I’ll get back to Ashman, but first let me take a tour through the literature, looking for definitions.

Relational and instrumental understanding

I found a 50-year old paper by Skemp which is strikingly relevant to today’s debates. Skemp distinguishes between relational understanding and instrumental understanding:

By the former is meant what I have always meant by understanding, and probably most readers of this article: knowing both what to do and why. Instrumental understanding I would until recently not have regarded as understanding at all. It is what I have in the past described as ‘rules without reasons’, without realising that for many pupils and their teachers the possession of such a rule, and ability to use it, was [what] they meant by ‘understanding’.

Skemp’s instrumental is our procedural, and his relational is our conceptual. He gives an analogy which is close to the description of my own experience above:

A person with a set of fixed plans [for navigating a town] can find his way from a certain set of starting points to a certain set of goals. The characteristic of a plan is that it tells him what to do at each choice point: turn right out of the door, go straight on past the church, and so on. . . .

In contrast, a person with a mental map of the town has something from which he can produce, when needed, an almost infinite number of plans by which he can guide his steps from any starting point to any finishing point, provided only that both can be imagined on his mental map.

This harmonizes with the experience of understanding a proof that I described above. For Skemp, relational understanding is a

conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point.

Skemp is candid in acknowledging his own biases, and in admitting that his earlier failure to appreciate instrumental understanding was a mistake. He makes an effort to play devil’s advocate for the notion, at the same time as arguing in the end for the superiority of relational understanding for the reason quoted above. He concedes that anything can be taught and understood in an instrumental way, and gives an amusing example from his own work in “modern mathematics,” the UK version of what was called the New Math in the US.

I was in a school which was using my own text, and noticed . . . that some of the pupils were writing answers like ‘the set of {flowers}’.1 When I mentioned this to the teacher (he was head of mathematics) he asked the class to pay attention to him and said: “Some of you are not writing your answers properly. Look at the example in the book, at the beginning of the exercise, and be sure you write your answers exactly like that.”

This foreshadows a point Ashman makes 50 years later, citing examples of supposedly conceptual problems that can be answered procedurally. The strongest point in Ashman’s post is the difficulty of measuring conceptual understanding.

The five strands of mathematical proficiency

The National Research Council’s report Adding it Up describes five strands of mathematical proficiency: procedural fluency, conceptual understanding, strategic competence, adaptive reasoning, and productive disposition. Their definition of conceptual understanding is “comprehension of mathematical concepts, operations, and relations.” They go on to say:

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. If students understand a method, they are unlikely to remember it incorrectly. They monitor what they remember and try to figure out whether it makes sense.

This echos Skemp’s notion of a schema. But notice a subtle shift here: Adding It Up has not made Skemp’s rich connected structure—knowing the layout of the town—part of the definition. Rather they have described that structure as a consequence of acquiring conceptual understanding. And they have given behavioral descriptions of conceptual understanding: integration, organization, ability to represent in multiple ways, ability to reconstruct when forgotten, ability to generate new knowledge. This is more than Ashman’s “feeling or impression.” And because these are behaviors, they are observable, and have the potential to answer the measurement question. The report hints at what such measurement might look like: “Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems.” Hold onto that thought about solving new and unfamiliar problems.

Procedural knowledge and conceptual knowledge

Ashman quotes a paper by Rittle-Johnson et al. to suggest that there is really no difference between procedural knowledge and conceptual knowledge, but he leaves out some context. Here is the full paragraph, with the bit Ashman quotes in bold.

There is a long-standing and ongoing debate about the relations between conceptual and procedural knowledge (i.e., knowledge of concepts and procedures). Although there is broad consensus that conceptual knowledge supports procedural knowledge, there is controversy over whether procedural knowledge supports conceptual knowledge and how instruction on the two types of knowledge should be sequenced. A review of the empirical evidence for mathematics learning indicates that procedural knowledge supports conceptual knowledge, as well as vice versa, and thus that the relations between the two types of knowledge are bidirectional.

Ashman takes that last sentence and runs with it.

Why would this be the case? Well, they would support each other if they were effectively the same thing. This could be an example of the jangle fallacy — the mistaken impression that two identical or near-identical things are different because they have been given different names.

But it is clear from the earlier sentences that the authors believe there is a difference between conceptual knowledge and procedural knowledge; indeed, they cite the Adding It Up definition approvingly. Two constructs can support each other without being identical.

Conceptual understanding and transfer

Ashman gives some examples of questions supposedly assessing conceptual understanding, pointing out that they could all be solved procedurally if students are given the right training. The examples are embarrassing and he is right to pillory them. Here he has rediscovered the point Skemp made 50 years ago. He makes the point that the real test of conceptual understanding is in the novelty of the questions. Here he has rediscovered the point that Adding It Up made 25 years ago, about “solving new and unfamiliar problems.” He then pivots to his own version of the jangle fallacy.

If you think that conceptual questions only work if students haven’t seen those question types before and if you think training students in methods to solve those specific questions is a form of cheating, you are really not grasping for the conceptual knowledge but for the transfer of learning.

Good point about transfer of learning being the behavior to observe. But what is the cognitive structure that would allow for transfer of learning? Ashman links approvingly to this chapter in the NRC report How People Learn, which has the choice quote, “Transfer is affected by the degree to which people learn with understanding rather than merely memorize sets of facts or follow a fixed set of procedures.” So it seems that conceptual understanding and transfer are, if not exactly two names for the same thing, two sides of the same jangling coin. Conceptual understanding is the cognitive structure that leads to transfer; transfer is the way you measure it. Ashman is right about that.

Assembling the sources, here is the definition everyone can agree with:2 conceptual understanding is the connected cognitive structure—Skemp’s schema, Adding It Up’s “integrated and functional grasp”—that lets you handle problems you haven’t seen before. Ashman is right to point out that it’s hard to measure, but wrong to treat the measurement difficulty as evidence that it doesn’t exist.

Ashman declares conceptual understanding a myth and says we should measure transfer instead. But his own sources say that transfer is conceptual understanding measured. Ashman’s remedy detects the thing whose existence he denies. He hasn’t refuted it, he has renamed it.

I thought I had discovered an amazing secret that I could use to impress people. In her wonderful essay Variables in Mathematics Education, Susanna Epp quotes the mathematician Jean Dieudonné:

When we solve an equation, we operate with “the unknown (or unknowns) as if it were a known quantity . . . A modern mathematician is so used to this kind of reasoning that his boldness is now barely perceptible to him.”

What a lovely description. It is indeed a bold move to represent an unknown number with a letter and then act on it with the same insouciance we would approach a known number. It would be great if we could get students to feel pride in that boldness. Here is the 9th century mathematician al-Khwarizmi in al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (“The Compendious Book on Calculation by Completion and Balancing”).

… what is the square which combined with ten of its roots1 will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. … Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3.

He is acting on an unknown as if it were known, showing Dieudonné’s boldness, but he doesn’t have the idea of using a letter to represent it, and must resort to the pronoun “it.”2 In modern terms, he is solving a quadratic equation by completing the square

We call x a variable. The word variable is a bit misleading, because the number we are looking for is not varying. (If it did that would make it harder to find the wriggling beast.3) Sometimes we use the word unknown, which is more accurate. Epp uses the word placeholder. But we are pretty much stuck with variable for the most part.

Once liberated by the idea of using a letter to stand for a number we can go further than al-Khwarizmi. He describes his method using specific examples. But we can represent the general quadratic equation ax² + bx + c = 0 by using letters for the coefficients. (Which we sometimes confusingly call constants. That’s a whole other post.)

In her essay, Susanna Epp talks about other uses of variables: describing functional relationships, expressing universal statements, dummy variables, and variables used as generic elements in discussions. However we use them, variables are just elements of a sentence we write in mathematical English, ways of referring to numbers we are thinking about. They are not as mysterious as we somehow manage to make them. Think about that.

The Hiebert & Grouws chapter I talked about last Friday pointed to two elements of instruction that cut across different pedagogical styles: explicit attention to concepts and struggling with important mathematics. In this post I want to concentrate on the first of these. We make things explicit by talking about them. So, somewhere in a mathematics classroom, there should be talk about meaning. I’m going to delve into two research papers, one co-authored by Grouws and one co-authored by Hiebert, and look for how meaning shows up in classroom talk.

Improving the traditional model

The Missouri Mathematics Effectiveness Project was a two-phase experimental study in 4th grade classrooms (Good & Grouws, 1979a, The Missouri Mathematics Effectiveness Project: An Experimental Study in Fourth-Grade Classrooms; 1979b, Experimental Study of Mathematics Instruction in Elementary Schools). The first phase studied 100 teachers in a relatively middle class district, and identified 9 teachers who were consistently effective and 9 who were consistently ineffective over two years, measured by gains for their students on the Iowa Test of Basic Skills. They developed a behavioral profile for each group through classroom observations, distilled a set of factors that distinguished the groups, and assembled them into a training program consisting of a 45-page training manual and two 90 minute sessions.

The second phase took place with 40 volunteer teachers in a different district with many schools in low socio-economic areas. Schools were randomly assigned to either receive the training and implement it, or to continue teaching as they had been. Students in the former group of schools generally outperformed students in the latter. “Treatment students moved from 26.57th to 57.58th percentile on national norms; controls moved from 29.80th to 48.81st.” The improvement in the control group suggests the possibility that teachers in that group sought to improve their teaching simply by virtue of being in the study, while continuing their traditional approach (the so-called Hawthorne effect)—they had been told that they would receive the training the following semester. In this case the possibility of a Hawthorne effect suggests that program was even more effective than the results show.

The program, called Active Mathematics Teaching, had the following characteristics:

(a) The program, in total, represents a system of instruction; (b) instructional activity is initiated and reviewed in the context of meaning; (c) students are prepared for each lesson stage so as to enhance involvement and to minimize student performance errors; (d) the principles of distributed and successful practice are built into the program; (e) teaching presentations and explanations are emphasized.

The lesson structure consisted of daily review, development, seatwork, and homework. The daily review and seatwork sections were fairly traditional: collecting homework, reviewing skills, some mental computation exercises for the daily review, and providing “uninterrupted successful practice” for the seatwork, with accountability by checking work at the end. The development section is where the meaning making comes in. In this section teachers should

1. briefly focus on prerequisite skills and concepts

2. focus on meaning and promoting student understanding by using lively explanations, demonstrations, process explanations, illustrations, etc.

3. assess student comprehension
   (1) using process/product questions (active interaction)
   (2) using controlled practice

4. repeat and elaborate on the meaning portion as necessary. [Emphasis added]

Product questions “ask students to provide an answer,” whereas process question ask student to “explain how the answer was or could be obtained.” Importantly for my main interest in this post, teachers are doing most of the explaining:

We feel that the presence of a few process questions in the development stage of a lesson are helpful (especially when a new principle is being introduced) because listening to a student’s explanation can help teachers diagnose inappropriate assumptions, etc., that students have made. However, we believe that most of the process development can be done through teacher modeling of process explanations rather than by asking students to respond to process questions. [Emphasis added]

The development section was the most novel element in the program; it was the one the teachers had the most difficulty implementing.

