Source:
National Research Council of the National Academies. (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press. (Chapters Introduction, 5-8).
Chapter 8 - Teaching and Learning Functions (Mindy Kalchman and Kenneth R. Koedinger)
1. How do the principles of How People Learn apply to the development of Functions?
2. How do we teach it?
3. Is it effective?
1. Addressing the 3 Principles
1.1. Currently, teaching instruction is very procedural based and does not take into account the 3 principles of HSL, especially not taking into account the importance of conceptual understanding. (p. 353)
1.2. Teaching, including instructional approaches and curricula, needs to reflect the HSL principles in order for students to be able to learn and understand the complex mathematical concept of functions. (p. 358)
1.3. Principle #1: Building on Prior Knowledge
1.3.1. Emphasizes the importance of students and teachers continually making links between students’ experiences outside the math classroom and their school learning experiences
1.3.2. Understandings students bring can be viewed in 2 ways and teaching should make use of both:
1.3.2.1.Everyday, informal, experiential, out-of-school knowledge
1.3.2.2.School-based or “instructional” knowledge
1.3.3. Students in this instructional approach are introduced to function in the context of a walkathon chosen because:
1.3.3.1.Students are familiar with money and distance as variable quantities.
1.3.3.2.They understand the contingency relationship between variables.
1.3.3.3.They are interested in and motivated by the rate at which money is earned.
1.3.4. It is critical to use a powerful instructional context, called a “bridging context”, which serves to bridge students’ numeric(equations) and spatial(graphic) understandings and link their everyday experiences to lessons in the mathematics classroom.
1.3.5. Walkathon context as introduction to functions in multiple forms.
1.3.5.1.Walkathon = real world
1.3.5.2.Table, graph for representations.
1.3.5.3.$1 for each kilometer equiv to $1 = 1*km equiv to y = x*1 (p. 359)
1.3.6. Slope
1.3.6.1.The topic is usually reserved for 9th grade mathematics and is part of students’ introduction to relations and functions in general and to linear functions in particular.
1.3.6.2.Defined as ratio of vertical distance to horizontal difference, or “rise to run”.
1.3.6.3.Slope = rise/run
1.3.6.4.Once the equation for a line has been introduced (y=mx+b), m is defined as the slope and is calculated using m = (y2 – y1)/(x2 – x1)
1.3.6.5.For students to understand slope in these definitional and symbolic ways, they must have a lot of formal knowledge, including: (p. 362)
1.3.6.5.1. Meaning of ratio, coordinate graphing, variables, subscripts
1.3.6.5.2. Skills such as solving equations in 2 variables and combining arithmetic operations.
1.3.6.6.Knowing the definition does not ensure actual understanding of the meaning of slope with regard to steepness or rate of change.
1.3.6.7.Younger students have intuitive and experiential understandings of slope that can be used as a basis for formal learning that involves conventional notations, algorithms, and definitions.
1.3.6.7.1. When given a word problem, students had a better chance of finding the slope without receiving instruction than students given the textbook definition of slope.
1.3.6.7.2. This is not to say that problems should be phrased in “student language”, as it is important for students to learn formal mathematic terminology and abstract algebraic symbolism.
1.3.6.7.3. The point is that using student language is a way to assess what knowledge students are bringing to a topic, and then linking and building on what they know to guide them towards a deeper understanding of formal mathematical terms, algorithms, and symbols.
1.3.7. Students’ prior knowledge acts as a building block for development of more sophisticated ways of thinking mathematically.
1.3.8. We may sometimes underestimate the knowledge and skills students bring to the learning of functions (p. 363) that may serve as basis for formalizing students’ thinking.
1.4. Principle #2: Building Conceptual Understanding, Procedural Fluency, and Connected Knowledge
1.4.1. The purpose is to simultaneously develop conceptual understanding and procedural fluency, and helping students connect and organize knowledge in various forms.
1.4.2. Students can develop surface facility with notations, words, and methods of domain study without having a foundation of understanding. This prevents students from utilizing mathematics and making number sense in the real-world.
1.4.3. Want students to understand the core concept of a functional relationship: that the value of one variable is dependent on the value of another.
1.4.4. Want students to understand that a function can be expressed in a variety of ways, which all have the same meaning (graphs, equations, tables, etc.)
1.4.5. Need to deliberately build and secure that knowledge so that a student can confidently tackle sophisticated problems.
