Q: Why do schools need standards? What are standards for? What do they have to do with curriculum?
A: A set of standards is a *minimum set of requirements* that a product must meet. Let's imagine a house; the building codes are standards. An individual standard says something like "a wall must withstand straight line winds of up to 75 mph" or "studs must be places at least every 24 inches." Standards tell *what* you need to do, but don't tell you how to do it. They are used so that in the event of a serious risk, the whole structure stays standing. The building code for homes applies whether building a tiny townhouse or a 5 bedroom multi-story rambler. They are meant to force people not to cut too many corners. Standards here in math education are similar: they tell you what must be taught to a student, when, year over year. They are still *minimum set of requirements*, the ones that need to apply to every classroom, every school, every student.
A set of building codes aren't plans for a building. They are not a complete design.
A design tells you what *this building* going to look like. It answers the question of what purpose this house serves. It shows you how the floors and rooms fit together, how space is utilized. Architectural plans are a design for a home. They explain what the floor plan is, what the foundation is shaped like, where the plumbing goes, where the electricity goes, how the building will maintain even temperatures, etc. It's a map for everything that needs to be built. A solid design takes into account the what the end result should be like and plans the structure accordingly. A good design is also relatively easy to build, because it anticipates what elements of the design depend on the other elements.
In math education, this design is the curriculum. The curriculum is the plan for each year of math education, starting with a foundation, then the walls, the plumbing, the electrical, the floors, the roof, the interior. Like a good design for a building, a good curriculum is easy to implement, anticipating not just the end product, but how to ease the building of the elements as they are put in place.
A design is necessary for a home, but we don't live in an architectural drawing. The home must still be built So there are builders. Clearly the builders need to know the standards, and build according to them, so the structure is safe. But they build according to the standards while the build up the plan, starting with the foundation, and then moving on through the structure. They complete the plan.
Teachers are the builders. They are building the house of mathematics that students will live in. Some teachers are like builders who just doing the efficient thing, putting the pieces up in order. Others are like exquisite craftsmen. The most experienced teachers are like the most experienced builders who are able to look at the plans and figure out what won't quite work in this context, and where changes should be made. They know how long the job will take, and where the hard parts are before they start. They make very few mistakes, and correct them before they cause more problems. A great math teacher is building above and beyond the minimum standards where and when she can; she is on the lookout for the pitfalls in the curriculum, ready to point them out and rework them; she is improving her own skills, employing new innovations, and producing beautiful results.
Bad standards mean the structure is unsafe. Unfortunately, this isn't always visible to the eye. The owner may not find this out until a minor tremor comes and the whole house collapses. Likewise, a bad plan means the house is not a comfortable or useful place to live, or worse, could be dangerous. If the house was designed incorrectly, then the second story may not be able to hold up the third story, and it could collapse. A truly terrible design usually can't be rebuilt, but needs to be torn down. And a bad builder means things go wrong and break in your home, and again, if bad enough, the errors could mean that the house collapses.
Likewise, in math education, bad standards means the whole structure of math that a student is building up is faulty, and sooner or later, it is going to collapse. But good standards aren't the same as a good design. Students themselves don't need to learn the standards, any more than a homeowner needs to know the building code. Schools need to have a design, a curriculum, for their teachers, the builders to work from. A bad design won't support another story being built on top, year after year after year. And a bad builder on a floor means nothing can be built on top of it until a great builder comes in and rips out the bad work and rebuilds.
Every part of the process matters. Good standards, good curriculum, good teachers.