Developing an alternative model

Hiebert & Wearne (1993, Instructional Tasks, Classroom Discourse, and Students’ Learning in Second-Grade Arithmetic) followed second-grade classrooms for 12 weeks learning place value and multidigit addition/subtraction. Two of the classrooms were “alternative” classrooms that had already broken with the textbook tradition; the study observes and characterizes them precisely. They had fewer problems per lesson, more time per problem, more questions asking students to describe and explain strategies, longer student responses, and higher performance and greater gains by year’s end.

The difference between the classrooms is illustrated vividly by comparing transcripts. This is from Classroom A, one of the traditional classrooms.

Teacher: “Look at Row A. See where it says 580 take away 234? Okay, what are you going to ask yourself?”
Student: “Can I take 4 from 0?”
Teacher: “Can you?”
Student: “No.”
Teacher: “What do you do?”
Student: “Go to the 10s, cross out the 8 and put 7.”
Teacher: “Go to the 10s, cross out the 8 and put 7. Good. What are you going to do with the 10 that you took away?”

Here the talk is about working through the procedure. Contrast with this transcript from Classroom D, one of the alternative classrooms.

Teacher: “Jane, can you come up and tell us how you did this? I didn’t see any blocks out on your desk, so I want to see how you did this.”
Jane: “I counted in my head.”
Teacher: “Where did you start?”
Jane: “I put 145 down and then I counted 135.”
Teacher: “How did you count 135?”
Jane: “I put a hundred with the 135 to make 200.”
Teacher: “She combined the 100 here with the 100 here and got 200. Then what did you do?”
Jane: “I put the 4 and the 3 together.”
Teacher: “Why did you put the 4 and the 3 together?”
Jane: “Because they’re the ones.”
Teacher: Oh, so you’re combining the 4 and the 3 for a very specific reason. Can you tell us about that again?
Jane: Tens. They’re 10s. So I got 7.
Teacher: Why didn’t you write the 7 down?
Jane: Because its going to change.
Teacher: How do you know it’s going to change?
Jane: Because it’s 5 ones, and 5 and 5 is 10, and that makes it 8.
Teacher: What’s the answer then?
Jane: (Writes 280).
Teacher: Who has another way we can do this?

In this exchange, the meaning talk is coming from the student, with careful solicitation by the teacher. I loved the teacher move after Jane incorrectly referred to the 4 and the 3 as the ones. Not correcting, but not allowing it to pass either. The teacher instigated a pause and a reflection—”Oh, you had a specific reason.”

The authors summarize the difference between the two groups of classrooms:

Students receiving the conceptually based instruction spent a greater percentage of time working on the rationale for procedures and examining the legitimacy of invented procedures. Students receiving the traditional instruction spent a greater percentage of time practicing taught procedures.

Two different classrooms, a common element, and a way forward

These two papers had very different research methodologies and dealt with very different traditions of teaching. What interests me is the common principle that emerged from both: explicit attention to meaning. It sounds obvious, but as I have described here and here, it’s not easy.

I have presented a contrast that seems to be a dichotomy. The answer to the question in “Who is taking care of making the meaning?” seems to be either the teacher or the students. But in reality, in a classroom where talking is happening at all, a more complex mix emerges.

The 5 Practices for Orchestrating Productive Mathematics Discussions describe such a mix.1 I’ll talk more about these in another post, but briefly, in a classroom where students are working on a problem, teachers anticipate what the students will do, monitor their work, select examples to discuss with the whole class, sequence the examples to build a coherent discussion, and then explicitly connect the different methods.

In these classrooms, teachers are conducting the orchestra and listening to the choir; teachers are bringing knowledge and seeing student thinking; teachers are making decisions on the fly every minute about which move to make next.

Many approaches to teaching are viable—explicit instruction, guided inquiry, some synthesis of both—if they share the common element of making meaning and attending to concepts.

In The Phantom Tollbooth Milo takes a journey through a land where meaning has gone missing. Milo doesn’t know what anything is for. The princesses Rhyme and Reason have been imprisoned. Words and numbers are estranged kingdoms: king Azaz the Unabridged and his brother the Mathemagician are at odds.

When I sent last week’s post to Jim Hiebert and Doug Grouws they replied gracefully, and Jim talked a little about his own intellectual journey. He said, “I assumed, from the beginning, that all children are driven (and well-equipped) to make sense of things, including mathematics.” I was a child like that, and I am driven now to help other children do the same.

I started this Substack with a series of posts on meaning in mathematics: on how Max found a theorem in fourth grade, on the fusion of concept and procedure in math facts, on the base ten system made manifest, on expressions as meaningful language, on more than you wanted to know about fourteen sevenths, on beautiful algebraic expressions, on seeing math with technology. And, of course, zombies. I considered the problem of the meaninglessness of mathematics for so many students and came up with Bill’s Three Excellent Research-Based™ Principles for helping students see mathematical objects as real.

I had a moment of clarity: you can love the journey too much and end up in the doldrums, like Milo, where nothing means anything, or you can love the destination too much and forget to include the meaning.

I started looking at the ed psych literature. I read a paper where meaning kept shifting. I read a paper which favored explicit teaching over unguided discovery and guided discovery over explicit teaching. I read a paper which defined explicit teaching and gave a history. I got some useful insights about cognitive load and worked examples and scaffolding from these papers, but the insights were not subject specific and were mostly from lab experiments. I worried about whether they would generalize to the complexities of a real mathematics classroom. I was delighted to find a paper that grappled with these issues, with a subject specific framework for curriculum design. But the subject was statistics.

So I turned to the math ed literature and found articles which were both specific to mathematics and embedded in the complexities of the classroom. I started with the TIMSS Video Study, read a report on the math wars1, and continued last Friday with a rewarding read of Hiebert and Grouws. This Friday I will continue that thread with two very different earlier papers, one by each of those two authors, and, continuing my lifelong quest to reunite estranged kingdoms in mathematics education, I will look for commonalities.

I’ve been writing a lot about the ed psych literature and needed a break, so today I am going to write about a synthesis of the math ed literature, The effects of classroom mathematics teaching on students’ learning by Hiebert and Grouws, 2007, a chapter in the second NCTM research handbook.1 I love this chapter because it is balanced, careful in its conclusions, and well written. In fact it is so well written that this post will consist mostly of quotations.

Definitions of skill and understanding

By skill efficiency, we mean the accurate, smooth, and rapid execution of mathematical procedures . . . We do not include the flexible use of skills or their adaptation to fit new situations. By conceptual understanding, we mean mental connections among mathematical facts, procedures, and ideas.

There are two things to note here. First, the exclusion of flexibility from the definition of skill makes a clean cut between the two notions. I’ve always thought that flexibility belongs more under the conceptual understanding column. “Accurate, smooth, and rapid execution” slots well into findings from the cognitive load literature. Second, the definition of conceptual understanding is about making connections. That’s something you can observe and measure more easily than some nebulous notion of whether the student possesses the concept.

Defined this narrowly and this precisely, the two goals are not in tension. You can study the effectiveness of various teaching practices for one or the other or both.

The article proceeds to discuss features of teaching that promote these goals, but first issues a caveat.

we do not expect the features of teaching that facilitate skill efficiency and conceptual understanding to fall neatly into categories such as “expository” or “discovery.” In fact, the features of teaching we describe do not fit easily into any of the categories frequently used to describe teaching: direct instruction versus inquiry-based teaching, student-centered versus teacher-centered teaching, traditional versus reform-based teaching, and so on. . . . we will argue that most of these categories, distinctions, and labels are now more confusing than helpful, and further advances in research as well as clarity of policy recommendations will benefit from abandoning these labels.

Hear hear!

Supporting skill efficiency

The authors review a number of studies and summarize them as follows:

teaching that facilitates skill efficiency is rapidly paced, includes teacher modeling with many teacher-directed product-type questions, and displays a smooth transition from demonstration to substantial amounts of error free practice. Noteworthy in this set of features is the central role played by the teacher in organizing, pacing, and presenting information to meet well-defined learning goals.

Notice the precision here: “teacher-directed product-type questions”—as opposed to process questions, I assume—and “error free practice”—not just any old practice, but practice designed to be within the capabilities of the student. Notice also the central role of the teacher, so refreshing in these days.

I like that the authors followed through on the commitment to avoid labels here. The package of teaching practices that they describe here looks a lot like explicit instruction, but it’s presented simply as a package, possibly one that could be combined with the principles for conceptual understanding that follow.

Supporting conceptual understanding

The authors identify two key features of teaching that support conceptual understanding. The first of these is that “teachers and students attend explicitly to concepts.”

By attending to concepts we mean treating mathematical connections in an explicit and public way. . . . This could include discussing the mathematical meaning underlying procedures, asking questions about how different solution strategies are similar to and different from each other, considering the ways in which mathematical problems build on each other or are special (or general) cases of each other, attending to the relationships among mathematical ideas, and reminding students about the main point of the lesson and how this point fits within the current sequence of lessons and ideas.

Well, duh, but it “becomes more interesting when one discovers that the claim is supported across a wide range of research designs and holds true across different instructional treatments or systems.” This decoupling of the feature from specific instructional approaches points to a basic truth.

Especially important here are the words “explicit” and “public”. This echoes the practicing connections framework I talked about here. You can’t assume that the connections will follow implicitly from students doing the math. You have to name them and design activities that lead to them. Here again the authors are agnostic on the instructional model:

We believe the evidence does not justify a single or “best” method of instruction to facilitate conceptual understanding. But we believe the data do support a feature of instruction that might be part of many methods: explicit attention to conceptual development of the mathematics.

The second feature that supports conceptual understanding is that “students struggle with important mathematics.” The authors are careful to say what sort of struggle they mean.

We use the word struggle to mean that students expend effort to make sense of mathematics, to figure something out that is not immediately apparent. We do not use struggle to mean needless frustration or extreme levels of challenge created by nonsensical or overly difficult problems. We do not mean the feelings of despair that some students can experience when little of the material makes sense. The struggle we have in mind comes from solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed (Hiebert et al., 1996). By struggling with important mathematics we mean the opposite of simply being presented information to be memorized or being asked only to practice what has been demonstrated. [Bold added]

I think everybody arguing about productive struggle today, on both sides of the debate, should just memorize this paragraph. The bolded phrase captures perfectly what I had in mind in What if the struggle isn’t productive? when I talked about the implications of limited working memory and suggested that problems should be designed at the edge of students’ knowledge and designed to activate that knowledge. This is Vygotsky’s zone of proximal development, which the authors refer to a couple of paragraphs later. The authors repeat their no-labels refrain:

Struggle is usually associated with student-centered or student inquiry approaches. But we can imagine teacher-centered approaches that provide targeted and highly structured activities during which students are asked to solve challenging problems and work through challenging ideas. . . . the possibility that appropriate struggle can be built into teacher-centered approaches is one reason why we think the old labels of student-centered versus teacher-centered instruction can be so misleading.

I’ve never liked the terms “student-centered” and “teacher-centered.” The center is the classroom, where students and teachers interact, and there are different ways of organizing that interaction.

From efficiency to fluency

Having separated skill efficiency from conceptual understanding for the purposes of analysis, the authors bring them back together in a section entitled “Teaching Features that Promote Conceptual Understanding also Promote Skill Fluency.” Following Adding It Up, they define fluency as possessing, in addition to skill efficiency, the ability “to adapt . . . skills to solve new kinds of tasks.” They conclude with a “plausible conjecture” that the two features of teaching they have identified as promoting conceptual understanding—explicit attention to connections and engaging students in struggling with the mathematics—also promote procedural fluency.

