Mathematical Sciences Digital Library

The following resources are included here:

Journal of Online Mathematics and its Applications – in its 7th yeah of publication, this journal is much too complex and advanced for use in high schools. Example.

Digital Classroom Resources – Again, too advanced for high school. Example: Fractal Maker

Convergence – Online magazine providing a wealth of resources to help teach mathematics using its history. Includes pictures of mathematicians and random quotes about maths. Interesting read for teachers. Example.

MAA Reviews – book reviews. For maths books.

Classroom Capsules and Notes – you have to subscribe in order to view these, and from the look of it and the rest of the site it might not be worth it if your focus is on high school students.

The Convergence online magazine may help high school teachers, but apart from that the site is aimed at a higher level.

Mathemagica Math Tools

The website as it was written didn’t exist, but I found this one.

A nonprofit subsidiary of the National Geographic Society, JASON connects young students with great explorers and great events to inspire and motivate them to learn science. Its core curriculum units are designed for 5th – 8th grade classrooms but are flexible enough to be adapted for higher or lower grades.

I did register for this site. It would be great for science teachers, but I only found one maths-related activity on Tornadic Air Pressure.

CLIME Microworlds

This site has games based on high school maths concepts such as factors and probability. I looked at this fraction game but couldn’t play it on my computer.

Each game also comes with a teacher page to help you incorporate it into a lesson. There are only a few games but it still could be worthwhile.

Manipula Math with Java

This site from International Education Software consists of over 250 maths applets, in 7 categories: middle school, trig, calculus, vectors, complex number, conics and miscellaneous. It is suitable for high school and uni students and originated in Japan, so you know it’s good

Example: Graphing y=asinb(x+c).

Why does every 2d shape have a perimeter except for the circle and elipse which have a circumference, and would it be of benefit to standardise?

The circumference is the perimeter around a closed curve. A perimeter is the sum of the lengths of the edges of a closed figure.

I would say that because a circle/ellipse only has one edge, and “perimeter” generally implies “add the sides” in school mathematics, the term “circumference” could mean a continuous perimeter (with no corners).

I reckon that when students are asked for the “perimeter” of a shape they just add up the side lengths, but the “circumference” of a shape has a special formula.

Then when you get to doing perimeters of semicircles and such, then you’re mixing them so that to find the perimeter you use half the circumference, plus the straight bit, coz remember you add all the edges to get the perimeter. I think “circumference” is a useful word and it wouldn’t benefit anyone to standardise.

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Ambiguous Words

LENGTH – the longest extent of anything as measured from end to end. Can refer to shapes, time, clothing, as well as meaning long (”remarkable length”) and difficult (”he went to great lengths to get that”). However, I think that a questions such as “Find the length of the line AB” doesn’t bring that much difficulty.

BREADTH – refers to width, or side-to-side length, or sometimes ” the measure of the second largest dimension of a plane or solid figure”, or just size in general. Extremely confusing and ambiguous.

WIDTH – the extent from side to side, or breadth. It would appear that “width” means the same as “breadth”, but in a more precise and well used way. Students would probably be able to guess what the “width” of something was more easily than they could find the “breadth”. Put both these words on a test and heads will explode.

HEIGHT – the distance upward from a given level to a fixed point, the extent or distance upwards, or the distance between the highest and lowest points. Evidently if you have a shape and rotate it, its height will be different. How do you know which is supposed to be the height? The area of a triangle is half the base times the perpendicular height. In this case the height is in relation to the side you use as the base.

DEPTH – “a dimension taken through an object or body of material, usually downward from an upper surface, or from top to bottom of something regarded as one of several layers”. It also applies to seriousness, complexity, emotions, colour intensity, pitch, and an unfathomable space (the depths). I always think of depth as going into the page when shown a drawing of a 3D shape. It is clearly confusing.

THICK – the state or quality of being thick, or the measure of the smallest dimension of a solid figure. One of the more thought-provoking definitions: is it always the smallest dimension? Does it mean the same as width? Or breadth? If the smallest dimension is the height, is that the thickness too?

All these words have slightly different meanings but it could be possible for them all to be the same. It wouldn’t be that much of a stretch.

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Number - 44 definitions, icluding:

Mathematics

1. A member of the set of positive integers; one of a series of symbols of unique meaning in a fixed order that can be derived by counting.
2. A member of any of the further sets of mathematical objects, such as negative integers and real numbers.
3. A symbol or word used to represent a number.
4. A numeral or a series of numerals used for reference or identification: his telephone number; the apartment number.
5. A position in an ordered sequence that corresponds to one of the positive integers: the house that is number three from the corner; ranked number six in her class.
6. One item in a group or series considered to be in numerical order: an old number of a magazine.
7. A large quantity; a multitude: Numbers of people visited the fair.  

