Common Core Conundrum: Absolute Value

I should begin by first disclaiming that while I'm generally pro-standards, I'm also somewhat agnostic about standards. How can that be? Without going into much detail, I believe (a) what we "count" as mathematics is socially determined and standards documents are just part of that determination, (b) I will never perfectly agree with a standards document, but the amount of agreement should not be underestimated, and (c) if there's any power to standards, it's in how they're implemented -- and "good implementation" is very likely to be seen as just "good teaching" under a different set of standards, or no standards at all. It is with this mindset that I watch with some amusement (and sometimes, disappointment) arguments against the Common Core State Standards because of some poorly designed accountability measure. I'm pretty sure poorly designed accountability measures would be a concern right now regardless of the standards in place.

Still, I occasionally see things in the Common Core math standards (CCSSM) that make me stop and wonder, "How are teachers going to deal with this?" One recent instance of that was with how the CCSSM addresses the topic of absolute value. As I looked through Discovering Algebra: An Investigative Approach, I saw the typical "distance from zero" definition followed by an investigation that included this rather unhelpful picture and caption:

(Clearly portrays Elvis? The middle guy looks like Matthew Perry joined a Vegas lion-taming act.)

From there, Discovering Algebra presents students with equations to solve, such as:

$$ | x - 2| + 7 = 12 $$

But do the CCSSM call for solving these kinds of equations? I went searching through the CCSSM for all mentions I could find of absolute value, and here's what I found.

Grade 6

CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers.
CCSS.Math.Content.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
CCSS.Math.Content.6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that –3oC is warmer than –7 oC.
CCSS.Math.Content.6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
CCSS.Math.Content.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
CCSS.Math.Content.6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

and

CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

If all goes according to plan, there shouldn't be a need to define absolute value in 8th or 9th grade algebra, as it will already have been introduced in 6th grade. I know there are concerns about content being "developmentally inappropriate" for lower grade levels in the CCSSM, but without any personal data to the contrary, I imagine students could come to understand absolute value along with understanding positive and negative numbers. The last standard above, from the moment I first saw it, has been a point of fascination for me. How will 6th grade teachers help students learn topics like interquartile range and mean absolute deviation? You need absolute value for mean absolute deviation, but I don't think that will be the hangup for those who struggle with that standard.

So where else is absolute value mentioned?

Grade 7

CCSS.Math.Content.7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

and

CCSS.Math.Content.7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Now we have absolute value as it relates to subtraction, and we get another mention of mean absolute deviation in the 7th grade statistics and probability standards. There's nothing yet about solving equations with absolute value or graphing absolute value functions.

Grade 8

Nothing. No mention of absolute value or mean absolute deviation.

High School

The CCSSM doesn't specify what specifically belongs in 9th grade versus other grades, but here's what I found across the whole of the HS CCSSM standards.

CCSS.Math.Content.HSN-VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

CCSS.Math.Content.HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

CCSS.Math.Content.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

The first of the three standards above, I think we can agree, goes well beyond what we'd expect to find in a traditional Algebra 1 class. The other two standards address the graphing of various kinds of functions, and depending on the activity might belong in either a traditional Algebra 1 or Algebra 2 course, or more likely, both. But do you notice what's not there? Solving equations with absolute value functions. Sorry, \( | x - 2 | + 7 = 12 \), but it looks like you didn't get invited to the CCSSM party. Even if you interpret the second standard to include solving absolute value equations, it only asks for an approximation or to use technology.

Conundrum

The researcher in me wonders what Algebra 1 teachers will do when they get to a lesson on absolute value. Will some just teach it out of habit? Will some think it was excluded due to a CCSSM oversight? Will some modify the lesson to emphasize the graphing of absolute value, but not solving absolute value equations? How many will not even notice it went missing from the standards? Will anyone be willing to kill the little darling?

