Thinking about your math instruction? Consider this...

First let me say, that because I have spent the last 10 years in elementary school (and before that 10 in middle school), much of what I'm going to write is from that perspective but know that this is not exclusive to elementary classrooms and it is something that middle and high school classrooms need to think about as well.

Most of us realize that the way we were taught math (sit and get or chalk and talk) is not effective for most students today (or then for that matter).  In a push for differentiated and/or individualized instruction many teachers have ditched the notion of whole group instruction and opted for 5 day a week small group instruction and/or centers (or workstations).    I applaud teacher's creativity in trying to help students learn more math content in a student centered conceptual way but....have we increased the application or transfer of that content through the sole use of small group instruction?  While we are presenting the material in a more engaging way (games, less worksheets, etc...), are we really teaching kids to understand math and apply or transfer those concepts more than the old traditional ways of math instruction?

The beauty of the Common Core and the thing that I believe ties all of the new standards together, be it Next Gen, C3, Math or ELA CCSS, is teaching in an inquiry based way so that students become problem solvers and critical thinkers.  Honestly, if you have attended on of my workshops, you have heard me say at least once (if not a dozen times), "If my kids only take away a few things from math instruction, I hope it is the Math Practices because if they are proficient with those habits of mind they can do anything"!

Consider this, how are we building those Math Practices in small group and centers?  I think some of that can happen there but I think that critical thinking and problem solving is best learned from each other in whole group settings as well as the teacher.  Often when teachers group students for centers they are placed in homogeneous groups.  But, I contend that students learn these critical thinking and problem solving skills for students of all levels.  Some of our "lowest" math students tend to "think outside the box" where some of our "high" math students are very linear in their thinking - both groups of students can benefit from each other.

So I'm not saying do away with small groups....I'm advocating for a BALANCE of whole and small group instruction.  There are benefits of both.  

What should that whole group instruction look like?  It should be filled with things like Number Talks, 3 Act Lessons and other intriguing problems to solve, open-ended problems, Noticing and Wondering, etc...  This should be a time where students are collaborating, working together, and sharing.    Most importantly this time should end with at least 10-15 minutes of whole group sharing of learning, questioning, and thinking aloud together.     Teacher's don't need a page of math problems for this -- they need 1 or just a couple of good ones!

This whole group time should not just focus on the content but the thinking as well.  I worked with a fabulous group of grade 5 magnet students (with a wonderful young teacher) for a 3 Act Lesson this year.  These kids had lots of math SKILLS!!!  They knew how to do math but, not surprising to me struggled to solve a problem that they generated in a 3 Act Lesson.  They were quick to "erase" their thinking if it didn't seem to work out.  When I noticed this, we stopped and had a quick whole group discussion about the value of analyzing what went wrong and just setting it aside in case they might need some of those calculations again (this is something we also talked about it in the conclusion of the lesson).    These students also did lots of calculations but didn't label much and therefore struggled when they had to explain what they did to remember why they did those calculations.  We talked, in the end about how solving this problem was a lot like doing a science experiment - if the math didn't work out we had to analyze what went wrong and then determine another path to take to solve the problem instead of just scrapping everything and doing something different or giving up.    Their teacher told me, after the lesson, that his kids didn't so much learn new math that day but they really learned the value of analyzing one's thinking and communicating in a more concise way, their thinking through what they wrote on paper.    He told me that the kids really enjoyed the lesson and learned so much that they could apply to future problems.

So, what should the teacher's role in these whole group sessions be?  The teacher should be the facilitator and questioner.  The teacher doesn't even have to have the "answer" to the problem - the teacher just needs to know how to question kids when they get stuck and when to stop and talk about the problem solving methods (as needed).  The teacher also needs to skillfully set up the class to help kids develop those habits of mind - whether to use pairs, small groups, and when to use protocols to cross pollenate those ideas.  

One of those ideas that I use to cross pollenate ideas is called "Send a Spy".  Often if kids are working in small groups I will stop them and tell the groups that it is time to "send a spy".  Sometimes I appoint the spy in each group and sometimes I let the group choose.  Each group send their spy to look at the work of other groups as they continue to work.  We have rules for this...no one can cover their work, the spies aren't allowed to write or talk until they get back to their group.   They only have about 2-3 minutes to do this - I usually play some "spy music", like the Mission Impossible Theme while this is going on.  When the music stops they have to return to their group and share what they learned.

These are just some of my thoughts on the structure of math time.  Honestly for me, in an elementary classroom I would have 3 days of small group instruction and 2 of whole group each week.

If you want to learn more about teaching problem solving be sure to read Max Ray-Riek's book, Powerful Problem Solving.

Agree...Disagree with any of this?   Want me to elaborate on any of this?   I'd love to hear from you!  Leave me a comment!

The Noticing and Wondering Revolution

When you do a read aloud with students what is the first thing you do?  I know that many of you, like me, will say things like, "what do you predict this book is going to be about based on what you notice about the cover?" or "take a look at the cover, what do you think this book will be about?".    What are we doing when we introduce a book this way?    First of all it activates prior knowledge and helps them make connections between things new information and things they already know.  Additionally it motivates kids to read or listen to the story because, through the predicting, peaking their curiosity. Students can't help but want to find out what actually happens.  Teachers know that these are all important things to do when we read aloud to kids and introduce a piece of text.  So, why don't we do this in math?

About 2 years ago I watched this Ignite talk given by Annie Fetter on using the Noticing and Wondering Strategy.  This quick 4 minute video has transformed how I think about teaching ALL subjects, not just math.    I have watched this clip no less than 50 times and every time I take something more away from it.

