The Types of Questions we Ask: which categories of questions should we focus on?

I think we can all agree that there are many different ways for our students to show what they know or understand, and that some problems ask for deeper understanding than others. In fact, many standardized math assessments, like PISA, aim to ask students questions at varying difficult levels (PISA uses 6 difficulty levels) to assess the same concept/skill. If we can learn one thing from assessments like these hopefully it is how to expect more of our students by going deeper… and in math class, this means asking better questions.

Robert Kaplinsky is a great example of an educator who has helped us better understand how to ask better questions. His work on Depth of Knowledge (DOK) has helped many teachers reflect on the questions they ask and has offered teachers examples of what higher DOK questions/problems look like.

In Ontario though we actually have an achievement chart that is aimed to help us think more about the types of questions/problems we expect our students be able to do. Basically, it is a rubric showing 4 levels of achievement across 4 categories. In Ontario it is expected that every teacher evaluate their students based on each the these categories. Many teachers, however, struggle to see the differences between these categories. Marian Small recently was the keynote speaker at OAME where she helped us think more about the categories by showing us how to delineate between the different categories of questions/problems:

  • Knowledge
  • Understanding
  • Application
  • Thinking

Knowledge vs. Understanding

Below are a few of Marian Small’s examples of questions that are designed to help us see the difference between questions aimed at knowledge and questions aimed at understanding:

As you can see from the above examples, each of the knowledge questions ask students to provide a correct answer. However, each of the understanding questions require students to both get a correct answer AND be able to show that they understand some of the key relationships involved. Marian’s point in showing us these comparisons was to tell us that we need to spend much more time and attention making sure our students understand the math they are learning.

Each of the questions that asks students to show their understanding also help us see what knowledge our students have, but the other way around is not true!

Hopefully you can see the potential benefits of striving for understanding, but I do believe these shifts need to be deliberate. My recommendation to help us aim for understanding is to ask more questions that ask students to:

  • Draw a visual representation to show why something works
  • Provide an example that fits given criteria
  • Explain when examples will or won’t work
  • Make choices (i.e., which numbers, visual representations… will be best to show proof)
  • show their understanding of key “Big Ideas” and relationships

Application vs. Thinking

Below are a few examples that can help us delineate the differences between application and thinking:

These examples might be particularly important for us to think about. To begin with, application questions often use some or all of the following:

  • use a context
  • require students to use things they already should know
  • provide a picture(s) or example(s) for students to see
  • provide almost all of the information and ask the student to find what is missing

Thinking questions, on the other hand, are the basis for what Stein et. al called “Doing Mathematics“. In Marian’s presentation, she discussed with us that these types of questions are why those who enjoy mathematics like doing mathematics. Thinking and reasoning are at the heart of what mathematics is all about! Thinking questions typically require the student to:

  • use non-algorithmic thinking
  • make sense of the problem
  • use relevant knowledge
  • notice important features of the problem
  • choose a possible solution path and possibly adjust if needed
  • persevere to monitor their own progress

Let’s take a minute to compare questions aimed at application and questions aimed at thinking. Application questions, while quite helpful in learning mathematics concepts (contexts should be used AS students learn), they typically offer less depth than thinking questions. In each of the above application questions, a student could easily ignore the context and fall back on learned procedures. On the other hand, each of the thinking questions might require the student to make and test conjectures, using the same procedures repeatedly to find a possible solution.

Ideally, we need to spend more time where our students are thinking… more time discussing thinking questions… and focus more on the important relationships/connections that will arise through working on these problems.

Final Thoughts

Somehow we need to find the right balance between using the 4 types of questions above, however, we need to recognize that most textbooks, most teacher-made assessments, and most online resources focus heavily (if not exclusively) on knowledge and occasionally application. The balance is way off!

Focusing on being able to monitor our own types of questions isn’t enough though. We need to recognize that relationships/connections between concepts/representations are at the heart of expecting more from our students. We need to know that thinking and reasoning are HOW our students should be learning. We need to confront practices that stand in the way of us moving toward understanding and thinking, and set aside resources that focus mainly on knowledge or application. If we want to make strides forward, we need to find resources that will help US understand the material deeper and provide us with good examples.