For years we've lamented math curricula in the United States that was "a mile wide and an inch deep." Given a fixed amount of time, going deeper means going narrower, and there is evidence that the CCSSM supports that (Porter, McMaken, Hwang, & Yang, 2011). For reasons I perceive to be mostly political, the Common Core State Standards don't go out of their way to specifically address content omitted compared to previous standards documents. In 1989, NCTM made a list of areas of emphasis and de-emphasis and many an argument in the math wars was waged using false dichotomies rooted in those lists. I'm sure those who wrote the CCSSM didn't want a repeat of those arguments, so now math teachers and those who support them are left to dig through the standards and find these omissions on their own.

(By the way, if you're looking for an interesting lesson for understanding and graphing absolute value, NCTM is offering one from a recent Mathematics Teacher for complimentary download: Angela Wade's "Teaching Absolute Value Meaningfully.")

References

Porter, A. C., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core standards: The new U.S. intended curriculum. Educational Researcher, 40(3), 103–116. doi:10.3102/0013189X11405038

A First Day Statistics Activity

I have the honor of again teaching our undergraduate statistics course in the School of Education, better known here as EDUC 4716 Basic Statistical Methods. Perhaps the most interesting thing about the course is that it's not required for any education programs, minors, or certificates. Instead, the course attracts students largely from the Department of Speech, Language, and Hearing Sciences (who don't need it to graduate, but do need it to apply to grad school or, more recently, to get certified) and sociology majors. So how does this course end up in the School of Ed? Probably due to the legacy we have in quantitative methods, thanks to people like Robert Linn, Gene Glass, Lorrie Shepard, and now faculty like Derek Briggs and Greg Camilli. Somehow all of their hard work and success filters down and gives a relative stats-hack like me a chance to teach undergrads.

Many of my students are upperclassmen and have spent much of their college experience avoiding math courses. In fact, on last year's FCQ (Faculty Course Questionnaire) my students' average rating for the item "Personal interest in this subject prior to enrollment" was a 1.8 out of 6 -- a response the university tells me is at the 0th percentile across campus. I like to think of this as a great opportunity in a "nowhere to go but up" kind of way, a chance for me to change the way students think of mathematics and see themselves as mathematical beings. Then again, it's hard to make big changes in only 15 class meetings of 2.5 hours each. If I'm going to make a difference, class has to get off to a solid start.

My opening activity this year started with the preparation of four simple index cards with different distribution shapes:

Four common distributions, clockwise from top left: normal, left skewed, right skewed, and normal.

I have the benefit of a small class of 14 students. So I cut my graphs into a total of 14 pieces:

14 pieces for 14 students. Note on the bottom I've provided the hints A, B, C, and D.

When class started, I mixed up the graph pieces and handed one to each student. Then I told the class to find the other people in class who had the graph pieces that aligned with theirs. Once they had a completed graph, form a group at one of the tables and discuss which of the following they thought their group's graph might describe:

• People born each month of the year
• Student GPAs at this university
• Student heights at this university
• Starting salaries of new graduates from this university

It took my class less than 3-4 minutes to find their groups and then I gave them another 3-4 minutes to discuss what their graph shape might describe. As a class, I had each group share their ideas and then we discussed them. Not everybody agreed initially about which shape matched which description, which led into important comments about how we might think about unbiased sampling of students and imagining different scales and labels along the horizontal axes.

So in less than 15 minutes I combined group-making, statistics, and active, student-centered problem solving into one activity. This activity also gets students thinking about distribution shapes, which I sometimes worry we ignore in the rush to calculate centers and spreads. If you're wondering how to adapt this for your classroom, I offer these suggestions:

• If you have a few more students, cut more slices.
• If you have twice as many students, consider making two of each distribution shape and scaling the x-axis to match one of 8 potential descriptions. (i.e., a normal distribution scaled for heights in inches could be distinguished from one scaled for SAT scores.)
• If you want to use this for Algebra 1, you can make graphs that describe things like, "Toni walked to the bus stop at 2 mph, rode the bus at 30 mph to the bike shop, then rode a bike back home at 12 mph." Such an activity begins CPM's Algebra Connections and was the inspiration for my activity.
• If you want to use this for Algebra 2 or higher, you can use graphs of functions that students will become familiar with (parabolas, cubics, hyperbolas, etc.). I don't think it's worth fretting over vocabulary at this point -- just give students an opportunity to think about how the functions behave and what phenomena they could possibly model.

