08. Teaching and Learning Functions

        Functions are used in every day life, even most people don't recognize them.  Algebraic tools express these relationships very efficiently.  These same tools allow much more complex problems to be solved.  Students often disagree with this description of functions - as they see the algebra behind them as hopelessly complex.  The teaching of functions usually violates all three principles of learning (learner, assessment, knowledge).  Students bring in a great deal of relevant knowledge (principle 1), however the link is rarely drawn.  Students rarely understand that when dealing with functions, they are being exposed to a whole new set of problems of dependency, which is a basis of principle 2, which states that both conceptual and procedural knowledge are important for learning.  For example, instead of just knowing how to create an equation from a problem, students must understand that problems can be represented in multiple ways and that these are showing the same relationship.  Students also need to develop the metacognitive abilities to monitor their own learning and understand to help them as they solve problems (principle 3).

Principle #1 - Building Prior Knowledge

        A "bridging context" means choosing a familiar topic to introduce a concept so that students can bring in relevant knowledge.  In this case, a walkathon was used.  The teacher introduces the rule, "for every mile I walk, I raise 1 dollar."  She then asks how much is made for 2 miles, and so on, slowly building a table.  Then they start graphing it together and building an equation, until ultimately they reach y=1x+0 (or the more generalizable y=ax+b) after talking about the slope (rate of change), initial value, etc.  The point here is that students already come to the problem with informal knowledge.  The point is to help them formalize it and this process needs to be shown, rather than just jumping straight to abstract formulas.  

Principle #2 - Building Conceptual Understanding, Procedural Fluency, and Connected Knowledge

        The goal is to get students to understand the core conceptual framework of functions, that one variable is dependent on the other, and that this can be represented with words, equations, graphs, and tables.  4 levels of understanding were identified in the development of function.  In general, level 0 contains seeing patterns, but no relation between spatial and numeric representations.  During level 1, students can find the common additive or multiplier in a sequence, and also record these into a table.  They can also, spatially, understand things like maintaining equal distances on the x-axis, and understanding that these are quantitative.  They can also make connections between the different representations, such as a line increasing by 1 just like the table.  In level 2, students can use the generalized functions y=mx+b and even y=x^n+b.  At level 3, they must understand different forms of functions, such as quadratics, differentiating the four quadrants with their relationships to one another, and in general an integrated sense of function.  

Principle #3: Building Resourceful, Self-Regulating Problem Solvers

        Teaching and developing students' understanding means not just thinking with procedures and concepts, but also about them.  This entails solving problems in more than one way.  It also entails checking the consistency of numeric answers using verbal representations.  Finally, it also involves teaching students to know when they've reached the limits of their understanding, and when they need support.  


        The following unit shows how instruction can be developed for deep understanding.  Compared with students in even 8th and 10th grade classes learning in traditional settings, the students in a 6th grade class outperformed them when using this approach.  

Curriculum for Moving Students Through this Model

        The curriculum requires about 650 minutes of instructional time to complete.  A single bridging context is used throughout, instead of multiple situations.  Emphasis is placed on moving around the multiple representations and their relationships.  Formal notation is progressively developed, instead of from the outset.  

Example Lessons

        3 lessons are explained: lesson 1 focuses on principle #1, lesson 2 on principle #2, and lesson 3 on principle #3.  Teh topics of slope, y-intercept, and quadratic functions are covered.  
        In example lesson 1, after students already understood how as miles goes up with money (from the walkathon example above) the term slope is introduced to denote the "up-by" amount.  This is related to stairs, and it is pointed out how the higher slope means steeper stairs.  After creating tables and graphs for the $1 amount, students develop other amounts greater than or less than y=x.  An algorithm for finding slope is not given, but students are asked to find the steepness or slope.  There is also some talk of using negative slope, such as by seeing it from the perspective of the donor.  Three forms of prior knowledge are applied.  First, the walkathon context.  Second, the "up-by" term to mean slope.  And third, initial numeric and spatial understanding.  
        In example lesson 2, y=intercept is taught.  This is typically introduced by substituting x=0, which helps students learn the formal process.  Instead, there is a focus on connecting procedural knowledge with conceptual knowledge.  The concept is introduced as a bonus for starting the walkathon.  Just like in lesson 1, they must build graphs for the function, however in this case they must take into account the initial value.  Different initial values are provided, with students asked to predict where each value will be relative to the first function given.  The point that the change involves a vertical shift in the graph is made for higher initial values is pointed out (and reverse for negative initial values).  A good follow up activity is asking students to invent 2 functions that allow them to earn $153 after 10 miles in tables, graphs, and equations.  
        ln lesson 3, operating on y=x^2 is taught.  Students work on a computer program that automatically graphs their functions.  They are challenged to make graphs fit the plotted points by manipulating the equations.  They can manipulate any part of y=ax^n+b, and are asked to describe what happens and why.  They are also asked to compare tables and graphs of equations such as y=2x^2 and y=3x^2.  Metacognitive activity is abundant in these examples, as students need to predict, error detect, and correct.  They are constantly adjusting what they need to do in order to solve the problems.  


        There is often in traditional education an "ungrounded competence", where students know the procedural techniques but don't really understand what is going on.  It is important to give students the deep conceptual understanding, yet still maintain the highest standards.  


The routine focuses first on identifying something interesting about an object or idea:

"I notice that..."

And then following that observation with the question:

"Why is it that way?"
"Why did it happen that way?"

I notice that  the principles of learning overlap quite a bit, in that it is hard to design a lesson for principle #1 without touching upon principle #2.  

It is that way because there is a deeper conceptual framework that makes students active in the sense making process.  To do this, the various principles come out.  Certainly they are a good guide to follow to get students to be active sense makers.