Today, I introduced a new expeditionary unit to one of my classes that revolves around a bakery selling cookies. In the introductory GroupThink (Grapple session, if you prefer) the problem’s complexity is reduced from what they will encounter, but meant to give enough information to still be realistic and be a pain-in-the-butt to guess-and-check. The idea with this structure is that by introducing a large problem like this, students will be able to dive in and engage right off the bat, but also realize that if they want to solve this problem most effectively and efficiently, they need to expand their mathematical toolbelt a bit further.
If you think ‘you don’t like math’, I encourage you to read along anyway – math is a subject of logical problem solving that gets coded in esoteric ways (mostly out of necessity, originally) and then taught in that same code rather than from the ground up (for the purpose of expediency of teaching… but of course it would be more expedient, in reality, to follow the axiom ‘go slow to go fast.’ But I digress). By reading the problem, you will have more insight into the situation and just maybe want to learn about some techniques that would speed the solution-finding process. Maybe you’ll become a math geek like me.
The problem is essentially as follows: a bakery wants to maximize their profit by baking the right combination of plain cookies and iced cookies. Every dozen plain cookies requires a pound of cookie dough, and takes a tenth of an hour to prepare. Every dozen iced cookies takes seven-tenths of a pound of cookie dough as well as two-fifths of a pound of icing. Iced cookies take fifteen hundreths of an hour to prepare.
The bakery also has some constraints: in the shop right now they have 110 pounds of cookie dough and 32 pounds of icing. They also only have 15 hours of prep time before they need to get the next batch out on the floor to sell. Finally, they can only fit 140-dozen cookies in the oven at time.
Plain cookies cost the bakery $4.50 to make and sell for $6.00. Iced cookies cost $5.00 to make and sell for $7.00. No matter what cookies are on the floor, the bakery is so good that they always sell out. What combination of cookies will maximize the bakery’s profit?
Without having learned how to create a simple system of linear inequalities and optimize with the slope of profit lines, the strategy to solving this type of problem using guess-and-check can be ascertained logically by a problem-solving adult: I could try some stuff, see if it holds a general pattern and make better and better hypotheses based on the pattern I see. For example, I could start by asking ‘what is the maximum amount of plain cookies I could make?’ Well, the answer is that with 110 pounds to work with divided by 1 pound per dozen plain cookies makes 110 dozen plain cookies. Then, I could run through and see if that fits the other two constraints – it comes in at 11 hours of prep time (check!) and all 110 dozen will fit in the oven at once (check!). This combination will make me $1.50 times 110 cookies – $165. Now, I doubt that is the maximum profit I could make, so let’s see what the maximum number of iced cookies I could make would be.
After checking on iced cookies and determining that by making 80 iced cookies, preparation time would limit the number of plain cookies I could make rather than the amount of dough I have. Well, maybe I should try other combinations to see if they make more money – ones that try to maximize the iced cookies within the constraints of still wanting to use as much of the available dough as possible while maximizing preparation time available. This method of answering this question is going to take a while and be quite frustrating the next day when the shop has the same problem to solve with different amounts of materials and constraints… but it’s totally possible to reason through logically.
The problem I’m having is more than half of my kids won’t reason through it logically, even though they can. And today, I want to provide the second explanation as to why this is happening, as it’s not always because kids are either taking on the strategy of denial or victim mentality. The second reason is that our society has changed our relationship to risk. What I mean by this is that not only are we sometimes teaching kids implicitly that they don’t need to solve their own problems because someone else will do it for them, we are also both implicitly and explicitly teaching them that they should not try to solve their problems, because solving them involves risk. And risk is bad.