Settling slope and constructive Khan criticism

This was co-written with Frederick Peck, a fellow Ph.D. student in mathematics education at the University of Colorado at Boulder and the Freudenthal Institute US. We each have six years of experience teaching Algebra 1 and are engaged in research on how students understand slope and linear functions. Fred shares his research and curriculum at RMEintheClassroom.com.

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Sal Khan (CC BY-NC-ND Elvin Wong)

The Answer Sheet has recently been the focus of a lively debate pitting teacher and guest blogger Karim Kai Ani against the Khan Academy's Salman Khan. While Karim's initial post focused mainly on Sal Khan's pedagogical approach, Karim also took issue with the accuracy of Khan Academy videos. As an example, he pointed to the video on slope. Specifically, Karim claimed Sal's definition of slope as "rise over run" was a way to calculate slope, but wasn't, itself, a definition of slope. Rather, Karim argued, slope should be defined as "a rate that describes how two variables change in relation to one another." Sal promptly responded, saying Karim was incorrect, and that "slope actually is defined as change in y over change in x (or rise over run)." To bolster his case Sal referenced Wolfram Mathworld, and he encouraged Valerie Strauss to "seek out an impartial math professor" to help settle the debate. We believe that a better way to settle this would be to consult the published work of experts on slope.

Working on her dissertation in the mid-1990s, Sheryl Stump (now the Department Chairperson and a Professor of Mathematical Sciences at Ball State University) did some of the best work to date about how we define and conceive of slope. Stump (1999) found seven ways to interpret slope, including: (1) Geometric ratio, such as "rise over run" on a graph; (2) Algebraic ratio, such as "change in y over change in x"; (3) Physical property, referring to steepness; (4) Functional property, referring to the rate of change between two variables; (5) Parametric coefficient, referring to the "m" in the common equation for a line y=mx+b; (6) Trigonometric, as in the tangent of the angle of inclination; and finally (7) a Calculus conception, as in a derivative.

(CC BY-NC-SA Raymond Johnson)

If you compare Karim and Sal's definitions to Stump's list, you'll likely judge that while both have been correct, neither have been complete. We could stop here and declare this duel a draw, but to do so would foolishly ignore that there is much more to teaching and learning mathematics than knowing what belongs in a textbook glossary. Indeed, research suggests that a robust understanding of slope requires (a) the versatility of knowing all seven interpretations (although only the first five would be appropriate for a beginning algebra student); (b) the flexibility that comes from understanding the logical connections between the interpretations; and (c) the adaptability of knowing which interpretation best applies to a particular problem.

All seven slope interpretations are closely related and together create a cohesive whole. The problem is, it's not immediately obvious why this should be so, especially to a student who is learning about slope. For example, if slope is steepness, then why would we multiply it by x and add the y-intercept to find a y-value (i.e., as in the equation y=mx+b)? And why does "rise over run" give us steepness anyway? Indeed, is "rise over run" even a number? Students with a robust understanding of slope can answer these questions. However, Stump and others have shown that many students -- even those who have memorized definitions and algorithms -- cannot.

(CC BY Amber Rae)

This returns us to Karim's original point: There exists better mathematics education than what we currently find in the Khan Academy. Such an education would teach slope through guided problem solving and be focused on the key concept of rate of change. These practices are recommended by researchers and organizations such as the NCTM, and lend credence to Karim's argument for conceptualizing slope primarily as a rate. However, even within this best practice, there is nuance. For instance, researchers have devoted considerable effort to understanding how students construct the concept of rate of change, and they have found, for example, that certain problem contexts elicit this understanding better than others.

