Monday, July 18, 2011

RYSK: Erlwanger's Benny's Conception of Rules and Answers in IPI Mathematics (1973)

This is the second in a series of posts describing "Research You Should Know" (RYSK).

In 1973, Stanley Erlwanger was a doctoral student at the University of Illinois at Urbana studying under Robert Davis (who taught many of us math as an advisor for Sesame Street) and Jack Easley when he published his landmark "Benny" article in Davis's new Journal of Children's Mathematical Behavior. (Now simply the Journal of Mathematical Behavior.) This and other Erlwanger articles became known as disaster studies (Spieser & Walter, 2004, p. 33) because they painfully reveal learning gone wrong, and they continue to impact the way we think about learning math and how we do research in mathematics education.

During the back-to-basics movement of the 1970s there was a push for programs that supported individualized instruction. One such program was Individually Prescribed Instruction, or IPI. IPI was designed for students to "proceed through sequences of objectives that are arranged in a hierarchical order so that what a student studies in any given lesson is based on prerequisite abilities that he has mastered in preceding lessons" (Lindvall and Cox, as cited in Erlwanger, 1973, p. 51). To measure that mastery, IPI relied heavily on assessments that were checked by the teacher or an aide, who would then have the opportunity to conference with the student and check for understanding. Erlwanger, however, saw a conflict inherent in the program: while the goals of IPI were "pupil independence, self-direction, and self-study" (Erlwanger, 1973, p. 52), teachers were supposed to have "continuing day-by-day exposure to the study habits, the interests, the learning styles, and the relevant personal qualities of individual students" (Lindvall and Cox, as cited in Erlwanger, 1973, p. 52). So is a teacher, with a class of students each working at their own pace, supposed to continuously monitor each individual student? How? The logical way to do this is to monitor assessment results and focus attention on strugging students. After all, if a student is passing the assessments and "mastering" objectives, how much could go wrong?

Benny was a twelve-year-old boy with an IQ of 110-115 in a 6th grade IPI classroom. Benny had been in the IPI program since 2nd grade, and the teacher identified Benny as one of her best students. By sitting down and talking to Benny about the math he was learning, Erlwanger discovered that Benny's conception of math was not only very rule based, but in many cases Benny's rules yielded wrong answers. For example:

  • Benny believed that the fraction \(\frac{5}{10} = 1.5\) and \(\frac{400}{400} = 8.00\) because he believed the rule was to add the numerator and denominator and then divide by the number represented by the highest place value. Benny was consistent and confident with this rule and it led him to believe things like \(\frac{4}{11} = \frac{11}{4} = 1.5\).
  • Benny converted decimals to fractions with the inverse of his fraction-to-decimal rule. If he needed to write 0.5 as a fraction, "it will be like this ... \(\frac{3}{2}\) or \(\frac{2}{3}\) or anything as long as it comes out with the answer 5, because you're adding them" (Erlwanger, 1973, p. 50).
  • When Benny adds decimals, he adds the number and moves the decimal point the total number of places he sees in the problem. So \(0.3 + 0.4 = 0.07\) and \(0.44 + 0.44 = 0.0088\). Benny's rule for multiplication is very similar: \(0.7 \times 0.5 = 0.35\), \(0.2 \times 0.3 \times 0.4 = 0.024\), and \(8 \times 0.4 = 3.2\). Because these are correct answers, that only served to reinforce Benny's rules about the addition of decimals.
  • Benny thinks different kinds of numbers should yield different answers: "2 + 3, that's 5. If I did 2 + .3, that will give me a decimal; that will be .5. If I did it in pictures [i.e., physical models] that will give me 2.3. If I did it in fractions like this [i.e., \(2 + \frac{3}{10}\)] that will give me \(2\frac{3}{10}\)" (Erlwanger, 1973, p. 53).

