Thursday, October 1, 2015

How's Your Number Sense? Not quite a quiz

Yesterday I talked about number sense in quite a bit of detail, and sure enough, several of my Twitter buddies began talking about it with me. Even more naturally, they all wondered how good their number sense was and where they stood compared to everyone else.
This is not a surprise. Every living thing that has eyes seems to love a mirror.
I've never met readers or students who learn a new idea without wondering if it applies to them. This is why so many medical students suffer from hypochondria, law students become fascinated with petty grievances to themselves and their families, and I'm told by a friend who went through a full set of factory training as an auto mechanic that it was at least a year before he could drive without hearing every little stray sound from the engine. And whenever I've found myself explaining number sense to the parents of my tutees at Tutoring Colorado, sooner or later the parents have wondered about their own number sense.
So I am guessing that you might be wondering how good your own number sense is. My quick answer is, "probably pretty decent, since you're reading this, and most people with really bad number sense won't read about math at all." Then again, some people will endure almost anything, even fractions, if they think it will benefit their kids.  Possibly you are even wondering if the whole problem is that your own number sense is deficient, so you never really learned real math, and now you can't help your kids. It's a bit like asking, before you start reaching for the victims and pulling on their arms, whether you yourself are standing on quicksand, and it's a very good question.
In that case, please take some comfort in this: helping your kids with Singapore Math will boost your own number sense tremendously. I often send my tutees home with Singapore Math-based projects to work on with their parents, and I've lost count of the number of times I've heard, "So while I was trying to help him I suddenly got it myself. I never got that before!", and sometimes the even more enjoyable, "She did fine. She got it before I did, and she was so proud of herself for being able to explain it to me." As you come to understand what should be happening in/with/for their number sense, you're going to rapidly improve or reawaken your own. You may also become a great role model for how to handle intellectual difficulty, and help the kid see that though knowing matters, learning matters more. 
So don't worry about how much number sense you have now. It's not a quiz. It's not a competition. There is no generally accepted scale for measuring raw number sense anyway; a good score might only mean you are a fast snail or a big mouse, a bad score might mean you're a slightly less beautiful eagle or a smallish whale. Most likely of all, it might mean I'm a poor questionnaire writer.
Nonetheless, let's see if this gives you a picture of where you are, and maybe some idea of where you want yourself or your kids to be.
This questionnaire is based on material I've used with older kids in the tutoring business. It aims to show how much you already use (or don't use) number sense in your approach to math, and I hope therefore clarifies what this number sense thing is all about. Please accept one hug, pat on the back, or small medal for voluntarily taking a math test in the hopes of helping your child. If that's not parental love, I don't know what else could be.
It will help to have some way of recording the results as you go along, so you might want to open a note window or grab pencil and scratch paper. There's a full explanation of the answers at the end, but I strongly suggest you do all the questions before reading through the answers. On the other hand, even if you get them all right, you will still want to look at the explanations to see if you were actually using number sense to get the answers.
Read each question carefully. Figuring before thinking is probably the leading warning sign of poor number sense.
Do not time yourself, or do anything else to give yourself an incentive to be fast with an answer rather than clear about why it is right.
For each problem, record two pieces of information: You might want to draw up a little table with 2 columns, "answer" and "NS level", and 20 numbered rows, if you're one of those people who likes to keep neat records. "Answer" needs to be the widest column. Here's one to copy to paste to your note window if you like:
Problem number
NS level




















