I stood next to the checkout lines in the grocery store for a half hour. I had a clipboard in one hand and a timer in the other, looking like the most conspicuous secret shopper ever. I was there trying to answer one question:
Which line is best?
Like many of you, I suffer from Acute Line Jealousy Syndrome (ALJS). No matter the line I select, I install mental clones of myself in all the other lines and curse those mental figments whenever they beat me to the register.
If youâ€™re a civilian, someone who doesnâ€™t teach or think about math for a vocation, itâ€™s possible intuition is your only guide here. If youâ€™re a math teacher, though, you know that mathematical modeling has your answer.
In the Common Core State Standards, mathematical modeling is the only standard that doubles as both a content area and a practice. The CCSS specify six events that are fundamental to the modeling experience. Let me tell you how the first five played out that day in the grocery store and how they often go wrong in our classrooms.
Identify essential variables.
I had only a question on my mind: â€œWhich line?â€ No one was there to tell me the information about the line that was most important. I could have looked at the age of the customer, how much produce the customer had, whether the customer paid with cash or check. None of those variables are inessential. But none of them are as essential as the number of items in the cart. That was my decision.
Now I had data for 60 customers in a table. Tables are a great model for organizing data but lousy for detecting general trends and making broader claims. So I used a graph model next. It looked linear, but I also had reasons to believe it should be linear.
That equation there reveals so much. The slope â€“ 2.96 seconds / item â€“ tells us how long one item takes to scan and bag. That y-intercept was the real revelation, though. Notice how even small amounts of groceries take close to a minute to transact. 41.17 seconds is all the time that isnâ€™t soaked up by the items. Itâ€™s hi, hello, howâ€™s your day, go ahead and swipe your card there, do you have a club card with us, and so on. 41.17 seconds! For every customer, no matter how many items they have.
Hold onto that fact.
Performing Operations & Interpreting Results
I can now decide which of the two lines above I should hop into. My gut says I should hop into the express lane because the express lane has great PR. But every single customer in the express line will take 41 seconds regardless of the number of items in their carts.
Express Lane = 2.96(3) + 41 + 2.96(5) + 41 + 2.96(2) + 41 + 2.96(1) + 41 = 196.56.
Regular Lane = 2.96(19) + 41 = 91.24
Here Iâ€™m performing operations. Those operations yield numbers that are meaningless without units. So I need to interpret them, realizing the unit of their outputs is seconds and that their implication is that I should stay away from the express lane.
The final modeling act asks us to validate our conclusions. In this case, that means hopping into the mathematically correct line and finding out if it itâ€™s actually correct. As the great modeler George Box said, â€œAll models are wrong, but some are useful.â€ This mathematical model will fail us occasionally, as it doesnâ€™t take into account (among other variables) price checks, produce, and method of payment. But validating conclusions means we have to see if math is better than our gut intuition.
Concluding Remarks Because Iâ€™m Bumping Up Against My Word Count
Look, this is pretty exhilarating stuff, right? The revelation that the express lane is a total time suck has paid off dividends ever since. And not just while shopping. The week after I posted my findings on my blog, I was invited onto multiple radio shows as far away as Australia. Two months later I was on Good Morning, America, racing one of their correspondents. (I lost.)
Mathematical modeling sings to people. They love the idea that math has the power to explain parts of the world they care about.
But the sad part is that in textbooks this exhilarating, multi-faceted process is reduced down to exactly two facets: performing operations and interpreting results. Read the equivalent textbook problem:
The amount of time it takes to get through a grocery line can be modeled by the equation T = 2.96N + 41 where T is the time and N is the number of items a customer has. The express lane has four customers in it with 3, 5, 2, 1 items each. The regular lane has only one customer with 19 items. Which line should you join?
The answer is in the back of the book.
For teachers, I hope youâ€™re as horrified as I am by the difference between textbook modeling and actual modeling. For me, the mandate is clear: get students to experience as much of the actual process as possible while helping only when the need arises. Invite them to guess the right line first. Ask them to speculate which information matters. Ask them to decide which information matters most. Ask them to draw a sketch of the graph before they plot exact points.
Modeling goes wrong when we do the most interesting parts for our students and leave them only the two most boring pieces â€“ performing operations and interpreting their results. Help your students experience the other aspects. Not only because there is great work available for competent modelers later in life, but because modeling is a lot of fun now.