1.4.6. Good teaching requires a solid understanding of the content domain and specific knowledge of student development.
1.4.7. Developmental model that this instructional approach is based on has 4 levels (0-3) each describing what students can typically do at a given developmental stage.
1.4.7.1.Level 0
1.4.7.1.1. Characterizes the kinds of numeric/symbolic and spatial understandings students typically bring to learning functions.
1.4.7.1.2. Initially numeric and spatial understandings are separate.
1.4.7.1.3. Initial numeric understanding – students can iteratively compute within a string of positive whole numbers. (p. 364)
1.4.7.1.4. Initial spatial understanding – students can represent relative sizes of quantities on graph and perceive patterns of qualitative changes in amount by left-to-right visual scan of the graph, but cannot quantify those changes.
1.4.7.2.Level 1
1.4.7.2.1. Students begin to elaborate and integrate their initial numeric and spatial understandings of functions.
1.4.7.2.2. Numeric understanding:
1.4.7.2.2.1.Students iteratively apply a single operation to, rather than within, a string of numbers to generate a second string of numbers.
1.4.7.2.2.2.Construct an algebraic expression for this repeated operation.
1.4.7.2.3. Spatial Understanding
1.4.7.2.3.1.Students progress from understanding graphs with verbal or categorical values along the x-axis to understanding graphs with quantitative values along the x-axis. (p. 367)
1.4.7.2.3.2.Use continuous quantities along a horizontal axis. Without this, students cannot see graphs as representing the relationship between 2 changing quantities. (p. 367)
1.4.7.2.3.3.Perceive emergent properties, such as linear or increasing, in the shape of the line drawn between points.
1.4.7.2.4. Integration of elaborated understandings (p. 365) – See the relationship between the differences in the y-column in a table and the size of the step from one point to the next in the associated graph.
1.4.7.2.5. Interpret algebraic representations both numerically and spatially.
1.4.7.2.6. Grasping the why and how the line on a graph maps onto the relationship described in a word problem or equation is a core conceptual understanding. If students’ understanding is only procedural, they will not be well prepared for the next level. (p. 368)
1.4.7.3.Level 2
1.4.7.3.1. Elaborate initial integrated numeric and spatial understandings to create more sophisticated variations.
1.4.7.3.2. Integrate understanding of y=x and y=mx + b to form a mental structure for linear functions.
1.4.7.3.3. Integrate rational numbers and negative integers.
1.4.7.3.4. Form mental structures for other families of functions, such as y=x^n + b.
1.4.7.4.Level 3
1.4.7.4.1. Integrate variant (e.g. linear and nonlinear) structures developed at level 2 to create higher-order structures for understanding more-complex functions, such as polynomials and exponential and reciprocal functions.
1.4.7.4.2. Elaborate understandings of graphs and negative integers to differentiate the four quadrants of the Cartesian plane
1.4.7.4.3. Understand the relationships of these quadrants to each other. (p. 366)
1.4.7.4.4. Without the development of a sound conceptual frame work for functions, students do not make sense of their work. For example, student’s graph has a negative slope and a negative y-intercept and his table had values without a constant rate of change for a function with a y-intercept of -10 and a positive slope. Student did not recognize the impossibility.
1.4.7.4.5. A teacher can help support integration of concepts by moving fluidly and rapidly between numeric and spatial representations. This helps students simultaneously build understandings of each representation individually, and in an integrated nature.
1.4.7.4.6. Integration helps students begin to understand and organize their knowledge in ways that facilitate the retrieval and application of relevant mathematical concepts and procedures.
1.4.7.4.7. If students’ spatial and numeric understandings are not integrated they may not notice when conclusions drawn from each understanding are inconsistent. (p. 369)
1.5. Principle #3: Building Resourceful, Self-Regulating Problem Solvers
1.5.1. Instruction should assist students not only with mathematical procedures and concepts, but also in thinking about procedure and concepts in reflecting on and articulating their own thinking and learning.
1.5.2. Encouraging students to reflect on and communicate their ideas about functions supports (p. 371) them in making connections among representations that are necessary for flexible, fluent, and reliable performance.
1.5.3. An especially important type of metacognitive thinking in mathematics is coordinating conclusions drawn from alternate mathematical representations.
1.5.3.1.Solve problem in more than one way.
1.5.3.2.Checking consistency of verbal interpretations of different representations. (Ex: Increasing versus decreasing slope.)
1.5.3.2.1. Encourages students to think about problems in (p. 372) multiple ways (strategies) and with multiple tools (representation).
