The other day Kristen Smith wrote a great post on her Substack about the limits of the instructional hierarchy (which I define below). Her point was that the instructional hierarchy is about knowing how to perform a procedure, but that she also wants her students to know why. Her example was quadratic functions:

By the end of the lesson students should be able to recognize that a pattern is quadratic because it includes a growing square and has a second difference that is consistent. This isn’t a procedure, and I’m not sure I would call it a skill either. It’s an understanding that is key because knowing why quadratic equations include a squared term leads to understanding how quadratic equations are solved.

I recommend the whole piece.

It got me thinking about another component of mathematical knowledge which I think of as knowing what: knowing what addition is, knowing what a ratio is, knowing what a limit is. Knowing what addition is is different from knowing how to do addition, and it is also different from knowing why the algorithm for addition works. Knowing what addition is comes before those things. In the Common Core that knowledge is encoded in the kindergarten cluster

K.OA.A Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

This is essentially a mathematical definition of addition and subtraction, at the kindergarten level. In today’s post I want to talk about mathematical definitions, and look at what the literature says about them. And in the process we will get a little tour of 20th century schools of psychology.

The instructional hierarchy

Let’s start where Kristen Smith started. The foundational document for the instructional hierarchy is the chapter “Systematic Instructional Procedures: An Instructional Hierarchy” by Haring and Eaton, in The Fourth R: Research in the Classroom (Merrill, 1978). From the introduction:

Learning may be conceived as the ability to perform new skills in progressively more complex situations. When a new behavior is added to a person’s repertoire, he/she must first acquire the behavior. Then he/she must learn to perform the required behavior fluently and to maintain the ability to respond when instruction is no longer provided. In addition, a person must be able to generalize and adapt the skill to fit new situations before it is fully integrated into his/her repertoire. These four levels of performance may be thought of as a learning hierarchy.

Phew. Very much in the Skinnerian behaviorist school, which refused to describe things in terms of internal mental representations. The whole chapter is full of words like behavior, response, reinforcement, and stimulus; there is no reference to ideas or concepts. Much of the research cited comes from Haring’s Experimental Education Unit at the University of Washington, which did applied behavior research on children with disabilities. There is only one citation of a mathematics study in the chapter, so I looked for a more recent study that did focus on mathematics and found Codding, VanDerHeyden, and Chehayeb (2024), which is explicitly about the instructional hierarchy in mathematics. It followed four students, successively applying indicated and contraindicated treatments for the acquisition and fluency phases of the hierarchy, using a multiple baseline design.1 The experiment supported earlier research about which treatments were effective for each phase.

All the students were struggling and in need of intervention. It should also be noted that SpringMath, an instrument developed by one of the authors, was used for screening, diagnostic assessment, and progress monitoring, as disclosed by the authors.2

The experiment did include some training on concepts for all students, including “rewriting number sentences into equations, finding missing numbers.” If this included equations with missing addends and sums on both sides then I would place them under the “know what” category, since such questions test knowing what the equals sign means. But we don’t know, and at any rate the conceptual treatment was not varied, so we don’t get any knowledge of its effect.

Neither the founding chapter in the instructional hierarchy tradition nor this most recent mathematical experiment has any acknowledgement of, method for, or language around knowing what a mathematical object is. Let’s move on.

Concepts as schemas

Another theory from the cognitive science literature that gets a lot of air time these days is cognitive load theory, which I talked about here. One thing I like about this theory is that, in contrast with the behaviorist tradition, it does admit the existence of mental constructs, in fact it embraces them in the idea of a schema, defined in one of the founding papers, Sweller and Cooper (1985), as

mental constructs that allow patterns or configurations to be recognized as belonging to a previously learned category and which specify what moves are appropriate for that category.

This sounds like what I am looking for! You could have a schema for addition as putting together, for ratio as a pair of quantities you compare multiplicatively, or for limit as the end of an infinite process. Unfortunately, when you look into the literature you get things like this, from Sweller (1994), explaining the concept of element interactivity:

Learning a simple mathematical procedure such as how to multiply out a denominator involves a large number of interacting elements. Assume a student is learning to multiply out the b in the equation, a/b = c. In order to learn this process, the student must simultaneously learn that the numerator on the left side and the denominator which is not shown on the right side, remain unchanged. The denominator on the left side is eliminated and appears on the right side as cb. Furthermore, if the student is to have any understanding of the logic of the manipulation, the full intermediate steps, ab/b = cb followed by cancellation of the b’s need to be understood and learned. All of these elements must be processed in an essentially simultaneous rather than serial fashion.

Something inside me died when I read this. By definition a divided by b is the number you multiply b by to get a. If you tell me that number is c, then you have told me that cb = a. One step if you know what division is. Yes, I get that Sweller is replicating Textbook School Mathematics here, a subject that Hung-Hsi Wu has eloquently described:

there has been a de facto national mathematics curriculum for decades: the curriculum defined by the school mathematics textbooks. There are several widely used textbooks, but mathematically they are very much alike. Let’s call this de facto mathematics curriculum Textbook School Mathematics (TSM). In TSM, precise definitions usually are not given and logical reasoning is hardly ever provided. [“Phoenix Rising: Bringing the Common Core State Mathematics Standards to Life,” American Educator, Fall 2011]

Lest you think Wu and I are advocating some horrible New Math formalism, here is the relevant standard from the Common Core:

3.OA.B.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

The high element interactivity that Sweller describes is manufactured by bad curriculum.3 One of these days I’d like to look more deeply into how cognitive load theory applies to knowing what, but for now it’s time to move on again.

Defining concepts in school mathematics

So, after moving on twice, maybe the correct conclusion is that I am asking the impossible. I should just resign myself to the fact that school mathematics is what it is, and researchers must study it as it is, not as I wish it to be. But it turns out that there is a line of research that recognizes the unique role of definitions in mathematics and the challenges in teaching them. In their 1981 paper Concept Image and Concept Definition in Mathematics with particular reference to Limits and Continuity, Tall and Vinner note that

Compared with other fields of human endeavour, mathematics is usually regarded as a subject of great precision in which concepts can be defined accurately to provide a firm foundation for the mathematical theory. The psychological realities are somewhat different. Many concepts we meet in mathematics have been encountered in some form or other before they are formally defined and a complex cognitive structure exists in the mind of every individual, yielding a variety of personal mental images when a concept is evoked.

Here are authors who both recognize mathematics for what it is and understand the complexities around getting it into students’ heads. They go on to say

The human brain is not a purely logical entity. The complex manner in which it functions is often at variance with the logic of mathematics. It is not always pure logic which gives us insight, nor is it chance that causes us to make mistakes. To understand how these processes occur, both successfully and erroneously, we must formulate a distinction between the mathematical concepts as formally defined and the cognitive processes by which they are conceived.

They go on to define the notion of the image of a concept as distinct from its definition:

The concept image consists of all the cognitive structure in the individual’s mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition.

The concept definition need not be the formal definition which is “accepted by the mathematical community at large”; it can be “the form of words that the student uses for his own explanation of his (evoked) concept image.”

The authors study situations in which the concept image and the concept definition come into conflict.

For instance, the verbal definition of a limit “sₙ→s” which says “we can make sₙ as close to s as we please, provided that we take n sufficiently large” induces in many individuals the notion that sₙ cannot be equal to s (see Schwarzenberger and Tall, 1978). In such an individual this notion is part of his concept . . . image, but not acknowledged by mathematicians as part of the formal theory.

There are no remedies offered for the misconceptions and difficulties described; rather the paper lays out what can happen when you don’t pay attention, in curriculum or instruction, to helping students know what something is.

Mathematics is built on definitions, and has survived without crumbling on that foundation for thousands of years. Maybe if we paid more attention to that foundation in school mathematics, we could help students build an understanding of mathematics that would survive for more than 12 years.

Last Wednesday’s post brought 87 new subscribers, the most I’ve ever had from one post. Welcome! I thought this would be a good time to give another retrospective of this Substack. First, although you came to this through a post on math ed research, you might be interested in the ones that are just about math.

Where’s the Math?

I love moments when a piece of deeper mathematics vividly appears in a seemingly routine classroom moment or K–16 course or curriculum.

• Max discovers a theorem. A story about a fourth grader who has an aha moment about subtraction.

• How do you know that 8 + 5 = 13?. What’s hidden inside a simple math fact.

• My abacus. The base ten system made manifest.

• Two stories about a foreign language and Parlez-vous Algebra?. Algebra as language.

• Math and zombies. A diatribe against lowest common denominators, with disagreements in the comment thread.

• More than you wanted to know about fourteen sevenths. Fractions as division.

• Moments of Clarity. Musings on productive struggle, plus the answer to the abacus question.

• Don’t just do something, stand there. A parliamentary debate about quadratic equations and an appreciation of the quadratic formula.

• Beautiful expressions. A celebration of algebra, brought to you by the letter x.

• Seeing with new eyes. Thoughts about technology, brought to you by the letter e.

• Think of a number. What is a variable?

• Think of an angle. The power of mathematical notation.

• Birth of a comma and What makes a line straight?. A remembrance of Dick Askey, who insisted on getting the math right about linear equations and straight lines.

• What is a ratio? and Where do ratios lead?. Setting the record straight about “solving proportions,” plus the 1500-year-old Rule of Three.

• Constants and variables. It’s not the letter, it’s the way it is used.

Reading the Research

I’ve been practicing what I call The Science of Reading the References. I’ve discovered some great papers and learned a lot, but I’ve also found that the citations don’t always support the claims.

Auditing the explicit-instruction literature

• What if the struggle isn’t productive?. A review of the paper everybody cites in support of explicit instruction.

• Yes, actually. A review of a paper that supports guided discovery.

• The right tool for the job. An attempt to bring the two sides together.

• I don’t think it means what you think it means. The one that brought 87 new subscribers!

Cognitive science and the mathematics classroom

• What are the cognitive constraints on problem solving?. Viewing the cognitive science literature as providing constraints on building rather than telling you what to build.

• Where was I?. In which I go down a citation rabbit hole about the worked example effect.

• Overgeneralization is the cause of all disputes in mathematics education. Looking at two experiments about explaining first versus problem-solving first that have opposite conclusions.

Traditions of teaching

• Making Connections. The TIMSS 1999 video study, with a focus on making connections problems.

• Connections and coherence. Looking at a proposed framework for making connections.

• Something fun for a change. A review of a chapter that brings together two traditions of teaching under a common set of principles.

• Making meaning. Continuing the exploration of the two traditions.

• Thin contexts. The mathematical purpose of contexts in word problems.

• What is problem solving?. A review of a seminal paper on problem-solving as a basis for instruction in mathematics.

• Meaning and meaninglessness. Musings on the dual nature of mathematics and what that means for a problem-solving approach.

• The residue of problem solving. Concluding the review of that seminal paper.

What the words mean

• What is conceptual understanding? Bill’s excellent definition of conceptual understanding.

• What is procedural fluency? Bill’s excellent definition of procedural fluency.

• Think about what you are going to call your recommended teaching strategy if you want teachers to use it. “Productive failure” and productive failure.

Is mathematics different?

• Cakes and Bicycles. Mathematical objects are really unreal.

• I did the research. How can we make them real?

• How different is mathematics?. Abstraction is the content.

