matthewdaly | Entries tagged with math

Well, I've been absolutely horrid at posting. I had a grand notion of reflecting on each of my classes after they happened, but, well, I didn't. Just a week left of classes yet, which will be followed by a few finals culminating activities, and then a month and a half of decompression.

One thing I didn't want to let go was my philosophy of (inclusive) education. I think this is like the third or fourth time I've had to write one, but this is supposed to be the final one that will allegedly work its way into my job applications and whatnot, and so this one was actually read and critiqued by my professor. He seems to get a kick out of it; as a school psychologist, he says that he has too much experience with seeing first-year teachers who are disillusioned by the reality of classroom management., so getting to ask ORLY at this phase in our professional development is a good move.

Anyways, I submitted enough drafts that my professor either ran out of questions or got sick of reading it, so here is my final draft which already got full credit. SPOILER ALERT: I am either planning to be a radical educator or I am the king of snowing my professors with the current generation of dynamic buzzwords.

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I affirm the inherent worth and dignity of all people, and I strive to use all of my talent, energy, and passion to provide a personally meaningful and practical mathematical education for all of my students. Every decision I make inside and beyond the classroom is, in some sense, a conscious reaction to my pursuit of this purpose. The following is a detailed but non-exhaustive list of specific strategies I undertake to achieve that ideal.

I define mathematics as “the systematic and objective process of making optimized decisions efficiently.” This includes learning how to perform numeric calculations, solving word problems, and other similar general tasks of primary and secondary mathematics. In the end, though, I believe that productive members of society must understand the range of decisions that they make throughout the day and the ways in which data is acquired, analyzed, and evaluated to make good choices and the ways in which both the choices and the processes are evaluated so that future decisions can be more easily and correctly rendered. For instance, if my students learn enough from about percentages and unit prices to pass a standardized test but not enough to be more strategic shoppers, then I will feel as if I have let them down.

To achieve that end, I believe in the principles of constructivist learning. As the mathematical tools students learn will be used by them throughout their lives, students must have an individual and intimate comprehension of those tools and how to apply them. When we teach a single perspective on a lesson and train students to apply it to solve a specific model of word problem, it should come as no surprise when students are unable to retain that knowledge past the unit test, much less when the lessons are built upon in future mathematics and science courses or in the real world. Only by giving students the resources and motivation to assimilate their knowledge into their own schematic understanding of the world will we provide authentic learning that the student can apply throughout his or her educational life and beyond.

Critical tools I use to showcase these diverse perspectives include small-group inquiry, peer learning, and a diverse array of multimodal media to explore problems in mathematics and the range of applications they have in the real world (focusing where possible on professional careers that require strong mathematical skills). An example that I often steer my students towards is the Khan Academy, which offers a thoughtful mathematical perspective that is often different from both my technique or the textbook's.

I believe that a teacher must be aware of and sensitive to the diversity of race, ethnicity, gender, religion, sexual orientation, gender identity, exceptionality, age, and socioeconomic status throughout our society. However, such a multidimensional cultural profile is only the first stage in forming authentic and respectful relationships with our students. Multicultural awareness is beneficial for roughly interacting with a group of people without unintentionally giving offense, but we must seek to destroy our stereotypes as quickly as we build them as we strive to know and serve our students as individuals. These plans require a foundation in a classroom culture of respect: my overt and consistent respect for all of my students and for the curriculum, my students’ respect for both me and for each other, and the students’ having respect for themselves as individuals, learners, and teachers.

To create this effective and inclusive classroom, I subscribe to the philosophy of culturally responsive teaching. My students have a diverse array of perspectives and differ in terms of the best strategies for educational success, and so I would be negligent to teach only in a way that benefited most of them or only in the way in which I was taught. For instance, lessons and projects should be customized to match the interests of a class to maximize their engagement, and vocabulary lessons should be structured around the linguistic strengths of each student. I also strive to take advantage of the diversity of perspectives and mastery amongst my students and use peer learning to further promote personal inquiry and self-constructed meaning, because there is no more effective and authentic teacher for a struggling student than the peer who just came to comprehend the material.