As you might guess, Benny got a lot of wrong answers and sometimes failed to achieve the 80% mastery mark on his assessments. It's clear that Benny isn't simply guessing and getting wrong answers -- his methods are consistent and he can confidently explain his reasoning. When Benny is wrong, he tries to change his answers until he gets ones that match the answer key, a process he called a "wild goose chase" (Erlwanger, 1973, p. 53). Because Benny's teacher/aide is only looking for answers that match the key (and trying to do so quickly), the emphasis is on the answer, not the reasoning. It was only Benny's persistence that resulted in him mastering more objectives than most of his classmates.

This style of learning led Benny to believe that math is little more than a collection of arbitrary rules and singularly correct answers: "In fractions, we have 100 different kinds of rules" (Erlwanger, 1973, p. 54). Erlwanger asked Benny where he thought the rules came from. "By a man or someone who was very smart. ... It must have took this guy a long time ... about 50 years ... because to get the rules he had to work all of the problems out like that..." (Erlwanger, 1973, p. 54). For both reasons of scholarship and concern for Benny, Erlwanger returned to the school twice a week for 8 weeks to work with Benny one-on-one. Unfortunately, despite Benny's eagerness to learn, Erlwanger found this to be too little time to change Benny's firmly-established view of mathematics and little progress was made.

What Benny Means to Theory, Research, and to Khan Academy

(It might be helpful to read yesterday's post about constructivism and the Khan Academy before reading this section.)

Erlwanger summed up the theoretical aspect in his conclusion:
Benny's misconceptions indicate that the weakness of IPI stems from its behaviorist approach to mathematics, its mode of instruction, and its concept of individualization. The insistence in IPI that the objectives in mathematics be defined in precise behavioral terms has produced a narrowly prescribed mathematics program that rewards correct answers only regardless of how they were obtained, thus allowing undesirable concepts to develop. (1973, p. 57)
Looking back at Benny in 1994, Steffe and Kieren summarized that
Erlwanger was able to demonstrate how Benny's understanding of mathematics conflicted with any "common sense" understanding of what would be regarded as "good mathematics." This was a crucial part of Erlwanger's work, because by demonstrating what a "common sense" view of mathematics should not be, Erlwanger was able to falsify (naively) the behavioristic movement in mathematics education at that very place where behaviorism has its greatest appeal -- at the level of common sense. (p. 72)
Prior to Benny, the large majority of research in mathematics education depended on quantitative methods -- using statistics to summarize and compare the performance of treatment and control groups. Erlwanger had opened the door to qualitative research, which essentially meant that researchers could now see the value of interviews, case studies, and similar methods. In other words, Benny showed researchers that they can, and should, talk to children.

Although we're approaching the 40th anniversary of the Benny study, anyone who has been paying attention to the debates regarding Khan Academy should be able to draw parallels between it and IPI and realize we're retreading a lot of the same water. In a recent Wired Magazine article about Khan, stories are told of students working individually, at their own pace, with their progress measured by a computer that judges answers right or wrong. The article highlights Matthew Carpenter, a fifth grader who has completed "an insane 642 inverse trig problems" (Thompson, para. 2). Carpenter has earned many Khan Academy badges, a sign of progress that pleases his teacher and amazes his classmates. Unfortunately, the article provides no evidence that Matthew Carpenter is not Benny. I, and hopefully everyone, sincerely hope he is not Benny. I hope he's developing a proper view of the nature of mathematics and developing solid mathematical reasoning and understanding. But I can't be sure, and maybe Carpenter's teacher can't be sure, either. While we sometimes can and do use behaviorist programs of instruction to learn, we can't rely on them to be sure that learning is happening the right way. That's Benny's lesson, and that's why we need to be critical (but not necessarily dismissive) of Khan Academy. People who fail to do so might be surprised with the results they get for all the wrong reasons.


Erlwanger, S. H. (1973/2004). Benny╩╝s conception of rules and answers in IPI Mathematics. In T. P. Carpenter, J.A. Dossey, & J. L. Koehler (Eds.), Classics in mathematics education research (pp. 48-58). Reston, VA: NCTM.

