Essential Questions Students Can Ask

I’m a private educator, and I specialize in early math interventions. I spend my days sitting across from young math students trying to untangle what’s got them confused.

Despite collaborating with students who — by nature of their ADD, ADHD or executive functioning deficits — need one-on-one instruction, I‘m always seeking ways to increase their autonomy. To that end, we work to fill learning gaps, increase confidence and build self-reliant habits.

Essential to this goal of fostering self-reliance is teaching students how to ask the right questions…or “essential” questions, if you will. That’s not an arbitrary word choice by the way. When I say essential, I mean it. Incorporating certain productive questions into a students’ universe can profoundly improve their facility with computations, resilience when problem-solving, and ability to progress independently in the classroom.

To wit, if there’s a refrain to my teaching, it goes something like this:

WHAT is this thing worth? WHY does it work? HOW do I do it? WHEN do I use it?

These inquiries are simple enough in theory, but in action, the content of what we explore, why it works, how to do it, and when to use each new strategy changes as frequently as the landscape of the material. A colleague once told me that math instruction was “ruthlessly cumulative.” Though that makes the subject sound heartless (which isn’t productive), it is a pretty accurate lens to view standardized math learning through. The reality is, if we miss a concept along the way, we’re in trouble; we’ll inevitably need to use it later. By learning to ask what, why, how and when as they go along, students lessen their chances of leaving learning gaps in their wake.

Let’s start by exploring questions of WHAT

Broadly speaking, when I ask students questions that begin with ‘what,’ those questions are addressing the value of numbers and how those values change depending on the math we’re doing. These questions are geared towards teaching students to think about what numbers are worth and how that value, that weight, changes when we operate.

Why is this necessary?

Here’s my theory: I think some early math students confuse numbers with letters. They unwittingly apply the rules of reading to math and run into trouble. Looking at the timeline, students learn what numbers are worth at the same time they are learning to read (in kindergarten and first grade), and they bring their reading strategies into their understanding of math. If a student learns that the letter that makes the first sound in bat, ball, and broom looks like this → b …then they can identify that letter and that sound on looks alone. Similarly, students learn what numbers look like. For example, if I hold up my thumb and forefinger, I’m talking about the value that’s represented by this symbol → 2. Students learn to write and read this number we call two. But after learning to associate values to numerals in kindergarten, students often forget that this thing we draw →2← is not really two. It’s the symbol that stands for two. The value itself can be represented by two pretzels, two fingers, two pencils, two quarters, two ANYTHING. Numbers are more than symbols we write on a page, more than digits we carry and borrow. Numbers have weight. They have worth. To be successful in math, students must always consider that worth, and how it changes as we operate.

Here’s an example:

When I meet new elementary students, I usually ask them to explain adding to me. Almost invariably, the exchange goes something like this:

me: So, what is addition?

them: It’s adding.

me: But what does that mean?

them: You add things. Like 2 plus 2 is 4.

me: Oh, I understand! That’s interesting. But can you explain what it is without using an example?

them: It’s when you plus things together.