Digit – any of the Arabic figures 1-9 and 0. Also any of the symbols of other number systems. Can also mean the breadth of a finger used as a unit of linear measurement.

Numeral - A word, letter, symbol or figure expressing a number.

Figure - A numerical symbol, and amount or value expressed in numbers, or a written symbol other than a letter. Plus 34 other non-mathematical meanings.

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Curve: A collection of points whose coordinates are continuous functions of a single independent variable.

Line: A collection of points whose coordinates are continuous functions of a single independent variable.

Straight line: A line traced by a point traveling in a constant direction; a line of zero curvature

Line segment: A portion of a line delimited by two end points; also, a line described by two sets of coordinates and the shortest path between them

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Invariably

Not able to change, the same every time. Is this what we mean when we use it? Is this what students understand?

Sum

“Sum” redirects straight to “summation” on Wikepedia, which also gives this:

• Sum (category theory), also referred to as “coproduct”
• Sum (country subdivision), an administrative division used in Mongolia and nearby regions
• SUM (interbank network)
• Soccer United Marketing
• Sum (Unix), a program for generating checksums
• Som (also spelt “sum”) meaning “pure”; the name of two national currencies:
  • Kyrgyzstan som
  • Uzbekistan som
• Sum (Mongolia), an administrative subdivision of Mongolia
• Spilka Ukraïns’koï Molodi (СУМ/SUM) Ukrainian Youth Movement
• Sum 41, a Canadian Punk band
• Direct sum of modules

Sum is also an acronym for Surface-to-Underwater Missile.

There are 15 definitions here.

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Order

Order has 56 definitions here.

Wikepedia gives 11 categories for what “order” might refer to. In the category of Mathematics:

Order in arithmetic, analysis and combinatorics

• Orders of approximation in Big O notation
• Orders of magnitude, a class of scale or magnitude of any amount
• Order of convergence, a measurement of convergence
• Order, or degree of a polynomial
• Order, or matrix (mathematics)#Definitions and notations
• Order (differential equation)#Types of differential equations, or order of highest derivative, of a differential equation
• Order in the Josephus permutation
• Weak order of permutations
• Twelvefold way#Ordered selections and partitions of the Twelvefold way in combinatorics
• Ordered set, a permutation, bijection or cyclic order
• Un-ordered subset or combination
• Ordered list, a tuple or sequence
• Long-range aperiodic order, in for instance pinwheel tiling
• Multiplicative order in modular arithmetic
• Z-order (curve) space-filling curve
• List of order topics in mathematics

Order in fractals

• Complexor, or complex order in fractals
• Orders of construction in the Pythagoras tree
• Order of extension in Lakes of Wada
• Order of Rényi dimensions

Order in graphs

• Graph order, the number of nodes in a graph [1]
• Ordered pair, including undirected and directed graphs
• Ordered triple, or mixed graph
• Glossary of graph theory

Order in mathematical theories

• Order (group theory), the cardinality of a group or period of an element
• Order (number theory)
  • Multiplicative partition, Unordered factorization of numbers
• Order in Ramsey theory, uniform structures in consequence to critical set cardinality
• Order (ring theory), an algebraic structure
• Order theory, which studies various binary relations known as orders
  • Dense order of rational and real numbers
  • Order topology, a topology of Total order for totally ordered sets
  • Partially ordered set, or poset
  • Total order, a generalization of the complete ordered field of real numbers
  • Glossary of order theory
• Set theory (music) encompasses ordered pitch and pitch classes (Musical set theory)
• Type theory encompasses: First-order logic, Second-order logic and Higher-order logic

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Net

23 definitions.

Wikepedia:

Net may refer to:

• Net (device), fibers woven in a grid-like structure
• Net (textile), any textile in which the warp and weft yarns are looped or knotted at their intersections
• Net (mathematics), a generalization of a sequence similar to a filter
• Net (polyhedron), an arrangement of polygons that can be folded up to form a polyhedron
• An electronic network connection in a netlist
• New Jersey Nets, a basketball team
• Net Television, a Maltese television station
• The Net (film), a 1995 film starring Sandra Bullock
• Cricket nets a safe netted environment for practicing cricket

Wikepedia also gives other definitions for “net” in computing and communication, for the Internet-related prefix net, .net or .NET, in business and finance, as an acronym, in broadcast and media and in science and psychology.