This is the training wheel phenomenon at work again – it’s just that the concept of risk is more fundamental to why training wheels exist in the first place. To go back to that original post, “The reality of what training wheels do is allow a user to have an experience by lowering or eliminating the inherent risk involved in that experience.” It is not uncommon for parents in today’s world to put training wheels on their kid’s bike despite being aware of the fact that they hinder the learning curve more often than not. The reason? Perhaps it is partially a function of the parent not having enough time to teach the kid by running behind them, sans any other scaffolding, as Sheldon Brown suggests. But it is almost certainly also because they fear the risk of crashing.
Of course, this brings me back to my classroom. I can’t fathom getting the classic question “how will I ever use this in my life” in response to this prompt. So with that line of reasoning off the table, how is it that students fear trying to solve a realistic problem with the use of logical reasoning and getting it wrong – or, more likely, coming close to the answer but never quite getting there? How does this fear cause the young brain to think “well, it’s better to just not start at all than to get the answer wrong.”? I think it’s time to delve into the concept of risk, both in terms of the biology at play and the longitudinal social factors and trends.
Risk is defined as exposure (someone or something valued) to danger, harm, or loss; however, it can also be thought of as an intentional interaction with uncertainty. Of course, it’s important to note that ‘exposure’ as well as levels of ‘uncertainty’ are subjective, relative terms. Risk is not the same risk for everyone. Again, that 30-foot kicker on a bike presents a different amount of risk to Johnny the 3-year-old down the street than it does to Bernard Kerr. Actually, to put it that way is inadequate, because for all we know Johnny may grow up to win the Red Bull Hardline – that is to say, that perception of risk and actual risk can (and should) change within an individual’s lifetime. In a paper published in Evolutionary Psychology, Sandseter and Kennair explain that “children develop fears of certain stimuli (e.g. heights and strangers) that protect them from situations they are not mature enough to cope with,” however, as children begin to mature, Sandseter and Kennair explain, they begin to want to engage in risky situations (through play) as a biologically-programmed response: by intentionally exposing themselves to the things they previously feared, the child (species, if we are thinking in terms of evolution) adapts themselves to having new competencies (bigger muscles, better problem-solving skills) that make the ‘scary things’ less scary. “Risky play, we will argue, is a part of the normal process that adapts the child to its current environment through first developing normal adaptive fear to initially protect the child against ecological risk factors, and thereafter risky play is a fear reducing behavior where the child naturally performs exposure behavior,” write Sandseter and Kennair. They continue:
If the child does not receive the adequate stimulation by the environment through risky play, the fear will continue despite no longer being relevant (due to features of the ecology no longer constituting a risk, and the child’s improved competencies due to physical and psychological maturation) and may turn into an anxiety disorder: fear responses toward imagined or exaggerated threats and dangers that reduce the individual’s ability to function despite the individual having developed the abilities to handle these situations.
They conclude, “it is suggested that we may observe an increased neuroticism or psychopathology in society if children are hindered from partaking in age adequate risky play.”
Well, the situation is no longer a ‘maybe.’ It’s happening to the current generation that we are teaching. Just think about the number of your students who feel anxiety while testing. Obviously this is at least partially because of the value our society places on tests… but the intensive efforts of recent generations of educators (especially the ones I have observed) to place less value on testing as well as some modern parenting strategies that ‘boost kids’ confidence’ and reduce the value of testing to the point of making it seem worthless doesn’t seem to reduce testing anxiety. I’ve taught in both extremes. Starting at one of the Basis Charter Schools, where testing is everything (and that’s why I left), moving to a Montessori High School (that basically rejected testing), I found that kids who took tests with much higher stakes felt less anxious about them. This, of course, grows another question in my mind based on the fact that all this anxiety has to – at least in some part – be related to not letting our kids experience risky play as kids. What if no matter how lovey-dovey supportive we are, no matter how much we tell our kids not to worry and that they are unique and extraordinary, our kids are still going to be low-level anxious, neurotic people unless they engage in risky play and figure all those things out themselves as youngsters?