In the domain of assessment, I tend to be a follower of the theory of mastery education. Particularly in mathematics, there is little sense in advancing a student before he or she has a solid understanding of all of the standards that form the foundation of the new class. However, I believe that students deserve a broad range of options to demonstrate those competencies and that we should be open to the realities that many students do not do their best work on multiple choice exams under time constraints. For these students, the most authentic testing accommodation we can offer is alternative assessment, like journaling, oral examination, or service learning. If the grading criteria are tied to established objectives like the CCSS, calibrated against standard assessment measures, and performed with a passion for student success, I am confident that the result will be a measure of achievement that will provide a relevant supplement to the traditional measures. And when you demonstrate in word and deed that the education program can be catered to meet each student's’ talents and interests, the result is bound to be an increase in motivation and engagement that will enable the greatest possibility of positive outcomes.

I think it all starts with word problems. This is contrary to the model of American math education, and maybe beyond. We teach you the skill and then teach you how to apply it.

You know what, though? I'm not the one who is backwards on this. Remember Psych 101? When a person is exposed to a new piece of knowledge, she will attempt to incorporate that fact into her worldview. If she can't find a schema for it or hasn't developed an advantageous strategy for assimilation, then she's barely better off than if she hadn't been exposed to the fact in the first place. On the other hand, if she can effectively identify a schema and assimilate the fact into it, then her worldview is expanded and she is embiggened by the process.

Not everyone has had Psych 101, I suppose, so I'll illustrate. Let's say that a young child sees a German Shepherd, which is the first time that she recalls seeing a dog that is as big as she is. She doesn't know what to make of it; perhaps she is inclined to think of it as a "big animal" like a pony or a goat, except that she has developed a sense of "dogishness" (that adults would call "canine") that this animal satisfies in many ways except size. That sense is her schema of "dog", and when an adult tells her that German Shepherd are also dogs, she is able to broaden her schema to understand that dogs can be big and perform tasks beyond companionship.

However, this model is exactly what we don't do in math education. The skill comes out of the textbook through the blackboard without giving any sense of how it works into your worldview. This is tragically short-sighted, since mathematics is actually the formalization of the modelling our ancestors did to solve their real-world problems. Sometimes we are so busy answering "how" that we forget to ever get around to "why". Some students can get it anyways, of course, but we're far too comfortable with thinking that when other students can't get it that the problem is with the students and not with the math instruction. Back when I was in high school, if 80% of students barely understood math and 10% of students mastered the concepts, we'd have enough understanding to sustain society's population of alphas and betas. We don't live in that world any more; machines are currently doing much of the work that used to be done by Deltas and the Gammas are next in line for reassignment.

Let me give you an example I use at the beginning of one of my pre-algebra lessons. I am leading a full-day field trip, so I made some packaged lunches for the adults and children. Each adult lunchbox contains seven crackers and five pieces of sausage, and each child lunchbox contains three crackers and two pieces of sausage. If there are ten children and four adults on the field trip, what is the total inventory of food in the lunchboxes? Go ahead and solve that problem: note that it is a moderately difficult arithmetic problem that requires a few steps. Still, in my experience, it is not beyond the intuition of the sufficiently motivated student to understand that they want to "dump out" all of the lunchboxes and gather all of the similar foodtypes together. I've got a follow-up question too -- did you come up with fifty-eight crackers and forty pieces of sausage as your solution, or ninety-eight food pieces? Whichever answer you came up with, why did you settle on that as the better answer?

So, once people have worked out a few problems like this in small groups, I show how you would solve this in "algebra language":

10 ( 3C + 2S ) + 4 ( 7C + 5S )
30C + 20S + 28C + 20S
58C + 40S

See, I don't need to drag students on a trip to appreciate how to simplify algebraic expressions using the Distributive Law and collecting like terms. They made that trip themselves. My job as a teacher is to point out some of the highlights they'd have spotted along whatever journey they made, and they're going to be able to retain and apply that knowledge effectively because they had already engaged their schema for solving pre-algebra problems.

