Why carpentry? One of the researchers' chose to research this topic primarily because he always had an interest in carpentry since he was a young boy. His interest in this area was sparked when he took a woodworking class in primary school. In addition, the measuring, cutting, use of carpentry tools, etc, and then seeing the final structure, simply fascinates him! He believes there's great satisfaction that comes with realizing that you built something tangible & functional, for example, a table or a chair. As such, it's no surprise that he does intend sometime to embark on carpentry as a hobby.
Our second researcher had spent childhood years as her father's assistant in the building of a deck and other carpentry projects, but had never made the transition from "assisting" to "do-it-yourself." She had long since been fascinated with the process of carpentry and the challenge inherent in carpentry tasks. Further, though a teacher of mathematics, she had also often considered adding a secondary certification in physics. Carpentry seemed to her to be the perfect application of both math and physics, and she looked forward to finding out more about the mathematical principles involved.
Context of the Study
Jason Chapman, a 34-year old independent carpenter and contractor, was the sole participant in our study and the research was conducted at the carpenter's residence. One of the researchers knew of Jason by virtue of living in the same neighborhood with him and his family for 5 years. While Jason has a master's degree in Biomechanics, and has taught college level classes in Health and Physical Fitness, he "just sort of picked up the trade - of carpentry - out of necessity". In 2005, he and his wife bought a house that had been foreclosed on by the bank, which was in a state of extreme disrepair. The couple's goal was to turn this ailing house into a family home mainly by virtue of their own "sweat equity" and they threw themselves into repairing roof leaks, refinishing hardwood floors, replacing doors and windows, and learning whatever they needed to know as they progressed. Jason found in the work a calling of sorts, and began putting more effort into learning the work in a larger sense, serving casual apprenticeships at the side of friends who worked in carpentry and general contracting. These were his first experiences in the field - he did not grow up with a "handyman" type father or take "shop" classes in high school. His first carpentry experience, and only one prior to working on his own home, was a tree house years earlier.
We based our research on a constructivist theoretical framework. According to Millroy (1992), "Constructivism is seen as a vital part of the conceptualization of the present research. The simple model of action and reflection on a problematic in the constructivist cycle has practical uses during fieldwork designed to recognize and describe the mathematical practices of a community of people, since problem solving is one of the occasions during which their mathematizing is demonstrated." She goes on to say that, "The identification of and engagement with, a problematic serves as a signal to the researcher to take careful note of all the interactions taking place."
One definition of constructivism is, "Constructivist learning is based on students' active participation in problem-solving and critical thinking regarding a learning activity which they find relevant and engaging. They are "constructing" their own knowledge by testing ideas and approaches based on their prior knowledge and experience, applying these to a new situation, and integrating the new knowledge gained with pre-existing intellectual constructs (Vanderbilt University, 1997). Another variation of its definition is that, "Constructivism is an educational philosophy which holds that learners ultimately construct their own knowledge that then resides within them, so that each person's knowledge is as unique as they are. Millroy cites in her work that:
"As pointed out by Confrey (1991), constructivists base their definition of the construction of knowledge on Piaget's principle of intellectual adaptation. Intellectual adaptation is an exchange between a person and his or her environment and involves assimilation and accommodation. Assimilation occurs when the cognitive system deals with environmental events in terms of its existing structures, and accommodation occurs when the cognitive system modifies itself as a result of environmental demands (Ginsburg & Opper, 1969).
Constructivism has, nonetheless, its critics who point out that there's no one "truth" as it were. Millroy writes that, "Constructivism is frequently criticized as being susceptible to relativism; skeptics claim that since there is no "certain knowledge "to appeal to, then everybody's knowledge must be equally valid (Confrey, 1985, p. 5)."
We sought to address the following questions in our ethnographic research:
What are the valid mathematical ideas found in the everyday woodworking activities of a carpenter?
How does a carpenter use mathematical ideas in his work?
How is his use of such ideas impacted by his math education?
Are there any useful, meaningful, and tangible lessons learnt that can be incorporated into teaching math?
Concise Literature Review
Our literature review was primarily drawn from the work of W.L. Millroy (1992). Millroy had conducted a six-month ethnographic study as an apprentice carpenter in Cape Town, South Africa, to document the valid mathematical ideas that are embedded in the everyday woodworking activities of a group of carpenters. Rather than simply being an observer in her fieldwork, instead, she believed that not only would she uncover better research but also fully experience and appreciate her research objectives by choosing to embed herself as an apprentice carpenter.
