Once a set of learning objectives have been settled on for an activity, problem, or project, what should the problem’s context be? Since linear equations model situations where there is a constant rate of change, common contexts for linear equation projects often include the following:

• Steepness, height, angle
Examples: road grade, hillside, roof, skateboard park element, tide height over the two weeks before (or after) a full moon, sun angle at noon over a six month period
• Estimating time to complete a task (setup plus completion)
Examples: mowing a lawn, painting a wall, writing a research paper
• Purchase and delivery costs of bulk materials
Examples: mulch, gravel, lumber
• Purchasing a service that charges by consumption
Examples: cell phone, electricity, water, movie rental, etc.
• Total earnings over time from differing wage and bonus plan structures
Examples: hiring bonuses, longevity bonuses
• Energy use over time
Examples: calories burned, electricity, heating oil, gasoline
• Game points accumulated over time
Examples: by a professional athlete, a team, a video game player
• Pollutant levels over time Read more…

A number of historically “good” math students seem to reach a point during their High School years where their feeling of mastery seems to be slipping away. While teachers usually expect more from a student with each passing year, student frustration usually arises from more than just increasing teacher expectations. I believe it arises because a tried and true study habit, memorization, is no longer enough to assure mastery.

### My experience

I used to read a math or science textbook in pretty much the same way I read anything: as quickly as I could. In fact, for math I often skipped the reading entirely as I had been shown how to do the new types of problems in class, so all I had to do was sit down and follow the procedure I had been shown – no need for all the verbiage.

However, this approach stopped working when I got to college. If my notes from lecture did not help me figure out how to solve a problem, I had to rely on the text part of the textbook for almost the first time. I learned that “believing I understood everything that happened in class” was a very different thing from being able to solve the problems assigned for homework.

After skimming through my math text, I often found that

The lesson plans I find most interesting, both to read and to teach from, have both “public” and “hidden” learning objectives.  The public objectives focus student attention and help interest students in the problem: they need to be short, to the point, and tightly related to the problem or project at hand.

The “hidden” objectives are the focus of teacher attention. They reflect the skills and concepts that the teacher hopes to see students grappling with, discussing with peers, and mastering over time while working on successive problems. If students are informed about a teacher’s list of objectives in assigning a task, students are likely to use only that list in their work. By not publicizing the teacher’s objective list, students are more likely to try a wider variety of approaches to solving a problem. I think the problem solving process starts with determining which concepts and skills seem relevant to the problem, therefore keeping the teacher’s objective list hidden helps students become better problem solvers.

The list below covers topics typically taught over a large percentage of the school year, so not all objectives are appropriate at any given point in the year. However, by the end of the year hopefully most of the following objectives will have been mastered by most students in a class:

I have seen three approaches to “Completing the Square”, as shown below. Each successfully converts a quadratic equation into vertex form.  Which do you prefer, and why?

### First Approach

This approach can only be used when you are working with an equation. It moves all terms that are not part of a perfect square to the other side of the equation to get them out of the way:

$y=2x^2+12x+10$

$y-10=2x^2+12x$

$\frac{1}{2} y-5=x^2+6x$

$\frac{1}{2} y-5+(\frac{6}{2})^2=x^2+6x+(\frac{6}{2})^2$

$\frac{1}{2}y-5+9=x^2+6x+3^2$

$\frac{1}{2}y+4=(x+3)^2$

$\frac{1}{2}y=(x+3)^2-4$

$y=2(x+3)^2-8$

### Second Approach

Grant Wiggins was the keynote speaker last night at the annual “Anja S. Greer Conference on Mathematics, Science and Technology” hosted by Phillips Exeter Academy in Exeter, NH. The focus of his talk was mathematics education, and the points that were noteworthy to me included the following:

Increasingly, schools and standards bodies are setting their goal for mathematics education to be the development of good problem solvers. Yet,
- few schools focus their curriculum on problem solving
- nationally, dismal percentages of students can successfully solve problems of types they have been taught to solve, let alone problems they are not familiar with
- a significant percentage of students hate their mathematics courses

We face some big questions that are challenging to answer:
- What is the problem with mathematics education today?
- What are we going to do to address it?
This is the problem that math teachers and curriculum designers must solve.