• The authority of mathematics. In every classroom there is a secret helper.

• Mathematics as a guide. An experiment showing the secret helper in action.

Where I stand

• Where I come from. In which I name my priors.

• If you want to know what I think, read what I write. What the title says.

Miscellaneous

• Where have I been and where am I going?. An older retrospective.

• Navigating Navigating the Math Wars. A review of a report.

• Death of a salesman. A joke.

• Subscriber poll. I know, still waiting on those posts about systems.

In one of my earliest posts on this Substack I looked at the paper that everybody cites in support of explicit teaching, Kirschner et al 2006, and documented both its strengths and weaknesses. Its weaknesses included citing three National Academies of Science surveys that directly contradicted its assertions. There are many things from the cognitive science literature that I find convincing, as I have said here. The one article of faith I am not convinced by is that you should explain things to students before having them work on problems, rather than the other way around. Today I want to touch on two more lines of research that are often cited for this claim.

The worked example effect

The canonical experiment is Sweller and Cooper 1985, where students were given instruction on solving slightly weird equations like

(to be solved for a) and then were treated to two conditions, either practicing more problems of that form or alternately practicing problems and studying worked examples. The latter group came out ahead, although the advantage was narrow and disappeared with equations that needed the same repertoire of moves but didn’t have the same form.1 It is odd that it gets cited so often in favor of explicit instruction, because both groups were given explicit instruction, and then the experiment compared different ways of consolidating what the students had learned. The experiment doesn’t answer the question of order.

In that earlier post I also talked about some different worked example research by Rittle-Johnson, Star, and Durkin:

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.

This research points away from explaining first; the whole purpose is to get students to look at the worked examples and think about them.

The worked example effect answers a question about practice; citing it for a question about order is a category error.2

Modeling

Another compilation often cited in support of explicit teaching is Rosenshine’s Principles of Instruction, in particular the fourth principle:

Provide models: Providing students with models and worked examples can help them learn to solve problems faster.

He goes on to say:

Students need cognitive support to help them learn to solve problems. The teacher modeling and thinking aloud while demonstrating how to solve a problem are examples of effective cognitive support. Worked examples (such as a math problem for which the teacher not only has provided the solution but has clearly laid out each step)3 are another form of modeling that has been developed by researchers. Worked examples allow students to focus on the specific steps to solve problems and thus reduce the cognitive load on their working memory.

My guess is that people who cite this paper in support of explicit teaching in mathematics are focusing on that sentence about worked examples. Rosenshine goes on to refer to the Sweller research I discussed above and makes, I think, the same category error between initial instruction and practice. But if you look at his three recommended readings under this principle you see that his conception of modeling is much broader than mathematics teachers explaining the procedure for solving a problem. He recommends Sweller on cognitive load theory, himself on student self-explanation, and Schoenfeld on problem-solving. I discussed cognitive load theory a little in that earlier post and gave some principles it implies for a problem-based approach:

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. 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.

As I’ve said before, cognitive load theory constrains curriculum design; it doesn’t do the design work. Explicit teaching is not the only model fitting those constraints. There is a gap in the proof. It reminds me of that famous cartoon which shows two mathematicians at a blackboard, with “then a miracle occurs” between steps one and three, and the colleague saying “I think you should be more explicit here in step two.”4

The other two readings Rosenshine suggests are revealing. The one about teaching students to generate questions is about reading comprehension, not mathematics, but it’s a useful strategy in mathematics as well. Students who are comparing worked examples are generating questions, and you can also give students a problem describing a situation and ask them for mathematical questions about it.5

The most striking reading is Schoenfeld, describing his problem-solving class, which would seem to be the opposite of explicit teaching. However, Schoenfeld is big on helping students develop heuristics for problem solving; he describes an expert working in public and backing out of dead ends. This could be what Rosenshine is thinking of when he talks about the teacher “modeling and thinking aloud while demonstrating how to solve a problem.” But if Schoenfeld’s approach is to be included under the principle of modeling, then that principle is broader than worked examples: a worked example is Schoenfeld’s modeling with the false starts taken out.

Rosenshine’s principles have become a foundational text for the people who support explicit teaching, but the foundation becomes weaker when you read his suggested readings.

The importance of curriculum design

The other day I restacked the following passage from an important post by Carl Hendrick, Retconning the Curriculum: Why The Science of Learning Has a Serious Design Problem:

The deeper and more uncomfortable truth behind all this is that the science of learning (specifically the findings around memory), was developed in conditions that usually look nothing like a classroom, in fact the classroom is in many ways hostile to the science of learning and memory.

As I remarked in the comments, the post

gets at something I’ve been struggling with as I read the research behind the science of learning. It puts constraints on curriculum design, just as physics puts constraints on building design. But it doesn’t tell you what to build.

It is not helpful to tell teachers how a lesson should go without considering how that lesson embeds into a year of lessons, how concepts and procedures build over time, and how the instructional architecture supports that building. And there is more than one way to do that work.

As always, I welcome your thoughts in the comments, particularly the ones telling me I am wrong! And if you have the argument that gets from the constraints to explicit teaching—the miracle in the middle of the cartoon—let me know.

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.

Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.

Rittle-Johnson, B., Star, J. R., & Durkin, K. (2017). The power of comparison in mathematics instruction: Experimental evidence from classrooms. In D. C. Geary, D. B. Berch, & K. M. Koepke (Eds.), Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts (pp. 273–296). Elsevier. link

Rosenshine, B. (2012). Principles of instruction: Research-based strategies that all teachers should know. American Educator, 36(1), 12–19, 39. https://www.aft.org/ae/spring2012/rosenshine

Rosenshine, B., Meister, C., & Chapman, S. (1996). Teaching students to generate questions: A review of the intervention studies. Review of Educational Research, 66(2), 181–221. https://doi.org/10.3102/00346543066002181

Schoenfeld, A. H. (1985). Mathematical Problem Solving. Academic Press. https://www.sciencedirect.com/book/9780126288704/mathematical-problem-solving

Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4(4), 295–312. https://doi.org/10.1016/0959-4752(94)90003-5

Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89. https://doi.org/10.1207/s1532690xci0201_3

Math teachers get a bad rap, but I didn’t realize how bad until I saw the current Broadway production of Arthur Miller’s play Death of a Salesman. The story follows Willy Loman, a salesman past his prime, who perhaps never even had a prime. I don’t think I’m giving away too much to say that he dies in the end. A key factor in his decline is the failure to launch of his son Biff, who was a star football player in high school, but was unable to accept a scholarship to the University of Virginia because he failed to graduate. So instead he got a series of dead end jobs and had lots of fights with his father, deepening the decline that led to his father’s death.

Guess why he failed to graduate.

His math teacher flunked him. He needed a 65 to pass but only got a 61.

There’s an interesting detail where he tells his father how he got caught making fun of the math teacher, Mr. Birnbaum, in front of the whole class. He crossed his eyes and imitated the teacher’s lisp, “The thquare root of thixthy twee is . . .” And in the middle of it, the teacher walked in. I assume here that Miller is inserting his own bad memories of math class. In his day they would have taught the method for estimating the square root of a number that is close to a perfect square:

The approximation in the middle uses the binomial theorem, which says that

when x is small. In this case we have x = −1/64 and n = 1/2. It’s a pretty good approximation: the true value is 7.93725 . . . . In general, if a number is a small distance b from a perfect square a², the same calculation gives

My mathematician friend who saw the play with us remembers being taught this trick in high school1; I didn’t have it in Australia.

So, evil math teacher causes death of a salesman.

I suppose I should take heart from the other Broadway production I saw recently, David Auburn’s Proof, where the brilliant but disturbed daughter of an equally brilliant and equally disturbed mathematician battles to convince a skeptical math graduate student (who also happens to be her lover) that the brilliant proof of some groundbreaking but unspecified theorem is hers. The graduate student finds it hard to believe she could have come up with it; it’s not explicit, but the inference is that part of the problem is that she is a girl. So yeah, math is good and all, but only for crazy people, and the ones who aren’t crazy are sexist.

Oh well. Let’s keep on working to change the world.

I’d love to hear about your favorite or least favorite portrayals of math teachers in fiction. Let me know in the comments.

Two Fridays ago I talked about Hiebert et al. (1996), which puts forward the principle that “curriculum and instruction should begin with problems, dilemmas, and questions for students.” The paper lists four potential outcomes or “residues”1 of problem-based instruction: specific problem-solving procedures, working out new procedures when they are needed, insights into the structure of mathematics, and positive dispositions towards mathematics.

I focused on specific problem-solving procedures, in particular on the finding that problem-based instruction does not shortchange students in that regard, neither does it boost them; I looked at three null findings with different designs supporting that finding. So why bother with a problem-based approach? The payoff is potentially in the other three residues. I studied all the cited papers, and one book. One thing that emerges is that problem solving as conceived of in Hiebert et al is very different from its caricature as “discovery learning.” This is clear from the banner problem of the paper, computing the difference in height between Paulo and Jorge, but also the problems in the cited research.

Meta-strategic problem-solving

In addition to specific problem-solving procedures, the authors posit that a problem-based approach to curriculum and instruction fosters meta-strategic problem-solving by giving students “the conceptual underpinnings and methods for actually working out new procedures when they are needed.” The citations supporting this claim are all about arithmetic, mostly whole number arithmetic, with one paper about decimals. As I read through them I discerned a thread of connections: students give meaning to symbols (numerals, multidigit numbers, decimals) through connecting them to concrete referents (base-ten blocks, counters), problem situations (the progression of word problems in the cognitively guided instruction tradition), or through the child’s own reasoning in games (the radical constructivism of Kamii). However they arrive at that meaning, they bring it to multi-digit addition and subtraction, creating their own base-ten strategies,2 arriving, through invention or instruction, at the standard algorithm or something close to it, and demonstrating the ability to use the schema they have built to solve problems they have not yet been taught to solve. No single paper in the citation list documents every link in this chain; each contributes one or two. The citation that comes closest to tracing the whole chain is Hiebert and Wearne (1996). The study that does trace it end to end, following 82 children for three years as they moved from invented strategies to standard algorithms and beyond, is one the authors could not have cited: Carpenter et al. (1998) had not been published yet.

Carpenter and his colleagues followed the children from first through third grade in three Wisconsin-area schools, in classrooms whose teachers were participating in a cognitively guided instruction development program. There was no prescribed curriculum; instruction varied from class to class, but generally featured word problems and whole-class or small-group discussion of alternative strategies. Each child was interviewed five times over the three years, on base-ten concepts, on addition and subtraction problems, and on extension problems that asked them to go beyond what they had been taught. About 90% of the children used strategies of their own invention at some point. The children who used invented strategies before learning the standard algorithms showed better knowledge of base-ten concepts, more success on the extension problems, and fewer of the buggy algorithms that plague multidigit subtraction. The design establishes an ordering, not a cause: children who invented first did better later, which is consistent with the residue claim but does not show that instruction built around invention produces it. The authors describe their study as providing “an existence proof that children can invent strategies for adding and subtracting.”

And existence proofs is mostly what we are looking at here. There’s an asymmetry between the math ed research and the cognitive science research as I have read them so far.3 The first mostly shows what is possible, the second mostly shows what is not, and as far as I can tell they haven’t met in the middle yet.

