For one example: in our last post on simplifying rational expressions, the process of turning a lengthy rational expression into a simpler one, Bill F writes:

Another benefit of evaluating both expressions for a set of values is to emphasize the equivalence of both expressions. Students lose the thread that simplifying creates equivalent expressions. All too often the process is seen as a bunch-of-math-steps-that-the-teacher-tells-us-to-do. When asked, “what did those steps accomplish?” blank stares are often seen.

Past a certain point, those operations are trivial. But it’s only past a point much farther in the distance that the *understanding* – these two rational expressions are *equivalent* – becomes trivial.

For another example: I left high school adept at graphing functions. I could complete the square and change forms easily. I knew how to identify the asymptotes, holes, and limiting behavior of those thorny rational expressions. But it wasn’t until I had graduated university math and was several years into *teaching* that I really, really understood that the graph is a picture of all the points that make the function *true*. This was difficult for me because graphs don’t often *look* like a bunch of points. They look like a *line*

That’s one reason I’m excited about the Desmos Activity Builder and this activity I made in it last week, Loco for Loci!

It asks students to place a point anywhere on a graph so that it makes a particular relationship *true*. Then it asks the students to imagine what *all of our points* would look like if we pictured them on the same graph. Then the teacher can show the results, underscoring this Very Big Idea that I didn’t fully appreciate my first time through high school – what we eventually think of as a continuous line is a picture of lots and lots of points.

Here is what happened when 300 people on Twitter played along:

“Drag the green point so that it’s the same distance from both blue points.”

Trickier: “Drag the green point so that it’s five units from both blue points.”

Whimsical: “Drag the green point so it is the same distance from a) the line of dinosaurs and b) the big dinosaur.” I really couldn’t have hoped for better here.

And then a couple of very interesting misfires.

“Drag the green point so that it’s four units from the blue point.”

“Drag the green point so that a line segment is formed with a slope of .5.”

You could run a semester-long master’s seminar on the misconceptions in that last graph.

Well.

Maybe more like ten quick seconds at the start of your Algebra class.

If you’d like to run this activity with your own students, here is the teacher link.

**Questions for the Comments**

- Obviously, I didn’t invite hyperbolas and ellipses to the party. Which other loci should have received the same treatment?
- Which Very Big Ideas did you only fully understand once you started math teaching?

**Featured Comment**

I find this sort of gap fascinating [my inability to conceive of graphs as a picture of solutions –

dm] especially because it is likely somewhere along the line you were at least told this fact (you might even be able to track exactly where). But it still didn’t stick! It’s as if being told just isn’t enough.

The description you give about graphs is something we have to hit early and often in CME Project, it’s one of the top 3 things to learn in the entire curriculum. It’s amazing how that can get lost in the shuffle, but it does, and where it gets lost is the algorithmic way of graphing a function or equation: the A does this, the B does that, etc — all of this ignores the deeper fact that under the hood, this is all a relationship between two variables x and y.

The other two of Bowen’s top three things to learn in Algebra, according to Bowen on Twitter, are:

- Variables represent numbers, so test numbers to test ideas and build equations.
- Rules for new stuff should respect existing rules.

**Featured Tweets**

Amazing, all the people unburdening themselves on Twitter of math they only understood once they began teaching. *What does it all mean?*

Simplifying rational expressions.

In particular, adding rational expressions with unlike denominators, resulting in symbolic mish-mash of this sort here.

I’m not here to argue whether or not this skill should be taught or how *much* it should be taught. I’m here to say that if we *want* to teach it, we’re a bit stuck for our usual reasons *why*:

- It lacks real-world applications.
- It lacks job-world applications. (Unless you count “Algebra II teacher.”)
- It lacks relevance.

So our usual approaches to motivation fail us here.

**What a Theory of Need Recommends**

We have to ask ourselves, instead, why anyone would prefer the simplified form to the unsimplified form. If the simplified form is aspirin, then what is the headache?

I don’t believe the answer is “elegance” or “beauty” or any of the abstract ideals we often attribute to mathematicians. Talking about “efficiency” gets us closer, but still and again, we’re just *talking* about motivation here. Let’s ask students to *do* something.

We simplify because it makes life *easier*. It makes all kinds of *operations* easier. So students need to experience the relative difficulty of performing even simple operations on the unsimplified rational expression before we help them learn to simplify.

*Like evaluation.*

So with nothing on the board, ask students to call out three numbers. Put them on the board. And then put up this rational expression.

