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Saturday, April 30, 2016

Math Workshop:Sharing & Reflection

Welcome to the final week of our Minds on Mathematics book study.  If you missed them you can go back and read Understanding Takes Time,  Shallow Versus Deep MathStarting Class and Mini Lessons & Work Time.  

This week are going to take a deeper look at ending class with sharing and reflecting when using a math workshop model.  


Perhaps the most important part of a math workshop model is the time for students to share.  It is so important to stop work time before the end of your math class period and give kids a chance to share.  This is the part that helps to solidify their comprehension and gives them a chance to practice metacognition which is thinking about their own thinking.  They get a chance to synthesize their understanding, check on their progress and make goals for the next day.  Teachers can gather important formative assessment data about what strategies kids are using and where to go next.  By communicating their thinking and listening to a variety of peers' solutions and ideas, students make connections and deepen their own thinking.  

My favorite way to structure sharing time is by bringing the whole group back together.  As I have been circulating during work time, I choose a few pairs to share their thinking.  I pick pairs who have different strategies and ideas to share.  Then I have students present their work while classmates ask questions or make connections.  When multiple groups present different ideas, we take some time to synthesize the learning and talk about which strategies were most efficient.  To add some variety to our days, I sometimes will mix the pairs up and have them share with another person or another pair.  With clear expectations and a lot of practice, my students have become more efficient at this portion of math workshop and it is easier for me to fit it all in to one class period.  In the rare case where work time extends beyond where I intended, we will start the next days class with sharing time.  


I love how kids can learn from each other during sharing, but I also love how kids can learn from themselves during reflection.  This is usually a quick but important part of math workshop.  I don't get to this every single day but at least a few times per week.  On any given day, we might reflect on behavior, or our skills as mathematicians, or what we have learned, or the process of solving problems.  We never do all of these at once, we usually just choose one.

The reflection I enjoy the most is the time at the end of a unit or a semester or school year when we have time to look a little more in depth at progress.  This is often when we will do some writing or comparing work from the beginning and end of a unit.  This is the deep reflection that makes all of my students feel like they are really learning.  It really helps to make their learning personal.  It also really helps with setting goals in an authentic way.

I love the reflection prompts printed on pages 162 & 163 in the book.  I think these would be well worth posting in my classroom to help us bring our reflecting to the next level.  I don't often do quick written reflections but with the questions and sentence frames presented here, I think I could get some valuable information from my students without adding a lot of time or stress to our day.

Thanks for following along as I read this great book!  Please share your thoughts about sharing and reflecting in the comments below!

Saturday, April 23, 2016

Math Workshop: Mini-Lessons and Worktime

Welcome to week 4 of our Minds on Mathematics book study.  If you missed them you can go back and read Understanding Takes Time and Shallow Versus Deep Math or Starting Class

This week are going to take a deeper look at the mini-lesson portion of math workshop and talk about what work time looks and sounds like. 


Mini-lessons during math workshop should be
- Short and focused (under 10 minutes)
- Whole group instruction
- Goal is fostering independence

"The more I explained the less my students seemed to understand.  The more sample problems I did for them, the sleepier they appeared." (Hoffer page 103)

This quote from the book perfectly sums up my past experience with teaching rather than listening. As I have shifted my practice from that of an expert giving out knowledge to that of a facilitator helping kids build their knowledge, this quote is no longer true.  Many of the ideas presented in this chapter were ones that I have been comfortable with for the past few years.  My mini-lessons look a lot like these.  As I read, I kept coming back to modeling thinking as something I really wanted to work on. 

My Own Experience with Modeling Thinking

 I immediately thought of my second graders and solving multi-step problems.  We are finishing the year up with more practice adding and subtracting 2 and 3 digit numbers and I wanted them to get the chance to practice these important skills while working on solving problems that involve more than one step.  I find that with second graders, story problems can be tricky because there are a lot of words on the page for those who are struggling readers.  I decided to use modeling thinking to help these kids out with multi-step problems.
In the spirit of keeping my mini-lesson mini, I presented one problem to students.  It was: Jonathan had 172 baseball cards.  He spilled his drink on part of his collection and had to throw away 58 of his cards.  For his birthday, his friends decided to surprise him with 75 new cards.  How many baseball cards does Jonathan have now?
I put the problem on the screen and had a student read the entire thing.  I then thought aloud about how overwhelmed I was with trying to figure out what happened to Jonathan and what I was supposed to do.  I decided aloud to tackle the problem one sentence at a time and to stop and think after reading each sentence.  I switched to a new slide on my screen that had one line come up at a time.  After reading each line, I stopped and thought aloud about what I knew.  I would then reveal the next line and repeat.  I did much of the thinking aloud but also had some students contribute to my think aloud.
When the problem was solved, kids shared what I did to make a challenging problem easier.  Then I sent them to work with a partner on 2 additional multi-step problems involving some combination of addition and subtraction.  They did an amazing job and really focused on what strategies made it easier. They finished by writing their own multi-step problems. Tomorrow I will start class with a think aloud on one of their problems and then they will pair up again and solve another few examples.  