Insights into the structure of mathematics

Implicit in the thread I described above is a dawning understanding of base-ten structure. My favorite example of this is Jazmin’s “opening a ten,” which I described in my first post on Hiebert et al 1996. Two studies are cited that explicitly explore this aspect of problem-based learning in arithmetic, Hiebert and Wearne (1993) and Cobb et al. (1991). I discussed the first paper a few months ago, and the second one in my first post. As I mentioned there, its comparative results between problem-based and traditional instruction need to be treated with caution. But setting that concern aside, it provides another existence proof. In the project arithmetic test there was a relational scale explicitly designed to assess “student’s conceptual understanding of place-value numeration and computation in nontextbook formats.”

The Relational scale included a total of 16 items. The computational items on this scale included 2 two-digit addition and 2 two-digit subtraction tasks presented in an everyday language format (21 more than 49 is ___). A two-digit missing-addend task was also posed in this format and 2 two-digit missing-addend tasks were presented as horizontal sentences together with 1 two-digit missing-addend word problem. The numeration items included two typical textbook tasks such as “How many tens in 28?” together with two noncanonical tasks such as “What number do 12 ones and 3 tens make?” (Cobb et al. 1991, p. 15)

There were also three items adapted from the work of Steffe (Steffe, Cobb, and von Glasersfeld 1988) and Kamii (1986). In one item, from Steffe, a number was presented in strips of tens and ones, with a hidden area that students were told contained 15; in another, from Kamii, they circled the dots in a collection of 23 that the 2 in the numeral stands for. To answer questions like these you have to see the base-ten structure behind the notation, not just produce an answer that the notation triggers.

Project students scored 1.00 out of a possible 2 on this scale. On its own that number tells us little; is half the maximum score good for second graders in early May? However, the nonproject group, who matched the project students on the test’s instrumental scale (1.28 versus 1.31 out of 2) and on the standardized computation test, scored 0.62 on the relational scale. Although I am cautious about seeing this as evidence that problem-based instruction is superior,4 I do see it as calibration. The relational items were measuring something beyond computational skill, and the project students had more of it. The parity on computation also constrains the confound that the project group consisted of volunteer teachers: if the project classrooms simply had stronger students or better teachers, we would expect them to be ahead everywhere, but their advantage was specific to the items sensitive to structure.

Another detail is telling. The test booklets were coded for whether students used the standard algorithm. Nonproject students used them on 82% of the items presented in vertical column format, 44% of those presented as horizontal sentences, and 19% of the relational computation items: they used the algorithm when the problem looked like one in the textbook, less so when the format departed from it, and were correspondingly less successful. Project students used it less and relied on invented strategies more, but the interesting point is that they were nearly uniform on the use of the standard algorithm across the three formats (31%, 28%, and 17%). That indifference to format suggests insight into structure to me.

The existence proof here is second graders who could read two-digit addition problems structurally, in formats they had never practiced, at no cost to their standard computation. What is less definitive, with an experimental design using volunteer teachers and no pretest, is that the instruction produced the difference.

Positive dispositions towards mathematics

I’ll be quick with this one since I’ve gone on long enough. Both Carpenter et al. (1989) and Cobb et al. (1991) reported differences in disposition between the treatment and control groups. Carpenter et al is the stronger experimental design but has the more marginal results; small differences in both confidence and reported understanding. Cobb et al, with the weaker experimental design, shows a larger difference in belief about the reasons for success in mathematics, with project students scoring lower on “students will do well in math if they use the same methods as the teacher or other students” and on success being the result of being “lucky, neat, or quiet.”5 I’d make the same point here I made last Wednesday: a problem-based approach allows the teacher to more easily see the student’s disposition, and try to improve it if necessary.

Why is the research so old?

The genre of research I’ve been discussing, small N classroom studies of arithmetic understanding with researcher-built measures, seems to have faded out around 2000. Some researchers moved in to professional development or textbook writing based on their work, others into design-based research. Mainstream math education research took a sociocultural or equity turn. The rise of the federally funded Institute for Educational Sciences put more emphasis on randomized controlled trials, which often addressed different questions, such as intervention for struggling learners. The sort of research I have described here has neither been replicated nor refuted, as far as I know, but I am still exploring.

References

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.-P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531. https://doi.org/10.3102/00028312026004499

Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20. https://pubs.nctm.org/view/journals/jrme/29/1/article-p3.xml

Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., & Perlwitz, M. (1991). Assessment of a problem-centered second-grade mathematics project. Journal for Research in Mathematics Education, 22(1), 3–29. https://pubs.nctm.org/view/journals/jrme/22/1/article-p3.xml

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12–21. https://doi.org/10.3102/0013189X025004012

Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393–425. https://doi.org/10.3102/00028312030002393

Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3), 251–283. https://doi.org/10.1207/s1532690xci1403_1

Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education, 1(2), 75–86.

Steffe, L. P., Cobb, P., & von Glasersfeld, E. (1988). Construction of arithmetical meanings and strategies. New York: Springer-Verlag.

I meant to write more last Friday about Hiebert et al 1996, considering the other “residues of learning” from problem-based instruction. But I decided I need to read all the cited papers first, so I’ve been doing that and will give you my thoughts this coming Friday. Today I want to pick up on a related thread, inspired by this sentence from the abstract of one of the papers, Wearne and Hiebert 1989, “Cognitive Changes During Conceptually Based Instruction on Decimal Fractions”:

The evidence suggests that students with varying levels of achievement can construct meanings for mathematical symbols and can use these meanings to solve both instructed and novel tasks.

In Mathematics as a guide I argued that mathematics is different from, say, science, in that the logic of the subject could carry a student through problem-solving. Science has empirical evidence rather than logic as its decider; scientific progress has been a story of overturning people’s natural assumptions about the way the world works. At one point it seemed only “logical” that the sun goes around the earth. But in mathematics there has been no such overturning; what was true 3,000 years ago remains true today.1 One critique that the explicit teaching camp makes of discovery learning in science is that discovery learning is based on a mistaken belief that students can learn science the same way scientists discover it; that the methods of scientific discovery can also be methods of student learning. That critique has some force for me, but it does not apply to mathematics. The way students solve problems in mathematics is the same as the way mathematicians prove theorems.

Or at least it can be the same. And therein lies the dark side of the point I made in Mathematics as a guide. The structural properties of mathematics that make it possible for students to reason—the logical connections between ideas—also make it possible for students to work through it without any reasoning at all, to follow what Skemp called rules without reasons. The very structure that can be benevolent for the learner can be treacherous for the teacher. A page of correct answers does not necessarily tell you what is going on in the student’s head. As I mentioned in Where I come from, I was first drawn to problem-solving as a method of teaching not because some theorist told me it was a superior method of instruction, but because watching students work on problems helped me see whether they were attaching “meanings for mathematical symbols.” It was teaching calculus as a graduate student to incoming freshmen at Harvard, who had all arrived with good grades in algebra, that revealed to me how far you can get without doing that.

On Friday I will continue looking at what the research says about whether problem-based instruction can help students “use these meanings to solve both instructed and novel tasks,” and what other residues of problem-based instruction there might be.

A while back I wrote about variables. Everybody knows that x and y are variables, also occasionally t. What about constants? Every algebra student knows that in the equations

and

the letters a, b, c, and m are constants. But what if I asked you to solve this equation?

What am I supposed to do with this? My mind goes all jaggy when I try to look at it. If I squint my eyes I can see it as a quadratic equation in a with coefficients x, y, and z. But it’s hard to keep seeing it that way, it’s like one of those pictures that keeps reversing. And it’s completely unfair. We have conventions, dammit! Letters at the beginning of the alphabet stand for constants, and letters at the end stand for variables.1

But what’s the difference between a constant and a variable, mathematically? They are all just letters standing for numbers. The difference is not in the letter itself, but in the way you use it in a question. In the first equation, the implicit question is, “given a, b, and c, what are the possible values of x?” In this question the constants are given and then you have to find the variables. If, on the other hand, I was trying to find the equation for a line that goes through two given points, I might be solving the simultaneous equations

Here the subscripts, by convention, turn the variables into constants, and m and b, normally thought of as the constants, become the variables to solve for. Again, the constants are given and the variables are what you have to find. The same letter can be a constant or a variable depending on the question you’re asking.

You can tie yourself into knots trying to define “variable” and “constant.” People talk about variables as representing numbers that vary; wait, numbers can vary? If you ask Google what are the constants and variables in the equation y = mx + b, it gives the conventional answer. Just for fun, I asked it what are the constants and variables in

In the equation b = xm + y, the constant is m and the variables are b, x, and y. The letter m stays the same as a fixed number, while the other letters can change to different values.

Constant

• m: A fixed multiplier or coefficient that does not change.

Variables

• b: The dependent variable that changes based on the others.

• x and y: Independent values that can vary or be chosen freely.

Kind of fun how it struggled with b and m: I guess the first is a variable because it sits by itself to the left of an equals sign, whereas m is a constant because it is a coefficient of x, even though it’s in the wrong position. Just to mess with it, I asked, “Are you sure m is a constant? It’s a funny position,” and it replied:

You are completely right to question that—I made a mistake in my previous response.

In standard mathematics, m and b are the constants, while x and y are the variables.

The equation you provided is a rearranged version of the classic slope-intercept form of a straight line, usually written as y = mx + b.

No it’s not a rearranged version of the classic slope-intercept form!2 I used to think it was funny that the version of Gemini that answers Google searches was so delightfully stupid. Now I realize it’s the version most students will see. Uh oh.

Claude Fable does much better. I asked it the same two questions, each in a separate fresh conversation, so it got the scrambled equation cold—no memory of the first question, and no prodding from me. Here is its answer about y = mx + b:

In the standard reading of y = mx + b:

• m and b are constants — for any particular line, they’re fixed numbers: m is the slope and b is the y-intercept.

• x and y are variables — they range over all the points (x, y) on the line.

The interesting subtlety is that m and b aren’t constants in the way that 2 or π are — they’re parameters: letters that stand for a fixed-but-unspecified number. Within one equation (one line), m and b hold still while x and y vary. But across the whole family of linear equations, m and b themselves vary — each choice of m and b picks out a different line. So which letters are “constant” depends on what you’re doing: graphing one line (m, b fixed; x, y varying) versus studying the family of all lines (m, b varying too, just on a different level).

And here is its answer about b = xm + y:

This is a trick question — and a good one. The answer is: you can’t tell from the equation alone.

Nothing about a letter makes it inherently a constant or a variable. That status comes from the context — from what we declare the letters to mean — not from the symbols themselves.

The equation is designed to short-circuit a habit. Students trained on y = mx + b learn to treat m and b as “the constants” and x and y as “the variables” as if that were a property of the letters. But b = xm + y has the same form as y = mx + b with the roles swapped: if we read it as (output) = (slope)(input) + (intercept), then x and y are playing the constant roles (slope and intercept), while m and b are the variables. Read with the usual letter conventions instead, m and b are constants and it’s just an odd rearrangement.

Both readings are legitimate until someone specifies the intent. So the pedagogically honest answer:

• By structural analogy to y = mx + b: x and y are the constants, m and b are the variables.

• By convention of what the letters usually name: the reverse.