Despite all we know from research, we should not be surprised that there's still no clear "right way" to teach slope. Mathematics is complicated. Teaching and learning is complicated. We should never think there will ever be a "one-size-fits-all" approach. Instead, educators should learn from research and adapt it to fit their own unique situations. When Karim described teachers on Twitter debating "whether slope should always have units," we see the kind of incremental learning and adapting that moves math education forward. These conversations become difficult when Sal declares in his rebuttal video that "it's actually ridiculous to say that slope always requires units*" and Karim's math to be "very, very, very wrong." We absolutely believe that being correct (when possible) is important, but we need to focus less on trying to win a mathematical debate and focus more on the kinds of thoughtful, challenging, and nuanced conversations that help educators understand a concept well enough to develop better curriculum and pedagogy for their students.

Khan Academy (CC BY-NC-ND Juan Tan Kwon)

This kind of hard work requires careful consideration and an open conversation, even for a seemingly simple concept like slope. We encourage Sal to foster this conversation and build upon what appears to be a growing effort to make Khan Academy better. Doing so will require more than rebuttal videos that re-focus on algorithms and definitions. It will require more than teachers' snarky critiques of such videos. Let's find and encourage more ways to include people with expertise in the practice and theories of teaching mathematics, including everyone from researchers who devote their lives to understanding the nuance in learning to the "Twitter teachers" from Karim's post who engage this research and put it into practice. This is how good curriculum and pedagogy is developed, and it's the sort of work that we hope to see Sal Khan embrace in the future.

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*Sal's point is that if two quantities are both measured in the same units, then the units "cancel" when the quantities are divided to find slope. As an example, he uses the case of vertical and horizontal distance, both measured in meters. The slope then has units of meters/meters, which "cancel". However, the situation is not so cut and dry, and indeed, has been considered by math educators before. For example, Judith Schwartz (1988) describes how units of lb/lb might still be a meaningful unit. Our point is not to say that one side is correct. Rather, we believe that the act of engaging in and understanding the debate is what is important, and that such a debate is cut short by declarative statements of "the right answer."

References

Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Heibert & M. J. Behr (Eds.), Number Concepts and Operations in the Middle Grades (Vol. 2, pp. 41–52). Reston, VA: National Council of Teachers of Mathematics.

Stump, S. L. (1999). Secondary mathematics teachers' knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf

RYSK: Stump's Secondary Mathematics Teachers' Knowledge of Slope (1999)

This is the ninth in a series of posts describing "Research You Should Know" (RYSK).

I think just about every Algebra 1 student I ever taught came to me from Prealgebra knowing what slope was. At least they thought they knew what slope was. They could usually echo the words "rise over run," and I admit that very early in my career I probably would have found that somewhat satisfactory. But with each new Algebra 1 class (I taught 14 sections in 6 years), my students' limited understandings of slope became more frustrating. Honestly, it wasn't until my last year of teaching that I really felt I had the kinds of problems, activities, and explanations to help students construct an understanding of slope that I was happy with.

In discussions with my graduate school colleagues Fred Peck and Michael Matassa, I found that my experience wasn't unique. We were interested in exploring slope further, and that led us to an article by Sheryl Stump. Her name seemed familiar, and as soon as I saw her picture I realized that I'd had lunch with her at a conference just a few weeks before. I suppose if I'd found the article earlier I could have talked to her about slope instead of swapping stories about our common Midwestern roots, but she's been kind enough to reply to my emails when we've wanted to know more.