Speiser, B., & Walter, C. (2004). Remembering Stanley Erlwanger. For the Learning of Mathematics, 24(3), 33-39. Retrieved from

Steffe, L. P., & Kieren, T. (1994/2004). Radical constructivism and mathematics education. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in Mathematics Education Research (pp. 68-82). Reston, VA: NCTM.

Thompson, C. (2011, July). How Khan Academy is changing the rules of education. Wired. Retrieved from

Sunday, July 17, 2011

Constructivism and the Khan Academy

Not long after sitting down at my computer this morning, there was this tweet from David Wees:
“How would you explain constructivism to someone not (well) versed in pedagogy? You have 140 characters. #edchat #BCed”
I took David’s challenge and what followed was a pretty good conversation with David Cox, Ira Socol, and Jennifer Borgioli. For the sake of clarity, yet with an attempt at brevity, I thought a follow-up post would be good here. My goal is to share the kind of knowledge that David asked for -- a short explanation for someone who might be new or unclear about these ideas -- so please excuse me if I don’t touch on some of the nuanced bits (and there are many, trust me!) of the theory.

Before we talk about learning theory, we should take a step back and talk about epistemology - the branch of philosophy concerned with the nature of knowledge. There are multiple epistemologies, but two are important here.

Objectivism: An objectivist epistemology holds that knowledge and meaning exists independently of the learner. It is not just believing that “a rock would be a rock if we were here or not.” Instead, it is a belief that the rock carries some meaning of what it is to be a rock, and when we study rocks we are discovering that meaning. Objectivism is sometimes called empiricism or externalism.

Constructivism: A constructivist epistemology holds that there is no objective knowledge. This doesn’t mean that there aren’t objects, but that the knowledge and meaning we associate with objects is constructed by us as we engage and interact with the world. Constructivism can take several different forms, depending on the importance placed on social and historical interactions.

Hopefully you can already see how different epistemologies can affect a person’s view of teaching and learning. Now let’s compare three learning theories associated with these epistemologies.

Behaviorism: Behaviorist learning theory is often associated with an objectivist epistemology. Human actions, including exhibitions of our knowledge, are viewed as behaviors that respond to a stimulus. The process begins with a transmission of knowledge from the teacher (which can be a non-human source of knowledge) to the student. If the response is the expected behavior, the student is rewarded. If the response is not the expected behavior, the student is punished. By stimulating the student with rewards and punishments, the teacher encourages the student to receive the transmissions of knowledge.

Information Processing: IP theory still applies an objectivist epistemology, but differs from behaviorism in that learning is seen as an inner cognitive process and not just a response to a stimulus. The brain is seen roughly as analogous to a computer -- it has memory and processing systems that serve to store and analyze information, although the analogy doesn't extend to understanding exactly how those systems actually work. Application of this theory in teaching generally involves heavy doses of repetition to ensure that knowledge is retained in memory.

Constructivism: Not surprisingly, constructivist learning theory is associated with a constructivist epistemology. Because knowledge is constructed by the learner, the teaching/learning process focuses on creating conditions for that construction to happen. There is no "transmission" of knowledge. Depending on the form of constructivism, the teacher might facilitate the construction of knowledge through the inclusion of contexts and social interaction.

The science of education would be much easier if we could prove that some theories never work and one works all the time. But we can’t. However, I don’t know of an educational psychologist that doesn’t think constructivist learning theory (in at least one of its variants) works better than those based on an objectivist epistemology. So why doesn’t every teacher do it? Or do it well? Teachers in classrooms have resource, time, and other constraints that makes constructivism more difficult than we all wished it was. Also, it’s not always clear cut which theory is being applied by a teacher. Suppose you were to peek into a classroom and see a teacher speaking to the entire class. Maybe the teacher is trying to transmit knowledge in a behaviorist/IP way. Maybe the teacher is trying to help students get into a certain frame of mind and is a constructivist. You can’t tell at a glance because the learning theories don’t always present themselves as extreme opposite ends of the spectrum. But when there’s controversy, we like to pretend that they do. Enter Salman Khan.

