How do you give feedback?

There seems to be a lot of research telling us how important feedback is to student performance, however, there’s little discussion about how we give this feedback and what the feedback actually looks like in mathematics. To start with, here are a few important points research says about feedback:

  • The timing of feedback is really important
  • The recipient of the feedback needs to do more work than the person giving the feedback
  • Students need opportunities to do something with the feedback
  • Feedback is not the same thing as giving advice

I will talk about each of these toward the end of this post.  First, I want to explain a piece about feedback that isn’t mentioned enough…  Providing students with feedback positions us and our students as learners.  Think about it for a second, when we “mark” things our attention starts with what students get right, but our attention moves quickly to trying to spot errors. Basically, when marking, we are looking for deficits. On the other hand, when we are giving feedback, we instead look for our students’ actual thinking.  We notice things as almost right, we notice misconceptions or overgeneralization…then think about how to help our students move forward.  When giving feedback, we are looking for our students strengths and readiness.  Asset thinking is FAR more productive, FAR more healthy, FAR more meaningful than grades!

Feedback Doesn’t Just Happen at the End!

Let’s take an example of a lesson involving creating, identifying, and extending linear growing patterns.  This is the 4th day in a series of lessons from a wonderful resource called From Patterns to Algebra.  Today, the students here were asked to create their own design that follows the pattern given to them on their card.

Their pattern card read: Output number = Input number x3+2
Their pattern card read:  Output number = Input number x7
Their pattern card read:  Output number = Input number x4


Their pattern card read:  Output number = Input number x3+1
Their pattern card read:  Output number = Input number x8+2
Their pattern card read: Output number = Input number x5+2

Once students made their designs, they were instructed to place their card upside down on their desk, and to circulate around the room quietly looking at others’ patterns.  Once they believed they knew the “pattern rule” they were allowed to check to see if they were correct by flipping over the card.

After several minutes of quiet thinking, and rotating around the room, the teacher stopped everyone and led the class in a lesson close that involved rich discussions about specific samples around the room.  Here is a brief explanation of this close:

Teacher:  Everyone think of a pattern that was really easy to tell what the pattern rule was.  Everyone point to one.  (Class walks over to the last picture above – picture 6).  What makes this pattern easy for others to recognize the pattern rule?  (Students respond and engage in dialogue about the shapes, colours, orientation, groupings…).

Teacher:  Can anyone tell the class what the 10th position would look like?  Turn to your partner and describe what you would see.  (Students share with neighbor, then with the class)

Teacher:  Think of one of the patterns around the room that might have been more difficult for you to figure out.  Point to one you want to talk about with the class.  (Students point to many different ones around the room.  The class visits several and engages in discussions about each.  Students notice some patterns are harder to count… some patterns follow the right number of tiles – but don’t follow a geometric pattern, some patterns don’t reflect the pattern listed on the card.  Each of these noticings are given time to discuss, in an environment that is about learning… not producing.  Everyone understands that mistakes are part of the learning process here and are eager to take their new knowledge and apply it.

The teacher then asks students to go back to their desks and gives each student a new card.  The instructions are similar, except, now she asks students to make it in a way that will help others recognize the patterns easily.

The process of creating, walking around the room silently, then discussing happens a second time.

To end the class, the teacher hands out an exit card asking students to articulate why some patterns are easier than others to recognize.  Examples were expected from students.

At the beginning of this post I shared 4 points from research about feedback.  I want to briefly talk about each:

The timing of feedback is really important

Feedback is best when it happens during the learning.  While I can see when it would be appropriate for us to collect items and write feedback for students, having the feedback happen in-the-moment is ideal!   Dan Meyer reminds us that instant feedback isn’t ideal.  Students need enough time to think about what they did right/wrong… what needs to be corrected.  On the other hand, having students submit items, then us giving them back a week later isn’t ideal either!  Having this time to think and receive feedback DURING the learning experience is ideal.  In the example above, feedback happened several times:

  1. As students walked around looking at patterns.  After they thought they knew the pattern, they peeked at the card.
  2. As students discuss several samples they are given time to give each other feedback about which patterns make sense… which ones visually represented the numeric value… which patterns could help us predict future visuals/values
  3. Afterward once the teacher collected the exit cards.