Ask students to evaluate the numbers they chose. It’s like an opener. It’s review. As they’re working, you start writing down the answers on Post-It notes, which you do quickly because you know the simplified form. You place one Post-It note beneath each number the students chose. You’re finished with all three before anybody has finished just one.

As students reveal *their* answers and find out that you got *your* answers more efficiently and with more accuracy than they did, it is likely they’ll experience a headache for which the process of simplification is the aspirin.

Again we find that this approach does more than just motivate the simplification process. It makes that process *easier*. That’s because students are performing the same process of finding common denominators and adding fractions with *numbers*, they’ll shortly perform with *variables*. We’ve made the abstract more concrete.

Again, I don’t mean to suggest this would be *the most interesting lesson ever!* I’m suggesting that our usual theories of motivating a skill – link it to the real world, link it to a job, link it to students’ lives – crash hard on this huge patch of Algebra that includes rational expressions. That isn’t to say we shouldn’t teach it. It’s to say we need a stronger theory of motivation, one that draws strength from the development of math itself rather than from a student’s moment-to-moment interests.

**Next Week**

Wrapping up.

**Featured Comments**

Another benefit of evaluating both expressions for a set of values is to emphasize the equivalence of both expressions. Students lose the thread that simplifying creates equivalent expressions. All too often the process is seen as a bunch-of-math-steps-that-the-teacher-tells-us-to-do. When asked, “what did those steps accomplish?” blank stares are often seen.

By a creating a “headache” using a theory of need, we’re really looking back to the situations that prompted the development of the mathematics we intend students to learn. We’re attempting to place students in the position of the mathematician/scientist/logician/philosopher who was originally staring down a particular set of mathematics without a clue about where to go and developing a massive headache from his hours of attempt. I love this idea because it transcends any subject and students learn the value of the learning process.

I feel like it’s a mathematical habit of mind. Mathematicians don’t like drudgery either. But what makes them different from a typical American math student is, rather than passively accepting the work as tedious and plowing ahead anyway, they do something about it. They look for a workaround, or another approach.

Mike:

]]>It is elegance, it is beauty, and I’m afraid I simply don’t buy the efficiency argument at all.

*Print*

**Missing the Promise of Mathematical Modeling**. My Spring 2015 article in*Mathematics Teacher*analyzes the meager opportunities for modeling in school mathematics and recommends some solutions.**The Checkout Line A Scam. (Or Is It? [Yes, It Is.])**My blog post for Heinemann illustrating modeling through the lens of grocery express lines, which are a scam.

*Video*

**Math Class Needs a Makeover**. A 11-minute talk I gave for TEDxNYED in 2010. Couple million views.**Fake-World Math: When Mathematical Modeling Goes Wrong and How to Get it Right**My TEDxNYED talk + five more years of study + 50 extra minutes.

Summarizing all of the above in a single paragraph:

]]>Modeling asks students a) to take the world and turn it into mathematical structures, then b) to operate on those mathematical structures, and then c) to take the results of those operations and turn them back into the world. That entire cycle is some of the most challenging, exhilarating, democratic work your students will ever do in mathematics, requiring the best from all of your students, even the ones who dislike mathematics. If traditional textbooks have failed modeling in any one way, it’s that they perform the first and last acts

forstudents, leaving only the most mathematical, most abstract act behind.

Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question.

Scott Farrar, on my last post on motivating proof:

]]>I think this latches onto the structure of the geometry course: we develop tool (A) to study concept (B). But curriculum can get too wrapped up in tool (A), losing sight of the very reason for its development. So we lay a hook by presenting concept (B) first.

Proof.

This is too big for a blog post, obviously.

**What a Theory of Need Recommends**

If proof is the aspirin, then *doubt* is the headache.

In school mathematics, proof can feel like a game full of contrived rules and fragile pieces. Each line of the proof must interlock with the others *just so* and the players must write each of them using tortured, unnatural syntax. The saddest aspect of this game of proof is that the outcome of the game is already known *every time*.

- Prove angle B is congruent to angle D.
- Prove triangle BCD is congruent to triangle ACB.
- Decide if angle B and angle C are congruent. If they are, prove why they are. If they aren’t, prove why they aren’t.
- Prove line l and line m are parallel.
- Prove that corresponding angles are congruent.

One of those proof prompts is not like the others. Its most important difference is that it leaves open the very question of its truth, where the other prompts leave no doubt.