Questions to Reflect on:

  • What did your last quiz or test or exit card look like? What is your current balance of question types?
  • What resources do you use? What balance do they have?
  • Where do you go to find better Understanding or Thinking questions?
  • What was the last problem you did that made you interested in solving it? What was it about that problem that made you interested? Likely it was a Thinking question. What was it about that problem that made it interesting?
  • Much of the work related to filling gaps, intervention, assessment driving learning… points teachers toward students’ missing knowledge. How can we focus our attention more toward understanding and thinking given this reality?
  • How can we better define “mastery” given the 4 categories above? Mastery must be seen as more than getting a bunch of simple knowledge questions correct!
  • Who do you turn to to help you think more about the questions you ask? What professional relationships might be helpful for you?

If you haven’t already, please take a look at Marian Small’s entire presentation where she labels understanding and thinking as the “fundamentals of mathematics”

I’d love to continue the conversation about the questions we ask of our students.  Leave a comment here or on Twitter @MarkChubb3

An Unsolved Problem your Students Should Attempt

There are several great unsolved math problems that are perfect for elementary students to explore.  One of my favourites is the palindrome sums problem.
In case you aren’t familiar, a palindrome is a word, phrase, sentence or number that reads the same forward and backward.

The problem itself comes out of this conjecture:

If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome.

Let’s take a look at the numbers 12, 46 and 95:

Palindrome adding1

Palindrome adding2

Palindrome adding3

As you can see with the above examples, some numbers can become palindromes with 1 simple addition, like the number 12.  12 + 21 = 33.  Can you think of others that should take 1 step?  How did you know?

Other numbers when added to their inverse will not immediately become a palindrome, but by continuing the process, will eventually, like the numbers 46 and 95.  Which numbers do you think will take more than 1 step?

After going through a few examples with students about the process of creating palindromes, ask your students to attempt to see if the conjecture is true (If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome).  Have them find out if each number from 0-99 will eventually become a palindrome. Ask students if they need to find the answer for each number?  Encourage them to make their own conjectures so they don’t need to do all of the calculations for each number.

Mathematically proficient students look closely to discern a pattern or structure. They notice if calculations are repeated, and look both for general methods and for shortcuts. (SMP 7)

Some students will notice:

  • Some numbers will already be a palindrome
  • If they have figured out 12 + 21… they will know 21 + 12.  So they don’t have to do all of the calculations.  Nearly half of the work will have already been completed.
  • A pattern emerging in their answers … 44, 55, 66, 77, 88, 99, 121… and see this pattern regularly (well, almost)
  • A pattern about when numbers will have 99 as a palindrome, or 121… 18+81 = 27+72 = 36+63…

Thinking about our decisions:
  • What is the goal of this lesson?  (Practice with addition?  Looking for patterns?  Perseverance?  Making conjectures? …)
  • Do we share some of the conjectures students are making during the time when students are working, or later in the closing of the lesson?  
  • How will my students record their work?  Keep track of their answers?  Do I provide a 0-99 chart or ask them to keep track somehow?
  • Will students work independently / in pairs / in small groups?  Why?
  • Do I allow calculators?  Why or why not?  (think back to your goal)
  • How will I share the conjectures or patterns noticed with the class?
  • Are my students gaining practice DOING (calculations) or THINKING (noticing patterns and making conjectures)?  Which do you value?

The smallest of decisions can make the biggest of impacts for our students!  

So, at the beginning of this post I shared with you that this problem is currently unsolved.  While it is true that the vast majority of numbers have been proven to easily become palindromes, there are some numbers that require many steps (89 and 98 require 24 steps), and others that have never been proven to either work or not work (198 is the smallest number never proven either way).

Some final thoughts about this problem…

After using this problem with many different students I have noticed that many start to see that mathematics can be a much more intellectually interesting subject than they had previously experienced.  This problem asks students to notice things, make conjectures, try to prove their conjectures and be able to communicate their conjectures with others…  The problem provides students with the opportunity to both think and do.  It offers students from various ability levels access to the problem (low floor), and many different avenues to challenge those ready for it (high ceiling).  It tells students that math is still a living, growing subject… that all of the problems have not yet been figured out!  And probably the most important for me, it sends students messages about what it means to really do mathematics!


Taken from Marilyn Burns’ 50 Problem Solving Lessons resource


Is That Even A Problem???

Ask others what problem solving means with regard to mathematics.  Many will explain that a problem is when we put a real-world context to the mathematics being learned in class… others might explain a process of how we solve a problem (look at what you know and determine what you want to know, or some other set of strategies or a creative acronym that we have likely seen in school).  Sadly, much of what most others would point to as a problem is not really even a problem at all.