In an attempt to address our own research questions, we look at what primarily Millroy findings were. First, what are the valid mathematical ideas found in the everyday woodworking activities of a carpenter? According to Millroy, "The results showed that many conventional mathematical concepts are embedded in the practices of the carpenters. They made extensive use of such concepts as congruence, symmetry, proportion, and straight and parallel lines in their everyday work. Furthermore, the carpenters' problem solving was enhanced by their strength in spatial visualization." In observing our participant, Jason, mathematical thought was not only evident, but prevalent throughout the observation. Specifically, we observed pattern, angle, measurement, count, cost calculation, problem-solving skills, and spatial reasoning ability, among others.
Rather than basing her research on a single theoretical framework, Millroy uniquely decides to use multiple approaches. She writes, "In order to avoid the "everything is mathematics" danger, I used several frameworks through which to view the activities of the carpenters." For instance, she cites "Bishop's work (1988a, 1988b), in which he describes the six "environmental activities" of counting, measuring, locating, designing, playing, and explaining and from which he claims that all mathematical ideas can be generated, provided a starting point." Based on her findings, Millroy asserts that not all activities observed agreed with Bishop's claims. According to Millroy, "However, finding activities in the workshop that fitted Bishop's categories was not enough to claim that such an activity was an example of mathematizing.
In addressing, how a carpenter uses mathematical ideas in his work, Millroy hypothesizes that, "the intuitive actions of the carpenters reveal a tacit knowledge of mathematics that is difficult for them, or for an observer, to describe explicitly." She refers to this as "mathematizing." She writes, "I shall use the word "mathematizing" to suggest the dynamic quality of this knowing-in-action. By observing their actions and performing the actions myself as an apprentice, by encouraging the carpenters to reflect upon their actions, by reflecting upon my own actions and by reflecting upon their reflections, I intend to describe the mathematizing implicit in the actions related to carpentry." Problematics have to be encountered for mathematizing to manifest itself. Our carpenter, Jason, ran into problematics in trying to attach accurately the top and bottom deck rails together using spindles and in solving the problematics, we observed him use logic and spatial reasoning.
When it comes to examining the ways in which, and extent to which, a carpenter's use of mathematical ideas is impacted by his math education, Millroy cited a study by Schliemann (1984) which targeted this very question. Specifically, Schliemann's study compared and contrasted a group of carpenters who had learned the trade on their own and had very little formal mathematics with a group of students in a carpentry apprentice program who had at least 4 years of formal mathematical training. The result of the study showed experience to be the better teacher - in one case in particular "each of the carpenters and apprentices was given a clear, well-labeled diagram of a bed and asked to find out how much wood would be required to build five such beds. Although most of the professional carpenters used suitable strategies to find a solution and displayed an understanding of the concept of volume, the apprentices' attempts were unsuccessful and the results obtained were absurd."
Regarding the search for useful and tangible lessons which can be meaningfully be transferred from the carpenter's workshop and implemented in the math classroom, Millroy chooses to focus on the question of "what it means to do mathematics." She goes on to say, "I have argued that the carpenters do mathematics in the workshop, according to the criteria that I set up in Chapter 1. Their mathematics has some unique characteristics that are different from the mathematics that we are accustomed to seeing in textbooks and that's how the culture of the workshop placed an indelible stamp on the mathematical ideas that were developed there. First, action is vital in their mathematics. There is tacit mathematical knowledge in their physical actions. Physical demonstrations formed part of active explanations, and there was movement and an involvement in the environment. For example, Jack's actions when squaring a box carried implicit knowledge that the diagonals of a rectangle are equal. Second, reflection on action often leads to the articulation of tacit knowledge. For example, Jack's reflections on the action she took when squaring a table led to his explanation of how to find the center of a rectangular box lid, while Clive's reflections on the actions he took when planing the edge of a plank and using his eye to check that the edge is straight led to his discussion on the concept of straight lines." Also, according to Millroy:
"I claim that the evidence provided in the present study suggests that mathematical knowledge can be implicit in physical actions. Acknowledgement of this claim requires a shift in the accepted epistemology of mathematics. The third unique characteristic of the carpenters' mathematics is that their explanations, discussions, and problem-solving activities were deliberately linked to concrete, contextual problematics that were used to provide physical props, mental models, and analogies." (Millroy, 1992)
The example of the carpenters' and apprentices' varying success with the problem of the bed frame, suggests that mathematics teachers seek to provide contexts for students that demand more strength and versatility from concepts than the symbolic manipulation that suffices in most classrooms at present. The results of the present research underscore the recommendations of the constructivist literature that students would benefit from being encouraged to construct and use multiple representations of their concepts." (Millroy 1992)
Examples of research of this kind are compelling because they raise fundamental questions about the usefulness of school-taught formal mathematics for people's lives. What is it about academic mathematics or about the way in which we teach mathematics in schools that leaves people unwilling or unable to apply that knowledge usefully in everyday settings? People are fully capable of devising innovative mathematical methods to solve the problematics and to accomplish the goals that they set for themselves. Perhaps part of the problem with school mathematics is that students are seldom given the opportunity to define their own problematics. Solving well-defined academic mathematics problems in the classroom does not adequately portray the kind of problem solving required in everyday settings, where the situation may be anything but clear-cut.
