This is the eighth in a series of Discussions about Learning and Mathematics. The Discussions are intended to provide an informative and sometimes provocative overview of the thinking of some key researchers in mathematics education and educational psychology and to be a forum for discussing these ideas.

From: krennin1@cc.swarthmore.edu (K. Ann Renninger)
Newsgroups: geometry.pre-college
Subject: Learning and Mathematics: Knowing, Lampert
Date: Wed, 08 Feb 1995 18:06:03 -0500

Magdalene Lampert, of the Institute for Research on Teaching at Michigan State University, advocates incorporating students' intuitive knowledge about mathematics into classroom lessons. Like Lesh, she encourages putting new concepts into familiar contexts so that students may more readily relate to the problems being investigated.

Article:

Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3(4), 305-342.

Overview:

Lampert describes experimental lessons and explains her view of the teacher's role in the classroom: to help students make explicit their ideas about analyzing and solving problems, to act as referee in arguments about the reasonableness of competing ideas, and "to sanction students' intuitive use of mathematical principles as legitimate." She requires that students decide in their own way whether something is mathematically reasonable, and sees instruction in mathematics content as inseparable from "building a culture of sense-making in the classroom, wherein teacher and students have a view of themselves as responsible for ascertaining the legitimacy of procedures by reference to known mathematical principles."

Lampert points out that because the social organization of the traditional elementary school classroom consists of large groups under one teacher, a sequential, standard curriculum within grade levels, and an assessment system that clearly distinguishes between right and wrong answers, it's easiest to teach mathematics primarily or even exclusively as computation, which then comes to be considered 'basic', the only kind of knowledge appropriate to teach directly, and the necessary foundation for principled knowledge.

She continues that it's often assumed--incorrectly--that if principled knowledge is to be acquired at all, "the brighter students will ascertain the principles that underlie the procedures without those principles ever having been the focus of instruction" (p. 312).

Lampert says that to counter this view, she almost always follows up an answer that has been offered without an explanation with a question intended to reveal "how the student figured it out." She does this for two reasons: 1) the way the student responds will show how the answer was arrived at and whether it's warranted, and 2) this approach will "develop a habit of discourse in the classroom in which work in mathematics is referred back to the knower to answer questions of reasonability. This habit needs to be developed because, in the traditional culture of classroom interaction, students have learned to rely on the authority of a book or a teacher to 'know' if their answers are right or wrong rather than asking themselves whether either the answer or the procedure they use to arrive at it makes sense" (p. 317).

Lampert questions whether and how the formal language of mathematical principles should be used in describing what students know and how they use what they know, and argues that "it is important not to create a separation between symbols and what they represent too early in the student's academic career, before the idea that mathematics is a system of principles that make sense has had time to be considered seriously" (p. 338).

Direct Quotes:

[In discussions about double-digit multiplication], "students used principled knowledge that was tied to the language of groups to explain what they were seeing. They were able to talk meaningfully about place value and order of operations to give legitimacy to procedures and to reason about their outcomes, even though they did not use technical terms to do so. I took their experimentations and arguments as evidence that they had come to see mathematics as more than a set of procedures for finding answers. Such discussions would not have been possible without their having learned a symbolic language for representing the steps of the procedure in ways that were meaningfully tied to operations on quantities" (p. 337).

"The experimental lessons described in this article stand as evidence that fourth graders can do mathematics and think mathematically. These students have the capacity to gather information, to organize it strategically, to generate and test hypotheses, and to produce and evaluate solutions. They can talk about what they are thinking, they can listen to and appreciate another student's procedures or way of understanding something, and they can invent problem-solving procedures that are both useful and sensible...

"What sort of help do children need from adults in order to do these things and to be confident of their ability to do them? I would suggest that they need to be asked questions whose answers can be 'figured out' not by relying on memorized rules for moving numbers around but by thinking about what numbers and symbols mean. They need to be treated like sense-makers rather than rememberers and forgetters. They need to see connections between what they are supposed to be learning in school and things they care about understanding outside of school, and these connections need to be related to the substance of what they are supposed to be learning. They need to learn to do computation competently and efficiently without losing sight of the meaning of what they are doing and its relation to solving real problems" (p. 340).