There’s been a lot written about Sal Khan and the Khan Academy over the past several months, including a recent article in Wired Magazine that became a large part of this morning’s discussion on Twitter. The idea of learning by watching videos isn’t necessarily behaviorist or solely an application of information processing theory, but it’s more easily seen as a medium for the transmission of knowledge, not construction, and the point-keeping for problems right and wrong also fits the stimulus/response model. Phrases such as "Khan and Gates both admit there’s no easy way to automate the teaching of writing" also point at behaviorism and IP. (There’s an underlying assumption here that if teaching can be automated, learning will be automated.) The Wired article quotes parents and teachers who are amazed at the progress their kids are making, measured by problems completed, modules finished, and badges earned. Are those students learning? Of course they are, but exactly what they are learning and how well they understand it is at the core of the debate.

Behaviorism and information processing aren't mentioned by name in the article. Perhaps they don't need to be; it’s a style of education that most all of us are familiar with and perhaps it doesn’t need much explaining. Constructivists are named as Khan’s critics, and the theory is described using terms such as "play around" and "fumbling around," the latter of which was probably an unfortunate choice of words by a constructivism supporter. Saying that "it’s better to give kids activities that let them discover the principles of math and physics on their own" doesn’t give enough credit to teachers in good constructivist learning environments. When done well, teachers don’t just "give" activities and students aren’t "on their own." Instead, there’s a careful orchestration going on and the teacher is with the students 100% of the way, asking questions, providing feedback, provoking the student to look at tasks in ways that help students construct deep understandings. Can a video do this? Obviously there are severe limitations -- not limitations that prevent all learning, but limitations that might be preventing the best kind of learning.

Look for an upcoming post about what happens when the instruction is based on objectivism but the student, a kid we'll call "Benny," constructs knowledge in his own, incorrect way. The "Benny" paper by Stanley Erlwanger in 1973 had huge ramifications for research and teaching in mathematics education, and has interesting parallels to learning via the Khan Academy.

I'd like to give great thanks to Jackie Hotchkiss for helping review a draft of this post. (Any final shortcomings are solely mine, of course.) When in doubt, talk to an educational psychologist!

Thursday, July 7, 2011

A Pretty Short Explanation of "Invert and Multiply"

I can't remember if I've seen this one before, but I thought it was worth sharing. I found it in a forthcoming book by Pamela Harris, at the end of an extended discussion about how to develop proportional reasoning.

We all have had students who know one rule for dividing fractions: invert and multiply. I learned it as "multiply by the reciprocal," while some of my students annoyed me with the vocabulary-loose "copy dot flip" (or flop). But why does it work?

There are many ways to explain it, including some that get into the gritty details of partitive and quotative division, but I think this explanation would satisfy most students and teachers. Let's start with a problem: \(\frac{3}{4} \div \frac{2}{3} \). I find this explanation is easiest to see if we write this as a compound fraction:
\[ \frac{\frac{3}{4}}{\frac{2}{3}} \]
The reasoning from here is simple: dividing by a fraction is hard, but dividing by one is really easy. But how do we turn \( \frac{2}{3} \) into one? By multiplying by it's reciprocal, of course. But we can't just multiply part of our problem by \( \frac{3}{2} \) without changing its value. The only thing we can multiply by without changing our value is one, and we can write one as something over itself, like this:
\[ \frac{\frac{3}{2}}{\frac{3}{2}} \]
So that's just a fancy way of writing one, and when we multiply the denominators we make a one:
\[ \frac{\frac{3}{4}}{\frac{2}{3}} \cdot \frac{\frac{3}{2}}{\frac{3}{2}} \rightarrow \frac{\frac{3}{4} \cdot \frac{3}{2}}{1} \rightarrow \frac{3}{4} \cdot \frac{3}{2} \]
Which explains "invert and multiply." It's just the result of wanting to divide by one instead of a fraction.