Work Time

The postulate and question of the day at the beginning of this chapter really helped me think through the big ideas about work time.

How can we facilitate thoughtful and productive work time for math learners?" 
Facilitating thoughtful and productive work time for math learners is something I have worked hard at developing over the past five years.  I think this is a strength in my classroom and in my school.  My challenge for next school year will be to make this thoughtful and productive work time work in a multi-age setting.  With our declining enrollment over the past few years, we have had multi-age classrooms but have been separating kids by grade for math class.  Next school year it is my goal to work with teachers to build capacity for truly multi-aging.  I think it will be a fun challenge to see how we can structure thoughtful and productive work time for such a diverse group of math learners.  
"Students learn most when they spend math work time thinking, talking, and making meaning of mathematics for themselves."
This quote sums up my teaching philosophy in one neat sentence.  To me, this is where the fun and the learning of mathematics takes place.  I know in my own education the math classes where I did the most talking were also the ones I did the most thinking and the ones where I finally had a chance to construct the meaning of mathematics for myself.  This nicely summarizes the constructivist ideas around learning and is what I strive to do each and every day in my classroom.  
"pages of mindless computation do not foster the construction of new knowledge. Learners need the opportunity to collect, generate, and frame their own problems and inquiries. The learner must be in the drivers seat." (Hoffer, p. 116)
This used to be so challenging for me.  I was very afraid that giving up the drivers seat meant giving up control of the situation and of my class.  It took years of seeing how other teachers managed their classrooms and employing the best management strategies before I was able to step back and really let my students be in charge of their own learning.  It is my goal to give the illusion of the classroom running itself.  I have high expectations for behavior and being on task and I am not afraid to spend the extra time making sure the backbone of classroom management is there.  Without excellent management skills, you can never be an excellent math teacher.  

Join us next week for the final installment of the Minds on Mathematics Book Study! 

Saturday, April 16, 2016

Math Workshop: Starting Class

Welcome to week 3 of our Minds on Mathematics book study.  If you missed them you can go back and read Understanding Takes Time and Shallow Versus Deep Math.  

This week are going to look at how to start a day of math workshop.

The Opening

The first part of math workshop is the opening.  This is a time to invite learners to make connections and establish purpose.  The book outlines 4 parts to a successful math workshop opening.

Welcome Learners

If you are teaching a self contained classroom, this is your chance to make a transition to math class.   You might play a math song, check out a math you tube video, have kids share a favorite memory of math class or have some way to get kids pumped up that math is about to start.  If your students switch rooms for math and this is the first time you are seeing those students that day, this is your chance to greet kids at the door and work on making those connections with students.  It is your chance to work on developing community.  

 Activate Prior Knowledge with an Opening Exercise

What do your students already know that can help them with the day's problems?  How can you ask questions in such a way that students are engaged and feeling capable?  You are trying to convey that students already know some thing that can help them and that they are capable of being successful mathematicians.  "Offer students problems that invite challenge by choice; Let the first question be something everyone will likely know, followed by questions of increasing complexity that may feed into one one another, reminding learners of the concepts behind the mathematics."  I love how the book presents these tiered openings and it is definitely one of my goals to be more intentional with choosing questions like these for my openings. 

The other way to activate student knowledge is by having them consider a concept.  They might write everything they know about division or provide examples of vocabulary words that are likely to come up during the day's problem.  

My struggle with the opening of math class is always making sure it doesn't take more time than I allotted or take over the class entirely.  This is something I am still working on.  

Learners Setting Purpose for this Lesson

This is more than just writing your learning target on the board and having students read it.  It is about having students set goals for themselves.  What are they good at? What do they need to work on?  It might involve the topic for the day such as fraction division or it might involved one of the math practices such as attending to precision

Managing Homework

I have stuck to my resolution this year and not given any homework.  This is a decision I am quite happy with and have no plans to return to giving homework.  If you do give homework and want to work it into your opening, the author offers several suggestions.  

Your turn!  How do you open your math class each day?  What are the essential components for you?  How do you make sure your opening doesn't take over the entire class?  Please respond in the comments below.

Join us next week as we look at the mini lesson and work time in our Minds on Mathematics book study.  

Saturday, April 9, 2016

Shallow Versus Deep Math

Welcome to our second week of looking closely at math workshop.  Get more details about my math workshop book study here.  

Deep Versus Shallow Math

In this week's reading, I was struck by the difference between deep and shallow math.  Here are some characteristics of each type of math.