• In truth: the equation itself doesn’t say — constants vs. variables is a choice the author of the equation makes and must communicate, which is exactly the lesson the question is built to teach.

Note that Claude, despite giving a much more sophisticated answer, made the same mistake as Gemini in thinking the scrambled equation was a rearrangement of the conventional one. Neither humans nor AIs are immune to the pull of notational conventions.

Have you ever noticed the alphabetical convention for constants and variables? Did the scrambled equation trip you up too? For those who teach algebra, how do you talk about constants and variables? Let me know in the comments.

As promised on Wednesday, today I’m going to talk about Ms. Hudson’s second grade class, where students became engrossed in finding the difference in height between Jorge, 62 inches tall, and Paulo, 37 inches tall (Hiebert et al 1996, "Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics"). The students had been working on two-digit subtraction problems using their own methods, but had not been taught the standard algorithm. Four students shared their solutions. Some counted up by tens and ones from 37, others took away tens and ones from 62. Some represented their work with drawings of dots to represent ones and sticks to represent tens, others used numerals. A key moment is when Roberto realized he had to break one of his sticks into 7 and 3:

I shrunk the big guy down by taking away the little guy from him [pointing to his drawing of Paulo and Jorge]. I took 3 10s from the 6 10s and 7 from this 10 [pointing to the 4th stick]. That leaves 3 and these 2 are 5 and 2 10s left is 25.

Maria wrote the subtraction in vertical form and had the same realization:

I subtracted Paulo from Jorge like Roberto did, but I used numbers. I took one of the 10s to get enough to take away the 7, so that was 3 and 2 more was 5 1s, and there were 2 10s left, so 25.

Ms. Hudson asked if anybody could tell the similarities between Roberto’s and Maria’s methods. Jazmin said:

They both had to open a 10 because there weren’t 7 1s to take away. So Roberto took his 7 from that 10 stick. He took 7 and left 3. And Maria took a 10 from the 6 10s and wrote it with the 1s and then took the 7 to leave 3.

What a lovely turn of phrase, “open a 10.” So many adults remember something about “carrying the 1,” without quite remembering why. Jazmin will remember opening a 10.

Making mathematics problematic

The authors propose a single principle for instructional practice in mathematics: making the subject problematic.1

Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students. We do not use “problematic” to mean that students should become frustrated and find the subject overly difficult. Rather, we use “problematic” in the sense that students should be allowed and encouraged to problematize what they study, to define problems that elicit their curiosities and sense-making skills.

There’s an important point here that I think got lost in the enthusiasm of the reform movement: a problem-based approach to instruction is not about having a particular type of problem. The problem doesn’t have to be “rich,” or “engaging,” or “real-world.” The Jorge and Paulo problem is one thin context away from being a naked subtraction problem. A problem-based approach is about what students do with the problem. It is about a classroom culture that elicits discussion, reasoning, and the sharing of ideas. It is about teaching practice that supports the development of that culture.

Importantly, this classroom culture does not entail the teacher stepping back. The teacher has two important roles: “providing information and setting tasks.”

Clearly students can benefit from having access to relevant information; they would make very slow progress if they were asked to rediscover all of the information available to the teacher. On the other hand, too much information imposed with a heavy hand undermines students’ inquiries. Our position is that the teacher is free, and obligated, to share relevant information with students as long as it does not prevent students from problematizing the subject.

The authors quote Dewey’s complaint that some teachers applying his ideas “had the mistaken impression that they were supposed to withhold information and ideas from students and simply let them explore,” and note that “there are some today who advocate such an approach.” With that word “obligated” they explicitly separate their position from unguided discovery.

Indeed, with the goal of having students become problem-solvers, the dichotomy between “telling students and letting them discover” collapses.

Allowing students to treat tasks as genuine problems may involve various configurations of sharing information and discovery. Teachers do not need to do only one or the other.

Another dichotomy that collapses is the distinction between artificial textbook problems and “real-life” problems.

The question of which are better turns out to be irrelevant. The important questions are (1) has the student made the problem his or her own, and (2) what kind of residue is likely to remain.

That second question, what kind of residue remains, is the big one in today’s debates. If we focus too much on having the students make the problem their own—a goal which I take as obviously desirable—do we lose the goal of lasting mathematical knowledge? The paper provides evidence about this and I’ll take that up next.

The residue

I liken the paper’s notion of the residue of instruction to the notion of a schema laid down in long-term memory in cognitive science research. And one type of residue discussed, “particular procedures that can be used for solving particular problems,” is the focus of much of that research. And here we run into a tension between the math ed research cited in this paper and much of the cognitive science research, particularly cognitive load theory, which claims that problem-solving gets in the way of laying down long term schemas, to the extent that it can leave no residue at all. In contrast, the authors cite research that belies that claim.

The strongest single study for this is Carpenter et al 1989, "Using Knowledge of Children's Mathematics Thinking in Classroom Teaching: An Experimental Study," a foundational paper in Cognitively Guided Instruction (CGI). Forty first-grade teachers were randomly assigned by school: twenty spent four weeks of their summer studying the research on how children think about addition and subtraction, while the other twenty got two two-hour workshops. A year later, with prior achievement controlled and whole classes as the unit of analysis, students of the two groups were indistinguishable on the standardized computation test. Where differences appeared, they favored the CGI classes: recall of number facts in interviews, and the harder word problems. Note that the variable manipulated here was what teachers knew, not what students experienced. The claim that students who problematize lose nothing on routine procedures depends on the inference that changing teacher knowledge changes the classroom. The study’s observation data support this, but it is an inference all the same.

The same pattern appears in Hiebert and Wearne’s own classroom studies, which I wrote about in Mathematics as a guide. There my interest was in the way students were guided by the mathematics, but those studies also found that students taught using a problem-based approach equaled the performance of students taught from the textbook on routine computation while working many fewer practice problems (Hiebert and Wearne 1993, "Instructional Tasks, Classroom Discourse, and Students' Learning in Second-Grade Arithmetic"). Classrooms were not randomly assigned, a caveat the authors note. But I see this as a cautionary amendment to strong versions of the claim from cognitive load theory.

Another study, Cobb et al 1991, "Assessment of a Problem-Centered Second-Grade Mathematics Project,” assessed a second-grade program. Ten teachers who had volunteered for the project used the program; eight comparison teachers used the textbook. Again, this experiment did not have random assignment nor any control for prior knowledge. The comparison teachers were the ones who hadn’t volunteered, a confound the authors note. So I’m going to treat the findings on conceptual understanding, which favored the treatment group by a large margin, with caution. But there was also a no-cost result: on the state-mandated standardized computation test the two groups were tied, with grade equivalents of 3.50 and 3.51.2

One could argue that the last two studies had a hidden boost for the treatment group in one way or another, but the null finding requires that the boost be equal to the hidden cost in each case, despite differences in design. Taken together with Carpenter et al we have a convergence of evidence which I find compelling.

In addition to specific procedures for particular problems, the authors also describe a meta-strategic problem solving skill, analogous to what the cognitive science literature calls transfer:

By working through problematic situations, students learn how to construct strategies and how to adjust strategies to solve new kinds of problems. What gets left behind are the conceptual underpinnings and methods for actually working out new procedures when they are needed.

I haven’t read the papers cited in support of this claim, so I won’t discuss this further here. And I’ll also leave the other two residues of learning the authors discuss, insights into structure and dispositions toward mathematics. Important as they are, these deserve their own post. My main focus here has been to consider the most common objection to problem-based instruction, that it shortchanges procedural skill.

What is the goal?

Skemp, in a paper I discussed here, distinguishes two sorts of understanding, relational and instrumental, his terms for what we would call conceptual and procedural today. He entertains, in a devil’s advocate way, the possibility that purely instrumental understanding is a valid goal of mathematics instruction. I doubt there are many people today who would take that position, but I do think the goals are balanced differently in different camps. I share the opening assumption of Hiebert et al:

We work from an assumption that understanding is the goal of mathematics instruction. In fact, we justify the practice of problematizing the subject by claiming that it is this activity that most likely leads to the construction of understanding.

Note they do not apply an adjective to the word “understanding” here. What I like about this paper, and the reason I have a preference for problem-based instruction, is that the principle of problematizing provides a tool for dissolving the boundary between conceptual and procedural understanding. I’ve been writing a lot about how cognitive science research constrains instructional design. These constraints do not by themselves furnish a goal; the principles described in this paper do that, in a way that is consistent with the constraints. Problem solving does not entail a particular type of problem nor does it dictate a particular instructional approach: it is what was happening when Jazmin opened a ten.

I made the following comment the other day in response to Dylan Kane’s scepticism about “contrived, fake-world” word problems.

Some “real-world contexts” are really real-world, but many are not, and that’s not the point of a context. The point is to map the mathematical activity to a meaning. “I bought 7 apples and ate 3, how many do I have left?” is not about apples. I could equally well use some non-real-world context, like pixies. OK, we don’t want to be eating pixies, but you get what I mean. The point of the context is not only to assess the student’s knowledge of a math fact, but to assess their understanding of subtraction as a process of taking away. I call contexts like this “thin contexts.”

Today I want to talk a little more about the role of contexts in mathematics problems. Here’s an example from the old Illustrative Mathematics task bank.1

Fred has some colored kitchen floor tiles and wants to choose a pattern using them to make a border around white tiles. He generates patterns by starting with a row of four white tiles. He surrounds these four tiles with a border of colored tiles (Border 1). The design continues as shown below:

Fred writes the expression 4(b − 1) + 10 for the number of tiles in each border, where b is the border number, b ≥ 1.

Explain why Fred’s expression is correct.

[Task continues with a generalization to starting with n tiles.]

The commentary to this task says

The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.

You could pose this problem as a purely geometric problem without the context, but you’d still have to talk about the squares as pieces getting added each time, and explain that you were focusing on the shaded part of the diagram. The context provides a compact way of avoiding those explanations: kids understand about colored tiles and borders. The 10 and the 4 in Fred’s expression are things you can point to.

Here’s another example from the cognitively guided instruction literature (Carpenter, Fennema, Franke, Levi, and Empson, Children’s Mathematics: Cognitively Guided Instruction, 2014):

Eliz had 8 cookies. She ate 3 of them. How many cookies does Eliz have left?
Eliz has 3 dollars to buy cookies. How many more dollars does she need to earn to have 8 dollars?
Eliz has 3 dollars. Tom has 8 dollars. How many more dollars does Tom have than Eliz?

All of these are about 8 - 3, but to young children “these are three different problems, which they solve using different strategies.” The second problem illustrates the relation between addition and subtraction, that 8 - 3 is the number you add to 3 to get 8. Again, you could just state that abstract fact, but would it stick? The context provides the substrate on which you can build that fact.

Not all contexts are thin. The problems I discussed last Friday are doing more than just providing a substrate. The problem about standard deviation relied on students having a notion of what it means for a player to score consistently in a basketball game. That context is more than just clothing. In the other problem, about efficiency of light bulbs, you couldn’t just swap the light bulbs for pixies. Here the contexts are real real-world contexts.

I think the sort of problem Dylan was complaining about is a problem with a thin context pretending to be real-world. I wrote about such a problem when I was talking about ratios the other day.

