1. Lattice Multiplication: Introduction

Remember the good old days of long multiplication? You multiply, bring down, and carry. There are so many new methods, and lattice is one. It makes complete sense logically once you understand it, but it is one of the least intuitive structures. Check it out:


It’s crazy looking, and definitely not natural. Let’s get there one step at a time. Let’s solve a problem using our old-school math, and then figure it out using a few different methods. Eventually we’ll do it using lattice.


867 x 39

Traditional Solution:

Displaying IMG_1061.JPG

Not too bad…so familiar and comfortable. Next we’ll look at basic multiplication using the area model.

3. Set Up Expectations and Routines, Not Rules

I’ve read in a few books and heard in many classes, as I have mentioned, that limiting rules to a select bunch is good. As you may guess, I disagree. I feel that starting with a list of rules is very limiting.


Also, setting up very specific consequences (if you do X, then Y will hapen [insert call home, office, etc] ) can get out of hand – if you don’t follow through, then you look bad. And if you do, then does the punishment really help the behavior in the long run? My guess: no.

Try this instead:


No rules. Instead, make a list of routines. What are your requirements? How do you want things to go? For example, turning in homework. Where, when, and how should students turn it in? Then teach them the process. For many teachers, the “I thought I turned it in, I gave it to you during class, etc” gets out of hand. Tell students how to do it, and that’s it. Then when there’s an excuse, you can respond simply: “Since you know the routine, I’m sorry but I can’t be responsible for losing it even if you did give it to me in the middle of the lecture as we transitioned to the quiz and 10 students were asking questions, which is when you decided to hand me a half-crumpled and ripped piece of paper.” Hopefully you don’t say all of that every time, but you get the point.

Routines are the background processes. Without them, nothing else gets done. It’s how you file into class and sit, it’s how to transition between classes, it’s how you turn it and pass back homework. Put the onus on students, tell them that it HAS to be done that way. Not because you’re strict and arbitrary, but because any other way will be too distracting and the class won’t function. I’ve heard these called non-negotiables. If you hand me homework mid-class, I will lose it. I don’t negotiate that. I need you to start and finish the warm-up in 10 minutes, since that helps me figure out how to frame the rest of the activity. Must happen.


Expectations are broader. Rules are strict and limiting, and don’t necessarily allow for flexibility in how different students approach and handle things. Expectation 1: Work when it’s time to work. 2. If you finish early, help other students. 3. Walk appropriately in the hallways. 4. Speak at the right volume for the activity.

“But students will try to take advantage of loopholes and play games and it will lead to arguments!”, I hear people screaming. Maybe. At first. But it leads to a broader discussion: what is really appropriate? Students know it’s not appropriate. Don’t argue, don’t discuss it right away. Tell them. “You do know that’s not appropriate, and if you want to talk about it, then see me ____ [lunch/recess/after school/study block].”

I am also going to add a possibly unpopular sentiment here: the middle of class when a behavior happens is NOT a teaching moment. Instead, you keep it in your back pocket. A student got upset and slammed his desk and walked out. You want to have a rational discussion about expectations? Good luck! (but not really, because don’t do it!) What to do: tell the student: “You know that you’re expected to work appropriately, and I feel you didn’t do that. I’ll see you at lunch/recess/after school/Friday advisory to discuss it.” NO argument allowed (you need to move on). What you MUST do is actually follow up with the student. The immediate consequence may be that the student now sits and works, maybe goes to the office to work away from whatever caused the scene, etc.

2. General Discussion on Behaviors

Here’s an interesting chart on behaviors:

It sums up 3 of the most common functions of behaviors (aka reasons for them): getting something, avoiding something, and sensory. In a typical classroom, the “disruptive” behaviors are, I’m guessing, probably about getting attention, or avoiding something (work?). If you’ve had similar experiences, you’ve probably heard that 90% of behaviors are escape.

But it’s interesting to think about different perspectives…we’re assuming here that students want to do well. But students still do these things….why?!? If you come up with a solid blanket answer, you’ll be golden.

We do have to consider that students are, in fact, people. Little, under-developed people. They make mistakes, like all of us. We do the same things. So why do we hold them to such high standards?? Because if we don’t, then they won’t be able to learn contexts for different behaviors. We hold them to a high standard – then, in real situations, the behaviors are internalized.

Luckily, positive reinforcement (rewards over punishment) is an effective technique for helping students not only understand expectations, but to meet it regularly and independently.

1. Positivity and Motivation: Basic Definitions and Assumptions

The topic of behavior management comes up a lot. For some it conjures the picture of the teacher standing guard over the class, ready to pounce at the first utterance. For others it’s the nurturing teacher, who loves her students dearly and goads them into compliance through other means.

Different methods work for different teachers. And for different reasons. Rather than compare methods, let’s work from the ground up. Reflect on any classroom you’ve been in – as a student, observer, teacher – and think about what makes it successful.