If students are to be able to solve problems of types they have not necessarily seen before, they need the ability to transfer their knowledge and skills to new domains. Yet, most of mathematics education today focuses on Read more…

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Adding a squared term to a linear expression, creating a quadratic expression in the process, seems like a relatively small change:

$3x+2$

$x^2+3x+2$

Yet, if this new term is part of an equation, the procedures that worked nicely when solving linear equations don’t work so well any more. Investigating what happens in such situations is useful, and leads to some new concepts and procedures.

If a quadratic equation is approached in the same way as a linear equation, it can sometimes be solved quickly:

$18=2x^2\\*~\\*9=x^2\\*~\\*\pm3=x$

Familiarity with square roots and how to solve linear equations are enough to solve this equation. By getting the variable all by itself on one side, the two possible solutions to the original equation are left on the other.

However, the following equation cannot be solved in the same way:

$0=x^2+2x+1\\*~\\*-1=x^2+2x\\*~\\*\dfrac{-1}{x}=x+2~~~~~~~~~(a)\\*~\\*-1=(x)(x+2)~~~(b)$

This equation has two “x” terms, and they are not “like terms” since one has the variable to the first power and the other to the second power. If a linear equation approach is used, moving the constant term to the other side, two un-like terms are left on the right… but what to do from here?

The 2012 Mathematical Sciences Institute (MSI) will be held July 16-20 at Tulane University. The MSI is modeled on the Anja S. Greer Conference held at Phillips Exeter Academy (mentioned below recently), and held every June. For more information, please click on the MSI link above.

I recently came across a start-up organization called the Peer Instruction Network. It sounds like it is seeking to expand on Eric Mazur‘s teaching approach, something which would be very interesting to me on the Mathematics side of things.  Check out their web site, and sign up to be included in their network if it sounds interesting.

I attended the Anja S. Greer conference at Philips Exeter Academy last year and highly recommend it to anyone interested.  Details of this year’s conference have just been published, so click on the image below for more information.

A recent eSchool News article by Meris Stansbury lists ten skills cited by its readers as being most important for today’s students to acquire:

2. Type
3. Write
4. Communicate effectively, and with respect
5. Question
6. Be resourceful
7. Be accountable
8. Know how to learn
9. Think critically
10. Be happy

The list is interesting to ponder. I would not argue that any skills on the list should be dropped, however I suspect we could have endless debates about what order to list them in or how to best group them. I am happy to note that all of the skills are beneficial in studying just about any subject or discipline.

Most folks learn to multiply increasingly complex quantities gradually over time, starting with constants in elementary school, and eventually continuing on to polynomials in high school.  As the quantities become more complex, students master “the distributive property”, “collecting like terms”, and perhaps even procedures made more memorable with acronyms like “FOIL”.

When learning to multiply binomials and polynomials, people often focus more on the process than the reasoning behind it – which can makes things feel complex. Once you understand the reasoning, multiplying polynomials will hopefully become straightforward. And with a small measure of melodrama, I will describe “FFFT!”, a trivial technique with a silly name that can help make multiplying polynomials easy.

Way back in elementary school, perhaps in first grade, you were probably taught to multiply two integers:

$8\cdot 7\\*~\\*=56$

From there, you were probably asked to learn your multiplication tables. It is extremely useful, particularly when studying algebra, to know your multiplication tables by rote. This “reflex knowledge” will help you work faster, rely on a calculator less, and verify that solutions are correct with greater speed and confidence. No matter how old you may be, no matter whether you are still in school or not, I recommend mastering any gaps your multiplication tables.

Having said that, I confess that Read more…

Steve Jobs spoke at the Stanford Commencement ceremonies in 2005. While his speech lasted only 15 minutes, it contains some wonderful advice – so I encourage you to click on this link to watch it. He will be sorely missed.

Suppose nobody had ever thought of measuring the size of an angle, and someone asked you “How can I describe the size of an angle?” What approach might you take in answering this question?

You might start by arbitrarily picking some angle, any angle, such as angle ABC in the image below, and call its measure “1″. All other angles could be