As I wrote in my earlier post about solving quadratic equations, introducing the quadratic formula in solving for the roots of a quadratic equation is not advisable because it does not promote conceptual understanding. All the students learn in using the formula is to substitute the values and evaluate the resulting numerical expression. I have seen test questions like “In , what is the value of a, b and c?” Where is the mathematics in this item?
Not teaching the quadratic formula in solving for the roots of a quadratic equation does not mean that the quadratic formula will not be part of the algebra lesson. It would be a good exercise at the end of the unit to ask students to derive a formula for finding the roots of because you will be talking about vertex and discriminant (if you think they need to know what a discriminant is as this will just add to the terms they need to memorize) in later lessons. However I suggest that you ask the students ‘to solve for x’than ‘derive the formula’.
Problem
Solve the equation .
Solution
Express the left hand side as product: .
Complete the square:
The rest as they say is pure algebra:
As you can see, deriving the quadratic formula is a beauty. Using it is not. Completing the square and factoring will do for students solving quadratic equations for the first time (ninth grade, for most countries). What is needed at this point is exposure to different problem solving context requiring representations of and solving quadratic equations.
Coming up in the next post is the meaning of this in graphs.
The original Bloom’s taxonomy include Knowledge, Comprehension, Analysis, Synthesis, and Evaluation. I was introduced to this when I was in college and I must admit it was not of much help to me in planning my math lessons. I just couldn’t fit it. The pyramid image was not of help at all and I think even created the now much ingrained deductive method of teaching. I think teachers must have unconsciously looked at it as a food pyramid so they give a dose of those of knowledge-acquisition activities first before providing activities that will engage students in higher-level processes
Lorin Anderson, a former student of Bloom, revisited the cognitive domain in the learning taxonomy in the mid-nineties and made some important changes: changing the names in the six categories from noun to verb forms and slightly rearranging them. The new taxonomy reflects a more active form of thinking of Creating, Evaluating, Analyzing, Understanding, and Remembering. I also like the inverted pyramid as long as it is not viewed like there is a strict hierarchy of the categories. In fact in my own experience I just make sure that all these are covered in a lesson as much as possible. The way to do this is to teach mathematics through problem solving or engage students in mathematical investigations. Still, the best framework will still be one tailored to mathematics. For me its my list of Mathematical Habits of Mind.
In searching for Bloom’s taxonomy I came across the image below – Bloom’s taxonomy for iPad. It’s a collection of iPad apps classified according to Bloom’s taxonomy. I found it cute so I’m including it here. This will come in handy once I have my own iPad and start creating math lesson for this device.
There is also such a thing as Learning Pyramid which compares how we learn things and the retention rate in our brain after 2-3 weeks.
Click here for source of image of Bloom’s Taxonomy for iPads.
Remember that classic maximum area problem? Here’s a version of it: Pam wishes to fence off a rectangular vegetable garden in her backyard. She has 18 meters of fencing wire which she will use to fence three sides of the garden with the existing fence forming the fourth side. Calculate the maximum area she can enclose.
This problem is usually given as an application problem and is solved algebraically. For example if x is one of the two equal sides to be fenced then the area is the function f(x) = x (18-2x). The maximum area can be found by graphing or by inspection. If students have done a bit of calculus already then they can use the first derivate to solve the problem. But with free technology such as GeoGebra, there should be no excuse not to make the teaching of this topic less abstract especially for Year 9 or 10 students. It need not be at the end of the chapter on quadratic but as an introductory lesson for quadratics. Here’s a GeoGebra applet I made which can be used to teach this topic more visually and conceptually. Below is the image of the applet. I did not embed the applet here because it takes a while to load. Click maximum_area_problem worksheet to explore.
Here’s my suggested teaching approach using this applet. Students need to be given a bit of time exploring it before asking them the following questions:
Pam wishes to fence off a rectangular vegetable garden in her backyard. She found fencing wires stored in their garage which she will use to fence three sides of the garden with the existing fence forming the fourth side. How long is Pam’s fencing wire? What are some of the sizes of gardens Pam can have with the fencing wires?
If you were Pam, what garden size will you choose? Why?
What do the coordinates of P represent? How about the path of P, what information can we get from it?
As the length of BD changes so does the length of the other two sides. What equation will describe the relationship between the length of BD and EF? between BD and DE? between BD and area BDEF.
What equation of function will run through the path of P? Type it in the input bar to check.
What does the tip of the graph tell you about the area of the garden?
The teachers at Level 1 can only tell students the important basic ideas of mathematics such as facts, concepts, and procedures. These teachers are more likely to teach by telling. For example in teaching students about the set of integers they start by defining what integers are and then give students examples of these numbers. They give them the rules for performing operations on these numbers and then provide students exercises for mastery of skills. I’m not sure if they wonder later why students forget what they learn after a couple of days.
Levels of teaching
Level 2 – Teaching by explaining
Math teachers at Level 2 can explain the meanings and reasons of the important ideas of mathematics in order for students to understand them. For example, in explaining the existence of negative numbers, teachers at this level can think of the different situations where these numbers are useful. They can use models like the number line to show how negative numbers and the whole numbers are related. They can show also how the operations are performed either using the number patterns or through the jar model using the + and – counters or some other method. However these teachers are still more likely to do the demonstrating and the one to do the explaining why a particular procedure is such and why it works. The students are still passive recipients of the teachers expert knowledge.
Level 3 – Teaching based on students’ independent work
At the third and highest level are teachers who can provide students opportunities to understand the basic ideas, and support their learning so that the students become independent learners. Teachers at this level have high respect and expectation of their students ability. These teachers can design tasks that would engage students in making sense of mathematics and reasoning with mathematics. They know how to support problem solving activity without necessarily doing the solving of the problems for their students.
The big difference between the teacher at Level 2 and teachers at Level 3 is the the extent of use of students’ ideas and thinking in the development of the lesson. Teachers at level 3 can draw out students ideas and use it in the lesson. If you want to know more about teacher knowledge read Categories of teacher’s knowledge. You can also check out the math lessons in this blog for sample. They are not perfect but they are good enough sample. Warning: a good lesson plan is important but equally important is the way the teacher will facilitate the lesson.
Mathematical Proficiency
The goal of mathematics instruction is to help students become proficient in mathematics. The National Research Council defines ‘mathematical proficiency’ to be made up of the following intertwined strands:
Conceptual understanding – comprehension of mathematical concepts, operations, and relations
Procedure fluency – skill in carrying out procedure flexibly, accurately, efficiently, and appropriately
Strategic competence – ability to formulate, represent, and solve mathematical problems
Adaptive reasoning – capacity for logical thought , reflection, explanation, and justification
Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (NRC, 2001, p.5)
I think it will be very hard to achieve these proficiencies if teachers will not be supported to attain Level 3 teaching I described above. No one graduates from a teacher-training institution with a Level 3 expertise. One of the professional development teachers can engage to upgrade and update themselves is lesson study. The book by Catherine Lewis will be a good guide: Lesson Study: Step by step guide to improving instruction.
