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Month: February 2014

How I used mathematics to choose my next apartment

The house my wife and I live in was recently sold, and so we have started looking for another apartment. Our current lease expires in a year and a half, and so we decided that, given how challenging the rental market is in NYC, that we should start looking right away. We also decided, somewhat arbitrarily, that we would attempt to find an apartment in the next six months, if only because we knew we would get sick of looking pretty quickly.

During the first 5 weeks of our apartment hunt, we found five apartments that we thought were worth looking at. Over six months, I extrapolated that we would get to see about 26 apartments that would satisfy our requirements. What we discovered, with each apartment, is that we basically got to see the apartment and then pretty much decide immediately if we wanted to apply to rent the apartment or not.

It turns out that there is a nifty mathematical algorithm that one can use to optimize one’s chances at picking the best apartment possible. We expect to have 26 apartments to look at, each of which we inspect and then either accept or reject immediately, and we want to optimize our chances of picking the best apartment possible, from the 26 that meet our minimum criteria. This exactly matches a solved problem in mathematics; the Secretary Problem.

In the Secretary Problem, where one has to decide on the best applicant between n randomly ordered applications, an optimal solution is to reject the first n/e applicants, and then accept the next applicant that is better than any applicant you have seen before. The proof of this particular solution is here. I couldn’t reproduce this proof if asked, and there are details in it which are fuzzy for me, but I am pretty sure I understand why it works. Informally, the first n/e applicants act as a sampling space, and this gives you information on how good applicants will be, and that n/e happens to be where you achieve an optimal amount of information on applicant strength, allowing you to make the best determination you can of which next applicant to choose, without raising the probability too high that you’ve already rejected the best applicant.

For our specific apartment hunting problem, with 26 total apartments to view, 26/e (e ≈ 2.71828) ≈ 9.6  10 apartments. So, my wife and I looked at 10 different apartments, and while we did this, I informally ranked the apartments based on the criteria my wife and I agreed were important to think about (space, apartment lay-out, cost, location, quality of neighbourhood school, commute time). The 11th apartment either she or I looked was superior in many ways on all of these criteria than any of the others we had looked at, so I whole-heartedly threw in my support for it, knowing that this specific apartment is most likely to be the best apartment we will see.

We’ve submitted an (overly lengthy) application for the apartment. Wish us luck.

 

 

One model for adding fractions

When I first started tutoring students, I often noticed that they struggled to add fractions. The addition of fractions just did not make sense to them. Part of this is caused by students having a weak understanding of fractions, and part of this is caused by them not understanding why the typical algorithms used to add fractions make sense.

Here is one model that I developed for myself, so that I could understand why addition algorithms for fractions make sense, and then use this model to help students make sense of adding fractions.

 

Step 1:

Draw a picture to represent each fraction you want to add. So if I want to add 2/3 and 3/4, I would draw the following, making sure to use rectangles which are the same dimensions, since these two fractions must represent a fraction of the same unit (in this case, the rectangle).

Adding fractions - step 1

Step 2:

One problem with the above drawing is that it is not the only way to represent fractions. Anticipating that I will want to be able to match units, I realized that it is convenient to draw my fractions with different orientations, as below.

Adding fractions - step 2

Step 3:

Now, I can’t add the fractions above because the individual pieces of each fraction have different units. I cannot easily see what the total area would be for the two shapes, so I need to divide the two areas again, so that I end up finding the area of each fraction with the same sub-unit of the rectangle (in this case, 1/12).

Adding fractions - step 3

Step 4:

Now that the units of each fraction are the same, I can simply count the total number of units in the first fraction (9) and add it to the total number of units in the second fraction (8), and I arrive at my answer of 17 units. I have to remember though that the size of this unit, relative to the original unit, is a twelth, hence my answer is 17/12 in terms of the original unit.

Adding fractions - step 4

 

What makes this model difficult to understand, and perhaps this lies at the heart of understanding addition of fractions, is that I have switched from counting in one unit (the whole rectangle), to counting in different units (quarters and thirds), to then counting in yet another unit (twelths), and then being able to see this final total in terms of the original unit (the whole rectangle).

This suggests to me that the prerequisites to understanding fraction addition involve an understanding of what a unit is, understanding how to compose and decompose units, understanding different ways of representing fractions, and how different fractions can be equivalent in size, but measured in different units.

 

 

What does effective mathematics teaching look like?

The definition of what effective mathematics teaching looks like very much depends on what purpose we assign to teaching mathematics. A classroom where the primary objective is to teach students a specific set of mathematical skills for them to use later will look much different than a classroom where the primary objective is to teach students how to think mathematically, although there is obviously overlap between those two classrooms. For a good description of the type of classroom which achieves the first goal but fails at the second goal, see When Good Teaching Leads to Bad Results by Alan Schoenfeld.

I will describe a classroom where the primary purpose of the classroom is to encourage mathematical reasoning, with a secondary benefit of students practicing mathematics skills they have developed.

 

What are the students doing?

  1. Students are engaged in the standards for math practice.

    The Common Core Standards for Mathematical Practice, which are similar in many ways to the NCTM Process Standards, are a useful tool for understanding the types of activities students should be engaged in within a mathematics classroom.

    In order to really do mathematics, students need opportunities to problem solving, to use mathematics they know to model processes, and to do all of this in the socia-cultural contexts of their classrooms. Both of these sets of process standards do an excellent job of defining what it means to do mathematics, but are flexible enough to allow for a variety of different activities to qualify.

