I love standards. I hate standards.

I guess it’s clear from the title of this blog that I have a love/hate relationship with the standards movement. Breaking down curricular content areas into grade level objectives -- in 3rd grade students in Texas are expected, among 40+ other math standards, to “use fraction names and symbols to describe fractional parts of whole objects or sets of objects” -- have done much for clarifying expectations for what students should know when. Enforcing these standards on a state, or even national, level insures that all students should be receiving a rigorous education. Classrooms, schools, states, and the nation can test students on their mastery of these objectives to identify where the instructional system is faltering. Curriculum developers and publishers can create instructional materials to support the expectations. When the system works, it should be beautiful.

Unfortunately, the reality rarely matches the plan. In fact, in some ways, the standards movement may have been more harmful than helpful. Part of the difficulty rests with the lack of a common national standard. It’s expensive to maintain a different set of standards for each state, along with different high quality assessments matched to those standards. Local control (and local standards) is not a bad idea, so long as the funding is available at each local level for successful implementation and support. It’s not particularly efficient for each state or district to develop standards, the particular assessment items to evaluate the acquisition of those standards, and the training and curriculum materials to support the implementation of those objectives. Funding shortages lead to short cuts and misalignments between what we say we expect and what we actually test.

But there’s another important problem with the current standards movement; it has led to the atomization of content and instruction. The list of grade level standards has become just that, a list to be checked off lesson-by-lesson. Instructional lessons carry a list of the individual standards they address. Check, check, check. Next lesson. The coherence that connects and makes sense of the path through the standards gets sacrificed in the urgent need to cover each individual objective. The learning expectation above about fractions cannot be taught successfully in isolation. Students need a coherent understanding of fractions and of number that unites the individual ideas, concepts, and skills described in the standards. Jumping from number lines for whole numbers to pizzas for fractions to place value grids for decimals may meet individual math objectives, but it can leave students feeling that each number form is a different number system. We’re just setting them up for later confusion. Standards are great but not at the expense of the path connecting them.

Good news. NCTM has recognized that 40 to 60 individual, equally-valued math objectives per grade level is a recipe for speed teaching. Each standard on the list gets 5-minutes. Better pay attention. The organization’s Curriculum Focal Points document attempts to identify the core standards upon which others are built. The Report of the National Math Panel reiterated the need to focus on the critical ideas and to build coherent curricula. Last year’s National Research Council report on science education moved in a similar direction. Hopefully, we can sustain the momentum, but there’s plenty of work to do to reconcile my rocky relationships with standards.

What Before How

I’m spending a lot of time these days thinking about multiplication. Keith Devlin, a Stanford mathematician, author, and NPR Math Guy, has created a swirling controversy through his Devlin’s Angle column on the Mathematical Association of America website (http://www.maa.org/devlin/devangle.html). Last summer Devlin sparked a debate by exhorting teachers to stop defining multiplication as repeated addition. His follow-up articles this summer fanned the flames and ignited a raging firestorm in the blogosphere. 4x3 can readily be rewritten by the repeated addition of 4+4+4. But what does repeated addition look like for, say, 3/4 * 5/8? It’s a challenge to articulate a definition that is both accessible to kids in elementary school and still true as the numbers become more complex.

I’ve enjoyed reading the unfolding arguments. It’s a healthy and important discussion, because determining what we should teach is essential before we decide how to teach it. Designing technology to more effectively instruct and engage students in a misconception or limited explanation shouldn’t be the plan. I fear that too much of the “innovation” and promise of technology doesn’t go deep enough into the roots of why our kids aren’t succeeding. It’s far too easy to take the existing curricular canon and put it into a glossy technological wrapper and be satisfied that kids “like” using it. We need to push ourselves further.

So, the search for a new way to define multiplication continues. When we have it, we can devise ways for technology to help visualize it, explore it, practice it at appropriate levels, and connect it to the content that came before and follows. It’s rigorous work, but if we get it right we can truly make a lasting and meaningful difference with kids’ understanding. It’s worth the extra effort.