PROBLEMS

"Finding the monetary value of combinations of coins was an important exercise in working in a context where the order of operations matters; because of the familiarity with how money works, everyone knows that you multiply first and then add to get the value of the coins. This familiarity gave students the opportunity to do mathematics confidently in an area where they would later be introduced to more abstract forms. Because of their intuitive knowledge about how coins worked, they would be unlikely to make the error of adding first and then multiplying, whereas this is a common error when students are using procedural knowledge alone to multiply" (p. 318).

"The second set of lessons uses stories and illustrations to link familiar computational procedures using numbers and arithmetical symbols with legitimate procedures for separating large numbers of groups into parts, counting the quantities in each part, and recomposing to find the total..." (p. 316).

"In my class, the most obvious result from our work on the coin problems was the initiative that many students displayed on other occasions for searching out different decomposition and recomposition strategies when faced with the task of counting groups of groups. This might be thought of as evidence that they had learned that there is a variety of legitimate ways to decompose numbers, operate on them, and recompose to count a total, which might be taken as an indication of principled knowledge" (p. 321).

"After three or four lessons... in which I used students' stories to do drawing and numerical symbolization representing the decomposition process on the board, I constructed assignments in which the children would do their own stories, numerical representations, and drawings on paper with decreasing amounts of teacher direction. In some of those assignments, students were directed to 'find the total' according to whatever decomposition and recomposition method they chose and then to find it again using a different method...

"Some of the children became quite interested in showing many different ways in which they could decompose one of the factors to find the partial products. They were using the language and drawings we had practiced to make a bridge between their intuitive knowledge about how concrete objects can be grouped for counting and the meaning of procedures using arithmetic symbols. By rewarding them for inventing reasonable procedures rather than for simply finding the correct answer, I was able to communicate a broader view of what it means to know mathematics and to learn something from what they were doing about how they would use mathematical principles in a concrete context. This 'game' of inventing multiple decompositions was an approach to doing mathematics that the class had derived from the lessons on coin problems" (p. 330).

Summarized by Maria Ong Wenbourne

From: hgehlba2@cc.swarthmore.edu (Hilary Gehlbach)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 8 Feb 1995 19:56:39 -0500

Lampert's arguments make a lot of sense to me. It seems that most teachers know that students gain a better understanding of a subject if the student is given the opportunity to verbalize or make sense of the information him or herself; for some reason, though, this technique is often neglected in the realm of mathematics. I had one wonderful calculus teacher in high school, though - she gave us essay tests rather than traditional math tests. While this was certainly unconventional, writing essays forced us to verbalize our understanding of limits, etc., thereby forcing us to make some kind of sense of these abstract concepts. By doing this, we not only understood calculus a lot better than most (I would guess), but this also linked our calculus to other disciplines - we were using a language other than numbers and symbols to express our math knowledge. At the end of the course, we had to do a research paper on a way in which calculus is used to solve real-world problems. Math was not an isolated subject for me that year - I saw connections between math and other disciplines.

I got so excited about calculus that I went to college and decided to continue my calculus. Unfortunately, the professor taught math in the traditional math way. I haven't taken a math course since.

From: hfreedb1@cc.swarthmore.edu (Hannah Freedberg)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 13 Feb 1995 20:03:17 GMT

I love this quote from Lampert: "Children ... need to be treated like sense-makers rather than rememberers and forgetters." I intuitively and intellectually agree with this statement, and am thrilled to realize that its ramifications extend far beyond the realm of mathematics education. Not only children, but all people (all learners) must be given this same respect whatever their field of endeavor. Making sense is what makes us human; it is a drive that cannot and should not be ignored. In it lies the source of our greates discoveries, and of our learning.