Current Mood: thoughtful

So I promised in an earlier post that I would talk about the anti-racist mathematical movement as I understand it (which is admittedly not well yet).

At a certain level, it seems to be a web of issues and I will mention them and then leave them without support or defense. One complaint is that mathematics is taught from the perspective of how it came to be understood in Europe, which often times ignores that virtually all of elementary mathematics was independently discovered by every culture in history (Chinese mathematics in particular often beat key European discoveries by a millennium or more) and that mathematical discoveries that Europeans knew were often the product of Arab, Indian, and Egyptian influences but we often don't highlight those contributions as such. This bleeds over into the same sorts of "dead white man" issues that literature and the physical sciences have faced over the years -- both that mathematical discovery is closed and that people of color have no talent for it anyways, which will discourage a student of color from mastering the material and furthermore from contemplating a career in a mathematical field. And that, in turn, is connected to all of the other ways that we fail to expect mathematical mastery (let alone excellence or prodigy) from students of color.

As I say, there is no lack of very important discussions to be had there. And my role in those various discussion would range from pulling out my cheerleader uniform to mildly defending the status quo all the way to heavy skepticism. I cannot help but become more informed as my own education continues, so perhaps I will someday come to know enough to speak on some of them in the future. In the mean time, I will speak of my first-hand experience with an issue here that I do know.

(For those who haven't been following along at home, my experience is as a math tutor for students studying for the GED and other related math tests at around the 8th grade level in the United States. My students are mostly black and Latino, nearly all female, I suspect virtually all living in poverty, and a significant number attempting to overcome learning disabilities and similar challenges.)

I will illustrate with an actual example that happened in the past week. About half of my students are taking a formal GED math class that my tutoring sessions are intended to supplement, and this past week they took the Official Practice Test. (A sufficiently high grade in this test would allow them to "graduate" to being allowed to retake the Math portion of the GED.) Here is a relevant portion of a question from that test. I won't display the rest of the question because the last thing I need is a cease-and-desist letter from the ACE, but trust me that an understanding of this sentence is a fundamental part of solving the problem. Again, I want to highlight that this is not a third-party product but an actual question generated by the American Council on Education that is a part of the gate-keeping process for GED diplomas.

"A restaurant menu lists 5 appetizers, 6 main dishes, and 4 desserts that are specialties of the house."

When I reviewed this question in my class, one of my best students shot up her hand and said "I'm sorry, but what does 'appetizer' mean? I'm sure I've seen the word before -- I mean, it's not like I've never been to a restaurant -- but I don't know if I've ever had one." And a few of us kind of talked out that it was like a plate of potato skins or chicken wings or shrimp cocktails that everyone at the table could share while they were waiting for the main dishes to be served, and she seemed to get it (although it was an embarrassing topic for her so I can't be certain).

But you see what happened there. If you're a middle-class white kid like me whose family ate out at sit-down restaurants on the average of once a week, you're answering an easier question than that student of mine did. Because we know that an appetizer isn't a specific sort of main dish, so we just multiply those three numbers together and move on. My student has to go looking deeper for contextual clues to figure out how to process these numbers. I can accept the argument that those clues are buried deeper in the problem, but there is a larger probability that she's going to miss those clues, and even if she does find them she will have less time and less morale than someone who "just knows" these non-mathematical facts.

And this is the sort of thing that you find quite a bit of once you're looking for it (and it is even easier when you have students who are comfortable enough to "admit" that it's their fault that they don't understand poorly-worded questions). How many days are there in June? What is the standard restaurant tip? What is an "at large" delegate? What does "reservoir" mean? Some of these questions are less unfair than others, and reading comprehension and setting up word problems are truly valuable skills that need to be tested. But when the word problems that you set up are biased against some classes or cultures, you really don't get to then dump on those classes and cultures for underperforming on the test.