me: Awesome! But can you try explaining it without using the word “add” or “plus”?

~~~crickets~~~

After we go back and forth a little bit, I usually pull out some colorful poker chips (because they can stack and be categorized by color) and ask them to show me what adding two numbers together looks like. Most students can do this confidently, and they enjoy it. They start with two small piles and push or stack them together. We then talk about that action. What was that thing they just did? It was combining. And that’s what adding is! Adding is combining. From that point forward, I will ask them often, “What is adding?” and expect them to answer “combining.” It’s an easy question to incorporate, and it underscores the fundamental action involved in addition. It reminds them to think about what happens to values when we add them (i.e., they accumulate), instead of just scribbling down and carrying digits on paper. Students learn that combining values creates something more, something heavier.

When adding is well in hand, depending on their grade and confidence, we’ll explore the other operations. What is subtraction? What is multiplication? What is division? We’ll perform these operations with the poker chips and discuss what happens to the values. These kinds of fundamental explorations can make a major difference in a student’s dexterity with all kinds of problems, especially story problems or those without clear instructions about what operation to perform, when intuiting that operation is part of the critical thinking skill being built or tested. Yon about 4th grade, we’ll also start asking questions about what the values of fractions and decimals are worth in comparison to whole numbers and how those values change when we operate. Middle school offers the opportunity to explore what integers, exponents, radicals and variables are worth and the chance to investigate the balancing act values do in expressions, equations and inequalities.

Next, let’s dive into the question of WHY

So, why ask why?

Most importantly, because it will keep your student from morphing into a copy machine. When students learn to ask why, they begin to care about why a computation or operation works, which means they’ll be less likely to rely on blindly following steps like an automaton.

Often, students learn how to do a computation on paper, and they follow the steps as if following a recipe. Except, when you follow a recipe, you get cake when you’re finished. You get to eat that thing you just made and marvel at all that went into making it. In math, many students don’t want to marvel. They want to follow the steps, finish, and move on. It’s robotic in a way, rather than participatory. Asking why the steps they’ve been taught on paper actually work reminds them that math is more than memorizing steps and that numbers are more than symbols we scribble on paper. Why we perform a computation is inextricably tied to what is happening to the values when we compute.

How does this look in practice?

Here’s an example: When a student is learning to regroup, they are taught to “borrow” in subtraction. If, for example, they are doing a problem like 54–26, they learn to cross out the 5, put a 4 on top, put a 1 in front of the other 4, then subtract from there.

But why does that crossing out and placing 1 stuff work? In this case, because we’re not actually borrowing 1 from the 5, we’re borrowing 10. The way that number (1) looks actually hides how much it’s worth (10).

***PSA: This happens ALL. THE. TIME. in math. What a number looks like and what a number is worth are different things. Please teach your student this and remind them as often as you can.***

The above borrowing lesson goes back to place value. If it’s something a student is really struggling to understand, I’ll pull out dimes and pennies to illustrate. We’ll make a stack of 5 dimes and 4 pennies and place them side by side. I’ll then ask them to take 6 pennies away from the stack of 4 pennies. They’ll tell me it’s impossible! Then I’ll ask them to borrow a dime from the stack next door. When they do this physical action, they can see that the ONE dime they are putting on their stack of pennies is actually worth TEN. They can then see that their new stack may only have 5 coins in it, but those 5 coins are worth FOURTEEN. They’ll notice or I’ll point out that the stack of 5 dimes now has only 4 dimes in it and is worth 40. That’s why regrouping works. We’re not just crossing out and carrying numbers around. We’re exchanging values with weight to make operations possible. It’s a balancing act, not a series of meaningless steps.

Now it’s time to consider the question of WHEN

This is a sneaky question most students forget about. They don’t think to ask, “Hey um…when do I use this thing you just taught me?”

This oversight isn’t a student’s fault. It’s an understandable byproduct of a lack of perspective.

When students are taught concepts or computation shortcuts, they learn each tip within the universe of a given chapter. Basically, they’re taught strategies for adding and subtracting fractions when they’re in a chapter on adding and subtracting fractions. They don’t have to focus on a question like ‘when do I make common denominators?’ because they’re making them in every problem. But what happens after they haven’t seen a fraction for months? What if they’re asked to do something like multiply fractions on a later assessment? Will they accidentally make the common denominators they learned when adding and subtracting fractions instead of just multiplying across the numerator and denominator?

Let me answer that question for you.

Yes, yes they will.

Students confuse strategies all the time because in the moment they learn them, they don’t have to worry about the when. Later isn’t impacting them now, and they lack the perspective to understand why when is important. So, they don’t worry about it, and the result is, that great concept that was really helpful and let them pass Chapter 3 with flying colors is completely forgotten by Chapter 5 and misused on every assessment that follows and not relearned until two years later when a friend reminds them the concept exists.

So how do we help them widen their perspective?

I regularly remind my students that math requires three steps of them in order to stay successful: Understand what they learn when they learn it, remember what they’ve learned, then use that knowledge later. (I call this Understand, Remember, Integrate.)

I let them know that my hope is to help them with the first step — Understanding — and the third step — Integrating — but that the second step — Remembering — is mostly on them. A big part of remembering is remembering when a strategy is used. Considering when to use each skill is a habit I try to help them build and one I encourage them (if they’re old enough) to constantly take notes on. We store those notes in a place where they can always reach them, usually at the front or back of a binder, and we add to them often. Then, when I’m not around or their teacher is busy, and they can’t remember whether they need those common denominators or not, they can check their own notes in their own writing before they raise their hand.

This teaches them to rely on themselves first before seeking help. It builds their confidence and fosters their resilience in the face of the ever-ending questions math will present them with. If we can teach them that finding answers isn’t impossible, then they don’t give up on asking. If we teach them they can find answers within themselves, then they begin to look there more often.

And finally, on to the question of HOW

This one is pretty straight-forward. When I ask students how to accomplish a computation, it’s usually the question they feel most confident answering because it’s the question that gets the most love in math class. How do we do long division? How do we balance equations? How do we simplify fractions?

I don’t think I need to give a detailed example here — you get the drift.

Yet, if asking how is intuitive, why do we need to discuss it?

Frankly, because it’s not the only question that matters and many students don’t realize that. How I find an equivalent fraction (multiplying the numerator and denominator by the same value) is an important skill to have tucked away in my toolbox, but understanding why that step works (because I’m really multiplying by 1), what it accomplishes (it helps me create common denominators) and when to use it (like when you’re adding, subtracting or comparing fractions that require common denominators) are all linked explorations. How is important, but it’s not the only question that counts.

So, how can we help students gain this perspective?

I think it comes down to being honest with them. We can teach them to value how, while also looking beyond and behind it. While teaching, I always compliment students when they remember how to do a computation, but I also prod them to tell me why that computation works, what it accomplishes and when they can use it. This ties the most obvious question that they’re most comfortable asking and answering- the question of how to do the math- to other equally important yet more inconspicuous inquiries. It widens their outlook, builds self-awareness, and teaches them to persevere in their curiosity, to ask tough questions even when they don’t think they have to because the reward is worthwhile.

To be honest, for many of my students, this takes a while. Some of them are shy and uncomfortable speaking up — asking questions is a lot to ask of them. Some of them are insecure and don’t want anyone to think they have questions. They build up a brick wall that is high and mighty and they’ll be damned if they’re going to let me chip away at it. Some students have learning differences that make it difficult for them to get what’s in their head into words, so the formation and identification of what questions to ask is hard. But in all these cases, every student can make strides by remembering that how do I do this isn’t the only question that matters.

Math is not a solo sport. It’s a team effort, and how needs its teammates: what, why and when.

If your student will let you, remind them of this…over and over and over again.