We initiated entry and collected data by way of visiting with the carpenter, Jason, at his residence. He was the only participant. Data collection was done via videotaping, interviews, field notes and observation. In order to study Jason's use of mathematical ideas in carpentry, the researchers first filmed the process of Jason building a portion of a railing for his home deck. While Jason was told that the job of the researchers was to tacitly observe his work, and that he should work as he normally would, he preferred to interact with the camera during the observation, doing his work as something of a demonstration. In fact, he clearly fell back upon his classroom persona to a certain extent, becoming quite comfortable with an instructional role throughout the filming - making the observation much like watching a segment of "This Old House" or a similar woodworking program with an instructional format.
We conducted interviews that ranged from informal conversations to formal structured interviews where we had a script that we followed. After observing the carpenter build part of his deck railing, we sat down and posed formal structured questions to him. All the while, we jotted down field notes during the entire time.
Mathematical thought was not only evident, but prevalent throughout both the observation and the interview. As he demonstrated the process of building the section of railing, Jason made dozens of decisions which were based on pattern, angle, measurement, count, and cost. First, he determined which boards to use based on not only quality of lumber, but also the specific dimensions required by both the situation (i.e. expected placement of the boards) and the relevant building codes. He discussed the strength of the board in terms of its flexibility in response to a lateral force as a reason for using the boards in a certain configuration. He then began to measure and count in order to determine how many spindles would be placed between each of the larger railing supports, and how those spindles were to be spaced in order to have an appearance of symmetry and visually pleasing proportion. Here, he again deferred to the maximum spacing allowed within the building code for purposes of safety, while pointing out that coming close to this maximum without surpassing it would ensure the most cost effective use of materials in terms of the spindles purchased. The next issue was the marking, then drilling of holes to set the spindles in the wood - both the diameter and the depth of each hole (actually a cylinder) was of issue, and Jason demonstrated the process of measuring and selecting the appropriate drill bit for creating the perfect resting place for the bottom of each spindle within the top and bottom boards. Finally, he showed the process of attaching the bottom and top rails to the vertical posts at either end of each section. Here the angle at which each screw was set became a crucial factor in how well attached the two boards would ultimately be. At one point during the filming, we departed from the specific task at hand to discuss some of the key decisions guiding the construction of the deck as a whole. Jason's deck is destined to be very large - 16' x 25' on one side and 12' x 25' on another. The joists were spaced a consistent 16" apart, running parallel to the exterior wall of the house, and the boards were laid perpendicular to these joists, each one running from the house on one end to the farthest edge of the deck on the other. Jason pointed out that these lengths - 16' and 12' - were chosen to maximize the efficient use of wood, as both are common lengths in which boards can be purchased. He pointed out that carpentry dictates allowing 5-7% waste in general for a project (when cuts are being made), so any "tricks" that can minimize this amount are ways to keep money in the carpenter's pocket. In fact, later in our interview Jason pointed out the fact that 20% of the time, part of a project is either done incorrectly, or simply not possible to be done in the way planned, so that a carpenter must "return to the drawing board". Here the carpenter's problem-solving skills - closely linked to his spatial reasoning ability - become a crucial factor in saving him time, materials, and ultimately, money.
