# The Quadratic Formula in Malta’s Learning Outcomes Framework

What I see as a deficiency of the Learning Outcomes Framework is that it does not specify learning outcomes in a usable way.

There are several references to quadratic equations in Levels 8–10, for example

Level 8
Number – Numerical calculations
18. I can solve quadratic equations by factorisation and by using the formula.

If a student from Malta comes to my university (and I have had students from Malta in the past, I believe), I want to know what is his/her level of understanding of the Quadratic Formula.

There are at least 7 levels of students’ competencies here, expressed by some sample quadratic equations:

(a) x2 – 3x +2 =0
(d) x2 – 1 = 0
(c) x2 – 2x +1 = 0
(d) x2 + sqrt{2}*x – 1 = 0
(e) x2 + x –  sqrt{2} = 0
(f) x2 + 1 = 0
(g) x2 + sqrt{2}*x + 1 = 0

These quadratic equations are chosen and listed according to their increasing degree of conceptual difficulty: (a) is straightforward, (b) has a missing coefficient (a serious obstacle for many students), (c) has multiple roots, (d) involves a surd, but no nested surds in the solution, (e) has nested surds in the answer, (f) has complex roots, although very innocuous ones, and (g) has trickier complex roots. Of course, another list  can be made, with approximately the same gradation of conceptual difficulty.

I would expect my potential students to be at least at level (d); but LOF tells me nothing about what I should expect from a student from Malta.

And one more comment: a comparison of the statements in the LOF Level 10:

I can solve quadratic equations by completing a square

and in the LOF Level 8:

I can solve quadratic equations by factorisation and by using the formula.

apperars to suggest that at Level 8 the Quadratic Formula is introduced to students without proof or proper propaedeutics which appear only at Level 10. In my opinion, this should raise concerns: at Level 8, this approach has a potential to degenerate into one of those “rote teaching”  practices that make children to hate mathematics for the rest of their lives.

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