Shallow Math

- Memorizing algorithms
- Applying an algorithm (usually a word problem found on the bottom of a page full of practice for that algorithm.
- Hunt & copy exercises
- Plug and chug numbers
- Not considering what the numbers mean
- About covering the content
- Teacher gives out knowledge

Deep Math

- Engaging, exciting, exhausting & inspiring
- Pushes learners out of their comfort zone
- Mental models
- An understanding of a concept that can be built upon later
- Discourse
- Challenging tasks
- Students wrestling to make sense
- Content understanding
- Teacher as a facilitator of learning

When I was in elementary and middle school 99% of the math I did would be classified as shallow math.  I was the queen of the plug and chug.  I thrived on algorithms and hated "word problems".  When I was in high school, it was more of the same until I got to Algebra 2 and was faced with new and challenging problems that no one had "taught" me how to solve.  This took my enthusiasm for and understanding of math to an entirely new level.  Math class became exciting and invigorating and for the first time I got to invent my own strategies for solving problems and compare them to my classmates.  It was such a dramatic and marked change for me that it really is what sparked my interest in becoming a teacher.

Now when I teach math, I try my best to keep most of what I do with my students at the deep level.  Math workshop provides me with a vehicle for giving kids support solving challenging tasks.  

Your turn!  Can you think of anything that is missing from these lists of shallow and deep math?  Where did most of your own learning take place? Please respond in the comments below!

Come back next week for part 3 of our Minds on Mathematics book study! 

Saturday, April 2, 2016

Understanding Takes Time

I know math workshop is for me!  Why?  Because I share these beliefs:
1) Students Are Capable of Brilliance
2) Understanding Takes Time
3) There is More Than One Way

Welcome to part 1 of our Minds-On Math Workshop bookstudy.  Here are some of my thoughts from this week! 

Students Are Capable of Brilliance

     My best teaching friend and Kindergarten teacher extraordinaire has this as her mantra.  Her students constantly outperform other Kindergarten students in the district and she is always being asked to share her secret.  Her #1 reason her kids do so well is because she holds them to very high standards.  She truly believes that all kids can learn and in many ways their teacher's attitude about their learning becomes a self-fulfilling prophecy.  Her students learn because she believes they can.  ALL OF THEM. 
     Every time I feel like giving up on a kid and just "Teaching him how to do it" (aka arithmetic without understanding) I remember my friend and how beliefs become her students.  All students can learn and we need to keep expectations high for our students.

Understanding Takes Time

     You can "teach" your students the standard algorithm for subtraction in ten minutes and have them practice it for an hour.  It will look like your students understand subtraction.  Next week or next month you will give them three subtraction problems and they will tell  you that they forgot how to subtract.   Worse yet, they might not tell you that.  They might keep missing a step in the procedure or do a step wrong repeatedly.  Now they are in a position where they think they know how to subtract and they have no idea all their answers are wrong. They don't know how to tell if an answer is reasonable because you didn't "teach" them how.
     Alternatively, you can spend an hour per day for three weeks guiding your students to develop flexible and efficient strategies and giving them opportunities to share these ideas with their classmates.  They will also have a chance to hear their classmates ideas and compare how they are similar or different from their own.  In the process of doing this, they will strengthen their understanding of addition, place value, estimation and inverse relationships.  Next week or next month, you will give them three subtraction problems.  They will solve them all mentally applying a strategy that is efficient for the numbers in each problem.  They may or may not use the same strategy for all three problems.
     The example above illustrates the difference between telling and guiding.  With telling, you are doing most of the talking and learning.  Sure it is faster but your students will lack understanding.  Guiding students to develop and refine strategies takes much more time upfront.  The students do most of the talking, and you get through fewer problems during a class period.   You might in fact spend 20 minutes talking about one problem.  It takes time but it also develops understanding.  Understanding is what will be there for your students next week and next month.  For me, it is worth the investment of time to produce understanding.

There is More Than One Way

     This is one I learned from my students.  There is one student in particular who I will always remember helping me see a new way to look at subtraction.  I was taught that there was one way to solve each math problem and it has taken me years of teaching and learning to undo that thought.  "With faith that each child, given time, has an innate ability to reason out a solution to a problem, even if their initial approach and strategy may differ from how we believe things "ought" to be done, we can begin to turn over the responsibility for learning mathematics to our students." This quote from the book really resonated this change for me and helped me see how my thinking has changed since I started my teaching career.  I know embrace multiple strategies and love that my second graders can currently solve a problem like 17-9 using six different strategies.  Most of them are very efficient and none of them involve counting! 

Do you share these beliefs?  How has your own experience in the classroom enhanced or changed your beliefs? Other thoughts about this weeks reading?  Leave your thoughts below in the comments!