Harris, P. W. (in press). Building Powerful Numeracy for Middle and High School Students. Portsmith, NH: Heinemann.

Monday, July 4, 2011

Not (Yet) Sold on Tablets in the Classroom

I keep seeing a lot of stories and posts about tablets (usually iPads) in the classroom. Some glowing with promise, others more skeptical. For example:

More Colorado schools turning to iPad to improve education (Denver Post)
Math that moves: Schools embrace the iPad (New York Times)
On tablets in the classrooms - are they really a necessity? (Manila Bulletin)
Digital Textbooks (Curmudgeon [not so much about tablets, but what we can do with them])

At one level, I get it. iPads are easy to use, have great battery life, are reasonably affordable, and have that Apple-ly shiny goodness that other people (but not me - sorry, Steve) seem to love so much. But if you have $500 to spend to put technology into a student's hands, why not buy a cheap laptop?

Let's do a comparison between an iPad, Chromebook, and budget laptop:

  • Price: $499
  • Fast enough for HD video playback
  • 10 hour battery life (to Apple's credit, this appears to not be exaggerated)
  • Front and rear camera
  • Instant on, simple administration
  • Input device: (Usually) fingers
  • No flash, some Google Docs limitations

Samsung Chromebook:
  • Price: $429
  • Fast enough for HD video playback
  • Up to 8.5 hours of battery life
  • Front camera
  • 10-second boot, instant resume, simple administration
  • Input device: keyboard and touchpad
  • Not a "real" notebook, limited to browser apps

ASUS Eee PC 1215B-PU17-BK (Some companies just can't name products, can they?)
  • Price: $437
  • Fast enough for HD video playback
  • Up to 8 hours battery life
  • Front camera
  • Slower boot, probably more complicated administration
  • Input device: keyboard and touchpad
  • A full-fledged computer, with all its benefits and hassles

In a recent conversation I had with Darren Yung on Google+, he said he'd rather have iPads for students than notebooks, citing administration hassles with the notebooks and the problem students have logging in. I've seen that personally - students forget passwords, leave caps lock on, or find other ways to not be able to log themselves into the OS. Darren also lamented the slow boot time of notebooks. He did, however, wish iPads worked as well with Google Docs as notebooks do.

So here's my argument: Logins are a hassle, but even if the OS has no login process, won't students still have to login to Google to get to Docs? In that case, the Chromebook and iPad seem pretty even. Also, notebooks are slow to boot. I agree, but I'm sure I couldn't type very quickly on an iPad, and the time lost typing would more than cancel out the time spent waiting for a notebook to boot.

I know there's more to this argument, but I think given the needs and wants described by Darren, I'd lean towards the Chromebook, with perhaps a carefully-yet-simply-configured (use auto login, for example) notebook coming in second. (This surprises me, as until now I had all but dismissed the Chromebook as a viable option.) I know I'm biased in two ways: (a) I taught in high school and now I'm a PhD student, so typing while reading web pages or PDFs is probably my primary academic computing activity, and (b) I'm not a fan of Apple. For (a), I'd like to personally see computers vs. iPads in the hands of elementary students to see which is more effective, and for (b) I'd feel the same way if we were talking about Android tablets. Now, because Apple imagery causes a religious reaction in the brains of Apple fans, I'm sure some readers will feel like I've just told them that they're praying to the wrong God, but hopefully even the most devoted followers can find my points reasonable.

Saturday, July 2, 2011

RYSK: Brownell's The Place of Meaning in the Teaching of Arithmetic

This is the first in a series of posts describing "Research You Should Know" (RYSK).

Prior to the emergence of mathematics education as its own field of research in the 1960s and 1970s, those who chose to study the learning and teaching of mathematics were generally either mathematicians (George Polya, for example) or educational psychologists. William A. Brownell was one of the latter, having earned his doctorate in educational psychology in 1926 at the University of Chicago. Brownell worked at a number of universities but is best known for his work at Duke University from 1930 to 1949 and eleven years as dean of the School of Education at the University of California, Berkeley. Brownell died in May of 1977.

