Elevators and Number Sense

Number sense should develop early, and what simpler way to do it then to start with elevators?

Elevator, Vicenza, Italy

Why elevators you ask? Well, I just returned from 2 weeks in Italy. Partly for work: training elementary math teachers in Vicenza, Italy on College & Career Ready Standards for UT Dana Center International Fellows and Department of Defense Education Activities; and partly for leisure: touring Venice, Cinque Terre, Florence, Tuscany and Rome with my husband, sister, and brother-in-law. The first thing I noticed was the elevators have negative numbers to indicate those floors below ground zero (i.e. what we usually call floor 1 or Lobby in the U.S.)   It’s not the first time I’ve noticed this – in England, in Paris, in Germany – all these other countries indicate on their elevators the ground floor to be 0, the floors above ground 0 are 1, 2, 3…. and the floors below ground zero are -1, -2, -3….

This way of numbering elevators makes sense. Much more sense than Floor 1, or Lobby and then Basement, Basement2 (or LL1, LL2) – which is our typical way of indicating the ground floor (1) and the floors below ground level (Basements/Lower Levels). If you were a young child living in these countries and taking the lifts (or elevators), you are regularly exposed to integer numbers – with a contextual connection that the ground floor of a building is ground 0, and the floors below the ground are negative numbers, and the floors above the ground are positive numbers. It may not even be explicitly explained to young children, though they would be using the terms ‘negative 1’ or ‘negative 2’ to go down below the ground floor. They will have this repeated exposure so when they are ‘officially’ taught about negative numbers in school, they have an immediate connection to prior knowledge about the numbers in an lift/elevator and can make a real-world connection. Negative numbers won’t be new or hard to understand because it’s just the numbers in the elevator. Or – the numbers of the temperature, because let’s not forget, these countries also use the Celsius temperature scale, where freezing is 0, and anything above 0 degrees is above freezing and getting warmer (positive) and anything below 0 degrees is getting colder (negative). The further from 0 in either direction, the warmer or colder you are – again, real-world connection and a contextual understanding of integers.

Number sense. Number lines. Integers. Real-world connections. Just from elevators and temperature scales.

This repeated exposure, informal as it may be, is developing an intuitive understanding of numbers and their real-world meaning. And when students are then exposed to number lines and positive and negative numbers more formally, in a school setting, they already get what that means because it is familiar to them. They can apply what they already know to ‘mathematics’. The formalization makes sense, and connections make sense, and understanding is that much deeper.  This is different in the U.S., where students often struggle with the idea of ‘negative’ numbers and number lines and the distance from zero because we are teaching them something new.  We don’t have a real-world exposure to negative numbers because we use LL or B1 to represent lower than 0, our ground floor is never called 0, it’s 1 or Lobby or G (ground). Our temperature doesn’t have 0 as the freezing mark – it has 32 degrees Farenheit. Think how much easier it would be to connect negative numbers (those numbers smaller than zero) to negative floors or negative temperatures. Freezing makes sense at 0. Negative temperatures are colder than freezing. Positive temperatures are warmer than freezing. 32 degrees – not quite the same one-to-one connection to a number line, is it?

Anyway – my point is that something as simple as changing the numbers on an elevator to integer representations would go a long way in helping young children develop number sense early on so that by the time they get to school, they already have a natural understanding of positive and negative numbers. Early on they would be exposed to the idea of 0 being the ground level, positive numbers mean higher floors or farther away from ground zero, and negative numbers mean lower floors, below the ground, and the further you go below ground, the more negative you get, the farther away from zero you are. Number lines would then be ‘recognizable’ because there’s a contextual connection. (If we could change our temperature scale to Celsius that would be great too, though that one is a lot harder to do).

Relabel elevator buttons to reflect numbers on a number line – a simple change that could go a long way in developing informal number sense in children.