The act of proving has many purposes. It doesn’t do us any favors to pretend there is only one. But one purpose for proof that is frequently overlooked in school mathematics is the need to dispel doubt, or as Harel put it, the “need for certainty“:

The need for certainty is the need to prove, to remove doubts. One’s certainty is achieved when one determines—by whatever means he or she deems appropriate— that an assertion is true. Truth alone, however, may not be the only need of an individual, who may also strive to explain why the assertion is true.

So instead of giving students a series of theorems to prove about a rhombus (implicitly verifying in advance that those theorems are *true*) consider sowing doubt first. Consider giving each student a random rhombus, or asking your students to construct their own rhombus (if you have the time, patience, and capacity for heartache that activity would require).

Invite them to measure all the segments and angles in their shapes. Do they notice anything? Have them compare their measurements with their neighbors’. Do they notice anything now?

Now create a class list of conjectures. Interject your own, if necessary, so that the conjectures vary on two dimensions: true & false; easy to prove & hard to prove.

For example:

“Diagonals intersect at perpendicular angles” is true, but not as easy to prove as “opposite sides are congruent,” which is also true. “A rhombus can never have four right angles” meanwhile is false and easy to disprove with a counterexample. “A rhombus can never have side lengths longer than 100 feet” is false but requires a different kind of disproof than a counterexample.

With this cumulative list of conjectures, ask your students now to decide which of them are true and which of them are false. Ask your students to try to disprove each of them. Try to draw a rhombus, for example, even a sketch, where the diagonals *don’t* intersect at perpendicular angles.

If they *can’t* draw a counterexample, then we need to prove why a counterexample is impossible, why the conjecture is in fact true.

This approach accomplishes several important goals.

**It motivates proof.**When I ask teachers about their rationale for teaching proof, I hear most often that it builds students’ skills in logic or that it trains students’ mind. (“I tell them, when you see lawyers on TV arguing in front of a judge, that’s a proof,” one teacher told me last week.) Forgive me. I’m not hopeful that our typical approach to proof accomplishes any of those transfer goals. I’m also unconvinced that lawyers (or even*mathematicians*) would persist in their professions if the core job requirement were working with two-column proofs.**It lowers the threshold for participation in the proof act**. Measuring, noticing, and speculating are easier actions (and more interesting too) than trying to recall the abbreviation “CPCTC.”**It allows students to familiarize themselves with formal vocabulary and with the proof act.**Students I taught would struggle to prove that “opposite sides of a rhombus are congruent.” This is because they’re essentially reading a foreign language, but also because mathematical argumentation, even the*informal*kind, is a foreign*act*. Offering students the chance to prove trivial conjectures puts them in arm’s reach of the feeling of*insight*which all non-trivial proofs require.**It makes proving easier.**When students try to disprove conjectures by drawing lots of different rhombi, they stand a better chance of noticing the aspects of the rhombus that vary and don’t vary. They stand a better chance of noticing that they’re drawing an*awful*lot of isosceles triangles, for example, which may become an essential line in their formal proof.

Resolving this list of conjectures about the rhombus – proving and disproving each of them – will take more than a single period. Not every proof needs this kind of treatment, certainly. But occasionally, and especially early on, we should help students understand *why* we bother with the proof act, why proof is the aspirin for a particular kind of headache.

**Next Week’s Skill**

Simplifying sums of rational expressions with unlike denominators. Like this worked example from PurpleMath:

If that simplified form is aspirin, then how do we create the headache?

**BTW**. For anybody not on board this “headache -> aspirin” thing, I want to clarify: totally fine. Thanks for contributing anyway. But please name your priors. Why that task instead of another? Some of these tasks you all suggest in the comments seem great and full of potential, but tasks aren’t generative of other tasks. I need fewer interesting tasks and more interesting *theories* about what make tasks tick. These kinds of theories, when properly beaten into shape, have the capacity to generate lots of other tasks.

**BTW**. Scott Farrar chases this same idea along a different path.

**Featured Comments**

I think this latches onto the structure of the geometry course: we develop tool (A) to study concept (B). But curriculum can get too wrapped up in tool A losing sight of the very reason for its development. So, we lay a hook by presenting concept B first.

We almost always do an always-sometimes-never to motivate a particular proof. Mine are usually teacher-generated (here’s a list of 5 statements about rhombi – tell me if they are always, sometimes, or never true). Then we prove the always and the never.

