Abstract
In this paper we share a classroom implementation of a task about basis in linear algebra, which was originally developed for research about the topic. The task asks students to construct an everyday situation that captures the definition of basis, and then to critique it mathematically. Using this task, the original research study uncovered learning resources from a group of undergraduate women of color. A mathematician who was not involved in the original study was given the opportunity to work with a group of underrepresented minority students in a linear algebra course. She was inspired by the findings of the study and decided to implement the task independently in her course. She shares how she did it, how her students responded to the task, and how it helped her further develop her understanding of an anti-deficit perspective in teaching mathematics.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 520-538 |
| Number of pages | 19 |
| Journal | PRIMUS |
| Volume | 30 |
| Issue number | 5 |
| DOIs | |
| State | Published - May 27 2020 |
Keywords
- Linear algebra
- anti-deficit teaching
- basis
- equity
ASJC Scopus subject areas
- General Mathematics
- Education
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In: PRIMUS, Vol. 30, No. 5, 27.05.2020, p. 520-538.
Research output: Contribution to journal › Article › peer-review
}
TY - JOUR
T1 - Everyday Examples About Basis From Students
T2 - An Anti-Deficit Approach in the Classroom
AU - Adiredja, Aditya P.
AU - Bélanger-Rioux, Rosalie
AU - Zandieh, Michelle
N1 - Funding Information: This material is based upon work supported by the National Science Foundation under grant numbers DUE-1246083 and 1712524. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. This material is based upon work supported by the National Science Foundation under grant numbers DUE-1246083 and 1712524. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. Table 1. Students’ everyday examples and their analysis Everyday context Students’ analysis City grids in cardinal directions for walking and biking, and intercardinal directions of subway train tunnels as two bases to get around New York City. Movement is constrained by the directions. Both bases are in ℝ 2. One can reach any point in the city (in ℝ 2) using both bases. If roads are blocked or stations are not in service, that can be a loss of a basis vector. In real life, walking is not constrained by the cardinal directions, and subway stations do not exist in every block. Categories of clothing in the closet (e.g., tops, pants, jackets) is a basis for a range of outfits. All the categories (vectors) are independent from one another. One can combine these vectors to make outfits. To ensure that all the vectors are of the same size, it assumes that all categories have the same elements. This is not true (e.g., people have more socks than pants). Combining vectors is confusing because it assumes a 1-to-1 correspondence between each item in the category with others, whereas one top can go with multiple pants. Bricks as a basis to create a courtyard. Different sizes of bricks would make a different courtyard. Different bricks can be added together in different combinations to make a space. The example is limited to two-dimensional space. Different items or ingredients (e.g., rice, beans, meat) as a basis for a Chipotle burrito bowl. Each item is independent, e.g., one cannot get lettuce from a tomato. One can choose to not use an ingredient to scale a vector to zero. Combining these items represents a linear combination. The quantity of each ingredient is seen as a scalar multiple of that vector. A vegetarian burrito bowl is a subspace of all burrito bowls because it excludes meat. A set of independent actions (e.g., brushing teeth, making the bed, getting dressed) as a basis to define a person’s morning. They are independent because one cannot achieve the result of one action by doing another or a combination of other actions (e.g., cannot brush teeth by making the bed). There is an order to when I perform these actions, thus they are not truly independent. These actions span my whole morning as basis vectors span the whole space. One can scale “breakfast” by eating a bigger breakfast, and scale “making the bed” by zero by not making the bed. Basic human needs (e.g., food and shelter) as a basis for human lives. “It represents the most simplified version that describes the larger body.” There are other non-essential or “luxurious” needs (e.g., economic and physical well-being) in the space. “The basis is the most minimal or the ‘starting point’ from which other components are derived.” Food and shelter might not truly be independent as one might trade one for the other. These basic needs do not span the whole “human needs” (e.g., they cannot be combined to guarantee an education). Three movement commands (move forward, turn right, and move up) as a basis to travel in a world. Revised: The four knobs in a stationary crane cab to get the crane jib into the proper location. “With these 3 commands you can ‘go’ anywhere in this world.” “We can’t use any