Some of the problems in the More Subtraction on the Number Line worksheets were outside the range of the number line shown to work out each problem. Got those fixed this morning, just end time for some end-of-the year subtraction review…

Most science tests are reading tests in a content area. The problem is, most students have a deficit of science vocabulary, which makes the tests much harder than they should be. Help eliminate this weakness by using science flash cards. Each one comes with a picture and a matching word. Have students give you a definition for the word, using the picture as a context clue.

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Science Flash Cards

It’s hard to get through the day if a student doesn’t know the names of school supplies. Help your preschooler/kindergartener learn what scissors, erasers, and pencils are. This can be a great way to start your classroom routines and establish classroom rules.

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School Flash Cards

This has increased my interest in category theory in an interesting way. As a programming language geek, I’m obviously fascinated by data structures. Category theory provides a really interesting handle on a way of looking at a kind of generic data structures.

Historically (as much as that word can be used for anything in computer science), we’ve thought about data structures primarily in a algorithmic and structural ways.

For example, binary trees. A binary tree consists of a collection of linked nodes. We can define the structure recursively really easily: a binary tree is a node, which contains pointers to at most two other binary trees.

In the functional programming world, people have started to think about things in algebraic ways. So instead of just defining data structures in terms of structure, we also think about them in very algebraic ways. That is, we think about structures in terms of how they behave, instead of how they’re built.

For example, there’s a structure called a monoid. A monoid is a very simple idea: it’s an algebraic structure with a set of values S, one binary operation *, and one value i in S which is an identity value for *. To be a monoid, these objects must satisfy some rules called the monad laws:

There are some really obvious examples of monoids – like the set of integers with addition and 0 or integers with multiplication and 1. But there are many, many others.

Lists with concatenation and the empty list are a monoid: for any list,
l ++ [] == l, [] + l == l, and concatenation is associative.

Why should we care if data structures like are monoids? Because we can write very general code in terms of the algebraic construction, and then use it over all of the different operations. Monoids provide the tools you need to build fold operations. Every kind of fold – that is, operations that collapse a sequence of other operations into a single value – can be defined in terms of monoids. So you can write a fold operation that works on lists, strings, numbers, optional values, maps, and god-only-knows what else. Any data structure which is a monoid is a data structure with a meaningful fold operation: monoids encapsulate the requirements of foldability.

And that’s where category theory comes in. Category theory provides a generic method for talking about algebraic structures like monoids. After all, what category theory does is provide a way of describing structures in terms of how their operations can be composed: that’s exactly what you want for talking about algebraic data structures.

The categorical construction of a monoid is, alas, pretty complicated. It’s a simple idea – but defining it solely in terms of the composition behavior of function-like objects does take a bit of effort. But it’s really worth it: when you see a monoidal category, it’s obvious what the elements are in terms of programming. And when we get to even more complicated structures, like monads, pullbacks, etc., the advantage will be even clearer.

A monoidal category is a category with a functor, where the functor has the basic properties of a algebraic monoid. So it’s a category C, paired with a bi-functor – that is a two-argument functor ⊗:C×C→C. This is the categorical form of the tensor operation from the algebraic monoid. To make it a monoidal category, we need to take the tensor operation, and define the properties that it needs to have. They’re called its coherence conditions, and basically, they’re the properties that are needed to make the diagrams that we’re going to use commute.

So – the tensor functor is a bifunctor from C×C to C. There is also an object I∈C, which is called the unit object, which is basically the identity element of the monoid. As we would expect from the algebraic definition, the tensor functor has two basic properties: associativity, and identity.

Associativity is expressed categorically using a natural isomorphism, which we’ll name α. For any three object X, Y, and Z, α includes a component α_X,Y,Z (which I’ll label α(X,Y,Z) in diagrams, because subscripts in diagrams are a pain!), which is a mapping from (X⊗Y)⊗Z to X⊗(Y⊗Z). The natural isomorphism says, in categorical terms, that the the two objects on either side of its mappings are equivalent.

The identity property is again expressed via natural isomorphism. The category must include an object I (called the unit), and two natural isomorphisms, called &lamba; and ρ. For any arrow X in C, &lamba; and ρ contain components λ_X and ρ_X such that λ_X maps from I⊗X→X, and ρ_X maps from X⊗I to X.

Now, all of the pieces that we need are on the table. All we need to do is explain how they all fit together – what kinds of properties these pieces need to have for this to – that is, for these definitions to give us a structure that looks like the algebraic notion of monoidal structures, but built in category theory. The properties are, more or less, exact correspondences with the associativity and identity requirements of the algebraic monoid. But with category theory, we can say it visually. The two diagrams below each describe one of the two properties.