Michael Paul Goldenberg and Michael Serra offer some very convincing criticism of the ideas in this post.

]]>Fake-World Math was the talk I gave for most of 2014, including at NCTM. It looks at mathematical modeling as it’s defined in the Common Core, practiced in the world of knowledge work, and maligned in print textbooks. I discuss methods for helping students become proficient at modeling and methods for helping them *enjoy* modeling, which are not the same set of methods.

Also, a note on process. I recorded my screen throughout the entire process of *creating the talk*. Then I sped it up and added some commentary.

**This Week’s Skill**

Here is the first paragraph of McGraw-Hill’s Algebra 1 explanation of graphing linear inequalities:

The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality. An equation defines a boundary, which divides the coordinate plane into two half-planes.

This is mathematically correct, sure, but how many novices have you taught who would sit down and attempt to parse that expert language?

The text goes on to offer three steps for graphing linear inequalities:

- Graph the boundary. Use a solid line when the inequality contains ≤ or ≥. Use a dashed line when the inequality contains < or >.
- Use a test point to determine which half-plane should be shaded.
- Shade the half-plane that contains the solution.

The text offers aspirin for a headache no one has felt.

The shading of the half-plane emerges from nowhere. Up until now, students have represented solutions graphically by plotting points and graphing lines. This shading representation is new, and its motivation is opaque. The fact that the shading is *more efficient* than a particular alternative, that the shading was invented to *save time*, isn’t clear.

We can fix that.

**What a Theory of Need Recommends**

My commenters save me the trouble.

Ask students to find two numbers whose sum is less than or equal to ten (or, alternatively, points that satisfy your 2x + y < 5 above). The headache is caused by asking them to list ’em all. The aspirin is another way to communicate all of these points — the graph determined by the five steps listed above. Rather than present the steps, have students plot their points as a class.

One problem I like is having each kid pick a point, then running it through a “test” like y > x

^{2}. They plot their point green or red depending on whether or not it passes the test — and a rough shape of the graph emerges.

John Scammell writes about a similar approach. Nicole Paris offers the same idea, and adds hooks into later lessons in a unit.

Great work, everybody. My only addition here is to connect all of these similar lessons with two larger themes of learning and motivation. One large theme in Algebra is our efforts to find solutions to questions about numbers. Another large theme is our efforts to *represent* those solutions as concisely and efficiently as possible. My commenters have each *knowingly* invited students to represent solutions using an existing inefficient representation, all to prepare them to use and appreciate the more efficient representation they can offer.

They’re linking the new skill (graphing linear inequalities) to the old skill (plotting points) and the new representation (shading) to the old representation (points). They’re tying new knowledge to old, strengthening both, motivating the new in the process.

**Next Week’s Skill**

Proofs. Triangle proofs. Proving trigonometric identities. If proof is aspirin, then how do you create the headache?

]]>Zak Champagne, Mike Flynn, and I are all NCTM conference presenters and we were all concerned about the possibility that a) none of our participants did much with our sessions once they ended, b) lots of people who might benefit from our sessions (and whose questions and ideas might benefit *us*) weren’t in the room.

The solution to (b) is easy. Put video of the sessions on the Internet. Our solution to (a) was complicated and only partial:

**Build a conference session so that it prefaces and provokes work that will be ongoing and online.**

To test out these solutions, we set up Shadow Con after hours at NCTM. We invited six presenters each to give a ten-minute talk. Their talk *had* to include a “call to action,” some kind of closing homework assignment that participants could accomplish when they went home. The speakers each committed to help participants with that homework on the session website we set up for that purpose.

Then we watched and collected data. There were two major surprises, which we shared along with other findings with the NCTM president, president-elect, and executive director.

Here is the five-page brief we shared with them. We’d all benefit from your feedback, I’m sure.

**Featured Comments**

Marilyn Burns on her reasons for attending conferences like NCTM:

I don’t expect an NCTM conference to provide in-depth professional development, but act more like a booster shot for my own learning.

Elham Kazemi, one of our Shadow Con speakers, tempers expectations for online professional development:

]]>I have a different set of expectations about conferences and whether going to them with a team allows you to go back to your own contexts and continue to build connections there. Can we expect conferences and the internet to do that — to feed our local collaborations? I get a lot of ideas from #mtbos and from my various conversations and conferences. But really making sense of those ideas takes another level of experience.