Data Analysis & Synthesis
The interview process led to some interesting discussion of Jason's use of mathematical ideas on the job. Many of the basic mathematical ideas present in carpentry were discussed. For instance, he claimed that fractions are always preferable to decimals in measurement, due to the markings present on rulers and tape measures. Another idea of interest is the fact that a carpenter considers a calculator another of his standard tools, and often has a calculator to quickly convert from one unit of measurement to another. Also, since the observers had noticed extensive use of both estimation and exact calculation, Jason was asked to describe the appropriate application of both. His comment on the subject was that, "the expression measure twice cut once is â€¦ well, more like measure four times, then get someone else to come double-check your measurement â€¦ then cut." In measurement, he said, exact is always necessary. Estimation was more acceptable in cases such as purchasing materials which could be used elsewhere if not necessary or "rough framing" a project versus putting on the finishing - and more exact - details. However, while the interview covered every possible mathematical facet of the job, much of the discussion hinged on the fact that Jason has, unlike many carpenters, the experience with higher mathematics which enables him to both recognize its applicability in a given situation and put it into practice.
When asked if he considered this a professional benefit, perhaps something which gives him a "leg up" on the competition, his surprising answer was that while his mathematical background does have an impact on the way he does things - a better math intuition about the way different elements of a structure interrelate, for instance, he made it pretty clear that it's not necessarily an advantage. He commented that "learning math promoted thought patterns that help me now to think about carpentry in a different way" but in the next breath points out that his less-math-aware peers are able to do the same things he can, they just may not be able to give a ready explanation of why they expect a certain structural element to work well. One example of this given by Jason involved his enjoyment of and on-the-job use of trigonometry: Because he really likes trig, any time he's working with a situation that involves rise versus run - roof pitch, length vs. height of a stair - he uses right triangle trig to figure out the necessary measurements. By contrast, he has a friend who does not have the math background, and Jason comments, "he would just look at it and trace it out on the board and say it's got to look like this to work." Jason believes, however, that his peers without the math background are not handicapped in any way when it comes to being good carpenters. He points out that they have developed a situational sort of math intuition that effectively replaces any gaps in formal mathematical study - in fact, Jason believes the more mathematically educated carpenter has no real advantage over the one who is not to the extent that mathematical education can be a hindrance: "Sometimes I think it causes me to over-think the situation, when they'll jump right in and just do it." Another area of carpentry that he points out as being mathematically intensive is the calculation of materials needed. Here, again, Jason may be more likely to use math, but his more experienced peers are able to do the calculation just as accurately. In fact, Jason lists this arena as one of the components of his job in which he has learned the most from his peers: "â€¦ there are ways to estimate material costs and material needs off a given measurement that you just kind of pick up from other carpenters." All in all, while it seems clear that carpentry is incredibly math intensive, it seems that its link to classroom mathematics is almost coincidental and even irrelevant in the course of daily practice.
The ramifications of our study for classroom mathematics are many. While our carpenter didn't seem to feel that a stronger tie between the classroom and the carpentry workshop would necessarily create better carpenters, it seems almost trivially obvious to recognize that a stronger tie between the classroom and the carpentry workshop could, in fact, create better classroom mathematicians by enhancing both the interest level and the likelihood of "buy-in" influenced by a sense of relevance. In an era in which "shop classes" or "industrial arts" classes are becoming difficult to find in our high schools, perhaps, it's not surprising that students seem to feel increasingly disconnected from mathematics. Particularly in the case of students who are not likely to pursue a college education - the very students our school systems seem to be in the process of marginalizing to the point of practically pushing them out the doors, the absence of any meaningful application for their mathematical skills, coupled with the fact that they don't expect to be using the math in later educational settings, makes the study of math completely pointless. Jason hit upon a very basic human desire when he pointed out that people generally have a great appreciation for carpentry in that "you've constructed something out of nothing and it looks really cool." One hundred years ago, our children were raised "learning to create" - create a garden, a barn, a table, a dress. Now, perhaps, we've moved so far away from this idea of the foundation of education that we're setting our children adrift with no foundation on which to build a sense of personal competency. Perhaps bringing carpentry into our mathematics instruction is a way of remedying that situation in some small part. As pointed out by Schoenfeld (1988, p. 2), mathematics educators need to use "broader definitions of mathematical understanding than 'mastering symbol manipulation procedures'," since there are "dangers to the narrow assessments of competency that are currently employed." In many classrooms, competence can be adequately displayed simply by imitating the procedures of the teacher and by repeating symbolic manipulations in learned algorithms."