1.5.3.2.2. Encourages students to draw conclusions that are not only quantitative, but also qualitative (verbal interpretations).
1.5.4. Must also create an atmosphere where students feel comfortable to explore, experiment, and take risks in problem solving and learning.
1.5.5. Teacher must also help students develop a tolerance for the difficulties mathematics sometimes presents.
1.5.6. Teacher must also help students learn when they’ve reached their limits in understanding and how to ask for help or find help on their own (peer, computer, calculator, etc.)
2. Teaching Functions for Understanding
2.1. In addition to knowledge of the content being taught and a development model of how students acquire understanding of the content, an instructional set of strategies for moving students along the developmental pathway and addressing obstacles and opportunities that appear most frequently on the way is required.
2.2. The below method has been successful in increasing understanding of functions for 8th and 10th graders. In fact 6th graders taught with this approach were more successful on a function test than 8th and 10th graders taught with conventional instruction.
2.3. Curriculum For Moving Students Through the Model
2.3.1. Encourages students to:
2.3.1.1.Build upon and apply prior knowledge. (p. 373)
2.3.1.2.Construct an integrated conceptual framework for understanding functions.
2.3.1.3.Apply metacognitive skills to their learning.
2.3.2. Used effectively for students in 6th, 8th, 10th, 11th grades.
2.3.3. Unit requires approximately 650 minutes of class time to complete.
2.3.4. Regardless of grade, it is recommended that you go through the whole sequence of lessons in order to develop and utilize deeper and proper understandings.
2.3.5. This instruction is different than traditional instruction in that it uses contextual bridges as opposed to different contexts for different situations. This allows for an easier understanding of relationships between representations.
2.3.6. This instruction also differs in that it focuses on multiple representations (co-presented) instead of singular ones. In traditional instruction, singular representations may be taught for weeks before different representations are related.
2.3.7. Engages students in the construction of functional notations and helps them build notations and meaning for constructs. Traditional instruction just gives students formal notation (p. 374) and then gives them procedures for finding the “answers”.
2.3.8. Over the course of instruction, students progressively formalize their initial notations until those notations correspond with conventional general equations (such as y = mx + b).
2.3.9. Suggests follow-up activities that allow students to remain situated in the context of instruction for the first part of the unit until they are confident and competent with the concepts on a more abstract basis.
2.3.10. Then when students move up to the computer environment, no new concepts are introduced initially. Students first have time to consolidate the individual concepts of the first part of the unit, then move on to more challenging activities that advance their thinking and understanding in the domain.
2.3.11. Students give presentations on a particular kind of function to their classmates in order to show their understanding and share their expertise in key characteristics and behaviors of those functions. (p. 375)
2.3.12. Example Lessons
2.3.12.1. Level 1
2.3.12.1.1. Introduction – Walkathon
2.3.12.1.1.1. Description – Students record in tables the money earned for each km walked and plot each pair of values for a variety of rules. An equation is formed using km and $ based on the rule of sponsorship.
2.3.12.1.1.2. Activities – Student pairs invent 2 of their own sponsorship arrangements for which their partner constructs tables, graphs, and equations.
2.3.12.1.2. Slope
2.3.12.1.2.1. Description – Introduced as the up-by (or down-by) amount between successive $ values in a table or graph. It is the relative steepness of a function and the amount by which each km is multiplied.
2.3.12.1.2.2. Lesson – big focus on prior knowledge(p. 379)
2.3.12.1.2.2.1. Approx 90 minutes (p. 379)
2.3.12.1.2.2.2. Suggest beginning with the graph and the table for the rule of earning $1 for every km walked. (p. 379)
2.3.12.1.2.2.3. Draw the students’ attention to the fact that the up-by amount corresponds to the mathematical concept of slope and that is a relative measure of a function’s steepness. (p. 379)
2.3.12.1.2.2.4. Use y=x as a conceptual landmark. (p. 379)
2.3.12.1.2.2.5. Challenge students to provide sponsor rules that are steeper, less steep than y=x. Encourage students to plot them on the same set of axes. Ask students to predict the steepness. (p. 379)
2.3.12.1.2.2.6. Ask students to invent other rules and make tables and graphs for them. (p. 380)
2.3.12.1.2.2.7. Ask students to summarize their findings. (p. 380)
2.3.12.1.2.2.8. Ask students how the steepness of a function can be quantified. (p. 380)
2.3.12.1.2.2.9. Introduce working with negative slopes by using the perspective of the person donating the $ (sponsor). (p. 380)
2.3.12.1.2.3. Activities – Students are asked to find the slope of several different functions expressed in tables, graphs, and equations.