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Base

46 definitions. Wikepedia gives numerous possible meanings, including

In mathematics:

• Base (mathematics), the number b in an expression of the form bⁿ, hence also the number of digits (known also as the radix) in a positional numeral system, for example, base 10
• Base (topology), a topological space
• Base (group theory)
• One of the sides of a trapezoid or triangle

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Vertex

–noun, plural -tex·es, -ti·ces

1.	the highest point of something; apex; summit; top: the vertex of a mountain.

2.	Anatomy, Zoology. the crown or top of the head.

3.	Craniometry. the highest point on the midsagittal plane of the skull or head viewed from the left side when the skull or head is in the Frankfurt horizontal.

4.	Astronomy. a point in the celestial sphere toward which or from which the common motion of a group of stars is directed.

5.	Geometry. a. the point farthest from the base: the vertex of a cone or of a pyramid. b. a point in a geometrical solid common to three or more sides. c. the intersection of two sides of a plane figure.

And from Wikipedia:

Vertex (Latin: corner; plural vertices or vertexes) may refer to:

Mathematics

• Vertex (geometry), a corner point of a polygon, node in a Triangulated irregular network or TIN
• Vertex (graph theory), a node in a graph
• Vertex (curve), a local extreme point of curvature

Physics

• Vertex (physics), a point where particles collide and interact
• Vertex (optics), a point where the optical axis crosses a lens surface
• Vertex algebra in conformal field theory
• Vertex function describing the interaction between a photon and an electron
• Vertex model in statistical mechanics, a discrete model of a physical system in which weights are associated with vertices of a grid graph.

Companies

• Vertex – Business Process Outsourcing, a business services provider spun off from United Utilities
• Vertex Pharmaceuticals, a biotech company
• Vertex Standard, a radiocommunication products company.

Music

• A song in In The Groove (video game)
• Vertex (album), an album by Buck 65
• Vertex (band), a band formed in 1996

Other

• Vertex (anatomy), the uppermost surface of the head of an arthropod or vertebrate

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Square

54 definitions.

And from Wikepedia, square may mean:

Mathematics

• Square (geometry), a polygon with four equal sides and angles
  • Unit square
• Square (algebra), to multiply a number or other quantity by itself
  • Square root
  • Square matrix
  • Square number
  • Perfect square
• Square wave, a basic kind of non-sinusoidal waveform

Entertainment

• Square (album), an album by Buck 65
• Square (band), a musical trio from Lincoln, Nebraska
• Square Enix, a video game company
  • Square Co., former video game company that merged with Enix to become Square Enix
  • Square Electronic Arts, joint venture between console video game developers Square and Electronic Arts
  • Square Pictures, a computer-animated film division of Square

Language

• Square (slang), either an unhip person or a cigarette
• Square character (■), a written character
• A modifier adjective, used when describing fielding positions in cricket
• An adjective or noun, used when describing two positions parallel with one another relative to the goal in football

Places

• Square Lake, Maine, an unorganized territory in Aroostook County, Maine
• Times Square, popular landmark in New York City
• The Square, a common name for public squares
  • Town square or Market square, an open area commonly found in the heart of a traditional town
• Tiananmen Square in Bejing China, the largest open-urban square in the world.

Other

• Square (building area), an Imperial unit of floor area
• Square (cipher), a cryptographic block cipher
• Square division, a form of military organization
• Square rig, a type of sail and rigging arrangement
• An astrological aspect of 90 degrees
• Carpentry square, a tool
• Infantry square
  • Broken square

ArcytechTABLE 16.2

Math Tools

This site includes resources for every area of maths you could possibly want, categorised into broad subheadings. An easier way to find what you need is to view the whole list of topics, each with how many resources it has written in brackets next to it. Each resource comes up with a small blurb when you place your mouse over it, so you know what you’re getting before you click on it, which is a help to find the better ones faster.

Most resources on this site require a fee for you to be allowed to print them. Many of the resources are simple worksheets but there are some interactive-type sites, eg Working With Fractions. Finding good ones might take a while because there is so much on this site.

There are lesson plans, stories, activities and technological tools available on the site and places to discuss, rate and review. The Getting Started page tells you what kind of resources are available on the site and how to search for them.