Modeling with Mathematics – Math Practice #4

Whether or not you teach in a state that has adopted the Common Core State Standards for Mathematics (or a modified version of them), the Common Core 8 Standards for Mathematical Practice should be something every math teacher fosters in their instructional practice. These practices are based on NCTM’s processing standards and NRC’s standards for mathematical proficiency in the Add It Up report. They are about helping students become problem-solvers, creative thinkers, communicators, users of multiple resources, and most importantly, able to apply what they know in multiple ways. That’s what teaching math should be about – helping students use what they learn in the world around them, now and in the future.

I am NOT going to get into a debate with anyone on the pros or cons of the CC Math Standards themselves.  That is a politically charged hot mess. Whether you are for or against these standards is irrelevant. How you teach and support student understanding – i.e. your TEACHING PRACTICES, is what makes the difference, NOT the standards you follow in your curriculum. No matter the content standards, they way you help students learn, understand and apply those standards is important and vital, and is what the practices are all about. Lack of how to incorporate effective practices is what I have found, from years of working with teachers, is one of the biggest deterrents in student learning. And, as evidenced in many articles and classrooms I have observed, there is a great deal of misunderstanding of the Standards for Mathematical Practice and how to incorporate them effectively.

In a previous post, I highlighted Mathematical Practice #5, Use Appropriate Tools Strategicallywhere I tried to explain what the practice meant and provide some examples. Today I’d like to do the same thing with Mathematical Practice #4, Model with Mathematics as I think this is one of the most misunderstood, or ‘misused’ practices. I myself, before doing an in-depth study of the practices, interpreted this practice wrong. In my mind, I thought it meant that I should be using manipulatives and ‘models’ (i.e. technology simulations, physical models, etc.) while teaching and I would therefore be modeling with mathematics.  That is part of the standard, but NOT the true purpose.  Remember, these standards for practice are what we, as teachers, are trying to foster in our students – meaning, we are trying to help our students model problem situations with mathematics to help them better understand it and/or solve it.

Let’s look at the actual standard.  I have highlighted key phrases that help clarify this standard:

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Notice the different ways “modeling” is represented: apply the mathematics; write an addition problem; apply proportional reasoning; use geometry; make assumptions and approximations; map relationships; use tools. This standard is not talking about physical models, but rather helping students look at a real situation (so context is important) and use mathematics to help understand it, explain it, make it simpler, relate it to something else, etc. Can they use physical manipulatives? Sure – geometry, tables, graphs, manipulatives – but all of those tools are to help them make sense and model the real world situation. As a teacher, what does this mean YOU should be doing to help foster students ability to model with mathematics? Well, for one thing, give them relevant and real problems to solve that are not “naked math” (see my previous post on this!) but contextual problems that force students to think, analyze and decide what mathematics will help them solve the problem. And then, provide them opportunities to collaborate, use a variety of tools, ask questions, and approach these problems in a variety of ways. Modeling with math means students apply mathematics and tools in ways that make sense to them so they can apply their mathematical understanding. That’s how YOU know they really understand the math, and that’s how THEY know math is relevant.

Obviously, the key here is making sure you provide relevant problems that “arise in every day situations” (as the standard emphasizes). Students learn and are more engaged if what they are doing is relevant.  And there are real problems every where.

lottery-abstract-illustration-dynamically-falling-balls-39580479So – here’s an every-day situation that arose just last night. There were 3 winners in the 1.6 billion dollar Power Ball lottery. Depending on the grade of your students, you could ask them a simple question – should these 3 winners take an annuity or take a lump sum? (Naturally – expect explanations and support for their answers)! In order to answer this question,  a lot of decisions need to be made concerning how to model the situation with mathematics. What, if any, equations would be helpful? Do they need a table to organize the data? What are the taxes and how will that impact the amounts?  These are big numbers – so could they make it simpler by using smaller numbers or proportions? Think of all the great mathematics that will happen and all the modeling with mathematics students will do within the context of this real-life problem. Think of the engagement. Think of the conversations. Hard to get that with a worksheet!