Apparently, it is not the case that only one set of teaching features facilitates skill learning and another set facilitates conceptual learning. In this case, two quite different kinds of features both seem to promote skill learning.

I’ll conclude with a quotation from earlier in the chapter.

Ausubel (1963) proposed a 2 × 2 matrix with rote versus meaningful learning on one axis and discovery versus expository teaching on the other axis. Ausubel contended that these dimensions were independent. Expository teaching, said Ausubel, does not necessarily produce rote learning, and discovery teaching does not necessarily produce meaningful learning.

Amen.

There’s a new report out, Navigating the Math Wars: A Practical Guide to the Divides and Debates Influencing Math Instruction, from the Center on Reinventing Public Education at Arizona State University. It is intended for state and district leaders trying to make sense of the current arguments about how mathematics should be taught, and it covers a lot of ground: a history of the wars from Sputnik through Common Core, five dichotomies that organize the current debate, a careful evaluation of the Science of Math movement on its own published terms, a state-by-state scan of post-pandemic math policy, and a set of recommendations for what to do next.

There is a lot to like in this report. The authors open with the refreshing observation that “many traditionalists and reformers occupy overlapping territory rather than opposite ends of an unbridgeable divide.” They cite the National Mathematics Advisory Panel’s 2008 finding that the conceptual-versus-procedural conflict is misguided and that the two are mutually supportive. They invoke Cohen, Jones, and Gibbons’s epistemic bunkers framing and apply it even-handedly to both camps. They evaluate the Science of Math movement on its own published terms, conceding that its public-facing materials sometimes outrun their citations and that it engages more fully with evidence against unguided discovery than with evidence for guided discovery (a tendency I discussed here). They acknowledge that research from special education generalizes only with care. The dichotomies that organize the body of the report are presented as deliberately pure positions that few individuals or organizations actually hold—a framing that is reasonable as a teaching tool, and one the authors are clear about the limits of.

Clarifying the Common Core

[Updated to reflect edits to the report since I wrote this post.] There was one sentence in the original version of the report that is open to misunderstanding, and it’s important. On page 12, in the section setting up the conceptual-versus-procedural divide, the authors describe the reform end of that spectrum—advocates who think students should grasp ideas like place value before learning formal procedures, and who worry that rote execution leads to fragile knowledge—and then write:

Some influential frameworks, including the Common Core and NCTM’s Principles to Actions, have reflected this concern, citing research that supports a conceptual-first approach based on better retention of procedures.

The standards do not endorse a conceptual-first approach. The front matter is explicit that “These Standards do not dictate curriculum or teaching methods.” I wrote to the authors and they agreed the statement was inaccurate as written—they explained that they had meant to include the Common Core on the fragile knowledge point, not on the concepts before procedures point. They have since updated the report to make this point clear.

The three aspects of rigor in the Common Core—conceptual understanding, procedural skill and fluency, and application—were intended as mutually reinforcing goals, not as a sequence. That choice reflects how mathematical learning actually works. Students develop fluency by using procedures to solve problems whose meaning they are also building, and they build understanding partly by reflecting on procedures they are practicing. Sequencing one before the other misdescribes the activity.

A dose of realism

The report’s chapter on what actually happens in classrooms, drawing on studies of more than 5,300 math teachers, deserves more attention than it will probably get. Worksheets remain widespread in early grades. Teacher-led lecture is common across grade bands. Cognitively demanding tasks are a top priority on roughly one-third of instructional days. Meaningful student discourse occurs, on average, once a week or less in middle school. Where explicit instruction is examined directly, implementation is partial or inconsistent. And in a finding nobody should be comfortable with, half of rural algebra teachers and seventy percent of early childhood teachers reported routinely using methods based on learning styles, a theory with no research support and no advocates on either side of the math wars.

This is a dose of realism for both sides. Both camps tend to argue from their model examples—the well-implemented inquiry classroom on the reform side, the well-implemented systematic explicit instruction on the traditional side. The classrooms most students are actually in look like neither. Both sides would do well to spend more time thinking about how to avoid their failure modes.

On Wednesday I asked how we can support a cultural script that keeps connection-making alive in math lessons. Why do I care so much about connections? First, mathematics is a uniquely connected subject. Everything is connected to what came before: counting numbers, fractions, rational numbers, and real numbers form one coherent number system, each one an augmentation of what came before. If, for example, students don’t see the connection between whole numbers and fractions through the number line, fractions appear to be some strange new beast with special rules for addition and subtraction. Second, the objects of study in mathematics are uniquely abstract—making them real is difficult, and making connections between different representations of them is one way to help. For example, functions become real once you see formulas, graphs, and tables as connected representations of the same thing. The different forms of an equation for a straight line become connected through the underlying concept of slope, so that you don’t have to memorize every form as a separate bit of information.

These examples illustrate that connections happen over time, in a well designed sequence of lessons. Making connections is not just a matter of what you do in an individual lesson—it is an aspect of coherent curriculum design. So I was intrigued by the time arrow in the diagram above that Jim Stigler sent me when he read my post on guided discovery a month ago, along with a paper, Practicing Connections: A Framework to Guide Instructional Design for Developing Understanding in Complex Domains, by Fries, Son, Givvin & Stigler. I love this diagram for many reasons. First, I love that it is two-dimensional, unlike so many arguments about pedagogy that try to fit everything on a one-dimensional spectrum. I love that the sweet spot is a convergence between Pr*d*ct*ve Str*ggl*1 and Explicit Connections. I love that it is in fact three-dimensional because it has that time arrow for Deliberate Practice, indicating that connection-making happens over time.

The practicing connections framework

Fries et al. describe a framework for curriculum design in relation to a statistics course the authors have developed, CourseKata. They argue that expert understanding in a complex domain is characterized by coherent, connected mental representations—schemas, in the cognitive-science literature—and that instruction should aim to help students build those. Three kinds of connections matter: connections to the world (contexts and practices the domain applies to), to core concepts that organize the domain, and to key representations used to think and communicate. Three principles guide instruction: make connections explicit, engage students in productive struggle, and give repeated opportunities to practice over time.

For their statistics course they chose as key representations verbal descriptions, visualizations, word equations, GLM (general linear model) notation, and R code. They add, “note that we did not choose algebra, in the form of formulas and equations, as one of our key representations, even though it is emphasized in most textbooks. Our reason for this is that our students do not typically find algebra to be readily accessible or useful, a consequence, we surmise, of K-12 mathematics education in the USA.”2

I’m not enough of an expert in cognitive science to judge whether the schema research that underlies this framework is the right synthesis of the evidence, and I’m not going to try. My main point here is that this is an example of what it looks like to pay careful attention to curriculum design and base it on an articulated instructional model.

Most of the pedagogical advocacy that I have been reading stops well short of this. It trades in Research-Based™ slogans and gets into futile chicken and egg arguments. The CourseKata folks have named the research, named the instructional principles, and built a curriculum that tries to follow them. You can check their work.

Subject-specific instructional design

The sweet spot in the diagram describes a single moment of instruction; the deliberate-practice arrow turns it into curriculum design—understanding developing over weeks and months of repeated connection-making at rising difficulty. What the paper doesn’t give you, and doesn’t really claim to give you, is what a teacher does in the room on Tuesday. The curriculum presumably supplies that.

But there’s something else worth noticing. The framework’s design choices are subject-specific. The connections and representations they chose, and the ones they decided to leave out, came from careful thinking about the structure of statistics—and that’s exactly where the curriculum work has to happen.

So I see three layers at work in any instructional design: (a) a pedagogical approach—the underlying view of how learning happens; (b) design principles for the subject—how the approach is operationalized in this domain; and (c) a lesson-level structure—what a teacher actually does in the room. The Fries et al. paper is strong on (a) and (b) and gestures at (c).

What I find missing from the current explicit-instruction trend isn’t the lesson-level layer—Rosenshine’s Principles of Instruction3 is vivid and widely distributed. What’s missing is the mathematics. The advocacy names the research behind explicit instruction and what a lesson should look like, but not how to graft those onto the structure of mathematics: what the important representations are, how to sequence them, how to make connections across topics, how the story of mathematics evolves over a unit or a year. The mathematics sits in the background, supposedly a known quantity. Whether this was also true of Engelmann’s Theory of Instruction—the direct-instruction tradition this trend descends from, about six decades old now—I don’t know; I haven’t studied it.

The reform tradition did take the mathematics seriously. NCTM’s Principles and Standards articulated a vision, not without contestation from mathematicians, and subsequent efforts at synthesis of the two sides followed, such as Cuoco et al.’s Habits of Mind and the Common Core. We built the Common Core on progressions precisely because we wanted to pay attention to the structure of the subject: articulating how the number system develops from whole numbers through fractions to rational numbers, how multiplication and division relate, how the properties of operations in arithmetic are a rehearsal for algebra. The reform curricula and their cousins of the 1990s and 2000s—Investigations, Connected Mathematics, Everyday Mathematics, IMP, CPM, Core-Plus, EDC’s CME Project, Math Expressions, and others—built curricula on such visions, with varying levels of explicitness about pedagogical and lesson-level design.

The point isn’t that all three layers must be written up as theory—that’s an academic exercise, not a curriculum-design imperative. The point is that all three are at work in any curriculum, and the test for any pedagogical recommendation is whether the work behind it has thought through all three. Especially, in mathematics, the middle one—is mathematics just a collection of bits and pieces, or is it a coherent whole about which the curriculum is telling a story?

Changing the cultural script

I remember in the reform era going to so many presentations that said, in effect, teachers have to change everything they are doing right now. That’s not how culture change works. Exhortation doesn’t change a script—the pressure to fall back on procedural walkthroughs is structural. Not even materials can, but they can help. One way we can support teachers is to provide them with materials that have all three layers articulated and connected. Frameworks operating at all three levels are partial answers because they can shape materials, can shape sequences across years, and can give teachers something to work from beyond exhortation. And a good design framework extends beyond curriculum materials to professional learning and supplemental products.

The argument worth having about teaching mathematics isn’t whether to model procedures or guide discovery. It’s whether the work behind your position connects all three layers—view of learning, subject-specific design principles, lesson-level practice. Focusing on the mathematics causes you to focus on finding the right tool for the specific topic at hand, be it ratios or completing the square, and recognizing that different tools do different work. That’s where common ground lies.

Earlier in my career I was involved in the calculus reform movement, an effort to make the undergraduate calculus course more balanced between procedural fluency, conceptual understanding, and applications. A catchphrase of that movement was the Rule of Three, that every concept should be introduced algebraically, graphically, and numerically.1 This idea was paralleled in the K–12 reform movement of the 90s and 00s. But it sometimes got degraded into a rather mindless routine—write a formula, draw a graph, make a table—that lost the main point: the connections. By explicitly making connections between the representations you helped student reify the concept.