Specifically with regard to math, the quote reminds me of my own struggles with math from grade school through college. I constantly badgered teachers with the simply question, "Why?" Why do we carry? Why do we use the quadratic formula to solve quadratic equation? Where do our methods come from? Sadly, I never received an answer. My teachers sometimes implied, sometimes directly stated that they couldn't explain or did not know. I became convinced that math was a subject in which there was no reason why. Things were simply done because they were done -- which seemed awfully naive and boring to me. Where was the sense-making I loved so much about my classes in English, Social Studies, even Science? I could not accept a discipline which did not question or revise its procedures.

In the past few weeks I have found myself finally understanding the appeal of math, as I have read and heard of its puzzle-like nature, of the principles, revisions, and multiple answers which are a part of doing math. I wonder where I would stand in relation to math today had I been exposed to these things in elementary school. If I had seen math as a sense-making discipline (not unlike my beloved English), I might have been interested to a magnitude I cannot now ever fathom.

From: lhunt1@cc.swarthmore.edu (Lucy Hunt)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 14 Feb 1995 05:26:14 GMT

Perhaps, the teaching of mathematics needs to be deconstructed a bit just as other areas of academic study have been. Indeed, Lampert suggests that students "decompose" their answers to math problems by suggesting more than one response. Her students have even turned this into a game, thereby enabling them to interact in a "habit of discourse." I like the idea of mathematics as a discourse; an exchange of ideas that would hopefully foster student's intuitive knowledge of math into procedural knowledge. Unlike in the traditional method of teaching math, this "discourse" or conversation between teacher and student, student and student, or student and self could help to cement the basic ideas underlying the procedural methodology into a student's repertoire of academic acrobatics.

Perhaps, if Hannah and my questions had been answered as we were muddling through mathematics we would have a different relationship with math today. Speaking for myself, I stay as far away from math as I possibly can. The closest I get to it is via a calculator. I have not taken a math class in over four years and I hope never to again. I am the perfect example of a student that was failed by a math system. I can trace back amongst my educational experiences to math trauma after trauma, and just once or even now I might like to know the answer to, "Why?"

From: hfreedb1@cc.swarthmore.edu (Hannah Freedberg)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 14 Feb 1995 19:31:30 GMT

I have a quick question for any of you out there who are currently teaching math on the pre-college level. How practically feasible do Hillary's, Lucy's and my requests for a new style of math seem to you? What is or has been your reaction to our demands for a reason "Why?" What kinds of practices would this involve? I'd love to hear your experiences.

From: roitman@oberon.math.ukans.edu
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 14 Feb 1995 16:35:48 -0500

Hannah, Hillary and Lucy, everything you're asking for seems to be both explicit and implicit in the NCTM Standards and supporting documents, and exists in many classrooms across the country. I visit a lot of classrooms as part of a teacher enhancement project I've been running for the last three years, and have seen a lot of group work, of discussions, of physical modelling, of kids both asking and answering why, of multiple reasons being encouraged, etc. etc. There are a lot of wonderful curriculum projects out there, and if you want I can send you a personal message with my personal favorites.

This is not to say that the situation is perfect. Far from it. In particular, professional development has lagged far behind curriculum development, hence some of the new and better curricula are being trivialized. Since professional development among curricula developers also is lagging, other curricula are being created which are inherently trivialized, especially by commercial publishers eager to jump on the Standards bandwagon (but not all commercial products are bad -- some are excellent). It is a very rich and interesting time for school mathematics right now, certainly a time in which your concerns are being addressed by many people in many ways.

You might want to look at some of the NCTM publications to get an idea of what is going on in the schools. In particular, the three journals for teachers (Mathematics Teacher for high school, Teaching Children Mathematics for elementary school, and a middle schol journal whose name I don't remember) would be a good introduction.

From: jehrenf1@cc.swarthmore.edu (Jane Ehrenfeld)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 15 Feb 1995 15:26:34 GMT

In article <199502090023.TAA28299@oak.cc.swarthmore.edu>, hgehlba2@cc.swarthmore.edu (Hilary Gehlbach) wrote:

"I got so excited about calculus that I went to college and decided to continue my calculus. Unfortunately, the professor taught math in the traditional math way. I haven't taken a math course since."

