Unlike the two most “friendly” arithmetic operations, addition and multiplication, exponentiation is not commutative. You will get a different result if you swap the value in the base with the one in the exponent (unless, of course, they are the same value):
The most significant impact of this lack of commutativity arises when you need to solve an equation that involves exponentiation: two different inverse functions are needed, one to undo the exponent (a root), and a different one to undo the base (a logarithm).
Just as there are many versions of the addition function (adding 2, adding 5, adding 7.23, etc.), and many versions of the “root” function (square roots, cube roots, etc.), there are also many versions of the “logarithm” function. Each version has a “base”, which corresponds to the base of its inverse exponential expression.
Inverse Functions: Logarithms & Exponentials
Logarithms are labelled with a number that corresponds to the base of the exponential that they undo. For example, the Read more…
If a student makes four errors in the course of answering ten questions, what is an appropriate grade? Presumably, it would depend on the severity of the errors and the nature of the questions. Consider how your approach to grading might vary if students had been asked to:
- match ten vocabulary words to a word bank, or
- define each of ten words, then use each appropriately in a sentence
- complete ten 2-digit multiplication problems, or
- solve ten multi-step algebra problems, each requiring a unique sequence of steps
- answer ten questions similar to what they have seen for homework or in class, or
- answer ten questions unlike ones they have been asked before
Would you label each answer as right or wrong, then use percentage right as the grade?
Would you assign a number of points to each answer (if so, out of how many points per question)?
Would you assign a letter grade to each answer (whole letters only, or with +/-)?
What would you consider a “D” set of answers?
What would you consider an “A” set of answers?
Would your answers vary depending on whether you had created the assessment yourself, or were using someone else’s questions?
Many math/science teachers seem to use a percentage approach (based on total points earned or number correct) more often than any other, particularly when their school defines its letter grades using a 0 – 100 scale. Teachers of other subjects also use this scale often, but less so for “free-response” questions. While a percentage approach can work well for some assessments, it can have unintended consequences for others.
Similar Right/Wrong Questions
When asking a series of similar questions, such as Read more…
What is a “system” of linear equations?
A “system of linear equations” means two or more linear equations that must all be true at the same time.
When represented symbolically, a system of equations will usually have some sort of grouping symbol to one side of them, such as the curly brace below, which is intended to convey that the set of equations should be considered all at once. For example:
When graphed, all of the equations in a system will be shown on the same set of axes, so that they can be compared to one another easily:
What is “a solution” to a linear system?
A solution to a system of linear equations is Read more…
The phrase “Flipped Classroom” is appearing with increasing frequency in publications and blog postings. Yet, it seems to mean different things to different people. Many of the references I see to flipped classrooms are made by people or organizations who have a vested interest in selling goods or services, which probably affects their view of the issues.
As proposed by Salman Khan in his TED Lecture, flipping the classroom involves using internet-based video to move “lecture” out of the classroom to some other place and time of a student’s choosing. Class time can then be used for student problem solving and group work. Dan Meyer and others have critiqued aspects of Salman Khan’s approach, with some such as Michael Pershan offering constructive ideas for improvements.
Eric Mazur, a physics professor at Harvard, has also been advocating a “flipped” approach - and for considerably longer than Salman Khan. His conception of “flipping” focuses on getting students to Read more…
What is the difference between a Problem and a Project? While it is difficult to draw a definitive line that separates one from the other, the attributes of each and their differences as I see them are:
- Require less student time to complete (usually less than an hour)
- Focus on a single task, with fewer than 10 questions relating to it
- Can involve open-ended questions, but more often does not
- Are often one of a series of problems relating to a topic
- Look similar to many exam questions
- Can be used to introduce new concepts (Exeter Math)
- Can be used as practice on previously introduced concepts (most math texts)
- Require more student time to complete (hours to weeks)
- Focus on a theme, but with many tasks and questions to complete
- Provide an opportunity to acquire and demonstrate mastery
- Ask students to demonstrate a greater depth of understanding
- Ask students to reach and defend a conclusion, to connect ideas or procedures
- Can introduce new ideas or situations in a more scaffolded manner
Why Use Problems?
- Short time to completion makes it easier to fit into a class plan
- Multiple problems allow Read more…
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.
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?
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:
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…
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?
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.
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:
- Communicate effectively, and with respect
- Be resourceful
- Be accountable
- Know how to learn
- Think critically
- 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:
- Listen actively
- Read more…
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:
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…