What are math facts for, anyway?
In reading, the goal isn’t to memorize phonics, it’s to find meaning in books. Does math have a similar goal in mind?
Today’s letter refers to posts from last week. To get the most context and the most out of today’s letter, check out last Friday’s letter as well as the discussion in the comments section of last Sunday’s post.
Too many students don’t possess solid foundational skills in mathematics, and it’s hurting their chances at full numeracy for life, not to mention a shot at higher math, a path to college, and a lucrative STEM career.
Lack of foundational math skills also just makes math class harder, and often makes kids want to avoid math, causing them to miss out on key opportunities including a higher-paying career. Once kids fall behind in these skills, taught in the early years of school—which include an understanding of number, quantity and counting, arithmetic operations, and a strong understanding of place value, plus a few others—it becomes harder and harder to catch up.
Math is “relentlessly hierarchical,” as researcher Amanda VanDerHeyden told me, and skills build upon skills. When kids miss some important skills along the way, which is most American kids, help catching up is hard to find in school. Schools that have recently revamped how they help kids with reading difficulties, including screening, plans for improvement, and dedicated staff, often don’t offer nearly the same level of support for math.
It’s become clear that one of the first things schools need to do to help more students achieve in math is put more focus on foundational skills, early on in a student’s school career. They need to be fluent in the operations, knowing addition, subtraction, multiplication facts by heart. And to do that, several things need to happen—classrooms need better curriculum that’s more aligned with both the most important skills as well as with evidence on how kids learn. Nearly every educator and principal I’ve talked with about math in the last year has said their math curriculum isn’t great, and teachers need to supplement to meet all the standards and fill in missing gaps in learning. But teachers also need better math training—data shows that teachers themselves are often math avoidant, especially early-elementary teachers, and need help solidifying concepts in their own heads before teaching them to students.
Similar to reading, there is an evidence base for better practices in teaching math to students, especially for students who struggle. Call it the ‘science of math’ or the ‘science of learning,’ or ‘evidence’ or whatever, this evidence exists, can play a much bigger role in classrooms, and has largely been ignored by much of the education establishment.
The work I’ve done reporting on math over the last year or so leads me to believe that all of the above is pretty accurate for a lot of elementary schools and students (certainly not all).
But the story doesn’t end there.
The conversation around improving math isn’t only about procedural foundational math skills; it also has to be about what students can do with those skills. I think that distinction might seem small, especially under the circumstances—so many students simply don’t have strong foundational math—but it’s vitally important. What’s the bigger picture here, and what do we want students to be able to do with all their math skills?
An op-ed in UK education magazine TES this week (disclosure: I’m currently a freelance reporter for them) by math educators and experts David Thomas and Peter Foulds addresses this question. In the quest for more evidence-based practices, they write in “How to fix the primary maths curriculum,” schools can’t lose sight of the larger goal: to understand math, to find pleasure in it, and to use it to solve problems.
It’s easier to see how foundational skills feed into the end goal for reading, they write, because “Everyone knows that the goal of English teaching in primary [elementary school] is to get children reading for pleasure.” Cognitive scientist Mark Seidenberg alluded to this same issue earlier this year in an essay and presentation about phonics and its passionate advocates, “Where does the science of reading approach go from here?”: the goal of teaching reading isn’t to learn phonics; it’s to find meaning in books. Phonics are in service of that goal. That doesn’t mean, of course, that schools should go back to balanced literacy and allowing children to guess at words (he wrote in no uncertain terms the disdain he has for that unscientific approach); but schools and educators should be wary of the other end of that spectrum—explicitly teaching every phoneme and blend might not be necessary (though it might in some cases, it really depends); the goal is to get kids reading fluently and also reading well for meaning.
Too much explicit instruction will hold kids back from the end goal, Seidenberg wrote. “Explicit instruction is important, and some of it is necessary, but it is not the only way people learn. The fact that some aspects of reading have to be taught doesn’t mean everything has to be. Most of the knowledge that supports skilled reading, even at the grade school level, is not explicitly taught.”
Thomas and Foulds at TES believe the same ideas apply to math. “We teach children to decode a set of graphemes. We get them to practice until they become fluent at decoding them. And, at every incremental level of fluency on this journey, we encourage them to read appropriate books for pleasure,” they write. “Maths should be no different. But currently, English is for pleasure and maths is for utility.”
They continue: “Too often, we do not spend nearly long enough on ensuring every child both learns and masters the absolute basics - number bonds to and within 10 and 20, bridging 10. Sometimes this is because we try to generate these through reasoning, rather than seeing reasoning as the thing we do with them.”
A lot in that op-ed rang true to me—for a while I’ve been thinking about how math has an end-goal problem that does nobody any good by ignoring.