Psychological features related to lack of risky play as kids can be seen in other areas of our society, as well. In The Fragile Generation, Lenore Skenazy & Jonathan Haidt explain that college campuses are becoming locations where “It no longer matters what a person intended to say, or how a reasonable listener would interpret a statement—what matters is whether any individual feels offended by it,” which of course “interferes with the process of free inquiry and open debate—the active ingredients in a college education.” In this realm, it’s perhaps not even risky play that was lacking in the childhoods of these college students – it was just unsupervised play. That is, play with other kids without an adult around. It is during these moments that young kids can learn to get butthurt and deal with it, move on. I know this is one of those things that sucks… we don’t want our kids to ever feel hurt – because we’ve been through that and it sucks. But we never saw the fact that it made us resilient as adults, saving us more misery now than it was for us on those playgrounds back then.
So what about our adolescents who were perhaps given training wheels as kids, but we now want to remove them before it’s too late? I think along with compassionately discussing why you are only going to encourage them from now on rather than help them (as discussed in the last post), I think you should seriously consider getting them to join a school sport that involves risk. While it’s not unsupervised time (not advised by the time they are adolescents) it is exactly the type of exposure to risk that allows teens with anxiety to learn pathways to overcoming anxiety. When you begin to fail (crash on a mountain bike ride, fall on a climbing route, etc), you begin to recognize that failure is not the end of the world – it’s actually a great learning experience. Parents and teachers can say this to anxious teens all they want, but it doesn’t compare to experiencing it firsthand. If the coach of your teens team is good, she will help to relate those failures to other aspects of life.
The second benefit of joining a risky sports team is that risk is one of the 17 identified ‘flow triggers’. This benefit would be a whole ‘nother post, done right, but for now let me say that this has several advantages. To enter flow, biologically defined, means to undergo several positive transformations in the brain. Neurohormones (dopamine, norepinephrine, seratonin, anandamide) begin to precipitate. These compounds serve to enhance pattern recognition, tighten focus, and make the experience autotelic (rewarding in itself). The brain also begins to shift from a characteristically Beta brainwave state characterized by focus on what needs to get done in the moment and low-level stress, to an Alpha pattern characterized by relaxed focus. Finally, the brain undergoes ‘transient hypofrontality,’ literally a shutting down of the prefrontal cortex. This is an energy-conserving mechanism in the brain – the task at hand is clear and beyond the need for conscious thought, so energy ceases to be used for those tasks. This phenomenon is the same thing that causes long time meditators (monks) to feel a sense of ‘oneness’ with the world – because the prefrontal cortex houses our conceptions of time and self, we begin to cease to be aware of them. Again, this topic deserves a full post, but the net result of all of these changes occurring is profound and can provide developing humans with authentic confidence and purpose in life.
I am lucky to live in a place and teach at a school where I could play a part in starting a climbing, skiing, and mountain biking team at our school. If your school doesn’t have such events, at the very least encourage your school to start an intensive fitness program (again – a whole post in itself) and get your teen to begin trail running. I know it seems crazy. I know it will take work, and present dangers. But as Leopold reminds us, “too much safety seems to yield only danger in the long run,” and with training wheels we’re playing the long game. We’re trying to “prepare our kids for the path, not the path for our kids.”
Elliot, Andrew & Todd Thrash. (2004). The Intergenerational Transmission of Fear of Failure. PSPB: Vol. 30 No. 8: 957-971.
Kotler, Steven. (2014). The Rise of Superman. New Harvest Publishing.
Robinson, Doug. (2014). The Alchemy of Action.
Sandseter, Ellen & Leif Kennair. (2011). Children’s Risky Play from an Evolutionary Perspective: The Anti-Phobic Effects of Thrilling Experiences. Evolutionary Psychology: 9(2):257-284.
Skenazy, Lenore & Jonathan Haidt. (2017). The Fragile Generation. Published on Reason.com accessed 12/19/2017 from http://reason.com/archives/2017/10/26/the-fragile-generation.