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Procedures for “Completing the Square”

June 29, 2012

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:



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

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





Second Approach

Read more…

Grant Wiggins on Mathematics Education

June 26, 2012

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…

Summary: Quadratic Equations

June 25, 2012

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



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:


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:


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?

Read more…

2012 Mathematical Sciences Institute

February 28, 2012

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.

Peer Instruction Network

February 12, 2012

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.

28th Annual Anja S. Greer Conference on Mathematics, Science and Technology

January 24, 2012

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.


Ten Skills Every Student Should Learn

January 9, 2012

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:

  1. Read
  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.

There are a few additional skills that I would advocate adding to, or being more explicit about in the above list:

Multiplying Polynomials and FFFT!

December 6, 2011

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…

Advice from Steve Jobs

October 5, 2011

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.

Angle Measures

October 2, 2011

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
Read more…

Mathematics and Multimedia Blog Carnival 15

September 26, 2011

Welcome to the 15th Mathematics and Multimedia Blog Carnival.

    According to Wolfram Alpha, the word “fifteen” is the 379th most common spoken word. It is also the age of most High School Sophomores. I invite you to lean on your caffeine and cuisine, then careen between the following keen carnival fifteen postings on your screen.

Connections between math and real life; use of real-life contexts to explain mathematical concepts

Bon Crowder (Math Is Not a Four Letter Word) describes Read more…

Next Math and Multimedia Blog Carnival: Article Selection Criteria

September 6, 2011

I am pleased to announce that Learning and Teaching Math will host the next Mathematics and Multimedia Blog Carnival.  If you have a post that fits in one of the categories below which you would like considered for inclusion in this issue, please use the link at the end of this post to submit it before September 23, 2011.

This Blog Carnival focuses primarily on articles that Read more…

Cost effective adult education: might it influence secondary education?

August 30, 2011

An article in The Washington Monthly titled “The College For-profits Should Fear” describes the founding and growth of Western Governors University. It uses an on-line model with some twists:

  • Course credits based on assessments completed. If you pass the final assessment, you get credit for the course… even if you just took the initial course assessment a few days earlier.
  • Tuition is charged per semester, not per course enrollment. This encourages students to complete as many courses per semester as they can, as it can save them money.

From what the article describes , this model seems most successful with older students – people who know what they seek, and don’t wish to waste time getting there. The WGU model is interesting for several reasons:

  • It has tuition levels that are around 40% or less that of other on-line programs, about $6,000 per year.
  • It employs full-time Mentors, who serve as a combination of guidance counselor, tutor, cheer-leader, and ombudsman for students. While they seem to provide a regular point of contact between students and the degree program, the article does not specify how many hours per week of such contact a typical student receives.
  • It uses industry-based standard assessments whenever possible as culminating assessments. The goals of the programs are therefore hopefully better aligned with the professional goals of the industries it is preparing students to enter.

The low tuition means that this model Read more…

“Teacher” is an inaccurate title

August 15, 2011

People, both as children and adults, are constantly learning new things. The more actively engaged in the learning process they are, the more likely they are to learn something well and retain that knowledge. So what exactly is the person “teaching” a course doing? Their title implies that they are somehow loading knowledge into student brains. While that may fit the assumptions behind the “lecture model” of instruction, that is not the way learning works.

So what title is appropriate for people who:
- Decide on, or create a sequence of topics and tasks that engage, but do not overwhelm
- Set the stage, pique student interest, then Read more…

Scheduling for Curricular Depth and Challenge

August 11, 2011

What was “the best” course you ever took? Probably one for which you had to work quite hard, one that you perceived as challenging from the outset, one for which you rose to the challenge. The course probably had a reputation as a tough course, so you probably added it to your schedule with care and made sure you did not take another really challenging course at the same time.

Major time commitments are regularly called for in schools: for musical or dramatic performances, athletic seasons, and some classes too. Could we improve the way such opportunities are scheduled so that  students can experience as many as possible each year without creating a killer workload for themselves at critical times during the year?


What if schools offered “challenges” that lasted for either half or a full semester? Each student could be required to be enrolled in two challenges at all times. A dramatic or musical performance would count as a challenge, as would a varsity sport, as would any number of academic and extra-curricular offerings. To qualify as a “challenge”, an offering would have to:

  1. Culminate in a public performance, presentation, or display of student work.
  2. Involve extensive Read more…