Problems that ask students to make connections don’t have to be difficult or complex. They can be as simple as

If (1, 5), (5, 4), and (4, 5) are on the graph of f, which of the following is true? Mark all that apply. (a) f(1) = 5 (b) f(5) = 1 (c) f(1) = f(4) (d) f(4) = f(5).2

The same process of degradation as I saw with the Rule of Three is described in Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study by Hiebert et al. The study filmed and coded 8th grade lessons, including about 80 in the US, and compared instructional practices and the structure and content of lessons across countries. It analyzed the teaching of thousands of mathematics problems. It categorized problems into using procedures, stating concepts, and making connections. The latter were defined as

Problem statements that implied the problem would focus on constructing relationships among mathematical ideas, facts, or procedures. Often, the problem statement suggested that students would engage in special forms of mathematical reasoning such as conjecturing, generalizing, and verifying.

The report gives the following as an example of a problem that would be classified as making connections.

Graph the equations y = 2x + 3, 2y = x − 2, and y = −4x, and examine the role played by the numbers in determining the position and slope of the associated lines.

Each problem was coded both for how it was stated and how it was actually taught.

Before I give the results I want to note that the report thanks the teachers who agreed to be filmed for “opening the classroom door,” and that the report is about a pattern across US lessons, not about any individual teacher’s practice. You can see some of the videos here as examples of the pattern.

About 17% of problems in US classrooms were coded as making connections problems. Less than 1% were solved that way (p. 103, fig. 5.12—the number rounds to zero in the figure).

Instead of solving these problems publicly through making connections, teachers and students . . . in the United States often solved them by giving results only [33%] or by using procedures [59%].

This finding is striking. But before we jump on it, let’s look at the example of a making connections problem that the report gave. It’s very open ended! I wouldn’t blame a student for not knowing what to do with it, and I wouldn’t blame a teacher for reducing it to a demonstration.3 A sharper problem along the same lines that would still be about making connections would be to give the three equations and the three graphs and ask students to match them up, without actually graphing the equations. You could make it an activity, individual or in groups or in a whole class discussion. Students would learn from hearing each other’s answers.

In The Teaching Gap by Stigler and Hiebert, a book based on the earlier 1995 TIMSS video study, the authors talk about different countries having different cultural scripts. A cultural script is a set of expectations about what a math lesson is supposed to look like, reproduced by teachers, students, administrators, parents, textbooks, and tests in a mutually reinforcing system. I would guess that the striking result above is a result of the US cultural script and its influence not only on teachers, but also on textbook writers. If teachers were always working with activities that were either procedural, or big and open-ended, then the pressure to keep things procedural would be stronger.

I assume everybody wants students to make connections in the mathematics they are studying, and I assume everybody wants a cultural script that is different from the one that produced that drop from 17% to essentially zero. But changing the script is not simply a matter of telling teachers they have to change everything they are doing. In my Friday post I want to talk more about how we can support a cultural script that helps students make connections.

For teachers reading, when you plan an activity designed to make a connection between representations or ideas or procedures, what makes it harder or easier to keep that work alive in the lesson? What pulls it toward just showing the steps or giving the answer?

On Wednesday I asked how we get a student to pay attention to something they can’t see yet, how we make it real for them. Anybody who has taught calculus knows the sinking feeling when you are working through the algebra behind finding the derivative of f(x) = x^n and you just know that students aren’t seeing the reality of the derivative as a limit of difference quotients that sits behind it.

In order to answer this question, I consulted three resources.

The Velveteen Rabbit

The Velveteen Rabbit, by Margery Williams, is a 1921 story about a stuffed toy rabbit that becomes Real through the love of a child. At one point the rabbit is talking to the Skin Horse, who has already become Real:

“Real isn’t how you are made,” said the Skin Horse. “It’s a thing that happens to you. When a child loves you for a long, long time, not just to play with, but REALLY loves you, then you become Real.”

“Does it hurt?” asked the Rabbit.

“Sometimes,” said the Skin Horse, for he was always truthful. “When you are Real you don’t mind being hurt.”

“Does it happen all at once, like being wound up,” he asked, “or bit by bit?”

“It doesn’t happen all at once,” said the Skin Horse. “You become. It takes a long time. That’s why it doesn’t often happen to people who break easily, or have sharp edges, or have to be carefully kept. Generally, by the time you are Real, most of your hair has been loved off, and your eyes drop out and you get loose in the joints and very shabby. But these things don’t matter at all, because once you are Real you can’t be ugly, except to people who don’t understand.”

Ever since first reading this story to my children over 30 years ago, I’ve been struck by how close the Skin Horse’s account is to the mysterious process of reification (the technical term for becoming Real) in mathematics: “it’s a thing that happens,” “it takes a long time,” and, most importantly, “it doesn’t often happen to [things that] have to be carefully kept.” You have to play with mathematical objects with abandon and for a long time for them to become real.

It turns out that there is a research paper by Anna Sfard that says pretty much the same thing.

On the dual nature of mathematical concepts

This is the title of a 1991 paper by Anna Sfard.

Sfard argues that mathematical objects have both an operational and a structural nature. She emphasizes that this is not a dichotomy but a duality, two sides of the same object. I said the same thing in my 8 + 5 = 13 post.

Sfard believes that the operational aspect comes first, both historically and in student learning. Students see a function as a process before they see it as an object. Sfard divides the process of reification—the move from the operational to the structural—into three phases. First, interiorization (getting fluent with the process), then condensation (chunking it so you can reason about it as a whole), and finally reification (the phase transition into becoming an object).

Reification is important because it enables you to start the process over to build higher order objects. As long as you are stuck seeing fractions as pairs of numbers that you do complicated things with, you can’t move on to seeing them as part of the number system. As long as you see functions as a matter of computing outputs from inputs, you can’t see them as objects you can take the derivative of.

“Reification, therefore, is defined as an ontological shift—a sudden ability to see something familiar in a totally new light.”

Or, as the Skin Horse said, “it’s a thing that happens.” And it happens through operating on the object, slowly, over time. Like playing with a beloved toy.

For Sfard it explains why mathematics is so hard to teach.

But here is a vicious circle: on one hand, without an attempt at the higher-level interiorization, the reification will not occur; on the other hand, existence of objects on which the higher-level processes are performed seems indispensable for the interiorization—without such objects the processes must appear quite meaningless. In other words: the lower-level reification and the higher-level interiorization are prerequisite for each other!

It’s a bit like the old joke that you can’t get a job without experience, but you can’t get experience without a job.

The hard work of teaching is figuring out how to navigate the vicious circle. The argument between explicit instruction and guided discovery becomes less important; it reduces to deciding where to enter the circle. The really difficult work of teaching is navigating the circle once entered.

The mathematician

That’s me. As a working mathematician I experience reification as intensely satisfying and somewhat mystical. There’s an old philosophical argument about whether the things that mathematicians study are discovered (the Platonist view) or invented (the Formalist view). I have always found that in order to do real mathematics research I have to be a practical Platonist, whatever my philosophical views. I have to believe that the objects I am studying are real. I don’t think I’d be able to prove any really good theorems about them if I didn’t see them that way. And let me tell you, the objects I have studied in my life are pretty damned abstract from anybody else’s point of view (e.g. my parents, my wife, my children). But they are real to me.

I told the story in Where I come from about what made mathematics real for me. Spending hours lying on my bed, grappling with the definitions in Spivak’s calculus. Doggedly solving all the problems. Falling in love with the beauty of the modern definition of the real numbers, limits, and continuity. Every advance I have made in my research in number theory has been the result of reification. And there is no limit to how high you can go in beautiful abstractions if you can climb the ladder described in Anna Sfard’s work, where reification forms the objects for interiorization at the next cycle.

The three principles

I was going to call these Bill’s Three Excellent Research-Based™ Principles for helping students see mathematical objects as real. But I suspect they are pretty idiosyncratic. I’ll be interested to see if anybody else finds them useful. Let me know in the comments!

1. Grappling. Students need to grapple with the mathematics. You can see this in the beautiful video Marilyn Burns pointed me to in her comment on my last post.

   Grappling is playing with the toy so hard the hair falls out. It is the operational side of Anna Sfard’s duality. It is me turning Spivak’s definitions around in my head. It can mean gaining deep procedural fluency, it can mean hands-on mixing of paints to learn about ratios, it can mean mastering the multiplication table, it can mean tackling a word problem without being shown how to do it. Grappling can’t be purely mechanical: it is not the shallow engagement with a toy taken off the shelf and put back again, it is not surface procedural fluency.

2. Doggedness. This is what takes a “long, long, time” in the Skin Horse’s words. It is navigating the vicious circle. It is my entire life in math research. It’s productive struggle, except that term has acquired baggage. It’s grit, except that term has acquired baggage as well. It’s perseverance. I like the term doggedness because, well, everybody likes a dog and admires the way they will patiently gnaw away at something.

3. Love. You knew I was going to bring this up. Look at the Velveteen Rabbit. True, Anna Sfard doesn’t talk about love, but she opens her paper quoting Poincaré in “obvious despair”: “How does it happen that there are people who do not understand mathematics?” For me it is the magnetic attraction that draws me back to the same problem over and over again, like a moth to flame.

These terms describe the way I experience mathematics research and the moments I live for in my students. And because they are terms that have not yet been captured by one side or the other, they help me see through the false dichotomies to underlying truths. They are how I try to make mathematics real for my students. I still remember the student who moved from grappling with what the derivative would look like on her calculator to anticipating what it would look like and using the calculator to confirm.

Helping students see mathematics as real is a step towards helping them see its beauty. As the Skin Horse says, “once you are Real you can’t be ugly, except to people who don’t understand.”

Next week I promise I will get back to what you are paying me for: reading research articles and making fun of them trying to figure out what their evidence really supports.

How would you teach a child to bake a cake? To ride a bicycle? For the cake I’d probably start by showing how to measure out the ingredients, and then invite the child to try it. Maybe let the child do some mixing. In other words, explicit instruction: start with modeling, then gradual release of responsibility. For the bicycle I’d start by getting the child to sit on the bicycle, get a feel for the pedals and balance, and try a short run while I ran alongside. In other words, guided discovery: start with a bit of productive struggle, then provide strategic support along the way. These examples are why I find the debate about these topics unhelpful, at least for mathematics, as I explained in my post about choosing the right tool for the job.

You could also try teaching about cakes and bicycles the other way—discovery learning for the cake, explicit instruction for the bicycle—and you’d probably be fine. You’d get a messy kitchen and a slightly bewildered child, but it would all work out in the end because the child already knows what a cake is and what a bicycle is. The object of the lesson is already a real thing for them. Not to mention that they probably want to make or ride one, but motivation per se is not my point here. Rather, my point is that the reality of the object you are teaching them about is a huge advantage.

And mathematics doesn’t have that advantage. Other disciplines are about things that students already see as real. Stories are real, as are the books that contain them; birds and falling stones are real; even historical events can be made real. But in mathematics, as the abstraction grows, objects become less tangible. Are quadratic equations real? Only if the lesson makes them so—they don’t arrive that way. Are ratios real? You can make them real with carefully chosen examples. There is something constant about what happens when you mix two parts red paint to one part blue, in whatever quantities. It’s called magenta. That’s the reality of the ratio.

The question is not how I would teach completing the square (probably by explicit instruction) or the concept of a ratio (probably by guided discovery), but how I get the student to see a quadratic expression or a ratio as a real thing in the first place.

What are your examples? What are mathematical objects you’ve struggled to see as real, or to get your students to see as real?

On Friday I want to dig into this issue. I want to think about how we get a student to pay attention to something they can’t yet see.

The number e

On Wednesday we looked at the formula for compound interest.