     

  2. Spending significant time solving rich mathematics problems.

    Routine problems with limited opportunity for investigation might be acceptable for students to use to practice skills they have learned, but they do not have the breadth necessary to allow students to do the inquiry necessary to learn mathematical reasoning. A significant amount of time in the mathematics classroom investigating, postulating, formulating, deciding, and analyzing mathematical situations is necessary if the habits of mind required for mathematical reasoning are ever going to be adopted by students.

    For excellent examples of rich mathematical tasks (some of which are used for assessment of understanding, and others are used more to prompt student thinking) see some of the web sites linked here.
     

  3. Students talk to each other about math.

    While there is definitely value in students spending at least some of their time thinking independently, there is tremendous value in students having opportunities to discuss mathematical ideas and problems with each other. The first is that it is through the repeated access to different linguistic and representative variations on an idea that we come to more than a superficial understanding of that idea. If I say words, you hear the words, and you might even think you can assign some meaning to those words, but it is only when you hear other variations on the formulation of the ideas behind the words, and see other representations (often physical or pictoral) of the ideas represented by those words, that you can come to a full understanding of the concept. For more information on the dangers inherent in a “linguistic-only” understanding of concepts, see Richard Feynman on Education in Brazil.

    There are other benefits of students talking to each about mathematics. One benefit is the person who describes their solution is either likely to see flaws in their reasoning (or at least receive feedback on those flaws) or in the articulation process of their reasoning, come to a better understanding of the concept. Another benefit is that instead of just one person in the room able to give feedback to students, every student in the room becomes a resource for each other. Finally, someone who has just learned a concept, and more importantly recently moved past their own flawed models of that concept, is often more able to explain the concept as compared to someone who learned the concept long ago, and no longer remembers their struggles with it.
     

  4. Students have the opportunity to revisit and reflect on mathematics they have learned.

    Human memory is limited. Essentially, our mind trims information from it that is not used frequently (or possibly archives it so that it is difficult to access).

    In a highly effective mathematics classroom, concepts students have learned are revisited, often as embedded practice within the current unit of study. For example, when students create graphs of linear functions by plotting points, they are also practicing plotting points. The skill they practice is used within the context of current problem solving. Note that this practice is only really effective once students have mastered the concept as practice without understanding leads to student confusion.

 

What is the teacher doing?

  1. The teacher uses formative assessment practices on a daily basis within their classroom.

    Aside from engendering the opportunities for “what students do” as described above, an effective teacher gathers evidence of their student’s learning in a systematic way. Formative assessment is a process through which a teacher assesses their students, and then uses this information to inform their teaching. It acts as a feedback loop within the cycle of teaching and learning. If we consider what is to be learned as being like a vast wilderness, then the curriculum the teacher follows is a map through that wilderness, and formative assessment is the process they use when checking their compass so as not to get lost.

    There are three basic indicators teachers can use to collect formative assessment information; what their students write, what their students say, and the body language students use that indicates how they feel. All of these are important markers for teachers.

    Unpacking formative assessment is not the goal of this blog post. For more information, I recommend reading Dylan Wiliam’s Embedded Formative assessment as a good starting place.
     

  2. Teachers must build a classroom environment where students want to talk about mathematics and have a growth mindset.

    Developing a positive and productive classroom culture is a critical component of effective teaching. Students must feel that their contributions to the classroom matter, and that they feel safe to make mistakes. Making mistakes, and learning from those mistakes is an important part of learning. The goal of mathematics classrooms should not be to prevent students from making mistakes, but to treat mistakes as opportunities for everyone to learn and to grow.
     

  3. An effective teacher uses questioning technique carefully and thoughtfully.

    Teachers ask a lot of questions. For example, in one classroom observation I did this year, a teacher asked 170 questions in a 40 minute period, which averages out to about 1 question every 14 seconds. Given that many teachers ask a great number of questions each class, improvements in questioning technique are therefore likely to improve overall teacher effectiveness, perhaps even dramatically.

    Good questions prompt students to think. Teachers with effective questioning technique do two things well; they have a set of generic questions prepared they can ask students to prompt their thinking which they use frequently enough that students begin to ask these questions of themselves before even talking to the teacher, and they actively listen to, and clarify their understanding of, student reasoning before responding.
     

  4. An effective teacher learns about the linguistic and cultural backgrounds of their students and adjusts accordingly1.

    Mathematics is a cultural activity. Therefore, as mathematics teachers, we are not only teachers of mathematics, but also teachers of the socio-cultural norms of mathematics. In order to do this effectively, we need to understand our students at a more than superficial level. A cautionary note here: This is an area where it is easy to fall prey to cognitive bias and judgemental attitudes. Teachers need to make their best effort to objectively understand their students’ cultures and their linguistic understandings and then make sense of how their students’ backgrounds impact what is effective for their students.
     

  5. An effective teacher uses technology to focus students on mathematical reasoning.

    Classroom technology, in an effective mathematics classroom, is used to support student’s mathematical reasoning. Rote practice exercises, even if administered via technology, do little to help students develop their reasoning skills, and because they lack context, have limited ability to help students develop connections between different areas of mathematics.

    Imagine a classroom where students are looking for connections between different forms of a quadratic function. They could plot these functions using pencil and paper, and then look for connections, but during the time students would take to draw the functions, they would lose track of the goal of the graphing. Every time we ask students to do another task in preparation for mathematical study, they lose active cognitive resources to keep track of the overall purpose of the task. Instead, in an effective classroom, the teacher would give students access to a graphing calculator or graphing software, and students would be able to focus on seeing connections between graphs, instead of creating the graphs.

 

What else would you add to this description of an effective mathematics classroom?

 

Reference:

1. Suggestion offered by Ilana Horn. See this tweet.