfl C. M. (1994). The impact of graphics calculators on students’ understanding of function. Journal of Computers in Mathematics and Science Teaching, 13(3), 252-264.

Yerushalmy, M., & Shternberg, B. (2001). Charting a visual course to the concept of function. In A. A. Cuoco (Ed.), The roles of representation in school mathematics: National Council of Teachers of Mathematics (NCTM) 2001 yearbook (pp. 251-268). Reston, VA: NCTM.

Zaslavasky, O. (1997). Conceptual obstacles in the learning of quadratic functions. Focus on Learning Problems in Mathematics, 19(1), 20-44.

Carlos:

At this discussion moment, it seems to be worthwhile to add something about Teaching Models concerning your question:

Look at Educational Algebra: A Theoretical and Empirical Approach. Springer, (2008), Chapter 10 mainly al incise 3 beginning, p. 262.

A more detailed version can be found in my ResearchGate publication: Aspectos Teóricos del Álgebra Educativa, Actas, Vol. 11.

Hi, Carlos!

In my experience teaching to high school students and undergraduate students I noticed that high school students use to think in a "linear way". So, for many of these students, almost all problems can be solved as a linear proportion (which I think that may be the cern of your question). This issue tends to be reduced when students go to the Faculty, since they get used to perceive nonlinearities in their everyday life.

I think that this issue can be solved if we started to bring nonlinearities (such as quadratic functions) to everyday life of elementary and high school students.

Consider, for instance, a ride to the subway. Bring the students to the inside of the train and let they "feel" the effects of the "quadratic function" embedded in the acceleration. After this experience, explain the meaning of acceleration through the use of a classic math function.

Thus, to break obstacles in the learning of quadratic functions, you may try to explore the previous conceptions of students and then bring mathematics to the inside of their everyday life.

Cheers!

Tarcisio.

Hello again Carlos,

Let me share some observations about some difficulties I have observed at different school levels.

Let's stay in the real values realm for now and with quadratic functions (no conics equations except that of parabolas with vertical axis).

Many don't realise that an equation of the form ax2+bx+c=0 is not a function and that the only things it represents are the values that make it true (it is a way to express them). They also don't see that this is a set (maybe empty) of at most two real values (that can be associated to zero, one or two "zero dimensional beings" represented by points on the real line). Taking a common non-zero multiple of the coefficients (or factoring the equation) does not change it's "truth set", and leads to a seemingly different but equivalent equation. These problems extend to inequalities where we deal with intervals on the real line.

In the case of f(x)= ax2+bx+c, they tend to think that f(x) is a variable and not the result of a procedural calculation with each value of x. So they don't see the difference in writing things like y=f(x). But that last statement establishes a covariation between two variable values in a functional way and can be represented by a two dimensional curve. So now taking a common multiple of the coefficients while keeping y does change the object, except for the particular values that make y equal to zero.

ax2+bx+c=0 can be written as ax2=-bx-c (or other variations) and we can decide to set g(x)=ax2 and h(x)=-bx-c to see these as two algorithmic calculations for which we want to know when they are equal (g(x)=h(x)). We could also set y=g(x) and y=h(x) to associate the calculated values with new variable y so as to have two functions, or two equations in two variables. We created something new, different from what we had first. Finding the values for which ax2+bx+c=0 in a one dimensional set is now replaced by finding the values of x when the two functions cross.

For the general case, implicit equations in x and y are difficult to graph because of the emphasis given to functions in early teaching.

The above answers assume that students understand Algebra. They often do not, let alone the notion of a function. There is a potent tendency for college students to interpret Algebraic equations as if they were natural number problems. So, it is seems as if the bottom problem is getting is getting students to recognize that an equation is not either a natural number problem and that there are bracket and order rules in math that are different than those in natural language. Actually, we do not write sentences with brackets and order rules, although they are hidden. One cannot do math without explicit use of these. 

Thank you so much for your answer. They have been very useful. 

Carlos:

In my researchgate publications list there is a Highschool lesson about iterative secuences of sets of objects and it’s second degree polynomials description.

This allows  to give algebraic formulation of polygonal numbers like Pitagoreans and Diophanto approach to them. 

This allows to introduce algebra as a generalization of arithmetic. There are very good materials about the roots of algebra in the U.K. Open University.