Compound interest pays more annually than simple interest. If you compound twice a year at half the interest rate then at the end of the year you get not only the full interest payment on the principal but also a half interest payment on the interest you were paid halfway through the year. So, at 100% interest, for example, your principal gets multiplied by (1 + 1/2)² = 2.25 instead of the doubling it would have gotten from a simple 100% addition. You might wonder how far you can go with this. If you compound every second, will you become rich? You would not be the first to wonder. Jacob Bernoulli wondered in 1690 and wrote about it.

“Quaeritur, si Creditor aliquis pecuniae summam faenori exponat, ea lege, ut singulis momentis pars proportionalis usurae annuae sorti annumeretur, quantum ipsi finito anno debeatur?”. “The question is: if some creditor were to put out a sum of money at interest, under the condition that at every moment a proportional part of the annual interest is added to the principal, how much would be owed to him at the end of the year?”

Here is the series he writes down, without derivation, for the year-end balance:1

The a is the principal and b the annual interest (expressed as a total amount, not a rate). Considering the case a = b, he found upper and lower bounds for this series and concluded that you couldn’t do better than tripling your investment at the end of the year, no matter how often you compounded. Then he moved on to thinking about a math problem that arises out of gambling.

This is all pretty complicated. To get an idea of what is going on, take out your phone or go to your browser or ask Siri to calculate (1 + 1/n)^n for n = 12 (compounding every month), n = 52 (every week), n = 365 (every day), n = 525, 600 (every minute) and n = 31, 536, 000 (every second). Even if you know how this is going to go you should give it a try because it is fun. I’ll put the answers in a footnote so as not to spoil it.2

It turns out that there’s a limit to how far we can push this no matter how frequently we compound. Notice how the decimal places stabilize. By the time w\e get to minutes and seconds it looks like the limit is 2.718 . . . . In other words, the best you can do is increase your principal by about 170%.

In 1748, in the Introductio in analysin infinitorum, Euler took the same series—with a and b both set to 1, the case Bernoulli had puzzled over—and calculated its sum to 24 decimal places, getting 2.71828182845904523536028. He named it e. Not because of compound interest; he was looking for the base that would simplify his logarithm formulas: when you choose the base that makes a certain constant equal to 1, all the equations clean up, and the base turns out to be this same number.

“Ponamus autem brevitatis gratia … constanter litteram e”—let us, for the sake of brevity, constantly put the letter e.

Euler had, among many other talents, a prodigious ability to make numerical calculations. He notes with wry satisfaction that the last digit, the 23rd after the decimal point, “is still in accord with the truth.” Today, anybody can do the calculation. I still remember the surge of astonished delight I felt when I successively punched into a calculator the numbers 1.1^10, 1.01^100, 1.001^1000 etc. and saw the familiar digits of e start to appear. Not that I didn’t know this would happen; but seeing it happen was new. All those 1s and 0s producing a magic number.

Graphing for the masses

Now I want to describe another eye-opening moment. In the early 1990s handheld graphing calculators came on the market, and the ability to produce graphs of functions moved rapidly from the lab to the classroom.

Playing around with my TI-81 helped me see functional behavior unfold in ways pencil and paper never could. In the old days, sketching a graph was a matter of imagination. You would find some key points on the graph of a polynomial function and draw a curve between them. Here is a graph of the function f(x) = 3x^4 - 4x^3 + 1 from the 1944 edition of Elements of the Differential and Integral Calculus, by Granville, Smith, and Longley, next to the real thing from Desmos.

Graphing calculators turned everything around. They would show you the reality directly. Expectations still played an important role in helping you choose the right window, but you were no longer relying on imagination or, if you were a textbook author, the vagaries of your publisher’s art department. A good example is comparing exponential growth with power growth. You know abstractly that exponential growth always wins. For example, a large power, say f(x) = x^10 (in red below), will eventually lose out to a slowly growing exponential, say g(x) = (1.1)^x (in blue). You can figure out where this happens numerically. But seeing it happen is different.

My second moment of seeing with new eyes came when I realized that the generic graph of an exponential function should be shown as the blue graph in the second of these two figures, going horizontally along the x-axis and then turning a sharp right angle and shooting up vertically, rather than the gently curving graph in the first.

Promise and peril

For me, the TI-81, and subsequent iterations of handheld and online calculators, are like having new eyes. They don’t tell me anything I don’t know already, but they show me things I have never seen before.

Of course, as calculators spread throughout the classroom, people started arguing about them, as they are wont to do. Some saw great promise, others saw great peril. The fear was that students would become mindless button-pushers. To my mind the mindless-mindful axis is orthogonal to the technology axis—mindless manipulation was a thing before calculators came along—and we should try to live in the first quadrant, where technology enhances rather than hampers learning.

The mountains and the birds were always there; the new lenses didn’t put them there, only let me see what was. The number e was always emerging in a table of compounding intervals, and the true shape of an exponential was always waiting in the right window. Technology, used mindfully, is the pair of lenses.

         n       (1 + 1/n)^n
----------     -------------
        12     2.613 035 290
        52     2.692 596 954
       365     2.714 567 482
   525,600     2.718 279 243
31,536,000     2.718 281 785

I had planned to continue my math ed article series this week, but I need a breather from the footnotes, and I suspect some of you do too. So I want to go back to an issue I brought up a few weeks ago in Parlez-vous Algebra? There I argued that many algebra students experience expressions the way I experienced the French insurance office: as a stream of stuff they are supposed to do something with, without any firm conviction that the stuff means anything. I’ve been thinking about that post ever since, and about one letter in particular.

At some point x decided to leave the boring life of the algebra classroom and went out into the world in search of fame and fortune. It didn’t just want to be unknown, it wanted to be mysterious. And it wanted to be capitalized. Madame X. X marks the spot. The X-Files. And it became an icon for algebra as a mark of seriousness. If you want to sound technically sophisticated and strategic in a business meeting, you talk about “solving for X.”1

None of this would matter if x behaved itself back home in the algebra classroom. But it doesn’t. Sure, some problems might give it a meaning—“let x be the number of apples”—but the meaning is thin, worn briefly at the start of the problem and shed the moment the manipulation begins. It is the bare equation problem I wrote about in Parlez-vous Algebra?. By the time students are a few lines in, x is just sitting on the page waiting to be solved for. It is not a number, or a quantity, or a count of anything in particular. It is a small italic shrug. And then we ask students to manipulate it, and are surprised when they move it around like a sticker without regard for operations or parentheses.

Here is an expression in which no letter is allowed to shrug:

This is the beautiful expression for compound interest. Its beauty comes not from glamor, but from the work it does. Every letter is famous, but its fame is earned; each has a job, and you can read the job from the first letter, a convention that rewards you for paying attention.

P is the principal, standing proudly and multiplicatively out the front. r is the annual interest rate, augmenting the principal faithfully every year for t years. n appears twice in a balanced way, saying, “apply the interest n times a year but only one n-th at a time.” Then 1 + r/n is the factor you multiply your balance by at the end of one period: you keep what you had (the 1) and add a little bit more (the r/n). You do that nt times over t years, because there are n periods per year. Here parentheses and exponential notation are called into duty to describe the repeated multiplication. And then you multiply the whole repeated-multiplication-by-one-plus-r-over-n business by P, because whatever happens to a dollar in this account happens proportionally to P dollars.

You can read that expression like a paragraph, as I just did. Every letter is carrying a small piece of a story about a bank and a year and some money.

This is the experience I want students to have with algebraic expressions in general: not a collection of symbols they need to rearrange, but a phrase in a language, written by someone who meant something by it. Sometimes the meaning is a bank account. Sometimes it is the area of a rectangle, or the time it takes a ball to fall, or the number of handshakes in a room. The expression is not the mystery; the expression is the answer to the mystery, written down so you can read it later.

Poor x was not always a shrug. When Descartes introduced it in La Géométrie in 1637—the appendix to the Discourse on the Method that invented much of the notation we still use—he gave x a specific job as a variable, along with y and z, standing in honest partnership with the constants a, b, c, . . . at the other end of the alphabet.

That is a respectable line of work. I would like to see x find meaning in that work again, next to P and r and n and t, and all the other letters that show up to an expression with something to say.

On Friday I want to talk about when technology enters the scene, starting with the appearance of graphing calculators in the classroom over 30 years ago, and musing on what happens when a beautiful expression meets a button that says ENTER. A famous mysterious number appears, and so do dangers. See you then.

P.S. If you are interested in the mathematics behind that image from Descartes, take a look at this article from the Mathematical Association of America.

I’ve spent the last couple of Fridays reading the references. I looked at Kirschner, Sweller, and Clark and found that its research base—for math at least—was narrow, its terminology slippery, and the three National Academy of Sciences reports it cited in support actually contradicted its thesis. I looked at Alfieri et al. and found a careful meta-analysis whose strongest finding was for guided discovery—an article widely cited in arguments against discovery learning that turns out, on a close reading, to support it. That has been fun, and I think it was worth doing, but debunking is easy and has its limits. At some point you have to stop saying what’s wrong with other people’s arguments and start trying to figure out where they are right.

So let me try.

I want to start by noticing that a lot of disagreements in this area come from different sides focusing on different problems. Often, both problems deserve solving. Everyone agrees that students should develop fluency with basic number facts so that working memory is freed up for reasoning with the facts rather than burdened with finding them. Everyone agrees that students should understand important mathematical concepts—what a fraction is, why the addition algorithm works, what an equation says. And everyone agrees that students should be able to apply concepts and procedures to solve problems with something like real-world complexity. These are genuinely different learning targets. What is involved in learning each one is different, and we should not expect a single way of teaching and learning to be best for all of them. We don’t drive nails with screwdrivers or pound screws with hammers.

So the question I want to ask is not whether explicit instruction is the right way or the wrong way, but what it is good for. And then, once we’ve answered that, what about everything else students need to learn?

What is explicit instruction, exactly?

To answer this question I went looking for an article that gives a definition and found “Explicit Instruction: Historical and Contemporary Contexts” by Hughes, Morris, Therrien, and Benson, published in 2017 in Learning Disabilities Research and Practice. The authors searched the literature for “explicit instruction,” “explicit teaching,” “explicit direct instruction,” and “learning disabilities,” and assembled what they found into five pillars:

1. Segment complex skills

2. Draw student attention to important features of the content through modeling/think-alouds

3. Promote successful engagement by using systematically faded supports/prompts

4. Provide opportunities for students to respond and receive feedback

5. Create purposeful practice opportunities.

The first and the last of these are sound principles for any approach to teaching; you would find them in a well-designed curriculum supporting a guided discovery approach. The middle three are where you see the specific shape of the model.

Think-alouds are the teacher performing the expert reasoning out loud while the student watches and listens. Compare that with the elicited explanations supported by Alfieri et al., where the student is articulating the reasoning. Both make thinking visible, but the direction is reversed. Faded supports presupposes that the teacher started by showing the student how to do it: you fade from full support to no support, which means you began with full support. In guided discovery the student starts with partial knowledge and builds toward fuller understanding, a different trajectory. And the language in pillar 4 is telling—providing opportunities for students to respond is not quite the same as inviting students into a conversation where information flows in several directions.

Put the five pillars together and what you have is a recognizable instructional model, a descendant of the traditional I-do-we-do-you-do approach. It is a model with a long tradition and it is worth asking when it works.