The recipient of the feedback needs to do more work than the person giving the feedback

Often we as teachers spend too much time writing detailed notes offering pieces of wisdom.  While this is often helpful, it isn’t a feasible thing to do on a daily basis. In fact, us doing all of the thinking doesn’t equate to students improving!  In the example above, students were expected to notice patterns that made sense to them, they engaged in conversations about the patterns.  Each student had to recognize how to make their pattern better because of the conversations.  The work of the feedback belonged, for the most part, within each student.

Students need opportunities to do something with the feedback

Once students receive feedback, they need to use that feedback to continue to improve.  In the above example, the students had an opportunity to create new patterns after the discussions.  After viewing the 2nd creations and seeing the exit cards, verbal or written feedback could be given to those that would benefit from it.

Feedback is not the same thing as giving advice

This last piece is an interesting one.  Feedback, by definition, is about seeing how well you have come to achieving your goal.  It is about what you did, not about what you need to do next.  “I noticed that you have switched the multiplicative and additive pieces in each of your patterns” is feedback.  “I am not sure what the next position would look like because I don’t see a pattern here” is feedback.  “The additive parts need to remain constant in each position” is not feedback… it is advice (or feedforward).

In the example above, the discussions allowed for ample time for feedback to happen.  If students were still struggling, it is appropriate to give direct advice.  But I’m not sure students would have understood any advice, or retained WHY they needed to take advice if we offered it too soon.

So I leave you with some final questions for you:

  • When do your students receive feedback?  How often?
  • Who gives your students their feedback?
  • Is it written?  Or verbal?
  • Which of these do you see as the most practical?  Meaningful for your students?  Productive?
  • How do you make time for feedback?
  • Who is doing the majority of the work… the person giving or the person receiving the feedback?
  • Do your students engage in tasks that allow for multiple opportunities for feedback to happen naturally?

PS.   Did you notice which of the students’ examples above had made an error.  What feedback would you give?  How would they receive this feedback?




Seeking Challenges in Math

I was working with a grade 7 teacher and his students a while back.  The teacher came to me with an interesting problem, his students were doing quite well in math (in general) but only wanted to do work out of textbooks, only wanted to work independently, and were very mark-driven. The teacher wanted his students to start being able to solve non-routine problems, not just be able to follow the directions from the textbook, and he wanted his students to see the value in working collaboratively and to listen to each other’s thoughts.

Our conversations quickly moved to the topic of mindsets. It sounded like many of his students had fixed mindsets, and didn’t want to take any risks.

For those of you who are not familiar with growth and fixed mindsets, students with fixed mindsets believe that their ability (in math for example) is an inborn trait.  They believe how smart they are in math is either a gift or a curse they are born with.  Those with growth mindsets, however, believe that their ability improves over time with the right experiences, attitude and effort.

When confronted with challenges, those with growth mindsets are willing to struggle, willing to make mistakes, knowing that they will continue to learn and grow throughout the learning process.  On the other hand, those that have fixed mindsets tend to avoid challenges.  They believe that struggle, making mistakes, and being challenged are signs of weakness.  Psychologically, they will avoid the feeling of discomfort in not knowing, as this threatens their belief about how smart they are.

Knowing this, we devised a plan to see whether or not his students were able to take on challenges.  We started the class by giving each student their own unique 24 card (see below).

24 card.jpeg

We explained that each card had 4 numbers that could be manipulated to equal 24.  For instance, the card above could be solved by doing 5 x 4 x 1 + 4 = 24.

We then explained that we would give them time to solve their own card (which had a front and a back), and that we would give them additional cards if they completed both problems.  We also explained the little white dots in the center of the card, 1 dot being an easy card, 2 dots being more complicated, and 3 dots being the most difficult.

As students continued to work, we noticed some students eagerly trying to solve the cards, and others starting to become frustrated by others’ successes.  After a few minutes, the first few students had completed both problems and asked for their next card.  We asked, “Would you like another easy card, or would you like to challenge yourself?” to which the vast majority asked for another easy card.  In fact, some students completed many cards, front and back, all at the easy level, never accepting a more challenging card (even bragging to others about how many they had completed).  Others, after giving up pretty quickly, asked if they could work with a classmate to make a pair.  While we were happy at first with this, none of the pairs had students working cooperatively together for most of the time.

Take a look at some of the challenging cards.  What do you do when confronted with something challenging?  Do you skip it and move on, or do you keep trying?