I think one of the reasons I liked Stump's article was that it focused on teachers instead of students. After all, the biggest reason I became dissatisfied with my students' understanding of slope was because my own understanding had grown more sophisticated with each trip through the curriculum. In her article, Secondary Mathematics Teachers' Knowledge of Slope (1999), Stump investigated the definitions, understandings, and pedagogical content knowledge of 18 preservice and 21 inservice teachers. Nearly all the inservice teachers had degrees in math or math education, including eight with math or math ed masters degrees, and their teaching experience ranged from 1 to 32 years.

Stump's review of previous literature on slope revealed a number of descriptions, including ratios, tangent, and -- because of its applications in physics, calculus, and other real-world applications -- the key concept of linear functions and rates of change. Stump also found everyday ways of thinking about slope, such as the downward slant of a hill from top to bottom. These different, yet related, descriptions of slope had led to misunderstandings in previous studies with students. No one had yet tackled this kind of research with teachers, so Stump designed and administered a survey and conducted interviews to understand what her study participants understood about slope. Her questions, "What is slope?" and "What does slope represent?" elicited responses that were sorted into seven categories:

• The category of geometric ratio included representations such as "\(\frac{\mbox{rise}}{\mbox{run}}\)" and "vertical change over horizontal change" and focused on slope as a geometric property.
• The category of algebraic ratio included representations such as "\(\frac{y_2 - y_1}{x_2 - x_1}\)" and "the change in y over the change in x, in which slope was defined by an algebraic formula.
• The words "slant", "steepness", "incline", "pitch", and "angle" were categorized as involving a physical property.
• Responses referring to slope as the rate of change between two variables were categorized as involving a functional property.
• The parametric coefficient category included references to m in the equation y = mx + b.
• A trigonometric conception of slope referred to the tangent of the angle of inclination.
• A calculus conception included mention of the concept of derivative. (p.129 )

Both the preservice and the inservice teachers in Stump's study averaged about 2.5 representations per teacher in their definitions. The geometric ratio representation of slope was easily the most common for both groups (83% of preservice, 86% of inservice). but preservice teachers most commonly (61%) used algebraic ratio as a second representation, while inservice teachers commonly (81%) described a physical property. Descriptions of slope using the parametric, trigonometric, and calculus conceptions were rare or nonexistent.

Stump then gave the two groups six math questions, each designed to test different understandings of slope. Both the first question, about rate of growth, and the second question, finding a linear equation given its parameters, were answered correctly by 100% of the teachers in both groups. Questions about slope as speed, read from a graph, were answered correctly by about two-thirds to three-fourths of teachers in each group. The most dissimilar performance came on a question about angle of inclination, answered correctly by 33% of preservice teachers and 67% of inservice teachers.

Next Stump asked the teachers, "What mathematical concepts must students have experience with before they can truly understand slope?" (p. 132). By a wide margin, both groups said a geometric representation was most important, but only three teachers in each group mentioned experiences with functional relationships. Similarly, when asked for real-world contexts for understanding slope, both groups tended to choose a physical property instead of a functional property. About a quarter of the teachers in each group didn't mention either, naming algebraic or geometric representations instead (p. 133).

Stump's teacher interviews allowed her to dig more deeply into teachers' understandings about how students learn about slope. When asked about student difficulties, almost all the inservice teachers referred to a calculation procedure, saying "they put the x's over the y's" or "the order in which they subtract them" (p. 139). Preservice teachers predicted similar difficulties with symbol manipulation. One preservice teacher said:

My guess is that some might be frightened off as soon as you introduce a mathematical definition or a formula for a line, like the slope-intercept of the equation. As soon as some people see equations, they just go nuts, especially with symbols instead of numbers. ... Not because they don't understand what slope is, but because they are not making the connection between the intuitive and even the not-so-intuitive idea of taking the ratio of this to this. Not making the connection between that and the symbolic abstract equation on paper. That's just a guess. I haven't had experience with that. (pp. 139-140)

In her discussion section, Stump acknowledges teachers' tendency to think of slope first as a geometric ratio, with a smaller majority commonly thinking of it as a physical property. Very few teachers -- less than 20% -- had a functional conception of slope. Stump continues:

