Friday, December 9, 2016

Let Them Make Mistakes

One of the pitfalls we find ourselves falling into as math teachers (I use the first person plural here because I have done this and I see others doing it too) is we facilitate our instruction with a goal that everyone gets it right.

It is easy to forget that the Everyday Math lessons we teach are structured in a way that allows and even encourages children to make mistakes.  In any lesson on any given day, we should expect students to make mistakes while they work, and especially when they think aloud.

Let's look at this in terms of the parts of a lesson.


For the Warm-Up portion of Everyday Math lessons, Mental Math and Fluency is a check-in opportunity.  This is not a time to make sure all students get it; it is a time to check for mastery. That's why there are three levels of difficulty for the mental math segment of every lesson.  If most students appear to struggle with the first level, don't go onto the next.  Mental math is a chance for you to have a snapshot of your students' levels of math fluency before you get into the bulk of the lesson.  What you see in mental math might impact how you facilitate the rest of your lesson.  Mental Math is not the time to clear up misconceptions, or to keep providing more examples until everyone gets it right.  "What do you see?" when flashing a Quick Look card, or "How do you see that?" are great questions to ask that will elicit formative information.  But asking every student to share a strategy, or making sure every student gets it right will take too long and disengage students.

Mental Math.

Mental math is your students' opportunity to get their feet wet, so let them.  They might make mistakes.  Give them opportunities to make those mistakes, and then let them talk to each other about their mistakes.   The Teacher's Lesson Guide gives you a little suggested script to follow up after the Math Message. Notice it never says, Keep quizzing children until everyone gets it right.  Instead, it usually offers a differentiation strategy with suggestions for scaffolding.  Your scaffolds should not be "hints," but rather sentence structures to help them grasp a strategy, or visual aides to help them understand how to use a tool.  We still want to give them opportunities to figure out the problem for themselves (and to occasionally make mistakes, even with scaffolds).

Math Journal work.

They can make mistakes here too!  But now their mistakes are visible on paper.  The journal work is an excellent opportunity for students to work together, check each other's work, and compare strategies.  Two partners have different answers?  Wonderful!  Have them see if they can come to a consensus.  The "growth mindset" in math requires that students make mistakes, and do it fairly often.  Every time a student truly discovers the root of his or her mistake, that student has gained significant mathematical understanding.


Mistakes are still encouraged here, while playing a game or working on math boxes. Here, mistakes are likely smaller and quickly resolved, but they still can and do happen.  This is where we want students to be able to catch and correct their own mistakes, either on their own or with the help of a partner.  If there are a lot of mistakes at this point of the lesson, that informs you that some extra practice or re-teaching may be necessary.

You may find that persistently allowing students to make mistakes actually saves you instruction time, because you are not so busy going from child to child making sure every student has gotten every part of the problem right, or demonstrated every strategy correctly.  Sometimes, that can postpone the entire class from proceeding to the next part of the lesson.  Instead, use those mistakes as learning opportunities, turn-and-talk topics, group consensus opportunities, and formative evaluation of their understanding for future instructional decisions.

Thursday, December 1, 2016

Whole Class Instruction: Know When To Say When

Much of our instructional time in math lessons is taken up with small group and partner work, which allows students to explore, experiment, debate and take risks with their mathematical thinking.  Whole group instruction is minimal in our Everyday Math lessons, but when it happens, there are some things to keep in mind.

Recently in a fourth grade classroom and I saw a masterful decision take place on the part of the teacher.  Students had been working on a task, and it came time to share out.  A student volunteered to share his strategy for dividing three 8" pizzas among two friends evenly.

"Well.. basically what I did was.. I started by taking the three pizzas, and then I.. well, they're each 8 inches in diameter but I don't think that really matters for this problem, but I thought I would mention that, and then I, um.. I cut the pizzas in half because then I would have a number of parts of pizza that I could distribute evenly for everybody, and then I had four pizzas instead of two, or rather four half pizzas... Wait... I think I made a mistake..."

The teacher then intervened.  Instead of asking the student to re-think his strategy, or to start from the beginning, or asking if there were any other students who could help this student with his thinking, or asking if there were any other students who had a different strategy, the teacher instead put a halt to the share-out and asked the rest of the class to turn and talk with a partner to share strategies.  After a couple minutes, students had another opportunity to share, and lo and behold:

"We both cut the three pizzas in half so there were six halves, which is also divisible by two, so there would be three halves for each person."