I think we all need to consider the real notion of what it means to problem solve…

George Polya shared this:


…Thus, to have a problem means: to search consciously for some action appropriate to attain a clearly conceived, but not immediately attainable, aim. To solve a problem means to find such action. … Some degree of difficulty belongs to the very notion of a problem: where there is no difficulty, there is no problem.

What Polya is suggesting here is that if we show students how to do something, and then ask the students to practice that same thing in a context IT ISN’T A PROBLEM!!!  A problem in mathematics is like the DOING MATHEMATICS Tasks listed below.  Take a look at all 4 sections for a minute:

math_task_analysis_guide  - Level of Cognitive Demand.png

My thoughts are simple… If a student can relatively quickly determine a course of action about how to get an answer, it isn’t a problem!   Even if the calculations are difficult or take a while.  The vast majority of what we call problems are actually just contextual practice of things we already knew.  Doing word problems IS NOT the same as problem solving!

On the other hand, if a student has to use REASONING skills, they are thinking, actively trying to figure something out, then and only then are they problem solving!!!

Marian Small has written a short article on her thoughts about problem solving:  Marian Small – Problem Solving

What are her main messages here?  Does this or Polya’s quote change your definition of a problem?

I’ve already written about What does Day 1 Look Like where I shared the importance of starting with problems.  So, why should we start with problem solving?  If we don’t start there, we aren’t likely ever doing any problem solving at all!


I also think that many hearing this might assume that this means we just hand students problems that they wouldn’t be successful with…  Ask everyone to attempt something that they wouldn’t know how to do.

Let’s look at an example:

Take a look at these two grade 8 expectations from Ontario curriculum:

  • determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies;
  • solve problems involving right triangles geometrically

Many teachers look at these expectations, think to themselves, Pythagorean Theorem… I can teach that… followed by explicit teaching on a Smartboard (showing a video, modeling how the pythagorean theorem works, followed by some examples for the class to work on together…), then finally some problems that students have to answer in a textbook like this:


This progression neither shows how we know students learn, nor does it even get students to be able to do what is asked!

Let’s take a step back and start to notice what the curriculum is saying more clearly:

  • determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies;
  • solve problems involving right triangles geometrically, using the Pythagorean relationship;

When we pull apart the verbs and the tools/strategies from the content, we start to notice what the curriculum is telling us our students should actually be doing that day!  Remember… these expectations are what OUR STUDENTS should be doing… NOT US!!!

Above I have colored the verbs blue (These are the actions our students should be doing that day) and tools/strategies orange (specifically HOW our students should be accomplishing the verb).

Now let’s take a quick look at how this might actually play out in the classroom if we are starting with problems like our curriculum states:

Let’s start with the first expectation.  Our curriculum often includes the statement “determine through investigation,” yet it is overlooked far too often!  Students need to determine this themselves!  We need to assess students’ ability to “determine the Pythagorean relationship through investigation.”

This doesn’t mean we tell students what the theorem is, nor does it mean that we expect everyone to reinvent the theorem… so what does it mean???

Well, it could mean lots of things.  Here is one possible suggestion…

maxresdefault (1)

Show the figure on the left.  Ask students the area of the blue square in the middle.  Possibly give a geoboard or paper and scissors for this task.  How might students come up with the area?  How many different ways might students accomplish this?

Share different approaches as a group.  (By the way, some calculate the whole shape and subtract the 4 corner triangles.  Others calculate the 4 black rectangles and divide by 2 then add 1 for the middle… others rearrange the shapes to make it make sense).

Now explore how the two pictures are similar / different.  What do you notice between the two pictures?

If the curriculum tells us to “determine through investigation” that is exactly the experience our students need to conceptualize the concept.  It is also what we need to assess.  This can’t be put on a test easily though, it needs to be observed!

That second expectation is quite interesting to me too.  At the beginning of this post we started talking about what is and isn’t a problem.  If our students have now understood what the Pythagorean Theorem is, how can we now make things problematic?

Showing a bunch of diagrams with missing hypotenuses or legs isn’t really problematic!

However, something like Dan Meyer’s Taco Cart would be!  If you haven’t seen the lesson, take a look:




Oh… and by the way… problem solving isn’t always about answering a question… really, at it’s heart, problem solving is about making sense of things that we didn’t understand before… reasoning though things… noticing things we didn’t notice before… making conjectures and testing them out…  Problem solving is the process of LEARNING and DOING MATHEMATICS!

So I want to leave you with a problem for you to think about: what does this have to do with the Pythagorean Theorem?