It took me about 8 of my 20 years of teaching to understand that the key to motivating students it to peak a student's curiosity.  Kids are born curious but over time they stop asking questions because we, as educators give them the questions that they have to answer instead of involving them in the process.  We know this about reading instruction, why don't we use this in math, social studies, and science?

So, how does noticing and wondering work?

In math Annie suggests giving the kids a stem of a problem (a word problem minus the question).  This could also be a graph, infographic, or anything that is a stimulus for doing some math.  Here is an example from the NCTM publication Teaching Children Mathematics from the Math by the Month section.

I took off the question (what could you buy?) and presented it to students just as you see the problem above.   I told students to take 2 minutes and on their own jot down things they "noticed" and "wondered" about this problem.  I think it is really important to give kids at least a few minutes to think, without talking, on their own (everyone needs processing time).  Then I have them in small groups or pairs just share out what they NOTICE (not WONDER at this point).

After kids have had time to think on their own then we do some whole group sharing and I record on a google doc that they can all see or chart paper.  This process levels the playing field.   Just like with reading you activate prior knowledge and help kids start to make connections with things they already know and helps some students be exposed to things that they themselves missed.  This is also a good assessment tool for teachers to see what they already know and what they do not know.  For accelerated students I believe this makes them slow down and process more - they often jump in and start crunching numbers without things about all of the details.  

I will tell you at this point I have them start sharing their "wonderings" with the whole group.  Every time I've done this problem kids have come up with the question that was with the original problem, on their own, and ones that are much harder.  I had a group of 2nd graders come up with these questions:

• What could you buy with this money?
• What is the least number of pieces of candy that you could buy with this money?
• What is the most number of pieces of candy that you could buy with this money?
• Could you buy just 1 chocolate coin for 10 cents instead of 5 for 50 cents? (Great question for grade 2)?

Some of the questions they ask are more "clarifying" type questions and we as a class or I as the teacher, answer those so we are on the same page.  This is true of the question about buying 1 chocolate coin for 10 cents above.   In one of the classes I was in I had a large range of students with several very high level students so I told them if they wanted to do this, they could but they had to show me the math to prove that they have the correct cost of one item.  

Next we agree on a central question, from their list, to focus on.   I always tell students if they get the class question done they are free to explore other questions as well.    I have been known, after kids have noticed and wondered for a while, to let students choose their own question and not have a central class question.  

Then we set off and work on the problem - you will find that kids are VERY motivated because this is their problem - they came up with the question and they are motivated to solve it.  Also, you won't find that you have kids just sitting their stuck because everyone has some information about the problem - even your lowest level students will have at least one way to start thinking and solving the problem.

My job as a teacher is now to question students.  I ask questions like, 

• tell me how you started to solve the problem?
• what does this calculation refer to, can you label it so that everyone know what you were calculating?
• are their other things you could buy with your money?
• how can you organize your thinking so that anyone who picks up your work can understand what you are purchasing and how you justified your work?

As a teacher you need to come up with questions to help students who get stuck or seem to have a wrong answer INSTEAD of giving hints.  Resist the urge to give hints - no one give hints in the real world - we want them to be independent thinkers.  Read Annie Fetter's post, One Example of a Bad Hint,  if you don't believe me. 

Then at the end of the time we have, I have students first share with a partner.  Sometimes I give students a simple rubric and they evaluate each other based on things like how they communicated their thinking or precision (practice 6).  Sometimes I evaluate them using a rubric and sometimes we put this work in their portfolio and then later on they choose the problems they want to be evaluated.  I always like to have some students share out with the whole group as well.

Goodness - I could go on and on about this.  I love this strategy and use it in ALL subjects, not just math.  In Washington County MANY of our elementary teacher use this strategy and can give you testimonials about it.  We use this strategy from our Cubs (age 3) all the way through our elementary grades.  I will tell you if you have read my post about 3 Act lessons you will see that this is always the basis for my 3 act lessons as well.

If you are interested in what I have gathered and created for Noticing and Wondering go to my google folder here.  I'm always happy to do a teacher workshop on this topic - contact me if you are interested!

Practices, Capacities, Dimensions...Oh My....

In my job in the last 7 years I dealt, primarily, with mathematics.  In my quest to "get up to speed" with the other 3 contents (for my new job) I found myself becoming a bit overwhelmed.  Elementary teachers who teach all four contents have quite a lot of Common Core Content to live and breath.  Not only do they have 4 sets of standards but also 4 sets of "habits of mind" or what they call Practices (Math and Science), Capacities (ELA), and Dimensions (C3/SS).  

My math partner and crime, Mary Ann, and I set out to help ourselves by looking at the similarities and differences between these habits of mind.  We first took the Math Practices (our comfort area) and then matched up the other contents were they were similar to math.  That still looked a little overwhelming so we, together, came up with a set of 6 general Habits of Mind that we felt could encompass all of the contents.  Now, we don't mean to replace the Practices Capacities, and Dimensions but simply create a more "teacher friendly" version that would help those teachers see the threads that tie all of the contents together.  

To access this as a pdf: https://drive.google.com/file/d/0B5W64FecuRP3LTdqTzRqNTk3U2s/view?usp=sharing

This certainly is not written in stone but is an attempt to make sense of the connections and those skills we want all kids to leave school with (the habits of mind).  If they master these things, the world will be at their finger tips - they will be able to solve any problem!  

What do you think of what we created here?