The upper (pentagonal) diagram must commute for all A, B, C, and D. It describes the associativity property. Each arrow in the diagram is a component of the natural isomorphism over the category, and the diagram describes what it means for the natural isomorphism to define associativity.

Similarly, the bottom diagram defines identity. The arrows are all components of natural isomorphisms, and they describe the properties that the natural isomorphisms must have in order for them, together with the unit I to define identity.

Like I said, the definition is a lot more complicated. But look at the diagram: you can see folding in it, in the chains of arrows in the commutative diagram.

I had the opportunity to receive evaluation copies of three iPad apps that I have included for a long time on my app wish list.  Each app currently costs $29.99 so they are relatively expensive as far as apps go.  I’ve been working with each of them for about two weeks and I’ll explain here what they do so you can make a decision on whether they would help your production.

Artemis (Director’s Viewfinder) HD

With Director’s Viewfinder, you select your camera and choose the lenses that you own.  The app shows you how much of the image you will see (referred to as Field of View) when using lenses of different focal lengths on your camera.  Most DSLR cameras crop the image because of the relatively small size of their sensor.  This app corrects for that cropping and shows you precisely what you’d be seeing with your lenses.

On a single screen you can see what the scene would look like when using any one of your lenses.  This is much easier than changing lenses  on a DSLR multiple times to find the best lens.

You can also take a picture of the scene at a particular focal length and record information about the focal length on the frame.  The captured frames can be used within the Storyboard app below to show you exactly what you’re going to see on screen or you can simply print them out or e-mail them to crew members.

Physical Director’s Viewfinders can cost hundreds of dollars and this app is far less than that.  If you are using a DSLR as your camera, this app is pretty much indispensable.  We are using it both for deciding which lenses we will need to purchase and to plan out our shots.

It’s been practical as well as educational.  I am used to shooting video with a fixed lens so I have a lot to learn about focal lengths.  I’ve fired up this app several times just for a refresher on how different focal lengths would affect the image.

Director’s Viewfinder from Artemis is not universal so you would need to purchase both the iPhone and iPad version if you need both.  The iPad real estate makes for a far better app experience but then again, the iPhone camera is generally easier to use and of better quality than the iPad’s camera.

Storyboard Composer HD is the app that I imagine would be most valuable to typical classrooms doing moviemaking.  You take a picture of a location (ideally the location where you’ll be shooting) and then you can insert people into the image.  You have the option of inserting men or women and positioning them forwards or to the side.  You can also easily simulate camera motion (pans and zoos).  We’ve been using the app in combination with Director’s Viewfinder…inserting images shot with Director’s Viewfinder into storyboard composer and completing sample storyboards there.

You can easily export PDFs of your storyboards which will be helpful for sharing boards with crew members.

Storyboard Composer HD is universal so you only need to buy it once for iPhones and iPads.  The bigger iPad user interface provides a better ease of use.  However, as I said, unless you have the newest iPad, photos you take with your iPad will likely be a little grainy and so the iPhone is preferable in that respect.

I still haven’t decided if it’s easier to storyboard using software or using paper and pencil but I am someone who has had much experience with the latter.  If someone has never storyboarded before this is an excellent tool for teaching them how to do it and it provides an advantage in terms of accuracy, sharing potential, and and clarity of vision.  It’s clearly the best software storyboarding tool I’ve used so far.

Movie*Slate (Clapperboard and Shot Log)

Whereas the other two apps are useful in the preproduction stages of moviemaking, Movie*Slate is for your production use.

Movie*Slate is a digital clapperboard and has many advantages over a chalk clapperboard in that it automatically advances the shot numbers and provides the time onscreen.  Although Final Cut Pro X has a feature to automatically sync audio and video captured from two separate devices, in the event that anything goes wrong it will be invaluable to have the accurate time information provided by this app.  The app also allows you to take notes after each take which will help in the editing process since those notes can be exported.

There are several bonus features like screens for focusing and setting exposure.  The app can also be greatly expanded through in-app purchases for sound and timecode which I haven’t tried.  Movie*Slate is universal but the small size of the phone probably makes for a less optimal experience if you need to be able to read the text on screen of your video camera.

Summary

It’s awesome when you find that there are apps that can transform the iPad into new tools.  All three of these apps are well designed and provide useful functionality for your production.  The apps work well as a team but if you can only afford one you must decide what your production needs are and what can be accomplished (albeit less efficiently) without the apps.

If you have any questions about the apps I’m happy to answer them below.  If you have movie making apps you’d like to suggest you can leave them down below as well.