As we go through this journey, we’re going to base our work on two assumptions:

  1. Students want to do well
  2. Teachers want students to do well

It sounds obvious, no? Think about the student who complains all day about being in school, and rejoices at the bell. Clearly he doesn’t want to be there. Or the student who always tries to get out of math – they have no interest in being there. Is it because they don’t want to do well, or feel like they just can’t (and turn it into a show)?

As we go down the rabbit hole,we’re going to encounter a lot of words that have a million meanings, depending on who says them. This is how I will be using them, and I’ll be talking about students generally (since that’s our topic). You can generalize from there.

  1. Behavior – something a student does.
    1. Example: talk, turn in homework, walk, run
  2. Consequence – whatever happens after/as a result of behavior
    1. A consequence is often used to mean punishment, but we’ll be talking about all kids. Positive, like a prize; negative, like losing recess; or natural, like getting hurt after jumping from a tall height. Positive and negative, when we talk about them, will be imposed by teachers. Natural consequences are natural.
  3. Reward – a positive consequence (special lunch, candy, positive call home, etc)
  4. Punishment – a negative consequence. (detention, losing recess, a call to a parent, etc)
  5. Black-list – a list of things NOT allowed (behaviors, or things like phones)
  6. White-list – a list of things that ARE allowed (behaviors, or things like water)
  7. Expectations – a list of things which are allowed, and SHOULD be there (like turning in homework, pen and pencil, etc)

It’s interesting to reflect on your own experiences as an observer in a classroom, and working with kids. Where have you seen any of these things in play? Many teachers use both rewards and punishments – but which happens more often, and more consistently? Which are more frequent and smaller, and less frequent but bigger?

Upcoming Series: Classroom Setup via Positivity and Motivation

As I prepare to take over my own classroom again (!!), I’ve been thinking about structuring expectations and routines. One piece of advice that I took away from my master’s program was a simple rule of thumb:

No more than 5 rules

I pondered this, and accepted it. I tried to use it as a guideline, since it limits how many rules you have to enforce.

But then I entered the world of special education, and it changed everything I thought I knew about behavior management.

In the upcoming posts, I will address my perspective on the following issues (maybe one post per topic, maybe they’ll just get addressed intermittently. We’ll see).

1. What is positivity and motivation? How does it benefit teachers and students as a framework?

2. Whole-class and individual behavioral guidelines (overview)

3. Behavioral expectations and academic expectations

What kinds of things have you done or seen that worked in structuring classrooms? The end goal is learning.

Arithmetic with Dots

Ben Orlin’s recent post on using arrays with multiplication made me reminisce on days past, in which I talked to other educators about foundations of mathematics. I’ve also started watching a great youtube lecture series on the history of math, which also made me rethink how I should be teaching math.

In tutoring, I’ve spent a lot of my time showing students how their current work is just their old work with a twist. Algebra is just arithmetic with a sprinkle of magic. (I’ve seen first graders do 1 + __ = 4). That’s a one-step equation in algebra, and students are forced to subtract one from both sides. WHY?!? Calculus is just algebra with very BIG and very small concepts added in. I spent a while showing a student that the equation of a tangent line to a curve is just point-slope form of a line. Oy. Then she got it.

So let’s look at how arrays can give us the basic operations, without needing any variables.

Addition: Hopefully obvious. 3 + 2 = 5. Count the dots.


Subtraction: A bit harder, because it involves “take away”. But otherwise obvious (once you know it, anyway).


That’s it for 1-dimensional. Now we move into 2D.

Multiplication can be defined as repeated addition (3 + 3 + 3 + 3 + 3 = 3×5) or as equal groupings (5 groups, each one has 3 in it). But instead of drawing numbers on a line, we go into arrays.

I’ll leave the details to Ben’s post.

Division is where I haven’t seen too much. We pester kids with the fact that it’s the inverse of multiplication using families (“If 2 x 5 = 10, then 10/5 = 2”). But where’s the JUSTIFICATION for it? Where can I SEE this happen?

The struggle later turns into remainders, and then remainders as decimals and fractions.
There are many great division strategies: standard long-division, partial quotients (one of my favorites), and many others.

But how do we show it? Well, just do multiplication backwards. Multiplication = repeated addition or equal groups. So division = repeated subtraction, or equal groupings.

Then let’s try 12 / 3.

even division

Not bad. What about 13/3?

remainder division

So, we’re left with remainder 1. That seems reasonable, right? But what about when we want students to represent the remainder as a *gasp* fraction? We can tell them that since it’s in 3 equal groups, we should put 3 as the denominator and then the remainder 1 becomes the numerator. But…WHY?!?

Well, we’re counting rows. How many rows did we get? 4. How many more ROWS do we have? Well…1 piece left over. OK, so if we JUST LOOK AT THE ONE ROW left there, all of a sudden we see it: there is 1/3 of a row! So 13/3 = 4 1/3.