What explicit instruction is good for

There are some pretty clear answers to this in the research literature.

Hughes et al. has a section called “Is Explicit Instruction Effective?” and the first paragraph of that section cites literature reviews that “identified explicit instruction as effective for teaching students with LD [learning disabilities] in the areas of math, reading, and writing.” Of the studies in this group, only one is specifically about math: “Mathematics Interventions for Children with Special Educational Needs” by Kroesbergen and Van Luit, a 2003 meta-analysis of 58 studies of math interventions for elementary students with special educational needs. It’s a careful piece of work and it finds that direct instruction is effective for basic computational skills with students who have special needs. The sample sizes are small and the advantage does not extend to problem solving—and interestingly, a third approach the authors call “self-instruction” has the highest overall effect size.1 But the finding for direct instruction and basic computational skills is real.

The second citation is a 2009 IES Practice Guide by Gersten et al., Assisting Students Struggling with Mathematics: Response to Intervention for Elementary and Middle Schools. IES Practice Guides are careful surveys of the literature, with a high bar for accepting studies and an explicit grading scheme for the strength of evidence behind each recommendation. This one recommends explicit and systematic instruction and gives that recommendation the highest evidence rating. And it is refreshingly clear about its own scope: the recommendation is limited to Tier 2 and Tier 3 support, and “the guide does not make recommendations for general classroom mathematics instruction” (p. 5).

So explicit instruction has evidence behind it for a specific job: securing fluency with foundational procedural skills, especially for students who are struggling and need focused support. That is not a small job. Fact fluency and procedural fluency matter, working memory is a real constraint, and a student who has to reconstruct how to add two fractions from first principles every time is not going to have much capacity left over for thinking about what the answer means. People in cognitive load theory have been saying this for years—I agree with them.

What about everything else?

The trouble comes when the same model is promoted as the right approach for every learning target in mathematics. Hughes et al.’s next paragraph does exactly that, expanding the claim from struggling students studying arithmetic to math students in general, and bringing out the big guns: four more IES Practice Guides. So I read each of them, looking for what they actually recommend.

• Siegler et al. (2010), Developing Effective Fractions Instruction for Kindergarten Through 8th Grade

• Woodward et al. (2012), Improving Mathematical Problem Solving in Grades 4 Through 8

• Frye et al. (2013), Teaching Math to Young Children

• Star et al. (2015), Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students

A pattern emerges here: each guide takes on a different learning target, and each picks tools that fit the target.

Siegler et al. (2010) is aimed at conceptual understanding of fractions. Its five recommendations are about building on informal understanding, helping students see fractions as numbers, understanding why procedures make sense, developing conceptual understanding of ratios and proportions before exposing students to cross-multiplication, and emphasizing fractions in teacher professional development. The emphasis is thoroughly conceptual, because the job is conceptual. The only mention of direct instruction in the guide is in describing a teaching approach “designed to address concerns about the limitations of direct instruction.”

Woodward et al. (2012) is aimed at one of the learning targets I mentioned above: applying concepts and procedures to problems with real complexity. Its recommendations include explicit teacher modeling and think-alouds and guided questioning and engaging students in conversations about their own thinking and exposing students to multiple strategies. The guide is explicit that no single instructional philosophy owns this territory:

When developing recommendations, the panel incorporated several effective instructional practices, including explicit teacher modeling and instruction, guided questions, and efforts to engage students in conversations about their thinking and problem solving. The panel believes it is important to include the variety of ways problem solving can be taught.

Frye et al. (2013), on mathematics for young children, does not recommend an instructional model at all. Its recommendations are about content areas, progress monitoring, helping children describe their world mathematically, and setting aside daily time for math. Where explicit instruction appears, it is in an appendix that found positive effects for semi-structured discovery learning and for structured discovery learning paired with explicit instruction on patterns and relations. Explicit instruction shows up as a partner to structured discovery, not as a standalone approach.

Star et al. (2015), on algebra—which I discussed in an earlier post—recommends using solved problems to analyze reasoning, teaching students to use algebraic structure, and having students choose among alternative strategies. The emphasis is on developing strategic thinking. The guide allows that explicit instruction may be necessary for some students, but adds a caution: “it is important to distinguish between providing explicit instruction and teaching only a single solution strategy and asking students to memorize the steps of that strategy.”

None of these four guides recommends explicit instruction as a general approach to teaching mathematics. I don’t think the authors were worrying about labels; they were just picking the right tools for the job. Conceptual understanding of fractions is a different target from securing fluency with single-digit facts, and problem solving with real-world complexity is different from either, and the research—when you read it—tells you as much.

The mechanism underneath

There is a nice confirmation of this from inside the cognitive load research itself. Here is part of the abstract from “The Expertise Reversal Effect” by Kalyuga, Ayres, Chandler, and Sweller (2003) (the same Sweller as in Kirschner et al.) with emphasis mine.

When new information is presented to learners, it must be processed in a severely limited working memory. Learning reduces working memory limitations by enabling the use of schemas, stored in long-term memory, to process information more efficiently. Several instructional techniques have been designed to facilitate schema construction and automation by reducing working memory load. Recently, however, strong evidence has emerged that the effectiveness of these techniques depends very much on levels of learner expertise. Instructional techniques that are highly effective with inexperienced learners can lose their effectiveness and even have negative consequences when used with more experienced learners.

In other words, the same cognitive science that tells us why foundational fluency matters also tells us that the techniques best suited to building that fluency are not the techniques best suited to every later stage of learning. Different learning targets need different tools, and the people who built the theory already know this. The trouble only starts when citations of the theory lose nuance and inflate its results.2

Where this leaves me

So here is the position I arrived at, which I don’t think is going to surprise anyone. Fluency with basic facts and foundational procedures matters, it reduces the mental effort required to execute basic steps, and the evidence is pretty good that structured, explicit practice—especially for students who need focused support—is a reasonable way to get there. Conceptual understanding matters, and the evidence there points to making connections among ideas, facts, and procedures explicit while students work on problems within reach but not yet understood. Application to complex problems matters, and that is a different job again, with its own pedagogy, and the field’s best guidance on it draws from several traditions rather than picking one.

None of these is in conflict with the others. The mistake is thinking that one pedagogy wins all the time, or that a finding about one target is a verdict about the others. If I have a quarrel with anyone, it isn’t with the researchers who did the careful work; it’s with the way their work gets used in broader arguments, where the narrow finding becomes a broad slogan and the careful caveats fall away.

So let’s can the slogans and start building a world for students and teachers in real classrooms. There is plenty to work on—about what guided discovery actually looks like in a classroom, about how the expertise reversal effect plays out as students move through a curriculum, about what making connections explicit looks like in practice—and I intend to address some of it on future Fridays. I invite you to join with me and bring your friends, especially the ones who disagree with you. Which learning targets do you think your own teaching, or your own children’s schooling, handles best, and which the worst? What does the right tool for the job look like to you? Let’s build this thing together.

The other day I read an article criticizing discovery learning by a Brave Debunker™ that cited the easily debunked paper by Kirschner et al.1 I thought of writing a snarky post about it. But then I thought, people in glass houses shouldn’t throw stones. I’ve been doing a lot of debunking myself, and plan to do more on Friday. It’s very easy in this business to fall into motivated reasoning and in-group thinking, and allow your own preconceptions to blind you to the flaws of research that supports those preconceptions. So I’m going to devote this Wednesday post to describing my own preconceptions and where they came from. If one can’t change one’s beliefs, one can at least be honest about their origins and one’s resistance to changing them.

When I was a first year undergraduate at the University of New South Wales I had two brilliant lecturers in mathematics, Alf van der Poorten, who taught linear algebra, and Jack Gray, who taught calculus out of a wonderful book by Spivak. They were lively, explained the reasons behind things, and I was mesmerized. It was that experience that led me to become a mathematician. And it also led me to believe that a sufficiently clear and beautiful explanation could light up a student’s mind the way mine had been lit up. When I arrived at Harvard as a graduate student and was assigned to teach sections of calculus I wanted to do that for my students. I thought up brilliant, clear lectures that not only taught the procedures, but the conceptual foundations of those procedures.

Then I discovered something quite shattering. When students came to my office hours for help, and I started questioning them to find out where their problems were, I realized that none of my brilliant lecturing had made it into their heads. Understanding the derivative as a limit of slopes of secants had turned into a mess of algebra with no numerical backing. Students could tell me that the derivative of x² was 2x, but when I asked what that meant at x = 3 they were baffled. That set me on a journey to where I am now. Later, as an assistant professor at The University of Arizona, I was invited by Deborah Hughes Hallett to join the Harvard Calculus Consortium, and got interested in writing problems that helped the teacher see what was going on in the student’s head, and helped the student see the living concepts behind the procedures. This was the beginning of work that culminated in co-founding Illustrative Mathematics and developing a curriculum designed around a problem-based instructional model.2

I do believe there is evidence supporting that model, and I’ll be talking about that evidence in coming posts. But if I’m being honest, the reason I have a predisposition towards that model is personal: watching students solve problems helps me understand what is going on in their heads. It would be hard for me to give that up even if you gave me a mountain of evidence that just telling them how to solve the problem was better for them. Fortunately I have not found that mountain of evidence yet. And yes, the use of the word “fortunately” here reveals my preferences, even as I hope that I would overcome them should the evidence demand it.

I had some great comments on last Friday’s post about the everybody-else-is-wrong article by Kirschner et al. One of them, by George Lilley, led me, via a wonderful webinar by Rachel Lambert, to an article by Alfieri, Brooks, Aldrich, and Tenenbaum whose title is the subtitle of this post. Its answer, surprisingly, is yes. I say surprisingly because, as Rachel Lambert explains, this article is cited in support of explicit instruction.

Part of the problem in this debate is that people are throwing around terms without precise definitions (see “minimally guided instruction”), so I appreciated this article’s attempt to do that. Alfieri et al is a meta-analysis of 164 studies covering math, computer skills, science, problem solving, physical/motor skills, and verbal/social skills. They ran two different analyses. The first one compared unassisted discovery learning with explicit instruction,1 and explicit instruction won. I assume that’s the result the proponents of explicit instruction like to point to. As I said last time, I don’t find this result surprising—and I don’t know any math curriculum that advocates unassisted discovery. Most curricula assume the teacher will do something other than sit around watching kids magically acquire knowledge.

The second study compared enhanced discovery with other forms of instruction, and this time enhanced discovery won. One form of enhanced discovery, guided discovery, won big, with an effect size of 0.5 overall and 0.29 for math.2 So yes, actually, discovery-based instruction does enhance learning.

Enhanced discovery versus other forms of instruction

Each of these categories is divided into subcategories, and the details are both interesting and a little confusing. I’ll give the details here and then try to draw some conclusions, particularly for math. Enhanced discovery is divided into

• generation, where learners “generate rules, strategies, images, or answers to experimenters’ questions,”

• elicited explanation, where learners “explain some aspect of the target task or target material, either to themselves or to the experimenters,” and

• guided discovery, which involves “either some form of instructional guidance . . . or regular feedback to assist the learner at each stage of the learning tasks.”