Upon arriving at the hotel, the staff gave us a map displaying the places of interest in the city, and told us that 5 centimeters on the map represented 600 meters in reality. Today we want to go to a park that is located 8 centimeters from the hotel on the map. How far from the hotel is the park?

Here the context needs to be thinned out; the decorations need to be removed. The problem still needs a context, but it could be just about a map, without the hotel staff and the places of interest.

On Friday I’m going to talk about one of my favorite math ed research papers, where a 2nd grade class got deeply engaged in discussing how to find the difference in the height of two children, Jorge and Paulo, who were 62 inches tall and 37 inches tall, respectively. Spoiler alert: it wasn’t a passionate interest in children’s heights that got them going.

Today I’m going to talk about two experiments that give different answers to a question that, for some reason, is at the core of the dispute between explicit and problem-based approaches to teaching mathematics: should problem-solving come before or after telling? Each experiment is part of a much larger literature; I’ve picked these two because they are clean and simple.

Productive failure

In Kapur’s 2014 paper, “Productive Failure in Learning Math” (Cognitive Science 38(5)), ninth-graders were given the following problem, before they had been taught anything about standard deviation:

Who is the most consistent Basketball player?

Mike and Dave are the top two players in a Basketball league. The table shows the number of points scored by Mike and Dave over the course of 20 games in the league.

An award has to be given to the more consistent player of the two. The decision has to be made mathematically.

Design as many measures of consistency as you can to determine the more consistent player.

Show all working.

Note that the mean1 for both players is 24, forcing attention to the spread.

One group spent an hour on the problem and was then taught how to calculate a standard deviation. The other was taught the method first and then spent the hour working on the problem. Both groups had the same teacher and materials and the same time on each phase; only the order differed. On a later test the two groups were equally good at calculating a standard deviation, but the group that had worked on the problem first understood the concept better and did better on unfamiliar problems that went beyond the lesson.

Explicit instruction

In Ashman, Kalyuga, and Sweller’s 2020 paper, “Problem-solving or Explicit Instruction: Which Should Go First When Element Interactivity Is High?” (Educational Psychology Review 32), Year 5 students (fifth grade) were given a booklet of problems, with these instructions:

This booklet contains some problems to try to solve. They are set in everyday situations so think how you would solve the problem in real life. You are not expected to solve all of the problems. Just do what you can.

A typical question asked them to determine which light globe was the most efficient (and/or which was the least efficient):

Both groups sat through the same lesson on how to work out efficiency. The only difference was whether they worked the problems before the lesson or after it. This time it was the group that received instruction first that came out ahead. In a first experiment, with 64 Year 5 students, they scored higher on questions like those in the lesson, with no difference on transfer questions (questions either different or more difficult than the originals). A second experiment, with 71 students, used problems with more connected quantities to hold in your head at once, and there the group that received instruction first was ahead on both the similar and the transfer questions.

This study was particularly clean, with an ingenious method for ensuring that both groups received the instruction portion together in one room; there was a third period of reading unrelated science material that occurred before instruction for one group and after for the other.

What would you do?

The thing that struck me about each of the studies was that there was a natural instructional strategy for each of these problems, and it was the one that the study ended up favoring. This is not to suggest there is anything wrong with the studies, nor do I mean to provide an explanation of the mechanism that led to the results; each paper has its own proposed explanation, and I’ll get to that below. But approaching this purely as a teacher, here is what I’d be inclined to do. I’ll be interested if people have different ideas in the comments.

For the standard deviation problem, I’m facing the fact that the formula for standard deviation is a beast. The basic idea is to average deviations from the mean, but for reasons that can’t be explained at this level you square the deviations before summing and then take the square root of the average. I’d want to separate out the basic idea from this twist. And the concept I’m asking students to work with, the notion of what it means for a player to be consistent, is natural; most students will understand that consistency entails getting pretty much the same number of points each time. They will be led to think about deviations, and some of them may come up with the idea of averaging the deviations (and this is indeed what happened with the experiment). Having them work on the problem first cordons off the mysterious part of the formula (square first then square root) from the basic intuition (average the deviations).

The energy efficiency problem is different. There I would worry that the concept I’m asking students to work with will be the issue, not the calculation. The idea of energy efficiency as a rate is, if not a beast, at least a fairly tricky animal. Rates are difficult. Students might think the bulb with the greatest output was the most efficient (this was a failure mode observed in the study); or maybe the one with least output because you aren’t using as much energy. Rather than have the students flounder around, I’d be inclined to at least give them a push in the right direction, if not just tell them outright that efficiency is measured by the ratio of output to input.

Each group of researchers has a hypothesis: in the productive failure research the idea is that the problem solving period primes students to receive the explanation; in the cognitive load literature the idea is that the number of elements students have to hold in their heads, and the interactivity between them, gets in the way of schema formation. These both seem like reasonable explanations to me for the problems studied; neither seems like a reasonable explanation for the problem the other group studied.

I’m not suggesting that therefore the experiments are rigged, or useless. Each provides a valuable existence proof; there are some problems where it is natural to go one way, and some where it is natural to go the other way. An accumulation of many existence proofs with different types of problems can amount to some general design principles for which types of problems are best for which strategies. And indeed, the literature on each side brings such an accumulation.

What I don’t see here is evidence that simply rules out one approach or the other on the narrow question of order. As I explained in this post, I do believe cognitive load theory places constraints on the problem-solving approach, but I don’t think it rules it out. Nor do I think that studies like the one I described here rule out explicit teaching. I have my preferences, for reasons I explained here. So far, from the research I have read, I do not see a reason either to abandon them nor to impose them on someone else.

One thing I’ve learned from writing these articles is that (a) precise definitions matter and (b) nobody pays attention to them anyway.

Take “productive struggle.” The concept was originally introduced by Hiebert and Grouws in their chapter in the 2007 Second Handbook of Research on Mathematics Teaching and Learning. They described two features of teaching that, in their judgment, the research links with student understanding:

• Teachers and Students Attend Explicitly to Concepts

• Students Struggle with Important Mathematics

Nobody is complaining about the first one.1 These days the second one, which acquired the term productive struggle, is under fire, mostly because it has been thoroughly misinterpreted. I’m going to quote their two paragraphs about this feature and I encourage both critics and supporters to read them carefully.

Our interpretation of the literature on teaching for conceptual understanding points to a second feature of teaching that consistently facilitates students’ conceptual understanding: the engagement of students in struggling or wrestling with important mathematical ideas. Unlike the first feature, this second feature might not be obvious to readers, so we first clarify what we mean by struggle, then elaborate the theoretical connection between struggling with and understanding mathematics, and finally review a small sample of empirical studies from this perspective.

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. [Emphasis added]

Another example of a term that, despite the best intentions of its authors, acquired a negative connotation, is direct instruction. Actually, there are two incarnations of this term, the uppercase one, Direct Instruction, which is a specific highly scripted program, and the lower case one, direct instruction, which is a set of practices along the same lines. Either way, it evolved and got rebranded as explicit instruction. To quote from Explicit Instruction: Historical and Contemporary Contexts (Hughes, Morris, Therrien, and Benson, 2017)

It is impossible to identify exactly when or why the term “explicit instruction” became more commonly used than “direct instruction.” . . . As to why, perhaps, as with many other educational “innovations,” there is a tendency to put “old wine in new bottles” to give the impression of being “cutting edge.” Or, possibly it resulted from some educators’ criticism of the term “direct instruction” based on a philosophical perspective about a method of teaching and learning that eschews a “teacher-directed or -centered” approach,2 and thus a newer term such as “explicit” may be more broadly acceptable or less emotionally charged.

“Explicit” was a smart choice—it echoes the first feature identified by Hiebert and Grouws.

As for productive struggle, there are other terms for the same general idea. “Desirable difficulties” is one, albeit at a different moment in the learning sequence.3 It’s another smart choice. Parents would love it: “I know you don’t like eating your vegetables, dear, but it is a desirable difficulty.” And the one that inspired the title of this article is “productive failure.” What were the authors thinking? Imagine standing up in front of a school board meeting with anxious parents in the audience and saying that you believe in productive failure.

Anyway, it turns out the research behind this idea is worth looking at, despite the name. I’ll talk about it on Friday, along with other lines of research around the core question I asked last week: what does the cognitive science research say about whether you should tell kids how to solve a problem before giving them a chance to try it, or go the other way around?

On Wednesday I said I would come back to this quote from Why Minimal Guidance During Instruction Does Not Work (Kirschner, Sweller, and Clark, 2006):

Solving a problem requires problem-solving search and search must occur using our limited working memory. Problem-solving search is an inefficient way of altering long-term memory . . . learners can engage in problem-solving activities for extended periods and learn almost nothing.

My interest in this passage is that it would appear to argue against my preference for giving students a chance to work on a problem before telling them how to solve it. To be clear, I’m specifically interested in the order here; does the telling come before or after the problem-solving?

Although there is no citation attached to this passage, it occurs in a section of the paper about the worked-example effect: the finding that novices acquire procedures more effectively by studying worked-out solutions than by solving the equivalent problems themselves. Cognitive load theory, at least in some incarnations, hypothesizes that this is because problem-solving search consumes limited working memory that would otherwise go to building schemas. In my previous post I discussed a paper that posits a specific type of problem-solving search as the culprit: means-ends analysis on problems with specified goals (find the angle X). Its recommendation is to give open goal problems to discourage that (find all the angles).

I downloaded all the papers cited in this section and read the abstracts, occasionally digging into the paper itself for more detail.1 I realized that most of them are about practice, not about initial instruction: once you have introduced a procedure, how do you practice it. One of the papers, Tarmizi & Sweller 1988, calls this phase the acquisition phase, as opposed to the initial explanatory phase.

For the practice phase, the worked example finding seems solid; studying worked examples is more effective than working through a bunch of problems in the traditional way. Still, these studies do not answer my question; they are about how you solidify schema acquisition after initial instruction. One of them, Carroll 1994, did use worked examples in initial instruction, but did it for both the treatment group and control group. The difference came in what happened after that: both groups worked through a sheet of 24 problems after initial instruction, with the treatment group seeing half of the problems as worked examples.

Another paper, Sweller, Mawer & Howe 1982, gives more evidence for the finding I discussed last week that means-ends analysis is inferior to what they call a history-cued process for solving problems, “in which people use previous moves to generate subsequent moves.” Some of the papers are about how you structure worked examples for the best effect.

Two of the papers are about the expertise reversal effect: although worked examples are more effective for novices, problem-solving is in fact superior for learners who have acquired more expertise. This is an important caveat when reading the explicit-teaching literature; a lot of it is about novice learners.

What do I get from all this? First, worked examples are useful. Second, cognitive load theory is important in instructional design, but it is not a barrier to the problem-based instructional model as I conceive it. (It is a barrier to unguided instructional models, but I agreed with that long ago.)

None of this, though, settles the question I actually started with—whether in initial instruction the problem-solving should come before or after the telling. These studies don’t test that; they test what happens once the telling is done. The order question has a literature of its own, which I will talk about next week.

The papers

Carroll, W. M. (1994). Using worked examples as an instructional support in the algebra classroom. Journal of Educational Psychology, 86(3), 360–367.

Chi, M. T. H., Glaser, R., & Rees, E. (1982). Expertise in problem solving. In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 1, pp. 7–75). Erlbaum.

Cooper, G., & Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79(4), 347–362.

Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38(1), 23–31.

Kalyuga, S., Chandler, P., Tuovinen, J., & Sweller, J. (2001). When problem solving is superior to studying worked examples. Journal of Educational Psychology, 93(3), 579–588.

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75–86.

Miller, C. S., Lehman, J. F., & Koedinger, K. R. (1999). Goals and learning in microworlds. Cognitive Science, 23(3), 305–336.

Paas, F. G. W. C. (1992). Training strategies for attaining transfer of problem-solving skill in statistics: A cognitive-load approach. Journal of Educational Psychology, 84(4), 429–434.

Paas, F. G. W. C., & Van Merriënboer, J. J. G. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: A cognitive-load approach. Journal of Educational Psychology, 86(1), 122–133.

Pillay, H. K. (1994). Cognitive load and mental rotation: Structuring orthographic projection for learning and problem solving. Instructional Science, 22(2), 91–113.

Quilici, J. L., & Mayer, R. E. (1996). Role of examples in how students learn to categorize statistics word problems. Journal of Educational Psychology, 88(1), 144–161.

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285.

Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89.

Sweller, J., Mawer, R. F., & Howe, W. (1982). Consequences of history-cued and means-end strategies in problem solving. American Journal of Psychology, 95(3), 455–483.

Sweller, J., Mawer, R. F., & Ward, M. R. (1983). Development of expertise in mathematical problem solving. Journal of Experimental Psychology: General, 112(4), 639–661.

Tarmizi, R. A., & Sweller, J. (1988). Guidance during mathematical problem solving. Journal of Educational Psychology, 80(4), 424–436.

Trafton, J. G., & Reiser, B. J. (1993). The contributions of studying examples and solving problems to skill acquisition. In Proceedings of the 15th Annual Conference of the Cognitive Science Society (pp. 1017–1022). Erlbaum.

Ward, M., & Sweller, J. (1990). Structuring effective worked examples. Cognition and Instruction, 7(1), 1–39.

Because I am a fan of what I call problem-based instruction, I’ve been looking for arguments against it. First, I should explain what I mean by it, and to do that I’ll quote from an exchange I had with Umes Shrestha on his excellent Substack in the comment section on why education debates turn futile.

It’s probably best to start with what I don’t mean. I don’t mean expecting students to discover mathematics on their own. I don’t mean giving them complex problems they aren’t ready for. I don’t mean withholding necessary information or guidance. I don’t mean never telling them the answer. I do mean giving them a chance to work on a problem before being told how to solve it. The problem should be chosen so that it is within reach and creates the need for the concept or skill I am trying to teach. And the teacher should make sure the punchline has landed, by direct instruction if needed. There’s a lot more to say, but that’s it in a nutshell.

As for how it aligns with how learning happens: my reading so far of the explicit teaching literature says the search for a solution method is just extraneous load that gets in the way of building a schema. I’ve been chewing on whether that’s actually true in mathematics. Maybe I’ll write about that on my Substack this week.

So here I am writing that article. The literature I was referring to is huge; today I’m just going to pick one article, coming out of cognitive load theory. Here is the tl;dr version of that theory as I understand it: our capacity for holding a bunch of connected things in our heads is small; our long-term memory is large; we cope with this by forming schemas, compressed bundles of associations that we are able to pull from memory as one thing rather than all the things in the bundle; learning is a matter of forming schemas and getting them into long term memory; instruction has to pay attention to the cognitive constraints on this process. This all makes sense to me. The idea of things coalescing into higher order things in an ever ascending hierarchy is how I think about mathematics.

Where I get stuck is the idea that therefore the best way to teach is to explain first and give students problems to work on second. This is the conventional wisdom of the explicit teaching camp.1

I first met the idea that searching for problem solutions impedes schema acquisition in the widely cited Why Minimal Guidance During Instruction Does Not Work (Kirschner, Sweller, and Clark, 2006), an article whose good, bad, and ugly points I have described here.

Solving a problem requires problem-solving search and search must occur using our limited working memory. Problem-solving search is an inefficient way of altering long-term memory . . . learners can engage in problem-solving activities for extended periods and learn almost nothing.

I’ll dig more into the citations for this on Friday. Here I want to focus on one article which explicitly links problem-solving search to schema acquisition, Development of Expertise in Mathematical Problem Solving (Sweller, Mawer, and Ward, 1983).2 There is a specific type of problem-solving search that the authors find impedes schema acquisition: what they call means-ends analysis. That is, for example, if you are given a problem in kinematics where you have to find the final velocity V from the given elapsed time t and distance s, under a constant acceleration a, you work backwards through the available equations from V to the given values, then work forwards to find V.3 This is what novices do; experts work forward from the given information because they see the structure of the problem and know where they are going.

Searching backwards induces higher cognitive load because you have to keep the goals (V) and subgoals (v) and, in more complex problems, subgoals of the subgoals, in working memory while you are searching, whereas working forward under the guidance of a schema allows you to forget each step once you have made it. The authors have a measure of schema acquisition which goes by the lovely name of Einstellung.4 They indirectly manipulate the backwards versus forwards analysis by varying the goal specificity. The clearest example of this is in their geometry experiments. Given a diagram, students are either asked to find a specific angle from the given angles, or to find all the angles they can. The latter inhibits means-ends analysis and forces a forward search. So varying goal specificity is their method for switching means-ends search on and off, and with it the load that such search imposes. I won’t go into all the details—there are 7 experiments, 3 on kinematics and 4 on geometry—but the punchline is in the last experiment, where the nonspecific-goal group showed both more forward-searching and more schema acquisition, and the two were correlated.

So this article presents evidence that means-ends analysis is cognitively demanding, and that therefore it would be a good idea to design instructional strategies that lead students to work forwards. The paper’s own example of this is telling students to find every angle they can, not to solve for a particular angle. But that’s exactly the sort of open-ended problem urged by proponents of guided discovery. I can guess at the riposte to this: students in an explicit-teaching classroom come equipped with the available moves—the theorems that derive one angle from another. They are making the forward search with that expert knowledge. They are not floundering. Fair point, but this goes back to the definition of problem-based instruction that I gave above; making sure they are equipped for the problems is one of my conditions. And there’s another point that I made here: in mathematics, the structure of the subject itself provides a scaffolding that guides the available moves independent of the teacher.

What I get out of this is a useful, friendly amendment to what I said at the beginning about the problems you should give students: The problem should be chosen so that it is within reach, encourages forward exploration rather than backwards search, and creates the need for the concept or skill I am trying to teach.

But it doesn’t talk me out of problem-based instruction.

The posts so far fall into three broad categories. Some are about the mathematics itself: what’s really going on with variables, or what makes a line straight, or where ratios lead. Some are about educational psychology and cognitive science: working memory and fact recall in procedural fluency, or the cognitive structure behind conceptual understanding. And some are about mathematics education: how students make meaning and how mathematics guides discovery.

Most of these cross categories; what I like to do most is weave things together. The mathematics, the cognitive science, and the classroom research are usually most interesting when they’re poking at each other.

So what would you like more of? If you have other ideas, please put them in the comments.

I’ll be taking this Friday off for the 4th of July weekend. Look forward to seeing you next week!

On Wednesday I talked about a concept, ratios, and a procedure, The Rule of Three. My impression from the history of The Rule of Three is that it was a pinnacle of learning in its day. In his short autobiography, Abraham Lincoln summed up his entire schooling this way: “I could read, write, and cipher to the Rule of Three; but that was all.”1

But ratios lead somewhere if you design the teaching of them with that in mind. Today I want to walk through one possible teaching progression on ratios. I also want to use this as a study in how to think about curriculum design. I’ve been thinking and writing a lot about the research on mathematics teaching and learning. If you are a teacher or curriculum writer designing a sequence of lessons, what are you to make of that research? What does it forbid, permit or require? Where is it silent?

Ratio tables: concept or procedure?

A ratio table is a conceptually dense object. It embodies many multiplicative and additive relations. If you multiply a row by a scale factor you get another valid row. Same if you add two rows, because of the distributive property. If you multiply a column by the appropriate unit rate, you get the other column; just one number, instead of a different scale factor for each row. This represents a proportional relationship between the two variables indicated by the column headers.

A student might use these relations to solve the following problem.

At a hardware store, they can cut a length of rope off of a big roll, so you can buy any length you like. The cost for 6 feet of rope is $7.50. How much would you pay for 50 feet of rope, at this rate?

On the other hand, the Rule of Three, essentially cross multiplication, cuts through all this complexity and gives a reliable procedure for solving the problem.

Reader Julie Pottinger warned me that people have strong opinions about how to solve such problems. I’ll come back to this at the end.

A possible ratio progression

If you mix 2 parts red paint to 1 part blue, you get magenta. It doesn’t matter how much paint you use, as long as the ratio is 2:1. The visible sameness of the color across different mixtures reveals some invariance between equivalent ratios. The IM grade 6 curriculum starts with this idea, using examples like drink mix, at first represented with discrete diagrams, and then with double number lines.2

The double number line makes the scale factor manifest. Each quantity gets its own line with its own scale, with teaspoons of drink mix counting by fours, cups of water by ones. The two scales are lined up tick for tick, so you can see the quantities scale together. It’s also where the unit rate first shows up: to find how much for one, you go to 1 on one line and read off the matching amount on the other.

Ratio tables come next. A double number line works well until the numbers get large or awkward. If you tried to solve a problem like the one about mixing soy sauce and vinegar illustrated in the table below with a double number line, the points would be hard to read.

By the time students meet tables like this, they are ready to extract an equation. The unit rate—the single number that carries you across each row, from one column to the other—is the constant of proportionality. Call that number k, and name the two quantities in the column headers x and y, and the table collapses to y = kx. Now a student can solve for any unknown without cross multiplication, without a diagram, double number line, or table. Algebra arrives as the last and most compressed representation in the sequence.

What does the research say?

Some of the choices in this design have research behind them. For example, there is research supporting starting with something concrete and fading towards the abstract.3 The idea that you can build conceptual understanding without giving up procedural fluency has support in the Hiebert and Wearne study I wrote about last week.4 The research doesn’t all push the same way, though. There is also good evidence for a quite different instructional design: model a procedure carefully, then give students plenty of practice on it while keeping the success rate high.5 Someone convinced by that evidence might reasonably begin with the ratio table and cross multiplication rather than save them for the end. And some of what is at stake here is not an empirical question at all—it is mathematical taste. The Rule of Three is a trick that buys you nothing later on, once proportional relationships grow into linear functions.

What does the mathematician say?

My main reason for liking this progression does not come from the research. It comes from the mathematics itself. I want to see the mathematics told as a coherent story; I want mathematical representations that have real mathematical meaning, where each is visibly a reworking of the one before.

In the transition from discrete diagrams to double number lines, we drop the individual objects while keeping the correspondence between the two quantities. What was “three of these for every two of those” as countable things becomes two coordinated positions.

In the transition from double number line to table, the lines stand up then disappear. What remains of the marked positions is the paired numbers. You lose the metric spacing but gain generality. You can have 100:150 next to 2:3 without drawing the gap, which is what you need for the next step to equations.