Brownell spent most of his career studying how children learn mathematics and authored a textbook series for the teaching of arithmetic. When the NCTM asked experts for nominations for articles to be included in their Classics in Mathematics Education Research, nine different articles were nominated. As the project leaders explained, "There was never a question whether to include an article by Brownell; the only issue to which of the many classic articles that he wrote to include" (Carpenter, Dossey, & Koehler, 2004).

In this article, Brownell defines the terms meaning of a thing and the meaning of a thing for something else. Or, in short, meaning of and meaning for. Using an example of the times, Brownell says that while he knows little about the workings of an atomic bomb (meaning of), he knows plenty about what having and using atomic bombs means for other things, like our culture (meaning for). In arithmetic, Brownell sees meanings of to be mathematical understandings, usually developed in classroom settings; meanings for are the connections to arithmetic we generally experience in every-day life outside of the classroom. Brownell insists that the distinction between the two is clear, although each type of meaning is relative, with varying degrees of complexity and depth.

Brownell saw, in 1947, too little teaching of arithmetic understanding. Common practice looked like this (you need only watch 20 seconds or so):

Through repetition, many students learned arithmetic. Many didn't. Again, learning was relative, not absolute, and Brownell never claimed that any students received zero understanding. What Brownell wanted was for students to see more meaning, which he organized into four categories:
  1. Meanings of whole numbers, fractions, decimals, percent, ratio, and proportion.
  2. Meanings of operations, including when to add, subtract, multiply, and divide.
  3. Meanings of principles and relationships of arithmetic, like the zero property, communitive property, etc.
  4. Meanings of place value that go beyond procedural "borrowing" and "carrying."
Brownell cited three sources for lack of meaning: (a) anecdotal evidence from adults who were poor at arithmetic, (b) poor military test results in arithmetic, and (c) the experience of secondary math teachers who receive students without arithmetical understanding. Brownell reasoned:
School personnel and, to some extent, the public at large are beginning to awaken to the fallacy of treating arithmetic as a tool subject. To classify arithmetic as a tool subject, or as a skill subject, or as a drill subject is to court disaster. Such characterizations virtually set mechanical skills and isolated facts as the major learning outcomes, prescribe drill as the method of teaching, and encourage memorization through repetitive practice as the chief or sole learning process.
Sound familiar? Of course. This kind of message has been echoed in NCTM materials at least since the Agenda for Action (1980) and common thinking can be found in articles such as Keith Devlin's Why We Should Reduce Skills Teaching in the Math Class.

So why didn't we (and don't we) teach arithmetic for meaning? Brownell anticipated four objections:
  1. Is it really necessary to understand meanings to learn arithmetic?
  2. Are these meanings too difficult for children to learn?
  3. Does it take too much time to teach meanings?
  4. Does learning meanings of and for arithmetic really improve ability?
To this, Brownell begins by asking why we teach arithmetic at all. If it is simply for students to be able to compute quickly and correctly, and nothing else, then the objections stand. However, if you believe that one of the goals of teaching arithmetic is to introduce to students a logical system of thinking, then meaning is necessary. Without it, computational efficiency -- which we should all still want -- will deteriorate once the drills are over. In this early day of research, Brownell apologizes for not citing "an impressively large body of competent research," but claims a strong case for meanings can still be made without it. Now, more than 60 years later, the research exists and the teaching of arithmetic looks very different. Students receive much better instruction about place value and use manipulatives like base-10 blocks, five-frames, and rekenreks to help their understanding.


Brownell, W. A. (1947). The place of meaning in the teaching of arithmetic. Elementary School Journal, 47, 256-265.

Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (2004). Introduction. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in Mathematics Education Research (pp. 1-6). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics [NCTM]. (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, VA. Retrieved from