Cutting Some Standards

I’d like to reduce the number of standards math teachers are required to teach. The less we have to teach, the better we can teach what’s left. Technology continues to advance. It doesn’t make math irrelevant because of computers or AI or whatever. But the same way we left slide rules behind decades ago, it’s time to let a few more things go so we can do what’s left better.

If someone gave me a magic wand to change K-12 math standards I would have two goals: reduce the time students spend practicing complicated calculations, and make math less sequential.

Here are three changes I would advocate for in the elementary/middle grades that are relevant to my current position:

Some people argue that students shouldn’t have to learn things that calculators can do for them. I disagree. Understanding arithmetic and knowing how to use different algorithms create fluency with foundational skills and an understanding of place value. But that doesn’t mean students should spend their time perfecting more and more complicated algorithms in third, fourth, fifth, and sixth grades. I think the whole-number algorithms should be part of the curriculum, but addition and subtraction fluency should end with numbers up to 100. Students should see how to extend the algorithm for larger numbers but shouldn’t spend time practicing it. Multiplication by hand should end at one digit by three digits and two digits by two digits. Those algorithms help students see place value at work, and nothing bigger is necessary. Finally, long division should end with one-digit divisors and three-digit dividends. The goal with each of these algorithms should be to help students understand how to calculate, to reinforce their understanding of place value, and to give them intuition so they recognize when an answer does or doesn’t make sense.

Next, fractions. Fractions are important. They are the foundation for proportions, slope, and more. But calculating with fractions often devolves into minutiae that real humans never use. The first thing we need are small, cheap, four-function calculators that can 1) perform operations with fractions, and 2) do so in an intuitive and easy way. Imagine a four-function calculator with an extra row of keys for entering fractions, and a display that formats those fractions accurately. Once we have those calculators, I think we should keep many fraction operations but reduce the complexity students need to do without a calculator. Adding and subtraction fractions should emphasize common denominators, denominators where one is a multiple of the others, and denominators of 2, 3, and 4. No more convoluted problems adding fractions with denominators of 6 and 8 or 7 and 10. Similarly, mixed numbers should play a much smaller part in the curriculum, with no mixed number operations with unlike denominators and no mixed number multiplication or division. Finally, division of fractions should be limited to divisors that are unit fractions. All of those other operations are fair game when calculators are allowed — where it just assesses whether a student knows how to use a tool, not whether they remember an algorithm most adults will never use. The goal is for students to understand how fractions operations work without getting lost in unnecessary calculations. Too many students never understand what fraction division actually means, or are too busy converting mixed numbers to improper fractions to think about what they mean. These narrower goals would help build number sense for fractions while giving students the tools to move forward even if they struggle with a few of the pieces.

Finally, where fractions are used. Fractions come up in the real world. I see fractions all the time in recipes. But in too many cases our current regime of tests and accountability throws fractions in everywhere. In proportions problems, as coefficients in complicated equations that require combining like terms and distributing, in systems of equations, and more. I think fractions should be radically reduced on contexts that are not intentionally assessing fractions. Some fractions are still appropriate, but they should primarily be unit fractions and fractions with denominators of 2, 3, and 4 to both match what is most likely to come up in the real world, and avoid assessing fractions skills when we really want to assess other concepts. Finally, calculators capable of operating on fractions should be allowed, always. The goal is to assess the skill in question, not a bunch of unrelated fraction and arithmetic skills. Fractions should come up — in particular, they are critical to understanding slope. Let’s not get rid of fractions everywhere. But we can be judicious in where fractions are appropriate, and give students tools so that fractions aren’t an unstoppable roadblock.

Math is sequential. That’s inevitable. But I worry that we overemphasize how sequential math is. Sure, plenty of things are hard to learn if you lack the right foundation. But it’s not true that, in order to learn 8th grade math, a student has to have mastered every skill in 7th grade math. Ubiquitous technology means we can outsource more (although not all) of those foundational skills to machines that can calculate for us. School should reflect this reality. Too many students fall behind in math and feel like it is impossible to catch up. These students constantly encounter barriers communicating that math isn’t for them. The less math class relies on previous learning, the more students we can engage.