Considering the importance of the study of functions for high school students, it is especially troubling that functional situations involving slope were missing from so many teachers' descriptions of their instructional practices. Their students may thus miss opportunities to make this important connection while forming their conceptions of slope. Rizzuti (1991) found that instruction that included multiple representations of functions allowed students to develop comprehensive and multi-faceted conceptions of functions. Based on the results of the present investigation, it is questionable whether the participating teachers could assist their students in developing such a rich conception of slope. (p. 141)

Finally, Stump asks some important questions for further study, such as, "When textbooks connect various representations of slope, do teachers emphasise those connections for their students? Can teachers learn to make connections even if textbooks do not emphasise them?" (p. 141). I don't think we really know the answers to those questions, but I do absolutely agree with Stump's closing recommendation: "Both preservice and inservice mathematics teachers need opportunities to examine the concept of slope, to reflect on its definition, to construct connections among its various representations, and to investigate functional situations involving physical slope situations" (p. 142). It's good to see that kind of work being done, such as with Fred Peck and Michael Matassa's teaching experiment research and curriculum on slope they shared at ICME-12.

References

Stump, S. L. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf

Functions of Functions

Mathematical functions are usually introduced formally to students somewhere around the end of Algebra 1 or maybe in Algebra 2. If my Algebra 2 final exam had included the question, "What do you know about functions?" I probably would have said: (a) You use \( f(x) \) instead of \( y \), and (b) if a graph fails the vertical line test, it's not a function. With all due respect to my high school math teacher, this would have been a lousy answer. I might have known some peripheral information, but not the core understanding. Sadly, the mathematical importance of functions is not that difficult conceptually, yet it's crucial to the majority of the content learned in Algebra 1 and 2. Yet students still struggle with what makes a function a function.

My first experience trying to teach the definition in a non-traditional way was using the "Cola Machine" problem in CPM Algebra (Math 1). The problem, several days into Unit 11, describes the following:

The cola machine at your school offers several types of soda. Your favorite drink, Blast!, has two buttons dedicated to it, while the other drinks (Slurp, Lemon Twister, and Diet Slurp) each have one button.

1. Explain how the soda machine is a relation.
2. Describe the domain and range of this soda machine.
3. While buying a soda, Mr. Hagen pushed the button for Lemon Twister and got a can of Lemon Twister. Later he went back to the same machine but this time pushing the Lemon Twister button got him a can of Blast! Is the machine functioning consistently? Why or why not?
4. When Karen pushed the top button for Blast! She received a can of Blast! Her friend, Miguel, decided to be different and pushed the second button for Blast! He, too, received a can of Blast! Is the machine functioning consistently? Why or why not?
5. When Loufti pushed a button for Slurp, he received a can of Lemon Twister! Later, Tayeisha also pushed the Slurp button and received a can of Lemon Twister. Still later, Tayeisha noticed that everyone else who pushed the Slurp button received a Lemon Twister. Is the machine functioning consistently? Explain why or why not. (Sallee, et. al., 2002, p. 375)

Most textbooks define functions approximately the same way: a relation is a function if there exists no more than one output for each input. Without a context, however, that definition might not carry much meaning, and relying solely on the vertical line test in a graph may not be helpful enough for many students. The soda machine problem, with its emphasis on consistency, gives both teacher and student a very approachable context within which to discuss what a function is and is not.