As soon as we were finished, we showed the class this video:

Watch the 3 minute video above as it ties in perfectly with the 24 problem from above.  We had a quick discussion about the video and why some of the students wanted to choose the easier puzzles.  The class quickly saw the parallels between the problems we had just done and the video.

While we had a great discussion about fixed and growth mindsets, it took most of the year to be able to get this group to see the value in collaboration, to focus on their learning instead of their marks, to be able to take on challenges and not get frustrated when they didn’t have immediate success.

Changing our mindset takes time and the right experiences!

I am really interested in why students who believe themselves to be “smart” at math would opt out of challenging themsleves.

Do any of your students exhibit any of the same signs as these students:

  • Not comfortable with tasks that require thinking
  • Eager for formulas and procedures
  • Competitive with others to show they are “smart”
  • Preference to work alone
  • Preference to work out of textbooks/ worksheets instead of on rich problems/tasks
  • See math as about being fast / right, not about thinking / creativity
  • Eager to do easy work that is repetative


So I leave you with some reflective questions:

What previous experiences must these students have had to create such fixed mindsets?

What would you do if your students avoided challenges?

What would you do if your students groaned each time you asked them to work with a partner?

How are you helping your students gain a growth mindset in math?

Can you recognize those in your class that have fixed mindsets?  Are you noticing those from different achievement levels, or just those who are struggling?

If our students find everything we do “easy” what will happen to them when they get to a math course that actually does offer them some challenge???



P.S.  Did you solve any of the 24 cards above?  Did you skip over them?  What do you typically do when confronted with challenges?

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


How to Set Up Positive Norms…

How will your students view mathematics this year?  What norms will you set up to help your students share your views of mathematics?

If you haven’t already seen YouCubed list of norms, you will probably want to take a close look:


Take a look at the following resource about how to set up positive norms in your class…Setting up Positive Norms in Math Class (linked is an expansion of each norm).  Which one(s) of these is really important to you?  Which one(s) challenges your thinking?

  1. Everyone Can Learn Math to the Highest Levels. Encourage students to believe in themselves. There is no such thing as a “math” person. Everyone can reach the highest levels they want to, with hard work.
  2. Mistakes are valuable. Mistakes grow your brain! It is good to struggle and make mistakes.
  3. Questions are Really Important.  Always ask questions, always answer questions. Ask yourself: why does that make sense?
  4. Math is about Creativity and Making Sense. Math is a very creative subject that is, at its core, about visualizing patterns and creating solution paths that others can see, discuss and critique.
  5. Math is about Connections and Communicating.  Math is a connected subject, and a form of communication. Represent math in different forms eg words, a picture, a graph, an equation, and link them. Color code!
  6. Depth is much more important than speed. Top mathematicians, such as Laurent Schwartz, think slowly and deeply.
  7. Math Class is about Learning not Performing.  Math is a growth subject, it takes time to learn and it is all about effort.

Thinking about what contradicts our current thinking is where learning can really occur!

While setting up classroom norms is a great thing to do, I think it is far more important to start the year off by enacting these norms rather than just discussing them or posting them.

The first few things you do in your year sets the tone for what you believe is important in mathematics!

My suggestion:

  • Start with something that allows your students to see their strengths
  • Start with something that is open enough to allow for differences in strategies and/or their answers
  • Start with something that focuses on the SMPs
  • Start with something that helps your students look for patterns, notice things, explore…
  • Start with something that allows your students to be creative
  • Start with something where your students can see the beauty in mathematics

If you want to share the norms listed at the beginning of the article, maybe experiencing a rich task/problem first might help students see what you mean and start to get excited about learning this year!


In the next few days I’ll share a possible example of what this could look like.


Never Skip the Closing of the Lesson

Once again, Tracy Zager has pushed us to think about our teaching.  In her recent talk at #TMC16 Tracy asked us to consider what it means to “close the lesson”.  Here is an example of a problem and a potential close, followed by some of my thoughts about how we should close any lesson.

First of all, give a problem that will help you achieve a specific goal.

Take this problem published in Marilyn Burn’s 50 Problem Solving Lessons resource:

If rounds 1 & 2 of a tug-of-war contest are a draw, who will win the final round?

Here is the full problem. Please take a minute to read through the problem and try to solve it for yourself. Which side will win round 3?  How do you know?  Are you sure?