Determining if a relationship is a function or not.

A relationship that maps one set to another can be confusing. Questions like, “What single element does 2 map to in the output set below?” are impossible to answer because 2 maps to *more than one* element.

By contrast, a function is a relationship with *certainty*. Take any element of the input and ask yourself, “Where does this function say that element maps?” You aren’t confused about any of them. Every input element maps to exactly one element in the output.

Pearson and McGraw-Hill’s Algebra 1 textbooks simply provide a definition of a function. Pearson’s definition refers to a previous worked example. McGraw-Hill has students apply the definition to a worked example immediately afterwards. Khan Academy dives straight into an abstract explanation of the concept. In none of these cases is the *need* for functions apparent. Students are *given* functions without ever feeling the pain of not having them.

**What a Theory of Need Recommends**

If we’d like students to experience the *need* for the certainty functions offer us, it’s helpful to put students in a place to experience the *uncertainty* of non-functional relationships first. Here is what I’m talking about.

Put the letters A, B, C, and D on your back wall, spaced evenly apart.

Ask every student to stand up. Then give them a series of instructions.

*Transportation*

If you walked to school today, stand under A.

If you rode your bike to school today, stand under B.

If you drove or rode in a vehicle today, stand under C.

If you got to school any other way, stand under D.

*Duration*

If it took you fewer than 10 minutes to get to school today, stand under B.

If it took you 10 or more minutes to get to school today, stand under D.

*Class*

If you’re in seventh grade, stand under A.

If you’re in eighth grade, stand under B.

If you’re in ninth grade, stand under C.

If you’re in any other grade, stand under D.

These instructions are all clear and easy to follow. Students are *certain* where they should go. Then give two other sets of instructions.

*Clothes*

If you’re wearing blue, stand under A.

If you’re wearing red, stand under B.

If you’re wearing black, stand under C.

If you’re wearing white, stand under D.

*Birthday*

If you were born in January, stand under A.

If you were born in February, stand under B.

If you were born in March, stand under C.

If you were born in April, stand under D.

Perhaps you see how these last two examples generate a *lack* of certainty. Students were lulled by the first examples and may now feel a headache.

“I’m wearing white *and* red. Where do *I* go?”

“I was born in August. There’s no place for me to stand.”

*Now* we gather back together and apply formal language to the concepts we’ve just *felt*. “Mathematicians call these three relationships ‘functions.’ Here’s why. Why do you think these relationships *aren’t* functions?” Invite students to interrogate the concept of a function in different contexts. Try to keep the focus on *certainty* – can you predict the output for any input with certainty? – rather than on the vertical line test or other rules that expire.

**Next Week’s Skill**

Graphing linear inequalities. It’s extraordinarily easy to turn questions like “Graph y < -2x + 5" into the following series of steps:

- Graph the line.
- If the inequality includes the boundary, make the line solid. Otherwise, make the line dashed.
- Test a point on either side of the line. Use (0,0) if possible.
- If that point is a solution to the inequality, shade that side of the line.
- If that point
*isn’t*a solution to the inequality, then shade the*other*side of the line.

Students can become quite capable at executing that algorithm without understanding its necessity or how it figures into algebra’s larger themes.

What can you do with this?

**BTW**

Kate Nowak encouraged me to look at other textbooks beyond McGraw-Hill and Pearson’s. She recommended CME, which, it turns out, does some great work highlighting this need for functions. It asks students to play a “guess my rule” game, one which has a great deal of certainty. Each input corresponds to exactly one output. Then the CME authors offer a vignette where a partner reports multiple outputs for the same input, making the game impossible to play. Strong work, CME buds.

]]>8 names are read per min at the CHS graduation, I will be here for another 36min.. slooooow.

— andreabonilla (@andreabonilla) June 9, 2011

My friend has taken a problem from the world that was personal to her, identified the variables that are essential to the problem, selected a model that describes those variables, performed operations on that model, and re-interpreted the result back into the world. And tweeted about it.

That is *modeling* – the process of turning the world into math and then turning math back into the world. My friend probably wouldn’t wouldn’t label her experience like that but that’s what she’s doing. That’s what people who do math in the world do.

We know how this looks in many textbooks, though.

The amount of time (t) it takes a number of graduates (n) to cross the graduation stage can be modeled by the function t(n) = n/8. How long will it take all 288 graduates to cross the stage?