The other forms of instruction to which enhanced discovery was compared include

• direct teaching, “explicit teaching of strategies, procedures, concepts, or rules in the form of formal lectures, models, demonstrations, and so forth and/or structured problem solving”

• feedback, in which “experimenters responded to learners’ progress to provide hints, cues, or objectives”

• worked examples, which “included provided solutions to problems similar to the targets”

• explanations provided, where “explanations were provided to learners about the target material or the goal task.”3

The authors were able to break down the overall effect for enhanced discovery into effects for each of its subcategories. Here generation on its own fared poorly (–0.15), which I don’t find surprising because it sounds a little close to unguided discovery. Elicited explanation fared well (0.36) and guided discovery fared very well (0.5).

On the “other instruction” side the study was not quite powerful enough to break out effect sizes, with the exception of worked examples, which on its own fared equally with enhanced discovery. This agrees with the findings I discussed last week. Thus, taking out worked examples would leave the other categories, including direct teaching, even worse off compared to guided discovery.4

Who’s doing the talking?

Explanations occur on both sides of the comparison, but there is a crucial difference. Under enhanced discovery we have elicited explanations, where students are explaining their thinking; on the other side we have explanations provided, where teachers are telling students what to do. In the fog of words people use, one thing that emerges fairly clearly from the proponents of explicit instruction is that they are pushing the traditional “I do, we do, you do” approach to teaching where teachers talk first, then the class discusses, then students work individually.

This research study does not support that position. Its evidence points more to having students try things and explain their thinking, with the teacher providing guidance and following up with a synthesis. There is still a role for explicit instruction in such an approach, but it doesn’t come first.

What does this mean for math?

An interesting aspect of both studies in this paper is that the effect sizes are smaller for math than for other subject areas. For the first study, unassisted discovery versus explicit instruction, we have an effect size of −0.16 for math, notably the smallest negative effect of any domain: compare science at −0.39, problem solving at −0.48, and verbal/social skills at −0.95. So even in the “discovery loses” meta-analysis, math was the domain where discovery came closest to holding its own.

In the second study, enhanced discovery versus other instruction, the effect size for math was 0.29—again on the lower end compared to other domains.

To me this suggests that math is a domain where instructional design quality matters most, which is an argument for a careful middle-ground approach to math instruction rather than extreme positions.

What does this mean in the classroom?

Most of the studies in this analysis were small-sample lab experiments, not year-long or multi-year studies of curricula, which is relevant to how far you can extrapolate these findings in the debates about what should happen in the classroom. I’m going to pick up on some of the classroom studies in future posts. To repeat the caution from Adding it Up:

. . . the effectiveness of mathematics teaching and learning does not rest in simple labels. Rather, the quality of instruction is a function of teachers’ knowledge and use of mathematical content, teachers’ attention to and handling of students, and students’ engagement in and use of mathematical tasks.

What I do take away from this study is that good classroom practice involves students exploring problems and explaining their thinking, and teachers providing feedback and guidance. Sounds a lot like problem-based instruction to me.

From the collection of the History of Science Museum, University of Oxford

In April 2003, Terry Bladen, president of the National Association of Schoolmasters/Union of Women Teachers, proposed that students should be allowed to drop advanced concepts such as quadratic equations at the age of 14. “Pupils should be numerate,” he said, “but numeracy can be divorced from mathematics. How often do the majority of people need or use mathematical concepts once they have left school?”

Two months later, the proposal was debated in parliament, on the initiative of Tony McWalter, a Labour MP who very much objected.

Tony McWalter (Labour MP): . . . A quadratic equation is not like a bleak room, devoid of furniture, in which one is asked to squat. It is a door to a room full of the unparalleled riches of human intellectual achievement. If you do not go through that door, much that passes for human wisdom will be forever denied you . . . .

. . .

Eleanor Laing (Conservative MP): Hear, hear.

Tony McWalter: Oh dear. I would like to have support from elsewhere as well.

. . .

Alan Johnson (Cabinet minister for education)

In preparing for this debate, the DFES conducted a straw poll involving a 16-year-old who had just sat maths GCSE, a head of maths and an experienced chemical engineer. The 16-year-old thought that quadratic equations were logical and fairly straightforward “because you substitute stuff into a formula”. He did say, however, that his opinion might have been influenced by having a good teacher. The head of maths said that quadratic equations formed an important step in students’ ability to solve equations, taking them from simple—one unknown—and simultaneous—two unknowns—and paving the way for more advanced work in mechanics and complex number theory. The engineer said that he did not use quadratic equations now, but had in the past in detailed design applications. Where he works, the chemists use them to explain multiple reactions.

This is not an April Fools’ joke. You can read the whole debate here.

I side with the bipartisan spirit of Tony McWalter and Eleanor Laing in this debate. The quadratic formula is a thing of beauty, and I invite you to just stare at it for a while.

I have given the form for the equation x² + bx + c = 0, where the coefficient of x² (normally denoted a) is equal to 1. And I have separated out the two roots; normally you have ± sign in there. You might recall from algebra class that one way to solve such equations is to find two numbers that multiply to c and add up to −b.1 I invite you to contemplate the two numbers displayed above and see if you can see how they are exquisitely designed to achieve exactly that goal. I’m going to put the answer in a footnote so as not to spoil the fun.2

For another puzzle, see if you can decipher the notation in the image above. It is a medallion from the 17th century bearing something hauntingly similar to the quadratic formula in the notation of that time. But why are there three formulas? What are those funny characters that look like ornate z’s?3 I think it is wonderful that a 17th-century devotee of mathematics created such a thing. It’s a bit like someone today wearing a t-shirt with E = mc² on it.

Thanks for playing! On Friday I’ll get back to math ed research, this time looking at an article on the other side of the debate from last Friday.

In my previous post I promised to engage with the sceptics of productive struggle, those in favor of direct or explicit instruction. A widely cited article for this position is Kirschner et al, and I’ll start by naming a couple of its stronger points.

The good

The authors talk about working memory, defined in this article as follows.

Working memory is the cognitive structure in which conscious processing occurs. We are only conscious of the information currently being processed in working memory and are more or less oblivious to the far larger amount of information stored in long-term memory.

A body of research shows that working memory is quite limited in capacity and duration, and that fact has important implications for instruction. When working memory gets overloaded, struggle becomes unproductive.

The article also cites a number of research studies that demonstrate the usefulness of worked examples in certain contexts, particularly with novice students and with routine tasks. I think this is a valuable finding, and I will return to it, and to more recent research on the topic, at the end of this article. That said, the discussion of the worked-example effect contains a significant concession: “the worked-example effect first disappears and then reverses as the learners’ expertise increases.” And even in the novice domain, the original 1985 study by Sweller et al found that the worked-example effect did not extend to even modestly varied problems. A 1987 follow-up (paywalled) did achieve some transfer, but only with extended practice periods and on narrowly similar tasks—and transfer depended on rule automation, which develops slowly. Recent research on worked examples goes in a significantly different direction from what Kirschner et al advocate, as I will discuss later.

The bad

The article targets “minimally guided instruction,” a term which expands and contracts to suit the argument. The title names the expanded version: “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching.” Spoiler alert: the article does not live up to the implicit claim here. When it comes to citing research, it retreats to a much narrower definition:

Studies conducted from 1950 to the late 1980s [compare] pure discovery learning, defined as unguided, problem-based instruction, with guided forms of instruction.

So now we are talking about “pure discovery learning” and “unguided” instruction. The word “minimal,” initially wide enough to include a variety of approaches with very different levels of guidance, has now contracted to zero. I think most people would agree that zero guidance is not appropriate. However, let’s look at the evidence cited. It’s narrower than you would expect from the sweeping claims of failure. On the areas most relevant to mathematics education, it is mostly about novices doing procedural tasks in algebraic manipulation, database software, statistics problems, and LISP programming.

The ugly

Perhaps aware of the narrowness of the research base, the authors bring in the big guns:

A series of reviews by the U.S. National Academy of Sciences has recently described the results of experiments that provide evidence for the negative consequences of unguided science instruction at all age levels and across a variety of science and math content. McCray, DeHaan, and Schuck (2003) reviewed studies and practical experience in the education of college undergraduates in engineering, technology, science, and mathematics. Gollub, Berthanthal, Labov, and Curtis (2003) reviewed studies and experience teaching science and mathematics in high school. Kilpatrick, Swafford, and Findell (2001) reported studies and made suggestions for elementary and middle school teaching of mathematics. Each of these and other publications by the U.S. National Academy of Sciences amply document the lack of evidence for unguided approaches and the benefits of more strongly guided instruction. Most provide a set of instructional principles for educators that are based on solid research.

I was surprised to see the Kilpatrick et al study in there, Adding it Up, because I remembered them saying this in the chapter on Teaching for Mathematical Proficiency:

Much debate centers on forms and approaches to teaching: “direct instruction” versus “inquiry,” “teacher centered” versus “student centered,” “traditional” versus “reform.” These labels make rhetorical distinctions that often miss the point regarding the quality of instruction. Our review of the research makes plain that the effectiveness of mathematics teaching and learning does not rest in simple labels. Rather, the quality of instruction is a function of teachers’ knowledge and use of mathematical content, teachers’ attention to and handling of students, and students’ engagement in and use of mathematical tasks. Moreover, effective teaching—teaching that fosters the development of mathematical proficiency over time—can take a variety of forms.

This passage is followed by four classroom vignettes, with discussion pointing out flaws in both directions and positive aspects of practices that fall between direct instruction and pure discovery.

I wasn’t familiar with the other two reports, but I took a quick look and was astonished to find that they directly contradict the claim of “negative consequences of unguided science instruction at all age levels and across a variety of science and math content.” I’ve placed the quotes in a footnote to spare you.1

So, to sum up, in support of their thesis, the authors cite three NAS reports that directly contradict their thesis. I tried to find a polite way of describing this citation strategy. The best I could come up with was that maybe, buried somewhere deep in these reports, is evidence that completely unguided instruction doesn’t work. Yup, I believe you, zero guidance is bad. But that is not the headline claim in this article.

The article concludes with a dazzling display of terminological pyrotechnics.

After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to intermediate learners. Even for students with considerable prior knowledge, strong guidance while learning is most often found to be equally effective as unguided approaches. Not only is unguided instruction normally less effective; there is also evidence that it may have negative results when students acquire misconceptions or incomplete or disorganized knowledge.

Count the labels: “minimal guidance,” “constructivist-based minimal guidance,” “unguided approaches,” and “unguided instruction.” Four different formulations in one concluding paragraph.

And notice how the qualifications accumulate even as the conclusion tries to sound definitive. The opening claims “no body of research supporting the technique”—a sweeping dismissal. But by the third sentence they’ve already walked it back to “novice to intermediate learners.” Then the next sentence concedes that for students with considerable prior knowledge, strong guidance is “most often found to be equally effective”—which is an admission that their preferred approach doesn’t actually win for knowledgeable learners, it merely ties. That’s a long way from the opening salvo. And for all the labels deployed, the article never describes what “direct, strong instructional guidance” actually looks like in the classroom.

The implications of limited working memory

The authors suggest that when working memory is overwhelmed by the search for solution methods, actual learning does not occur. I thought this was the single strongest point in the article, and it deserves to be reiterated. Advocates of the reform approach to teaching sometimes make the same mistake I am attributing to the authors of this article: starting with an approach that works some of the time and raising it to the level of a universal mantra—never tell, always ask—that discourages teachers from providing important information as students solve problems.