Once students understand the problem and are given time to write their solution (individually or in pairs) the learning isn’t over. In fact, while answering the problem might require thinking and writing an answer requires decisions about representations, much of the learning hasn’t happened yet!  Really, students have just shown what they already understand…new learning happens in the close!

Closing the lesson:

Step 1 – Sharing Different Solutions

Since this problem is open, allowing for different strategies, there is a huge potential for interesting discussions.  For example, some students will create an answer similar to the one below.  Using this sample there are several things that we can bring up in conversation with the class.  Notice the student(s) created equations using symbols and equal signs.  Also, they did something really interesting in round three (notice the brackets and the arrows).  Having a conversation about substituting would be really helpful for many students.  Many might not realize that this problem would be easier to solve if only acrobats and grandmas were considered.  Through substituting Ivan for 2 grandmas and 1 acrobat, the final round ends up with 5 grandmas and an acrobat against 4 acrobats.  Substitution requires us to really understand the equal sign and what balance means!


Other students will create things more like the picture below.  They will use numbers to represent values for each of the figures.  Again, conversations with the class could be about how they chose the 1 = 1.25 at the top.  Or, how this information could be used later.  Again, the importance of balance and the equal sign could be brought up.

Others might do things like the sample below.  Here the students have decided not to draw a picture, but instead to represent the players with letters.  For many students, their first introduction with algebra is where letters are all of a sudden thrown into some equations.  However, these students have realized that it is far easier to represent the items with letters than draw the pictures.  The conversation here could be quite useful in bringing about the need for letters!  Also, noticing that their strategy is similar to the 2nd sample above might be helpful since while the strategy is the same, the values are not.  How are the two student samples’ values different?  How are they the same?  Is one right or are they both right?


Step 2 – Making connections / discussing big ideas

Once a few samples have been shared, we need to make sure our students are making connections between the samples.  The learning from the problem needs to made clear for all students.

We can do this by asking students to notice similarities and differences between samples, or by taking a few minutes writing down a few simple things we can take away from the problem.  Either way, our students need to be involved here and the generalities we can draw from the problem need to be clear.

For example, things that can be discussed here:

  • The equal sign represents balance
  • We can substitute things in equations we don’t know with things we do know
  • Symbols or letters can be used in equations to represent either a variable or an unknown

Step 3 – individual practice

Now that our students have had time to grapple with this problem, and have discussed what we can learn from it together, we need to continue the learning.  Part of the close includes time for students to do something with their learning.  Here are a few possible ways we can include individual time to practice what has been learned.

  • Provide an Exit Card or journal prompt that asks your students to show what they have learned (linked are a variety of types of exit cards you can choose between)
  •  Have your students create their own problem using 3 rounds where the first 2 are a tie and the 3rd round needs to be figured out (possibly have students switch problems with a classmate and find the solution)
  • Relate the problem to some practice questions that will help continue your students’ thinking.  Practice with some of the early Solve Me Mobile puzzles might work nicely here.

Some General Advice about the Close

  • Predetermine which students you want to share.
  • Have a goal in mind for each person/group sharing.
  • Ask them specific questions, or ask questions of the rest of the class that helps you achieve your goal.
  • Avoid general questions like “tell us what you did” (we don’t want the sharing to become a show-and-tell, we want rich discussions about specific things from their work).
  • Use the 5 Practices for Orchestrating Productive Conversations (found here, here, and here) to make sure your close involves real discussion of the meaningful mathematics.
  • Practice is important, but students still need to think and make choices.
  • Closing the lesson is about bringing the learning together as a group, then individually.  The SHARED experiences here are where our students can learn WITH and FROM each other so they will be ready to work independently.
  • The majority of the learning takes place in the close!
  • Closing a lesson takes time, but skipping the close is the biggest waste of time!

Want more information about the close?  Take a look at this monograph: Communication in the Mathematics Classroom

So I leave you with some final reflective questions:

  • How often do you close the lesson?
  • What obstacles stand in your way to closing your lessons regularly?
  • Does your close look like a show-and-tell where students are not listening to those talking, or do you have rich discussions going on between many members of your classroom?  How can you help increase the level of discourse in your classroom?
  • How is this different than closing the lesson by taking up homework?  Does this allow more opportunities for those that might struggle, or who don’t identify with mathematics yet?
  • What goals do have this year with regards to closing your lessons?

Some say they can’t afford the time to close the lesson.  I say you can’t afford not to!