Unlike reading, where reading a book or other text for meaning is the end goal, or other pursuits we often compare math skills to, like playing the piano or playing basketball, which also have specific performance end-goals that make the practice meaningful, and that you do alongside the skill practice (no one, for example, would suggest students only shoot freethrows for a couple of years before trying a basketball game; you play games alongside getting better at individual skills, they are for a purpose)—the goal for math learning can be much fuzzier, more ill-defined.
This end-goal problem is why I wanted to interview instructional coach Neily Boyd last week. Boyd has been an instructional coach and math teacher for some time, turning around a Nashville school’s math scores in a couple of years, as well as training teachers in math across the country. She has boots-on-the-ground experience with teachers in classrooms. Boyd knows the research on learning inside and out, and is a strong proponent of science of learning concepts like spaced and retrieval practice, using knowledge about working memory and cognitive load to help inform instruction, and others.
But, like Seidenberg, Boyd is also wary of going overboard with too much explicit instruction and procedure, or maybe only explicit instruction and procedure, worried that the larger goal of what students are supposed to be able to do with math could be lost. As math education professor Christian Bokhove recently wrote on Twitter, ”procedural and conceptual understanding go hand-in-hand at all ages,” and Boyd leans toward thinking that the basketball game should always be played alongside the building of skills—which in math means letting kids use the math knowledge they already have to solve problems, and then stretch their thinking one step, then two steps further, making meaning out of the procedures and understanding the relationships between numbers.
She calls this effort productive struggle, which I often associate with something I’ve seen and heard about quite often—teachers offering no guidance at all to students to let them “struggle” with a problem that’s too hard for them. But what Boyd is describing, where students have both the necessary knowledge and teacher guidance, would fall more into the category of “effortful thinking.”
Understanding both the procedures as well as the relationships between numbers makes it easier to use math knowledge for solving problems. I thought Thomas and Foulds put it perfectly here:
“The expectation… isn’t to be able to do the things in it once, or in controlled conditions, but to practice them to fluency. Pupils need to be able to use them, not to do them. Fluency in maths is more like blending a phoneme into a new word than saying it when the teacher holds up the corresponding card.”
Fluency is reached when you can use the knowledge to do something meaningful—but when it comes to math, what is that thing? What’s the ballgame?
In general, in my experience I think the UK is years ahead of the US in their evidence-based reforms, and this op-ed makes that perfectly clear: the US is still hung up trying to convince educators and schools to merely learn about evidence in teaching and learning practices. The UK, already armed with a strong base of research knowledge, is forging ahead to make math both doable and pleasurable, which in the end will make it more worth doing.
I’d love to hear what you think about this. What’s the goal here? What do we want students working toward, and how are they learning to do that? Are students doing freethrow practice alongside playing lots of basketball games?
In Virginia - and the rest of the country is little different - the standardized math test starts in the 3rd grade and is given every year. In the 3rd grade the test is 40% without calculator and 60% with calculator. Each year the non-calculator portion decreases until in the 8th grade (and ALL of high school) the test is 100% calculator active. More significant, there are NO NUMERICAL CALCULATIONS ON ANY NON-CALCULATOR PORTION OF ANY TEST! Because the pressure on teachers to have a high pass rate on the standardized test is IMMENSE, they teach ALL elementary students to do all numerical work on the calculator. The result is we have been producing innumerate students for almost twenty years. I taught high school math for 36 years and the deterioration of number sense in the students I taught was dramatic. An example: In a class of 30 Algebra I students the topic was factoring an expression. This required them to choose the correct pair of factors of 63. When asked the entire class could come up with only one pair of numbers that multiplied to give 63; 1 and 63. Clearly nobody knew their multiplication tables for 7 or 9; not to mention the idea that if 3 divides both 6 and 3, it must divide 63. If you look up Cognitive Load Theory, you will find research that says our short-term working memory is very limited - we can hold four facts for about a minute - and when it takes more than a minute to determine this one piece of information needed for a multi-step process like factoring, the high probability is the student will experience cognitive overload, for which the most common result is the student quits trying - a result ALL math teachers are quite familiar with! Since there is no high school diploma available for a student who can't pass Algebra I, the most likely result was a significant portion of my Algebra I class would never graduate.
The answer is disturbingly simple - remove the calculators from the elementary classrooms during instruction of numerical calculating skills.
Math for elementary students is indeed for solving problems. Those problems can include questions about quantities, relative distances or rates, areas, and volumes of relatively simple solids/containers.
There are LOTS of opportunities to solve this type of problems within a framework of everyday cooking, travel, shopping, and house/yard work. Teachers could learn to provide example problems using situations common to their area (eg. raking leaves in suburbs or rural areas, walking 5 blocks (where a block is X feet or yards, or meters long) in big cities, planning a party menu for X people with $Y to spend, etc.)
Not to mention explicitly teaching kids to translate rates into distance/time for all kinds of time increments, and how to calculate a rate in different distance or time units from those given.