Mustn’t, may, you decide, must

I group the research’s implications into four categories: things you must not do; things you may do; things the research is silent on, where you have to fall back on your own judgment; and things you must do. The one clear example I know of in the first category is unguided discovery learning, which has been pretty thoroughly disposed of.6 The fourth category includes a few things like giving students timely feedback, distributing the practice on a topic across several lessons rather than massing it into one, and keeping an eye on cognitive load as you design a task.7 Neither list fixes the order in which you take up topics. That is a matter of professional judgment, including judgment about the structure of mathematics itself, and it belongs in the third category. The second category, the things the research permits, is by far the largest, and the progression I have walked through lives mostly there. Hiebert and Wearne show that you can pursue conceptual understanding before teaching procedures without losing fluency, but the research lets you teach this way; it does not oblige you to.

What happens in the math wars, I think, is that people make their choices in the second and third categories, the permitted and the matters of judgment, and then quietly relabel them as the first and fourth, the forbidden and the required. A defensible choice becomes the only acceptable one, and the alternative becomes malpractice, or the “pedagogy of poverty.” Meanwhile teachers are stuck in the middle.

Going back to Julie’s warning about the polarizing nature of cross multiplication, my own opinion is that working towards the unit rate as a unifying principle is worth the trouble. It connects the work on solving ratio problems with later work on linear functions and even later work on instantaneous rates of change, and it makes cross multiplication unnecessary. That said, cross multiplication is efficient and gets the job done and, as I showed last Wednesday, has a venerable history.

I’m interested to know how you teach this topic or how you think it should be taught, particularly if you think I’m wrong about cross-multiplication! Let me know in the comments.

I had an amusing exchange with Gemini the other day. I asked it how to solve ratio problems. It gave me four types:

Finding a missing amount — you know one part and want the matching value of the other; set up a proportion. Boys to girls is 3:5; if there are 15 girls, how many boys? (9)
Dividing a total amount — you’re given a total and a ratio, and need the value of one share. Divide $600 in the ratio 2:3. ($240 and $360)
Using part-to-total ratios — the ratio is of parts, but the question asks about the whole. Dogs to cats is 1:3; if there are 24 animals total, how many cats? (18)
Working with differences — you’re told how much more one quantity is than the other, instead of a total. Apples to bananas is 2:3; if there are 4 more bananas than apples, how many apples? (8)

Then I asked it if there was one unifying concept behind all of these, and it responded:

Yes, absolutely. Every ratio problem can be solved using a single unifying concept: the ‘Value of One Share’ (or the Unit Rate).

Then it started going on about boxes, as in boys get 3 boxes, girls get 5 boxes, 15 girls means 3 in each box, 3 boxes of boys means 9 boys. As Dave Barry would say, I am not making this up. I finally figured out that it was talking about scale factors; scaling the ratio 3:5 up by 3 gives you 9:15. But that’s not the same as the unit rate! I pointed this out and it said, in typical LLM fashion,1

You are completely right, and that confusion is entirely my fault. Your definition of a unit rate—like ‘girls per boy’ or ‘miles per hour’—is the correct mathematical definition.

I have a conjecture about the source of the confusion, and it is at least 1500 years old. But first, let’s make sure we understand the difference.

Scale factors and unit rates

I found a hilarious example of a ratio problem on the internet a few years ago:

Upon arriving at the hotel, the staff gave us a map displaying the places of interest in the city, and told us that 5 centimeters on the map represented 600 meters in reality. Today we want to go to a park that is located 8 centimeters from the hotel on the map. How far from the hotel is the park?

Right. And we intend to get to the park by walking in a straight line, through buildings, because we have that superpower.

Here are two real-world problems2 using the same numbers.

1. If you can make 600 meringues with 5 lb of almond flour, how many meringues can you make with 8 lb of almond flour.

2. If 5 bottles of burgundy cost $600, how much do 8 bottles cost?

It is natural to solve the first problem using a scale factor.

You are scaling the recipe up by 8/5, so you multiply the top row by 8/5 to get the bottom row:

For the second problem it is natural to calculate the unit rate (or unit price), $600/5 = $120 per bottle.

We multiply the first column by 600/5 to get the second column:

There are two sorts of multiplicative relationship in a ratio table; scale factors relating the rows, giving equivalent ratios, with a different scale factor for each row; and a single unit rate relating the columns, the same unit rate applied to every entry in the first column giving you the second column. The move from rows to columns, from scale factors to unit rates, is an important development, a precursor to the idea of a proportional relationship between the columns with a constant of proportionality relating them.

But what if you don’t care about this interconnected schema of ideas. What if you just want to be able to do the math?

The Rule of Three

The two problems above had the same numerical answer, and that’s because

Or, put symmetrically,

For any two rows in a ratio table, the two diagonal products are equal. So we can forget about scale factors and unit rates and just cross multiply to find a missing number. This is called The Rule of Three. The website where I found the problem above was using it to illustrate the rule:

The Rule of Three has a venerable history. It appears in an 18th century textbook, Francis Walkingame’s The Tutor’s Assistant (first published 1751):3

The Single Rule of Three Direct teacheth, by three numbers given, to find out a fourth, in such proportion to the third as the second is to the first.

RULE. First state the question; that is, place the numbers in such order, that the first and third be of one kind, and the second the same as the number required; then bring the first and third numbers into one name, and the second into the lowest term mentioned. Multiply the second and third numbers together, and divide product by the first; the quotient will be the answer to the question.

And in a 5th century Hindu manuscript, the Âryabhaṭīya of Âryabhaṭa (499 CE):4

In the Rule of Three, the phala (“fruit”), being multiplied by icchâ (“requisition”) is divided by pramâṇa (“argument”). The quotient is the fruit corresponding to the icchâ.

The Rule of Three was described this way in the 12th century:

As the being, who relieves the minds of his worshippers from suffering, and who is the sole cause of the production of this universe, pervades the whole, and does so with his various manifestations, as worlds, paradises, mountains, rivers, gods, demons, men, trees, and cities; so is all this collection of instructions for computations pervaded by the rule of three terms.5

What a comedown for procedures, from being regarded as divine to being regarded as second-class, as they often are today!

The tension between procedure and concept

I’ve written before about the fusion of procedure and concept. But it doesn’t have to be that way. The Rule of Three is what Skemp described as a rule without reasons. It dissolves the distinction between scale factor and unit rate into a symmetrical cross-multiplication property. Skemp considered the possibility that there was really a second type of mathematics taught in schools, the rules-without-reasons type, and entertained as a devil’s advocate the thought that it was all some students needed. And certainly in the days when computations were done by hand that could make sense. There was a time when executing algorithms fluently and accurately without knowing why they worked could probably get you a well-paying profession. But I don’t think we live in such times any more. Computational ability lives in the air we breathe; the need for understanding unit rates is more urgent than it was 1500 years ago.6

On Friday I’ll consider the implications of this for teaching.

A while back I wrote about zombies in the math curriculum, things that live on for no good reason.1 Today I want to write about a ghost; something that is there but nobody sees.

Take yourself back to your high school math class and imagine your teacher asks you to graph a linear equation, say y = 2x − 3. Depending on how old you are, you would plot a few points—say (0, −3), (1, −1) and (3, 3)—on real paper with a real pencil and draw a line through them with a real ruler,2 or you would enter the equation into some sort of graphing device and have it draw the line on a screen.

Pause and think about the difference between these two methods. In the first case you are drawing a Euclidean straight line. You only need two of the points. I would always choose the two farthest apart and get a little spark of pleasure when the line went through the other point. In the second case, the device is plotting actual solutions to the equation, pixel by pixel. These are completely different things! And one of them, the Euclidean straight line, is the ghost here, in the sense that it was the missing piece Dick Askey was complaining about in my Wednesday post.

Let’s forget about the equation and focus on the Euclidean line, passing through a coordinate plane. There are other ghosts here: the horizontal and vertical lines parallel to the axes. Pick one of each and they make a right triangle with the line we drew. If we make a different choice we will get another triangle, and it turns out to be similar to the first. We can see this using the theorem about transversals of parallel lines. Any two horizontal lines in the plane are parallel.3 So by the corresponding angles theorem for a transversal of parallel lines, the bottom angles of our slope triangles are the same. The same is true for the vertical lines and the top angles. (See the diagram on the left below.) So we can use the theorem that says if the corresponding angles of two triangles are congruent then the triangles are similar.

So what have we proven here?

Theorem 1: The slope triangles on a Euclidean line in the coordinate plane are all similar.4

There’s another lovely proof of this fact using dilations that Hugo Rossi sent me, encoded in the diagram on the right above. I’ll let you figure it out.

It turns out that the converse of Theorem 1 is also true.

Theorem 2: If you have a set of points in the coordinate plane with the property that all the slope triangles you make from them are similar, then it is a Euclidean line.

I won’t go into the details because you might find it fun to think about for yourself. I’ll put a hint in this footnote.5

So let’s go back to the equation now. To show that its graph is a Euclidean line, we need to show that the slope triangles formed by any two of its solutions are similar. Then we can invoke Theorem 2 on the set of solutions. How do we know that the slope triangle between two different solutions, (x₁, y₁) and (x₂, y₂), is similar to the slope triangle between any other two solutions. Well we know that the slope between any pair of solutions is 2.

If (x₃, y₃) and (x₄, y₄) is another pair of solutions, the slope between them is 2 as well, so the two slopes are equal,

and cross-multiplying shows that the corresponding vertical and horizontal sides of the slope triangles are in proportion:

Each slope triangle has a right angle between its horizontal and vertical sides, so by the side-angle-side criterion the two triangles are similar.

So the solutions to a linear equation satisfy the conditions of Theorem 2, and must form a Euclidean line.

I went through all the steps here because the algebra and geometry sing together in a beautiful harmony, and I wanted you to see it. That harmony has been a binding chord in my own research. The spark of pleasure I felt as a high school student when the line went through the third point was the ghost of Theorem 2 at work. Dick Askey spent his career refusing to let such things stay unseen.

There’s a grade 8 standard in the Common Core about linear functions:

8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

This is a story about that comma after the word “function.” Dick Askey was a dogged critic of the NCTM standards and the reform curricula of the 1990s and 2000s. He cared deeply about school mathematics and wanted it done right. When I was serving on the writing group for the mathematics standards, he was on the feedback group, a duty he took seriously. In October 2009, soon after the work had started, he sent me an email:

If there is anything deep about linear functions in school mathematics it is why the graph of a linear function is a straight line. This should eventually come from slope triangles.

By a slope triangle he means the right triangle formed by two points on a straight line, with horizontal and vertical sides and hypotenuse lying along the line. Note the direction of implication here: linear function to straight line graph. There are really three objects in play: linear function, linear equation in two variables, and straight line graph. A draft of the standards circulated to the states in February 2010 connected linear equations and graphs:

The graph of a linear equation in two variables is a line. If the equation is in the form y = mx + b, the graph can be obtained by shifting the graph of y = mx by b units (upwards if b is positive, downwards if b is negative). The slope of the line is m.

The first public draft, in March 2010, added a standard connecting all three:

Understand that a function is linear if it can be expressed in the form y = mx + b or if its graph is a straight line.