The decision to intervene in the initial student's share-out was masterful because of the timing of the intervention.  She realized the student had not fully thought out his conclusion, or was not yet confident enough in his strategy to present it to the class.  Sure enough, by the time the student had said, "Well.. basically what I did was.." he had lost most of his peers' attention.  Eyes started turning away, pencils were being picked up, body language across the room said, almost universally, "We're not paying attention anymore."

The student's hesitation alone had lost the confidence in his peers, and many had decided it was not worth their time to listen.  The class had been well trained not to be distracting when distracted themselves, so there was little noise or even whispering going on, but it was clear very few, if anyone, was listening intently to the student share out his temporarily confused thought process.

The teacher had masterfully gauged student engagement and transitioned to an interactive partner review of the problem.  We all know what would likely have come next if she had allowed the student to continue, or if she had called on another student to follow up.  A visible or audible distraction from somewhere in the room, perhaps a spoken redirection or two from the teacher, and a prolonged share-out with even fewer attentive students.   Instead, students turned to a partner and rehearsed their thinking out loud together, interactively.  When it came time for a volunteer to share out, the result was clear, precise and succinct.

We always want to elicit mathematical thinking from our students, and we have a tendency to want to hear multiple students share a variety strategies all in a quick whole group share-out, so that little lightbulbs might turn on around the room as new understandings are shared and discovered by all. Unfortunately, it is a challenge for elementary and middle school aged children, and even adults sometimes, to articulate mathematical thinking on the spot.  And listening to someone think out loud is not often that engaging, especially for those of us who are not primarily auditory learners.  If what we hear doesn't make instant sense, we shut down.

So a couple of important lessons can be learned from this that pertain to whole-group instruction.

Firstly, student attention is short-lived in a whole group setting unless the topic is purely engaging. Even though a teacher might not be lecturing, sitting and listening while student after student attempts to articulate a strategy or defend an answer to a problem will not yield a great deal of new understanding.  Moreover, it will often lead to distracted students and sometimes misbehavior. Masterful teaching moves I have seen include expertly timed turn-and-talks, whole class physical transition to a new location (such as moving to the floor from their desks or vice versa), and transitioning to the next stage of a lesson.  If students are not comprehending, they are not going to invest their energy in listening.  If they are bored, they are going to struggle to keep focused. Something has to change.  No matter how animated and excited we get in the front of the room, it still often feels like a losing battle:

"Oh! Great thinking!  Did everybody hear what Timmy just said?  Timmy can you say that again so everybody can hear?  Ok, everybody, listen to Timmy's awesome strategy.. Suzie, you too.. and Billy, that means you.. Ok Timmy, go ahead.. but wait until everybody is quiet..."  

Ugh.  Not fun for you, not fun for them.  Sometimes a student just hits the nail on the head and you want everyone to absorb the learning opportunity, but if it took too long to elicit that sweet morsel of demonstrated knowledge, beware.  We might expend less energy typing it up and inserting it into 23 fortune cookies than we would trying to get every student to devote his or her complete attention to it.

Another lesson to be learned about whole group instruction involves giving directions.  In problem solving tasks, directions are often complex and require considerable effort on the part of students (and sometimes teachers) to understand.  It becomes very important in those cases for the teacher to be the one to read or deliver instructions slowly and carefully, with all the right intended emphases.  We often want to give students opportunities to read aloud and take ownership, but instructions to problem solving tasks are not appropriate for those opportunities.  Even some of the best student readers still need work on their delivery and might not execute the right emphasis.  Some read very fast, others read very quietly, others read with a monotone, still others insert careful but perhaps awkward space between words.  All of these can make it difficult for their peers to understand, and thus result in students abandoning the effort of paying attention.  "Two.. brothers.. go.. to.. lunch.. and.. share.. three.. pizzas.. equally.. how.. much.. pizza.. does.. each.. brother.. get.." By the fifth word, chances are wandering eyes and body language will begin to present visible evidence that students are not eagerly anticipating the task.  A confident adult reader is important when giving instructions and sharing important details for mathematics problem solving tasks.

Whole class instruction can be engaging, and it can be effective.  It is important to be keenly aware of student engagement and to be practice teaching moves that maintain engagement or re-engage students. Then your energy can be more devoted to listening to what students are saying, gauging student understanding, and confidently moving forward with facilitation of learning.  Knowing when to say when will keep whole class instruction limited, and leave more time for partner and group work, as well as those all-important games and hands-on activities.