Here students would simply perform operations on real-world-flavored math while the important and interesting work is in turning the world *into* that math and turning that math *back* into the world.

Here is an alternate treatment, one that has students modeling as the practice is described in the Common Core.

**Show this video**.

Ask: “If I want to set an alarm that’ll let me take a *long* nap until just before my cousin Adarsh crosses the stage, how should I set the alarm?”

By design, it’s a short video. I’d like it to be boring enough to provoke my friend’s modeling but not *terminally* boring.

By design, it lacks mathematical structures because we’d like students to participate in the process of developing those structures. They won’t do that unassisted.

Before we get to the algebraic model, we can ask some important and interesting questions.

**How long do you think it will take my cousin to graduate? Just estimate.**

I asked that on Twitter and received the following estimates:

These guesses interest us in a *calculation* and also prepare us to evaluate whether or not that calculation is correct.

**Sketch the relationship between the number of graduates and time.**

Asking students to sketch the relationship, rather than plot it precisely, asks them to think relationally (“how do these two quantities change together?”) rather than instrumentally (“how do I plot these points?”).

Many students will *assume* the data is linear. But this prompt may invite some students to consider the possibility that the data is non-linear.

**Collect data. Model the data. Get an answer.**

Ask students to create a table of values. Ask students to plot the data in Desmos. Regress the data. Give them the graduation program. Calculate an answer.

I plotted the first ten names and modeled their times with a linear equation. (“Time v. names read” was my model, though commenter Josh thinks “time v. number of syllables read” would be more accurate.) The calculation for cousin Adarsh’s 157th name is 19 minutes. I would be foolish to rely on that calculation, however.

**Ask your students to “Assume your answer is wrong, that something surprising actually happens. Anticipate that something and fix your mathematical answer.”**

George Box: “All models are wrong, but some are useful.”

This is where we turn the math back into the world. This is where we make some math teachers uncomfortable, admitting that the world and the math don’t correspond exactly and that the *math* needs modification.

Watch all of these math teachers make *exactly* those modifications in the comments of the preview post. They perform mathematical operations and then proceed to describe why the results of those operations are *wrong*.

- Scott: “Add the bit of time prior to starting and a few seconds for a switch in readers as tends to be customary in larger groups like this … “
- Sadler: “14 minutes and 10 seconds but given that it is better to wake 10 seconds early than miss it, I would submit 14 minutes.”
- Scott #2: “You would probably want to set your timer a little earlier so you are fully awake when your cousin’s name is called.”
- Julie Wright: “As an embittered W, I am aware that there is lots of ponderous gravity for A’s and B’s, then everybody gets bored and speeds things up.”

**Validate (or invalidate) the answer.**

Commenter Mark Chubb, at the end of his modeling cycle: “Can’t wait to see Act 3.” Act 3 is the *reveal* in this task framework I call three-act math. It isn’t enough for Mark to simply read the answer in the back of the book or hear it from me. He wants to see it. So:

If you built a linear model from the first ten names, your answer winds up too large. Instead of 19 minutes, my cousin graduates at 17:12, *sooner* than the math predicted.

Why?

In the video, you can hear the validation of Julie Wright’s hypothesis above. The A’s and B’s get a lot of pomp, and then the commencement reader races through the rest.

Many congratulations to Megan Schmidt for her guess and to Scott and Kyle Pearce for their calculation. They all put down for 18 minutes. Special mention also to aga bey for 16.3 minutes. That commenter’s method? “I took the average of all submissions upthread.” Strong!

Again, if mathematical modeling requires the cycle of actions we find here, our textbooks typically only require one of them: performing operations. The purest mathematical action. The one that is often least interesting to students and the least useful in the world of work. So let’s offer students opportunities to experience the complete modeling cycle. Not just because those are the skills that most of the fun jobs require. But because modeling with math is fun for students *now*.

**Featured Comment**

]]>I ran into this when working up an exponential growth problem for my son’s precalculus class. The CDC had data on the number of Ebola cases which could be modeled with an exponential growth curve at the time. However, the math needed correction because of a sudden increase in cases. The CDC readily admitted they believe the cases were unreported by a factor of 1.5 to 2.5. Thus, a human eye on the data to recognize that and make an adjustment was necessary.

Later, when the curve could be modeled nicely by a logistics curve, the equation was still incorrect in predicting the end of the epidemic. As teachers we would like to be able to button everything up and wrap it in a bow, but the real world seldom works that way.