This function formulation persisted as late as a May 23 draft, and it didn’t sit well with Dick because it could be read as suggesting that the fact that the graph of a linear function was a straight line was obvious, or part of the definition. Two days later, after Dick called me and reiterated the point he had made in October (forcefully), the standard was changed to the form of 8.F.3 given above. There are two important changes. First, the clause before the comma is saying that there is an act of reasoning in going from the equation to the function. A linear function is defined by its constant rate of change; you have to show that the equation exhibits that property. Second, the all-important comma makes it clear that the graph being a straight line is not part of the definition, but a separate fact. That comma is doing the work Dick asked us to do back in October, without burdening the standards with an extended exegesis.

In the end, after a year of voluminous feedback, Dick was satisfied with the standards. Well, no, he could never be completely satisfied, but he accepted them as moving significantly in the direction he had been pushing all his life. One of my proudest moments was when he came up to me at an NCTM meeting in July 2010, after the standards had come out, and thanked me.

On Friday I’ll dig into the mathematical connections, including the reasoning with similar triangles that Dick was referring to. You might want to try it yourself before then!

One of the common criticisms of guided discovery approaches to teaching in science is that the practice of science and the practice of teaching science are fundamentally different, because of the distinction between novices and experts. Scientists bring to their work a set of connected schemas centuries in the making; the job of science education is to help novices build those schemas, and they won’t do that by replicating scientific discovery. Greg Ashman gave a great example recently; he had asked his students why satellites stay in orbit, and they answered, quite reasonably, rockets. The fact that satellites are in fact continually falling around the earth is not something students can discover; you just have to tell them.1 I’m pretty sure that advocates of discovery learning are not advocating that you don’t, but it’s a fair caveat nonetheless.2

Mathematics is different. For one thing, I think that students are sometimes doing exactly the same thing that mathematicians do. I described one such moment in Max discovers a theorem. For another, as I hinted last Wednesday, the structure of mathematics is itself a guide for students. Problems in mathematics arrive with a hidden scaffolding that reveals itself to students as they climb, provided they have been given the tools to see it. I want to discuss an example of that today, from a 30-year-old study by Hiebert and Wearne.

The study

Hiebert & Wearne 1996 studied multi-digit addition in and subtraction in grades 1–3. Their goals were manifold and complex.3 Here I want to pick out one thread related to this idea of mathematics as a guide.4 I was interested to see that the authors provided two parallel definitions of what they meant by conceptual understanding of this topic. The first was the standard definition as a schema in the mind of the learner connecting “ideas, actions, facts, and procedures as well as among various representations.” The second takes mathematics as its guide.

From a mathematical point of view, we defined understanding . . . as the construction of connections between the key ideas of the base-10 number system . . . connecting this structure of groupings with the written notation. In other words, we focused our definition on the ideas behind place value and their connections with written numbers.

This definition makes understanding answerable to something outside the learner’s head and outside the teacher’s authority. Understanding in this case isn’t just having lots of connections, it’s having connections that track the base-10 structure, and the arbiter of whether they track it is the structure of the mathematics.

The study followed 6 classes over three years, from grade 1 to grade 3, with a final assessment in grade 4, all in one suburban-rural school. They studied a random sample of 72 students, 12 from each class. The classes were divided into two groups: four got an “alternative” approach where students invented and discussed their own procedures for place value and multidigit addition and subtraction, and two got conventional textbook instruction that demonstrated the standard algorithms. Several times a year each student sat for an individual interview built around two kinds of tasks: understanding tasks (e.g., how many teams of 10 from 64 children; showing 32 two ways with 1- and 10-point chips) and skill tasks (addition and subtraction story problems), plus a “demonstrate” task asking students to show why their written algorithm works with chips.

Before any instruction on regrouping, the only children who could solve a problem like 28 + 35, where the ones spill past ten, were those who already understood the base-10 structure. They invented their own regrouping procedures, reading the next step off the structure itself, while the children who lacked that understanding scored essentially zero—and for subtraction stayed there through two more years of instruction. After instruction everyone could carry out the standard algorithm, but only the understanders could say why it worked, showing it with the chips where the others could not.

The authors give a careful and richly detailed account of those three years and conclude that conceptual understanding plays a crucial role in stimulating and guiding the development of procedural skill, and that the alternative instruction, by not short-circuiting that relation, fostered both more understanding and a tighter connection between the two.

But my main point is something different. I am less interested in the contest between the two kinds of instruction than in what the understanders were doing when they invented those procedures, and in what exactly they had been given that let them do it.

Tools to see the structure

Here I’ll follow a particular thread in the alternative-instruction classes. The grade 1 instruction is documented in detail in a companion paper from the same project (Hiebert & Wearne 1992). The alternative instruction was designed on the principle that external tools could help students develop internal connections between concepts and procedures. In particular, the researchers wanted to “build connections between the key ideas of place value, such as quantifying sets of objects by grouping by 10 and treating the groups as units . . . and using the structure of the written notation to capture this information about groupings.” They followed four design principles:

First, external representations (physical, pictorial, verbal, symbolic) were used as tools for demonstrating and recording quantities, acting on quantities, and communicating about quantities. Second, once a particular representation was introduced (e.g., base-10 blocks) it was used consistently to allow students to practice using it as a tool and to become familiar with the uses it afforded. Third, the representations were used to solve problems as well as being analyzed as interesting artifacts in their own right. Fourth, class discussions focused on how the representations could be used and on how they were similar and different.

What I see here is the teacher explicitly equipping the students to be guided by the mathematics. This is not explore-your-own-adventure, it is preparing students for a specific journey.

The January lessons began by posing problems of finding how many objects there were in large sets, mostly sets between 50 and 100. Class discussions and suggested strategies began with counting by ones and shifted to counting more efficiently by grouping and counting by twos, fives, and eventually, tens. One kind of object investigated was Unifix cubes. These were eventually grouped into quasi-permanent bars of 10. Two-digit numerals were introduced as efficient ways of recording the size of sets.

This follows the idea of using external tools for internal representations. The move to freeze groups of 10 into a single bar is explicitly related to the idea of using a digit in the 10s place to represent the number of 10s. Having the students physically handle the manipulatives seems to have been important; in the textbook-instruction classes a beans and sticks representation was used, but more often as a representation shown by the teacher than handled by the students.

You can see the base-10 representation at work in this transcript from a lesson in April, near the beginning of the addition and subtraction unit.

One of the alternative-instruction teachers began the lesson by distributing sets of base-10 blocks (sticks and small cubes) to each student and writing 12 + 35 = — on the chalkboard.

T: Who knows how to show the number sentence with the blocks? What would you do? [Students put out various arrangements of blocks.]

S: Take 35 little ones and 12 little ones and put them together.

T: That’s one way. What’s another way?

S: Put one 10 and two little ones for the 12 and three 10s and five little ones for 35.

The teacher then discussed the two alternatives.

And the payoff:

T: Everybody show 12 on one corner of your desk and 35 on the other corner. [Pause.] What would we get if we added all these blocks together?

S: Thirty-five.

T: Look at your blocks.

S: Thirty-seven.

T: How many sticks do you have?

S: Four.

T: How much does that make?

S: Forty.

T: And how many little ones?

S: Seven.

T: So that’s 47. Could someone make up a story about that?

[Student makes up a story]

T: When we put the blocks together, did we have enough left over ones to make another stick of ten?

S: No.

T: We had seven little ones. How many more would we need to have ten?

S: Three.

Although this calculation does not require regrouping, and the teacher makes use of that fact, notice how the teacher anticipates regrouping with that question at the end about how many more we would need to have ten. Between that April lesson and December of grade 2, nobody taught these children regrouping. However, the ones who understood the structure were already solving 28 + 35 in the interviews, some as early as December of grade 1. Nobody showed them the method; the mathematics did.

Being explicit about mathematical structure

What I see here is a melding of the concerns in the debate between explicit instruction and guided discovery. The teacher introduces the tools—the base-10 blocks and the “little ones”—and models the procedure of addition, by having the students put them at each end of the desk and move them together. The instruction is explicit about the scaffolding. But it is open as to the way students climb the scaffolding; which procedure a child builds for adding two numbers, and how they explain it to the others. The structure of place value is encoded in an object a child can hold. Ashman’s students couldn’t come up with the falling satellite because there’s no scaffolding in the phenomenon that a novice can see. Regrouping is different: it was latent in a structure the children had been equipped to see, and they climbed to it before anyone told them anything.

This is what I was getting at earlier when I said that the definition of understanding based on mathematical structure makes it answerable to something outside the learner’s head. The arbiter is the structure of the base-10 system, and the tool is what brings that structure within a six-year-old’s reach.

As I said in an earlier post, I do this writing to figure out what I think, and for the last three posts I’ve been groping my way. I’d welcome your corrections and guidance in the comments!

In last Friday’s post about what makes mathematics different, I failed to mention one obvious difference. Reader Dev Sinha commented, “I do think there is some difference in kind, not centered in language but stemming from the fact that the basis for validity in mathematics is logic rather than evidence/data.” Also, Jim Hiebert emailed me that the “uniqueness of mathematics lies in its logic. The logic is always the decider in debates about correctness.” He pointed me to an article by Inagaki, Hatano, and Morita, “Construction of Mathematical Knowledge through Whole-class Discussion,” which I want to talk about today.

The experiment

The following problem was given to 11 classes of 4th- or 5th- graders, about 300 students in all. The classes had been taught to add fractions with like denominators and to add decimals, but not yet how to add fractions with unlike denominators.

Taro drinks 1/2 liter of milk at breakfast and 1/5 liter at supper. How many liters of milk does he drink a day?

Students were given three answer alternatives.

Students first chose privately, in writing, among the three answers and wrote down their reasons. The choices were tallied on the board by a show of hands, and a few supporters of each answer stated their thinking before the whole class discussed it. Each student then chose again (they were free to switch) and named the classmate whose ideas had been most persuasive, recalling them as best they could. After that, in 6 classes, the experimenter announced, without explanation, that Z was the preferred answer.1 In the other 5 classes he skipped this step. Students were immediately given a post-test (solve the original problem from scratch, writing the expression and the answer) and a transfer test on two new sums with unlike denominators, 1/2 + 1/3 and 1/4 + 2/5.

The researchers coded the classroom discussions, recording how many students chose each answer, the various reasons given for the answers, who spoke up and who was silent, how many students changed their answer on the post-test, and which students the others found most persuasive. They discuss the results under three headings: how the classroom discussion proceeded, arguments for and against each alternative, and arguments that induced public conversions. The answers on the post-test moved strongly towards the preferred one, but the headline result for my purposes here is that there was no significant difference on either the post-test or the transfer test between the classes that were told the answer and those that were not, with the exception of one class where none of the students gave a reasoned defense of the preferred answer, probably because time was short.

Who is the authority?

In the classrooms where the teacher gave the preferred answer, authority was flowing from the teacher. What about the classrooms where the answer was not given? Where is authority located there? In part the answer is the students nominated as most persuasive, but that begs the question of why they were regarded as persuasive.

The most ‘popular’ nominees . . . in these 11 classes were, with one exception, vocal supporters of Z (the most appropriate solution) . . . Moreover, supporters of Z were on the average nominated more often than the supporters of X or Y.

Among the most-nominated students, 69% had given arguments about why you need a common denominator or how to make one, versus only 13% of the least-nominated. Those students were convincing because conviction was possible; the mathematics itself was the authority. The logic is the decider.