Questions, comments?


How Bad IS Standardized Testing? AKA Understanding vs Skills

As I delve deeper into one-on-one tutoring of my own, as well as working in different classrooms, I run into common conundrums. How do we get kids to truly understand what they’re learning?

Standardized tests provide a measuring stick of sorts – they tell you what standards they’ll assess, and educators are theoretically responsible for making sure students can meet those standards. Not bad, in theory.

But the standards are often phrased in terms of understanding…how do you measure understanding? I can measure behaviors – can you match one thing to another, can you enumerate each step of a calculation, etc.

How do you measure understanding? Is it applying skills (behaviors) you’ve learned to novel situations? Is it being able to articulate the process in words/writing? How do you measure something so complex?


The longer I work in education, the more I notice that what people are really discussing under different guises is the use of motivation. How do we motivate students? How do we motivate teachers and administrators?

Everyone responds differently to different things – I am motivated by academic success, making my close friends and family happy, and food.

What motivates you? Have these motivations played any part in your education?

[more detailed explanations and analyses of what I’m talking about to come later]

Math Motivation

Lately, a lot of focus in what I’ve seen in math has been around motivation. It’s a key term in education, and more or less a standard to which people are help – who is supposed to motivate kids to learn, and who is held accountable for it? Some say parents, teachers, districts, kids themselves, etc. So how do we motivate them? I’ve seen a few ways:

1. Make the work more interesting/relevant/practical/etc.

This method relies on the kid engaging with the work because it just makes sense to engage with it. It’s interesting, fresh, and usually much more concrete than solving a number problem.

2. Make their lives dependent on it.

You know those posters…”Think You Won’t Need Math?” then it lists jobs that need math…right? I agree that all jobs, in fact all areas of life rely on math in some way. However, I think that motivation stops being relevant around middle school. I’m an avid math educator, and I think many of the skills you learn in math could help in life. I really do. But who needs trig unless they plan on going into math or science fields? Why does everyone need to know trig? Why not teach them how to work collaboratively, so that the kid who knows trig and hates writing can work with the writer who hates trig? 

**Side note: I hate these posters.

3. Miscellaneous

Some teachers make the subject interesting by showing their own interest. Some just engage students personally (or in a similar way), and motivate them to get an education (and students accept that math is part of that). 

Is just one of these ways enough? Do students need a blend, or can we assess the levels of motivation they have coming in and work off those? Some students will show up and learn no matter what you do…do they need to be “motivated” further?

Math Example – The Chinese Room

Dan Meyer recently posted an article entitled The Chinese Room. The moral of the story is: if I give you something written in Chinese and a manual for writing down other Chinese characters, are you literate in Chinese? Now, you have to write the actual response (assuming the manual is correct), and to someone who is actually literate the answers must be correct. So…are you?

It’s an interesting debate, and one I’ve seen many time working with students. The idea is a bit abstract at first, but this post is dedicated to the same idea, outlined in a particular example: Linear Equations
Consider the two equations:

(1) y = -(3/4)x + 24


(2) 3x + 4y = 24

Mathematically, they’re equivalent. But do they mean the same thing? Same slope, same intercept…same interpretation?

When I’ve seen it taught, one of the fundamentals is changing between the 2 primary forms of equations (slope-intercept, standard form). But why?

Equation (1), how we often write them, to me signifies change over a given interval. You start with 24, and every 4 units it decreases by 3. Dollars per hour? Cookies per day? The graph ultimately shows how much you have (or owe, if you like negatives). I think of it as filling in the blank “after ___ days you will have ___ dollars” (change units as you please). There is an answer to be found, depending on what you’re looking for. Traditional algebra.

Equation (2), though, is something totally different. I see that, usually, as comparing costs. I sold x-ray goggles and yams, making $24 total. The goggles cost $3, and the yams $4. So, how much did each cost? Here, the 2 variables each represent their own object. There is no change over time. So, what does the slope mean? For every goggle, the price of yams goes down? Not quite. It shows how much each could cost. Each ordered pair is a possibility. There is no one answer you’re seeking. Uncertainty, conditional information, no answer.

Reflecting on my own experience, I have basically taught students to turn all equations into form (1) and go from there (unless you’re graphing using intercepts – but that’s another show). So, I’ve been teaching them to read and write Chinese according to the post title. I haven’t promoted true understanding.

Have you done what I’ve done, or how have you been able to get students to the heart of it?


**Additional note:

Equation (1) can also use, with reason, the negative values in quadrants II – IV. -1 for x means the day before. -1 for y means you owe someone money. Negative = directionality.

Equation (2), though, in this case does not allow for negatives. Unless buying x-ray goggles entitles you to a refund.

Another bit of algebra caught up in this…has anyone used these as a motivator for discussion about different ways to use negatives?