Don't give up Hilary -there's still hope! I am currently taking Calculus (the easy version) with Professor Grinstead and I see him using methods that attempt to do exactly what you and Lampert find essential to math education. We must write all answers in full sentences and explain all our work, he uses lots of car examples to demostrate limit concepts, etc. For the students in the class (who I assume are all more verbal than mathematical) this is an excellent approach. This class has also helped demonstrate to me the necessity of a teaching style such as the one that Lampert advocates and practices.

I think that one of the most important points that comes out of what Lampert is saying is that math IS interactive, and must be taught as such to all students -- in a way that parallels the interactive nature of other subjects (English, Social Studies, etc.). I think that this would not only be beneficial to more verbally-minded students, but would also serve to help mathematically-minded students have a dialogical relationship with math (as opposed to merely learning and understanding rules that do not change and over which they have no power).

From: speltz1 (Sam Peltz)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 20 Feb 1995 17:40:24 GMT

Although I very much agree with Lampert's suggestion about making math a more accessible subject for young learners by allowing a forum for their intuitive mathmatical knowledge to play a part of the learning of formal principles, I have a question about one particular quote in the article.

In her article, Lampert argues that "it is important not to create a separation between symbols and what they represent too early in the student's academic career, before the idea that mathematics is a system of principles that make sense has had time to be considered seriously" (p. 338). I followed and backed Lampert up until this point. What I do not understand is why it would not be advantageous to introduce students to the formal mathmatical symbols early on and incoporporate students' intuitive knowledge into their understanding of these symbols. I do not see how this introduction would result in a separation between symbols and their definitions.It seems to me that this process would give students a better understanding of these symbols and actually cement their meanings rather than dichotomize the symbols and their meanings. Also, this process seems as though it would result in a faster formation of a system of mathmatical principles for students. ...and the sooner this system is formed, the sooner students will be able to reflect on this system as it relates to their everyday life, and pre-existing math knowledge.

I may just be way off base and have misinterpreted what Lampert was suggesting. If that be the case, I would be interested to hear feedback on what other people think Lampert is really stating.

From: hot@SOE.Berkeley.Edu (Henri Picciotto)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 20 Feb 1995 16:03:59 -0500

> I have a quick question for any of you out there who are currently
> teaching math on the pre-college level. How practically feasible do
> Hillary's, Lucy's and my requests for a new style of math seem to you?
> What is or has been your reaction to our demands for a reason "Why?"
> What kinds of practices would this involve? I'd love to hear your experiences. > Hannah Freedberg '95

This is not a "quick question"!

I strongly agree with Judy Roitman's comments on this: the kind of changes you are suggesting are happening in many places, under the leadership of the NCTM and the various NSF-funded projects and some enlightened publishers; the biggest hurdle is teacher training, without which these changes will remain limited to very few classrooms; the new curriculums are necessarily uneven, but represent an important and exciting break with the kind of math you experienced as a student. (I am involved with curriculum development and teacher training, as well as teaching. Creative Publications and Terrapin published some of my curriculum creations.)

I have been teaching in what seems to me the style you suggest, at the high school level at the Urban School of San Francisco. The math program at my school has changed drastically over the past ten or so years. Collaborative work, discussion of "why", hands-on labs, "real world" applications and modeling, use of technology and writing, and so on have mostly replaced the "teacher lecture + quiet seat work" of yesteryear.

This is entirely feasible, though it takes a while to get good at it, and initially is met with substantial resistance. Our strategy to maximize support for the changes among administration, students, and parents, has been to start with changes in pedagogy, which lead to greater student engagement and success, and only when those are starting to be established to proceed to content changes, which are more anxiety provoking for teachers and everyone else.