In the answer column, write the correct answer, if you can see how to get it. I've used the current Colorado fifth-grade math standards (i.e. the last year before middle school in a state that has average math scores and happens to be located immediately around me) to devise the questions, so there is always a way to the answer through elementary school math. You are very welcome to use any higher math you know, however (and often that will be much easier). If you don't see any way to the correct answer, write "guessed", "?", or something else to remind you of your process.
In the NS level column, write a letter from the list below; what level of number sense did you use to solve the problem? You should be trying to work at the highest level of number sense you can, so you should probably read through the levels first:
a.      You just knew the answer, and why it had to be right, right away. For example, most people can correctly answer "Which is bigger, 1448+5 or 1448+6?" right away, without calculating, because they notice that the only actual difference is the one between 5 and 6, so it feels like they "just know." They would record "a" for their level.
b.     You could arrive at the right answer after some thinking about it, but you didn't have to calculate. For example, whether or not you know how much a quadrillion is, you can probably answer "What is half of six quadrillion?" by thinking of an analogy (what is half of six dollars? half of six sheep? half of six gallons?) and referring to a math fact. That would be a "b." If you actually had to do something to compute half of six, either because you don't have that in memory or because you don't see that "half of six" is the same number no matter what the units or the multiplier are, then you would record "c" or below.
c.      You calculated correctly and got the right answer. For example, most people would calculate to answer, "What is 162 divided by 6?", by long division, mental short division, or factoring, and would record "c" for it. If, however, right after doing that you realized either "oh, wait, I know 6x25=150 and 6X2=12 so it had to be 27" that might be more like "b". (Notice this is truly an honor system; the difference between "b" and "c" is more about how much calculating you had to do than about not doing any or having to calculate every tiny step. Are you mostly thinking, or is the pencil or calculator really busy?).
d.     You thought you knew how to calculate but then realized you weren't getting the right answer, or you got confused in the middle of the calculation, or you couldn't decide which of several possible calculations to do.  "d" is probably best described as "I used to know that, I think."
e.      You can see there must be a way to calculate this, but don't know or remember enough to see how to do it yourself. In other words, you're pretty sure the answer is in there (without my having to tell you it is -- though there's one trick question where the answer is there's no answer), but you really don't have any idea how to go in there and drag it out. Most people who know what a cube root is will concede, for example, that there must be some way of finding the cube root of 864 without a calculator or spreadsheet, but they wouldn't know where to begin, so they'd put a question mark for the answer and an "e" for number sense. (There's not actually any problem that hard below, by the way).
f.       You have no idea at all; don't even see how an answer could be calculated. This isn't the same thing as the terms being unfamiliar; those should be marked "unfamiliar" or "didn't know the words" in the answer space. For example, since most people don't know what a hyperbolic cosine is, if I asked you to calculate one (I won't!) you would write "unfamiliar" in the answer space and leave NS level blank. On the other hand, if the problem is that it costs 72 cents each to make the first gallon pitcher of lemonade, and each successive gallon is 13% cheaper, you always sell exactly one gallon at $1.00 per glass on a sunny day with 80 degree temperatures, you sell an extra quart for every degree the temperature goes above 80, and sales double for every 20% price reduction, if the temperature is 91 degrees, how much lemonade should you make and at what price should you sell it for maximum profit?, (about a college sophomore level economics problem -- don't worry, nothing like that below either). Then if you see there's a way to get an answer, even though you couldn't do it yourself, that's an "e." If you don't see any way that anyone could get any answer at all, give it an "f."
Record the highest level of number sense you could have used, whether or not it was your first thought. This second score is about the highest level of number sense you can work at, not about what level you usually work at. (Though if you notice you're always calculating first and then number-sensing afterward, that information might be useful or interesting also.)
Again, record BOTH your answer (if any) and the level of number sense you were able to approach the problem with (whether you got the right answer, or any answer at all). We'll be looking at both the answers and the NS level at the end.
All right, grab your pad and let's begin. (Sorry about the weird formatting of what follows; I haven't mastered all the nuances of getting math notation to work in Blogger's interface. I decided to prefer size and readability to style, as well as to staying up all night figuring it out.  If you have elderly eyes like mine, click on any panel and it will pop up as  a separate, easily enlarged window).

The answers, and how people with number sense might know them without calculating. 

You can count the answers right/wrong in any conventional way you like. The problems were taken mostly from the Grade 5 advanced standards with some additions from the Grade 6 regular, so if you got 14 or more right, you're about on par with what we expect of a brainy, well-trained 11 year old in Colorado.
For number sense, count the frequency of a, b, c, d, e, and f. (If you're ambitious you might even do a histogram).  If you have ten or more a's and b's combined, that looks like pretty good number sense to me; if most of your answers are c's and d's, you probably have fairly good number sense but learned math procedurally, so you may have to work on your own number sense to coach Singapore Math well. e's and f's mean you probably really need to work on your own number sense at the same time you are trying to help your kids. Be sure to admit you're trying to figure it out together -- seeing you struggle and catch on may very well be exactly the model the kid needs.