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?