Early in my teaching career I thought Algebra 1 basically boiled down to two big ideas: solving equations and graphing lines. If students could do those two things, I felt pretty good about them passing my class. Now I see the big ideas of Algebra 1 differently and basically aligned with Colorado's revised standards for high school mathematics. The six expectations listed for Colorado's high school algebra standard can be summarized as follows:

1. Functional representations (equations, graphs, and tables)
2. Function behavior (qualitative)
3. Function transformations (parameters and parent graphs)
4. Equivalent expressions, equations, and inequalities
5. Solving equations, inequalities, and systems of equations
6. Mathematical modeling using functions

Four of the six expectations explicitly mention functions. The last expectation, mathematical modeling using functions, represents (for many math educators) the ultimate goal for math instruction: to give students the mathematical power to describe and understand the world around them. Instead of just solving and graphing, the big idea of high school algebra is functions, with most linear function work in Algebra 1 and most non-linear function work in Algebra 2.

I think it's a mistake to delay the explanation and definition of functions until late in Algebra 1 or later. Algebra 1 and younger students can understand the cola machine problem or other, similar contexts. Suppose the class plays an "exchange" game. Student A gives the teacher three triangles in exchange for two squares. What should Student B expect to get in exchange for his three triangles? What exchange would represent a function versus a non-function? For something based more in the real-world, this conversation could be set in the context of currency exchange. Also, a helpful model for learning functions might be function machines, such as:

Function machines are helpful models for learning functions, function composition, inverse functions, and even solving equations. They help stress the input-output relationship in a way that words or equations alone might not. Students will also naturally expect a single output for each input. The real challenge for Algebra 1 or younger students is to present a variety of equations that aren't functions, or else risk having students think that every two-variable equation describes a function.

Patterns can also be used to teach functions. If students are given the sequence 2, 4, 8, …, some students are likely to predict 14 as the next term (adding 8 plus 6, the next consecutive even number), while other students might predict 16 (the fourth power of two). Because the fourth term could reasonably be two different values, we can't establish a functional relationship to describe the sequence. This could even be an example worth graphing to discuss the meaning of the vertical line test:

Our calculators also help us distinguish functions from non-functions. If you try to graph a circle on a Texas Instruments graphing calculator, you have to enter two functions: one for the top half of the circle, and one for the bottom half. Therefore, the written forms of the equation of a circle (such as \( x^2 + y^2 = 1\) or \(y= \pm \sqrt{1-x^2} \)) can't be functions. The graphing calculator is also a good tool for discussing the square root function, and why its graph must only be half of a parabola's inverse if we want it to be a function.

There two key obstacles that are likely to remain in a student's way of understanding functions. First, students will continue to focus the vast majority of the time and effort on functions, and there are too few real-world examples of useful non-functions to help distinguish the two. Non-functions are less powerful and less common, but without them we risk having students who casually accept that any equations with an input-output relationship is a function. The second obstacle is notation, and it's not an obstacle we can likely avoid. The change to function notation, such as using \(f(x) = 3x - 2\) instead of \(y = 3x - 2\), is not only difficult to explain as something other than an arbitrary change in symbols, but includes two aspects that are directly contrary to a student's prior knowledge. Now a letter (such as the \(f\) in the preceding example) is no longer a variable, but a name with no numerical value by itself. Function notation also uses parentheses to represent something other than multiplication, adding more work to the already overloaded duties mathematicians place on these two simple arcs. The power of function notation is the preservation of the input (\(x\)) with the output, but it's confusing that the output (\(f(x)\)) reuses the input variable, uses a letter that isn't a variable, and uses symbolism for multiplication of variables that's no longer multiplication. Perhaps a question like this could help students realize the power of the notation:

1. If \(y=16\), \(y=49\), and \(y=64\), what might be an equation relating an input, \(x\), to the output, \(y\)?
2. If \(f(4)=16\), \(f(7)=49\), and \(f(-8)=64\), what is \(f(x)\)?
3. Are the two questions above the same? Which one is easiest to understand? Why?

References
Sallee, T., Kysh, J., Kasimatis, E., & Hoey, B. (2002). College Preparatory Mathematics 1 (Algebra 1). (L. Dietiker, Ed.) (2nd ed., Vols. 1-2, Vol. 2). Sacramento, CA: CPM Educational Program.