Algebra Context

Algebra Context

Assessment Center Exercises

Candidates' mastery of the content necessary for accomplished teaching in their field is assessed by means of a computer-based assessment consisting of six individual 30-minute exercises taken at a designated testing center. The board contracts with a vendor to establish centers throughout the country so that 90 percent of candidates do not need to travel more than 60 miles. Candidates can take the assessment center exercises between July 1 and June 15. The NBPTS posts scoring guides on its website to assist candidates in understanding what will be expected of them; these describe several sample exercises and provide discussion of how responses are scored. These assessment exercises focus primarily on the candidates' knowledge of the content important for the area in which they seek certification, although some questions cover pedagogical strategies.

For the middle childhood generalist certificate, the six exercises measure the ability to support reading skills, analyze student work, and integrate the arts, as well as knowledge of science, social studies, and health. For the middle childhood and early adolescence mathematics certificate, the six exercises measure candidates' understanding of six areas (algebra and functions, connections, data analysis, geometry, number and operation sense, and technology and manipulatives) with an emphasis on the capacity to draw inferences and apply knowledge to real-world circumstances.

For example, in a sample exercise for the middle childhood generalist certificate, the candidate is asked to imagine that he or she is working with a group of third grade students of mixed ability and to read a passage and a transcription of a student's oral reading of the passage. The candidate then responds to four prompts that explicitly ask him or her to identify errors, cite examples from the student's text to support analysis of the student's skills, describe strategies for addressing the errors, and provide a rationale for using these strategies. An example from the early adolescence mathematics assessments shows a similar assessment approach in a different context.

The scoring guide for this exercise, which covers data analysis, explains that candidates must present: a complete and accurate graphical representation of a given set of data; a meaningful interpretation of the data based on the graphical representation; an appropriate and accurate alternate graphical representation of the data; and a meaningful, accurate, and distinct interpretation of the data based on its alternate graphical representation. For the exercise, the candidate is provided with some data on high-grossing movies. The candidate is asked to create a box-and-whisker plot to display the data, to discuss the skewing of the data, and then to produce an alternate representation of the data and answer a question about it. Learning things is not limited to the scentific area. Instead it also has relations with some other things like speaking a language or using software, including Rosetta Stone English and Rosetta Stone French. If you have a creative mind, you will make all your own differences in the end!

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the name linear algebra?

Hello i was wondering,
why is it called linear algebra? what does linear mean in this context??

thanks

linear algebra grew out of the study of systems of linear equations, like this one:

ax + by + cz = u
dx + ey + fz = v
gx + hy + kz = w

each equation can be thought of as a plane in 3-dimensional space. if two planes intersect,

they intersect in a line (hence the term "linear"). so the above equations can be thought of as:

find 2 lines the 3 planes intersect in, and then see if these two lines intersect. if so,

we have a point that is on all 3 planes.

such a system can be thought of as a matrix equation:

[a b c][x]....[u]
[d e f.][y]....[v]
[g h k][z] = [w]

where the matrix takes the triple (x,y,z) to the triple (u,v,w) by matrix multiplication.

as linear algebra developed, it was discovered things like "triples" and "matrices"

had properties that could be abstracted apart from their numerical representations,

and thought of as "vectors" and "linear transformations".

a group of vectors with suitable properties is called a "vector space".

an example is R^3, which one can visualize as being a space like the world we live in.

in this context, a 2-dimensional "subspace" is a plane, and a 1-dimensional subspace

is a "line" (to be precise, in linear algebra we require these go through the origin).

so vector spaces are higher dimensional things like lines.

why abstract it? well, normally, to represent a point in space, you choose a way of identifying it

by choosing a "coordinate system". but the point in space is the same "place" no matter

which coordinate system you are using. by considering vector spaces, rather than just

matrices and n-tuples (the n-dimensional version of pair(double), triple or quadruple),

you see what information you can get in a "coordinate-free" way. for example, when you choose

a basis for a vector space, you can do this a lot of ways. but no matter which way you do it,

the number of elements in the basis will always be the same. that is, this number does not

depend on WHICH basis you choose, so it is an intrinsic property of the space.

this number is called the dimension of the space.

so in 3 dimensions, we can identify a point, anyway we like, but one thing is certain:

it takes 3 coordinates, no matter how we choose these coordinates.

similarly it takes 2 coordinates to place a point in a plane, whether you use (r,θ) or

(x,y). there is no getting around this: one number is not enough information to tell us

which point we mean.

it turns out that many mathematical situations involve some property like this:

L(ax + by) = aL(x) + bL(y) (this happens with integrals and derivatives, for example).

since this is the same property that matrices have, this is called a "linear relationship".

short version: the whole thing goes back to studying how planes and lines act.

Linear Algebra: Rule of Sarrus of Determinants