National Library of Virtual Manipulatives

This site allows you to download a free trial desktop version of the software. Resources are ordered both in topic areas (Number & Operations, Algebra, Geometry, Measurement and Data Analysis & Probability) and in grade levels (K-12), so browsing through them is easy. It offers interactive resources rather than lesson plans.

Project Interactive

This site offers 90 lessons divided into topics, for students in years 3-12.

It also includes over 100 discussions, which are extremely useful for preparing for students’ questions.

Interactive activities are also categorised into topics, and there are over 100 of those too. You can search for topics within the site, which makes it easy to find whatever you’re looking for.

Educational Software Components of Tomorrow (ESCOT) Project

This site has about 40 interactive problems but is very technological and the site itself is not interesting. The font is small and it is unappealing. The activities are well written with reflections and extra questions so if you were used to using technology the activities might be very good.

Arcytech

This site offers only a few maths interactive activities. Only about three would be relevant for a high school classroom. You can only use these if your system and browser are Java capable.

Count On

This site is brightly coloured but finding your way around is not as easy as with some of the other sites. There are fun games but little is written about the content so finding a relevant one would be tricky. The site offers The Handbook for Leading Mathematics Teachers and a section on misconceptions, so it might prove useful for a teacher, but doesn’t have anything that could be used for classroom activities (not easily found ones anyway!)

“…the majority of students give up in despair and conclude that mathematics is just mystical gibberish.”

The first reading this week was unbearably boring and took me about 4 days to read coz I kept falling asleep/finding something else to read. It was something about things like internal chatter and the importance of discussion in the classroom, scaffolding, homonyms and appropriate communication.

The second reading was a better read. It focused on the language of mathematics and why it is so difficult for students to understand. I think I liked this reading because it was directed at uni students like myself as well as school students.

“Paradoxically, the very clarity and lack of ambiguity in mathematics is actually a stumbling block”. In spoken language, we have to interpret things from crude, simple phrases and infer the meaning, or decode waffling descriptive paragraphs to find the important bit. Mathematical language involves neither of these things. We have been conditioned to think in descriptive ways and infer our own meanings and therefore reading and writing the concise sentences such as are needed in maths is nigh impossible.

The concept of a “genus” was not something I’d ever known about, and it actually cleared up some of the confusion. After reading this article I am now more aware of what to look for in mathematical proofs and theorems. This is the first step towards finally getting some understanding of what I’m trying to learn. I can see that the layers of abstraction are cluttering up the page and are too much for an untrained brain to cope with.

I think the most important thing I got from this reading was just the fact that there ARE NO ASSUMPTIONS. EVERYTHING you can use/need to know is written in the theorem. There IS nothing else. I never realised this before. This has to be made explicit to students because everyone subconsciously tries to infer more meanings or links to everything.

A Mathematician’s Lament

http://hayds.edublogs.org/2008/03/06/lockharts-lament/

For anyone who reads my blog, but hasn’t read that one, you should.

“Mathematical thinking needs to be based, not on the symbols that are used, but on the ideas that the symbols come to represent” (Booker, 2002).This quote jumped at me while doing the readings this week. This is the first time I’ve thought about the symbols we use in maths being more than just a shorthand for something. Booker presents mathematical symbols as concepts that you learn from experiences with them, rather than having a specific meaning that you can give a name to. While the article focused around elementary maths and things that you would expect students to learn in primary school, such as the concept of zero and how to use the place value system, this is often not explained to children properly and so can become a big issue in high school.It was also illuminating to read this week that there are proportionately more dyslexic people in English-speaking countries because of the way our language is put together. Combining difficulties with our complicated sound-phoneme system with mathematical jargon and symbols is a problem for many students. It is also a good point that in maths we tend to not use unnecessary wording, and hence every word is important, which makes it different from any other sort of reading. Because we use as few words as possible, maths books are full fo dense sentences that need to be carefully read a few times, then broken down into more manageable pieces in order to construct meaning from them.In the vein of last week’s readings, this week’s readings also included the problems with prepositions and other issues with word problems. The interesting thing in the first reading was comparing English word problems with the same problem in Mandarin. English confuses students by having the algebra not correlate with the words. For example, expressing “There are 6 times as many men as women” should be m=6w, but many students will read and then write w=6m (reversal error). In Mandarin they don’t have this problem, because the words tell them: “Male members are female members 6 times”.

-prepositions, homonyms etc

-5 steps to answer – unfair to grade based on word problems

-nonsensical questions – have the skill to know if a numerical answer is erasonable for what they think they’re answering

-recognition and realisation

-calculators hard to use, individuals need help when push the wrong button