2.3.12.1.3. Y-Intercept
2.3.12.1.3.1. Description – Introduced as the “starter-offer”, a fixed starting bonus students receive before the walkathon begins. It affects only the vertical starting point of the numeric sequence and graph. It does not affect the steepness or shape of the line.
2.3.12.1.3.2. Lesson (p. 381)
2.3.12.1.3.2.1. Theory-based instructional design in connecting students’ factual/procedural and conceptual knowledge. (p. 381)
2.3.12.1.3.2.2. Instead of using x=0, it focuses on other numbers to develop a better understanding. (p. 381)
2.3.12.1.3.2.3. Approx 90 minutes (p. 381)
2.3.12.1.3.2.4. Bonus or additional amount of money sponsored before the walkathon begins – “starter-offer”. (p. 381)
2.3.12.1.3.2.5. Students are asked to construct tables of values and then graph them, then formulate an equation. (Ex. $5 bonus, and $1 for every km) (p. 381)
2.3.12.1.3.2.6. Have them verbally describe the relationship to help formulate the equation. (p. 381)
2.3.12.1.3.2.7. Offer new starter amounts with $1/km and ask students to predict where each new function will be on the graph, then construct the new tables, graphs, and equations. (p. 382)
2.3.12.1.3.2.8. Students are asked to describe patterns and characteristics they see. (p. 382)
2.3.12.1.3.2.9. Want students to see that the only effect of changing the starting offer is a vertical shift in the graph. Students tend to confuse slope and y-intercept. (p. 382)
2.3.12.1.3.2.10. Negative y-intercepts are introduced using the concept of debt. (p. 382)
2.3.12.1.3.2.11. Allow students to be informal in their notation until they are able to consolidate and formalize the concepts. (p. 383)
2.3.12.1.3.2.12. In lessons on non-linear functions, the starter-offer idea is also applied. (p. 383)
2.3.12.1.3.3. Activities - Students invent 2 linear functions that allow them to earn exactly $153 after walking 10 km. Students record the slope and y-intercept of each function and explain how the y-intercept of each function can be found in its table, graph, and equation.
2.3.12.1.4. Curving Functions
2.3.12.1.4.1. Description – Nonlinear functions are introduced as those having up-by amounts that increase (or decrease) after each km walked. They are derived by multiplying the km (x) by itself at least once. The more times x is multiplied by itself, the greater is the difference between $ values and thus the steeper the curve.
2.3.12.1.4.2. Activities – Students are asked to decide which of 4 functions expressed in tables are nonlinear and explain their reasoning. Asked to write an equation for and to sketch the label the graph of each function. Students were asked to come up with a curved-line function for earning $153 over 10 km. (p. 376)
2.3.12.2. Level 2 and 3
2.3.12.2.1. Computer Activities
2.3.12.2.1.1. Description – Use spreadsheet technology and prepared files and activity sheets to consolidate and extend the understandings about slope, y-intercept, and linearity (level 2). Work in all 4 quadrants to transform quadratic and cubic functions and explore their properties, behaviors, and characteristics of exponential, reciprocal, and other polynomial functions.
2.3.12.2.1.2. Lesson – big on metacognitive abilities (p. 387)
2.3.12.2.1.3. Activities – Students change the steepness, y-intercept, and direction of y=x and y=x^2 to make the function go through preplotted points. They record the numeric, algebraic, and graphic effects of their changes. They invent functions with specific attributes, such as parallel and y-intercept below x-axis, or inverted parabola that is compressed and in the lower left-hand quadrant.
2.3.12.2.2. Presentations
2.3.12.2.2.1. Description – groups of students investigate and then prepare a presentation about a particular type of function. This stimulates discussion and summarization of key concepts and serve as a partial teacher assessment for evaluating students’ post-instruction understanding about functions.
2.3.12.2.2.2. Activities – Groups of students use computer generated output of graphs, equations, and tables to illustrate a particular type of function’s general properties and behaviors. Students give presentations about their function and share expertise with their classmates. (p. 377)
2.4. Alternate development models
2.4.1. Jasper Woodbury Series – more complex, real-world contexts
2.4.2. Cognitive Tutor Algebra (previously called PUMP) – uses multiple real-world contexts and computer support.
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