Overall, this has been smashingly successful. Our enrollment in elective classes beyond the required three years of math has skyrocketed: trig, modeling and statistics, calculus, and an advanced topics course. This is a lot for a small school which used to have a single pre-calculus course, with a handful of students.

I don't think our students are necessarily better at any of this than they were, but their relationship to the subject is much better. They realize it's something they can do, and they sometimes enjoy it. We don't have as big of a split between math / science types and others.

Our program is completely untracked, but since we're a private school we have less of a range of students than many public schools. Nevertheless, the range between an A+ and a C- at our school is enormous, and we have many students who would not have been in a college preparatory course at another school, and yet who manage to get quite far. Many if not most of the students who take calculus never thought they would have gotten this far if you had asked them during their freshman year.

At the other end of the spectrum, our top students do go on to math, science, or engineering majors at prestigious universities, so they clearly do not suffer from the changes.

Nevertheless, there are questions. For example, only a few of our students are able to write a decent proof in tenth grade, and the algebraic manipulation skills of most are quite shaky even by 11th or 12th grade. I think these are important questions to think about, and that is where much of my mental energy is going these days. Within the framework of the kind of math promoted in the NCTM Standards, how do we make sure we bring symbolism, proof, and mathematical structure to our students?

--Henri

From: hfreedb1@cc.swarthmore.edu (Hannah Freedberg)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 21 Feb 1995 02:53:56 GMT

Thanks Henri, for sharing your insightful advice and inspiring experience. Can I ask you (or anyone else out there) what kinds of answers you have found to the question you pose above?

Thanks, Hannah

From: gmartin@kimiyo.ed.Hawaii.Edu (W Gary Martin)
Newsgroups: geometry.pre-college
Subject: Re: Learning and Mathematics: Knowing, Lampert
Date: 21 Feb 1995 19:43:37 -0500

Sam Peltz says:
"In her article, Lampert argues that "it is important not to create a separation between symbols and what they represent too early in the student's academic career, before the idea that mathematics is a system of principles that make sense has had time to be considered seriously" (p. 338). I followed and backed Lampert up until this point. What I do not understand is why it would not be advantageous to introduce students to the formal mathmatical symbols early on and incoporporate students' intuitive knowledge into their understanding of these symbols. I do not see how this introduction would result in a separation between symbols and their definitions. It seems to me that this process would give students a better understanding of these symbols and actually cement their meanings rather than dichotomize the symbols and their meanings. Also, this process seems as though it would result in a faster formation of a system of mathmatical principles for students. ...and the sooner this system is formed, the sooner students will be able to reflect on this system as it relates to their everyday life, and pre-existing math knowledge."

The problem is with creating a symbol system which exceeds the needs of the students. Learning to use the symbol system can quickly overwhelm the making of mathematical meaning. Symbol systems should be an embodiment of the understanding that students have, at a time that it is useful to them as a "short-cut" way of expressing something. To the degree to which the formalization outstrips the students' construction of mathematical meaning, the emphasis will shift to manipulating the symbols."

Example: When learning to find the area of a parallelogram, the formula is often given almost immediately. Students thus focus on using the FORMULA without really understanding what's going on with parallelograms--i.e., why one focuses on one side and the height to that side rather than just using the two sides as with rectangles--with frequently disastrous results. Students will indiscriminately plug numbers into the formula without any sense of what is going on (i.e., find ANY two numbers). On the other hand, if students have experiences finding areas of parallelograms non-formulaically, they will come to realize that it can be transformed into an equal-area rectangle having one side of the parallelogram and the "height" of the parallelogram. Thus, to find the area of a parallelogram, multiply a "base" by its corresponding "height", or perhaps at some point A=b*h. The formula is a "short-cut" for a deeper understanding.

Why the hurry for a formal symbolic representation? If the meanings are there, representing them symbolically can be addressed at an appropriate time. On the other hand, if symbols are prematurely introduced it can actually make it harder to get at the meanings because the students think they "understand" (i.e., how to use the symbols) when in fact they don't (i.e., what the symbols represent)."

W. Gary Martin, University of Hawaii