Their conclusion from the limitations of working memory is that “when dealing with novel information, learners should be explicitly shown what to do and how to do it.” For genuine novices encountering truly novel procedures, that makes sense. You can’t teach a child to count without modeling the counting procedure. Completing the square is a very clever but subtle trick; it makes sense to me just to show algebra students that trick. But the authors’ description of the interaction between working memory and long term memory gives me other ideas. Think of an explorer making their way through unfamiliar territory. Long term memory is full of knowledge about trees and rocks and rivers and bogs that the explorer can draw on. The explorer does not need to be “explicitly shown what to do.” They just need to be in a place that is interesting and exciting enough to make them want to go on, but not so alien as to frighten them.

What are the implications of the interaction between working memory and long term memory for curriculum design? First, a student solving a problem needs to have firmly held prior knowledge to draw on as they explore a solution space. This suggests that problems should be designed at the edge of that knowledge and lessons should be designed to activate it.2 Second, because different learners have different reserves of firmly held knowledge in long term memory, they need different strategies. Problems should be designed with multiple entry points so that students with different reserves can all engage. And there is an important role for the teacher in monitoring student work and providing appropriate guidance. Rather than “never tell, always ask” the teacher makes judgements about when to tell and when to ask. And the teacher can help students with different reserves learn from each other, like a band of explorers, sharing knowledge.3

Worked examples

Now let me return to the second strong point of the Kirschner et al article, the worked-example effect. As I mentioned, the evidence for this is drawn from a narrow set of procedural tasks with limited transfer.4 But the finding that novices can learn procedures more efficiently by studying worked examples than by flailing through problems on their own strikes me as sound. The question is what to do with it.

Kirschner et al see worked examples as a way to reduce cognitive load—show students the solution so they don’t waste working memory searching for one. But more recent research by Rittle-Johnson, Star, and Durkin takes worked examples in a very different direction. Instead of showing a single solution to reduce the burden on the student, they present two different solutions to the same problem side by side and ask students to compare them. Which method is more efficient? Why does each one work? When would you choose one over the other?

This is not cognitive load reduction. This is using worked examples as objects for analysis and discussion. And the evidence is strong: students who learned through contrasting worked examples made greater gains in procedural flexibility—the ability to choose adaptively among strategies—and sometimes in conceptual understanding as well. That’s precisely the kind of transfer that the original Sweller et al studies failed to produce.

I was involved in the development of the IES Practice Guide on algebra, which rated the recommendation to “use solved problems to engage students in analyzing algebraic reasoning and strategies” as having strong evidence. Notice the language: not “use solved problems so students don’t have to think,” but “use solved problems to engage students in analyzing.” The worked examples are not replacing student thinking; they are giving students something rich to think about.

Here is the irony. The strongest evidence-based use of worked examples—carefully designed, presented in contrasting pairs, with structured opportunities for analysis and discussion—looks a lot like the kind of instruction that Kirschner et al would dismiss as constructivist-based minimal guidance. It manages cognitive load, yes, but through thoughtful task design, not by eliminating the need for student reasoning. It is, in fact, a form of productive struggle.

I had a moment of clarity in my mathematics research the other day. I was facing and dreading a horrible calculation. Then I saw a clear path through the thicket. Such moments are rare and beautiful in the working life of a mathematician. Or, indeed, in the life of a student of mathematics—I believe that Max experienced such a moment.

This set me to thinking about productive struggle. Everybody loves productive struggle—well, actually, not true. Half the people love it and half the people hate it. I’ll have more to say about that in my Friday post. But for now let’s think about the “productive” part, which sometimes gets lost. People think that it will come for free if you just push hard enough on the struggle. And when it doesn’t, they start validating whatever comes out of the struggle, correct or not, confused or clear. This is what leads to the fetishization of mistakes. Student mistakes are valuable evidence of what the student is thinking, and can provide opportunities for learning—but they are not moments of clarity. They are not the end goal.

Productive struggle needs careful nurturing, and we need to be honest with ourselves about whether the “productive” part has been achieved. It takes careful attention to the role of the teacher in guiding students through the struggle as they work on a problem, and it takes carefully designed instructional routines to bring about a synthesis of student learning at the end.1

A lot of the debates in mathematics education boil down to whether you love the journey—the “struggle” part of productive struggle—or love the destination—the result of the “productive” part. That result should be a moment of clarity, not an error or a moment of confusion. Those who favor reform methods of teaching tend to fall into the first camp; those who favor direct instruction tend to fall in the latter.

We should all try to live in the intersection of those two distributions. But how? How does a teacher nurture productive struggle without tipping into aimless struggle? How do they land a moment of clarity without simply handing students the answer? What does it look like in a lesson when the journey and the destination come together? I’d love to hear from teachers who have felt that click in their classrooms, or from students who remember a moment when the fog lifted.

On Friday I’ll take up the other half of that throwaway line—the half who hate productive struggle. They have some good points, and I think engaging with those points honestly is more useful than dismissing them.

When she was thirteen, my daughter came into the kitchen with a question.

Daughter: Dad, fourteen sevenths is two, right?

Me: Yes, that’s right.

Daughter: Good, I just wanted to make sure it wasn’t division.

Me (puzzled): How did you see it if not by division?

Daughter: Oh, I just wrote 14 = 2 × 7 and cancelled the 7’s.

Me: But if 14 = 2 × 7 , doesn’t that mean that 14 divided by 7 is 2?

(Long pause)

Daughter: Oh, yeah. (Flees to her room.)

I never got a chance to probe any further what she was thinking, because she didn’t want to talk about it any more. But over the years I’ve speculated about what might have been going on in her head.

My guess is that her view of the problem was exactly what is indicated by the phrase “fourteen sevenths.” That is, she was imagining some sevenths of something, say a cake, and visualizing how much you would have if you had 14 of them. Her reasoning could be represented symbolically like this:

But then she remembered that fourteen sevenths could mean 14 divided by 7. The question that perplexed me at the time was: How could she think of division as something different from what she had done? What picture of division did she have in her mind?

There are two ways you can think of 14 divided by 7. One is: how many times does 7 go into 14? The other is: how do you share 14 things among 7 people?

Now, the first way of thinking of the division problem closely parallels the way my daughter was thinking. She was thinking of grouping the 14 sevenths into two groups of 7 sevenths, each of which made a whole, for a total of 2. This is pretty close to grouping 14 into two groups of 7. Still, they are not the same, so I can see how she would think that what she was doing “wasn’t division.” Reconciling the two ways of thinking amounts to understanding that multiplying by 1/7 is the same as dividing by 7, which is not completely obvious.1

Another possibility is that she was thinking of the second model of division, and that would be even more confusing. At first sight, grouping 14 things into two groups of 7 seems very different from sharing 14 things among 7 people. If I close my mind’s eye for a moment, I can get myself quite confused on this point.

How do you reconcile the sharing model with her way of solving the problem? One way would be to compare two different methods for sharing 14 cakes among 7 people. On the one hand, if you remember that 7 × 2 = 14, you can simply dole out 2 cakes to each person. From this point of view, 14 divided by 7 is 2.

On the other hand, you can take a more cautious approach and start by dividing each cake into sevenths, so each person gets a seventh from each cake. At the end of this process each person has received 14 sevenths. So, from this point of view, 14 divided by 7 is 14 × 1/7. Case closed. My daughter’s way of thinking is division!

Why was I so confused by her confusion? I think it’s because I have a habit of mind that collapses all these different ways of thinking into one. When I see 14 ÷ 7, I think: that’s the number that gives 14 when you multiply it by 7. That’s it. That’s all it is to me. And from that single vantage point, everything my daughter did looks the same. She found that 2 × 7 = 14, so 2 is the number. If you take 14 × 1/7 and multiply it by 7, you again get 2 according to my daughter’s original reasoning, so 14 × 1/7 is also that number. If you think of 14 ÷ 7 as sharing 14 things among 7 people, then by definition 7 shares make 14. So 14 ÷ 7 is also that number.

So 2, 14 × 1/7, and 14 ÷ 7 are all the same because they all answer the same question: what do you multiply by 7 to get 14?

This is why I couldn’t see the problem. From where I was standing, “fourteen sevenths” and “fourteen divided by seven” weren’t two things that happened to be equal. They were two descriptions of the same thing. But my daughter hadn’t yet climbed to that vantage point. She had two different mental pictures — one of collecting pieces of cake, one of dealing out whole objects — and no reason to think they should give the same answer. The fact that they did was, for her, a coincidence that needed checking rather than an inevitability.

Phew. Perhaps you can see why my daughter hurried back into her room before I explained the problem to her. But I have found over the years that this sort of mathematical detective work is necessary if I want to help my students. Like real detective work, it involves trying to interpret their often incomplete and inaccurate eye-witness accounts of their encounters with mathematics in terms of the reality of the mathematics itself. It is difficult but rewarding work.

If you have stories of similar detective work, please put them in the comments.

I googled “how to add fractions” and got this answer:

To add fractions, ensure they have the same denominator (bottom number). If denominators differ, find a least common denominator (LCD) to convert them into equivalent fractions. Once denominators match, add the numerators (top numbers) and keep the denominator the same, simplifying the final answer if possible.

I was horrified. I thought we killed the least common denominator with the Common Core. But it’s baaack! And still as stupid as ever. OK, I know that a lot of thoughtful teachers still teach the LCD and believe in it. But I’m about to pick a fight. With the zombie, that is, not with the teachers.

Fractions are all about the measurement interpretation of number—as a length on the number line, or a length on a ruler if you like. Adding fractions with like denominators is like adding inches or meters; you are just measuring things in units of 1/3, say, or 1/4, or whatever units you like. Adding fractions with unlike denominators depends on a beautiful but difficult idea: if you want add 2/3 and 5/4 you subdivide the units in each fraction according the denominator of the other—thirds into 4 parts and the fourths into 3 parts, each time getting twelfths, because 3 × 4 = 4 × 3. You get 4 twelfths for each third because you divided it into 4 parts and you get 3 twelfths for each 1/4 because you divided it into 3 parts. So

It works every time!

Zombie: Excuse me sir, but there’s a more efficient way to do that. You didn’t need to use a denominator of 60. You see, you could just have divided the tenths into 3 parts to get thirtieths, and the sixths into 5 parts to get thirtieths, and added the thirtieths. You see, there’s this thing called the least common multiple, which in this case we call the least common denominator, because . . .

Me: More efficient? In the time it took you to explain this to me I’ve done five more problems.

Zombie: Yes, but your answer isn’t simplified!

Me: Well neither is your answer of 62/30! And who cares?

Including the least common denominator in the procedure for adding fractions is a waste of precious time in the elementary curriculum. It obscures the beautiful idea, which is hard enough for kids as it is. Worse than that, it is positively damaging if you want students to see the connection between arithmetic and algebra. In the formula

there is no “find the LCD” step. The formula will be familiar to students who haven’t been pestered by the zombie. Students who have been trained on LCD could be genuinely confused when they see this formula in algebra class because it doesn’t match the way they were taught. This just reinforces the perception that I talked about in my last post that algebra is meaningless.1

I’d love to hear other people’s examples of zombies in math education: topics that are useless, or even worse than useless, but refuse to die. Please nominate your candidates for those blank tombstones and I will respond on Friday. (Please also feel free to disagree with me about the LCD!)