of the 3 commands to create another” (independence). One can use a negative scalar. Unsure about the interpretation of “negative/reverse” step. One knob is for positive and negative movement. And the other three knobs for upward, forward, and to the right. A selection of basic paint colors in a paintbox as a basis for different colored paintings. One can “describe any painting by describing it as a combination or mix of some paints from the paintbox.” “Isn’t exactly able to capture how a vector in the basis could be multiplied by any scalar and still be included in the space.” “Vectors can be of infinite length, but you can’t have an infinite amount of blue paint.” It can describe different spaces as a result of using a different basis (e.g., a strictly red painting or black and white painting). Some colors can be combined to make another color (blue and yellow to make green). “The paintbox is limited to only a few colors, while in reality there is an endless amount of vectors that could form an infinite amount of basis.” Length, width, and height measurements as a basis to determine appropriate location for an object in a 3D space Revised: Mapping locations of different items (e.g., light fixtures) in home decorating. “We can theoretically describe the location of any part of the room using a coordinate system leg (height, width, length).” The example is limited because it requires that the basis is orthonormal because the measurements are perpendicular to one another. The room can be described using a non-orthonormal basis. One measurement cannot be replaced by another. “Negative width and length would mean that something is behind the walls of the room; negative height would imply that something is located below the floor of the room.” Realistically there is no reason to use negative coordinates, because “one could easily reset the coordinate system so that any object in any given room in the house could be represented with positive values.” Different majors as a basis for a college. The majors in a college are all “different” and thus are independent. But since some majors are similar, they might not be “completely linearly independent.” You can double major to make up your college education (i.e., linear combination of majors). Going to graduate school can be seen as multiplying the vector by a scalar as you can have 3 times the knowledge as from your major. The three primary colors (red, yellow, and blue) as a basis for any other color. “You can mix different amounts of these colors (and different ratios) to produce any other color.” “Any color we mix represents a vector in the span of our basis.” “The ratio of each color is the scalar we multiply to each vector.” “You can’t get 1 primary color from the other 2.” This captures linear independence. It does not capture dimensions. It is supposed to span R 3 but that is meaningless with colors. Publisher Copyright: © 2019, © 2019 Taylor & Francis Group, LLC.
PY - 2020/5/27
Y1 - 2020/5/27
N2 - In this paper we share a classroom implementation of a task about basis in linear algebra, which was originally developed for research about the topic. The task asks students to construct an everyday situation that captures the definition of basis, and then to critique it mathematically. Using this task, the original research study uncovered learning resources from a group of undergraduate women of color. A mathematician who was not involved in the original study was given the opportunity to work with a group of underrepresented minority students in a linear algebra course. She was inspired by the findings of the study and decided to implement the task independently in her course. She shares how she did it, how her students responded to the task, and how it helped her further develop her understanding of an anti-deficit perspective in teaching mathematics.
AB - In this paper we share a classroom implementation of a task about basis in linear algebra, which was originally developed for research about the topic. The task asks students to construct an everyday situation that captures the definition of basis, and then to critique it mathematically. Using this task, the original research study uncovered learning resources from a group of undergraduate women of color. A mathematician who was not involved in the original study was given the opportunity to work with a group of underrepresented minority students in a linear algebra course. She was inspired by the findings of the study and decided to implement the task independently in her course. She shares how she did it, how her students responded to the task, and how it helped her further develop her understanding of an anti-deficit perspective in teaching mathematics.
KW - Linear algebra
KW - anti-deficit teaching
KW - basis
KW - equity
UR - http://www.scopus.com/inward/record.url?scp=85067879482&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85067879482&partnerID=8YFLogxK
U2 - 10.1080/10511970.2019.1608609
DO - 10.1080/10511970.2019.1608609
M3 - Article
AN - SCOPUS:85067879482
SN - 1051-1970
VL - 30
SP - 520
EP - 538
JO - PRIMUS
JF - PRIMUS
IS - 5
ER -