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! 

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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! 

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Why Don't We Get Math?

     We are less than a week away from the launch of our new book study of Minds on Mathematics.  I had a few minutes today to dig into the book and wanted to share some of my thoughts about the introduction.  

Why Don't We Get Math?

-Cultural: It's Okay Not to Get It

   This is the one I see the most in my school and in my life.  I see over and over again an acceptance among adults that math is hard and it is okay not to get it.  I know of several well educated adults who almost brag that their fifth grader does math they don't understand.  I have never heard a well educated adult bragging that they couldn't read at a fifth grade level.  As the author states, "this social acceptance of mathematical illiteracy is a huge barrier to our children's progress and preparation for life beyond our classrooms." 

- Pedagogical: Math is Memorizing

    When I was in elementary school math was all about memorizing.  The parents of my students are very much in the same place.  For some of us, memorizing worked.  For others it didn't even come close.  No matter where parents fall on this spectrum, most parents believe that math is memorizing.  They want to help when their student is struggling but their idea of helping is to remind kids of the steps to a procedure.  

- Individual: Math Ability is Innate

    This is true even in my own family.  I remember being good at math from an early age and my Mom always being surprised and wondering where my math ability came from.  More than any other subject in school, people seem to have this idea that math skills are inherited or passed down genetically.  This is where the excellent research on growth versus fixed mindset can really help understand how ability is the product of effort.  

Minds-On Math Workshops

Minds on math workshop aims to challenge these beliefs and provide students with a "learning experience in which students are challenged to grapple with their thinking and understanding about math in light of new information and challenges, to make meaning for themselves." Minds on math workshop is about leaving behind decontextualized arithmetic and giving the kids a change to construct meaning for themselves.

After reading the introduction, I can see this book aligns well with what I am already doing and the things I have experienced as a math teacher over the past 10 years.  I am excited to read more and further refine my teaching practice!  Want to join me? Check out the posting schedule below!  Need more information, get it here! 

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Fluency with Comparing Fractions

   I spend a lot of time in grades 3-5 working on developing strategies for comparing fractions.  I have written before about the 5 strategies my students use to compare fractions.  I also shared my favorite free computer game for kids who are working on fluency with comparing simple fractions.  Today I was working with an intervention group of fifth graders and had a few minutes at the end of our session.  I knew that I needed something quick and fun so I grabbed my regular old deck of playing cards.  

   We split the deck and each flipped two cards and made a fraction.  The first card flipped became the numerator and the second card the denominator.  This allowed us to make all kinds of fractions.   The person with the largest fraction won all of the cards.  Some of our fractions were less than one, some were equal to one and some were more than 1.  It was a great way for students to practice comparing fractions.  Out of the 5 strategies for comparing fractions, I saw students use comparing to a benchmark most frequently during this game.  It was a great way to reinforce the idea of one half, one whole and improper fractions.  I also saw students using common numerators to compare fractions and quite a bit of unit fraction reasoning.  My favorite use of unit fraction reasoning was when a student used the idea of unit fractions and the distance from one to compare 10/11 and 11/12.  "They are both missing one piece.  11/12 is missing 1/12 and 10/11 is missing 1/11.  Since 11/12 is missing a smaller piece, it is greater."         

     After a bit, we changed up the game and had each person flip 2 cards, use their smallest card as the numerator and their largest card as the denominator.  This made all our fractions less than or equal to one.  It made it so that the fractions were a bit closer together and required more critical thinking.  After a few minutes playing that way, we reversed it and had folks put their higher card as the numerator and the lower one as the denominator.  This was a great way to really reinforce the improper fraction, mixed number connection.  

This entire game took under 10 minutes and was a great way to reinforce some big fraction ideas!  I love how it requires no prep and is easy to differentiate!  By pulling out some of the cards in the deck, I could make this game based on friendlier fractions which would be great for third and fourth graders.

Book Study Launch

Looking for some motivation as the school year comes to a close?  Want to try a new structure for math class this spring and/or next school year?  Join us for a 5 part book study on Minds on Mathematics: Using Math Workshop to Develop Deep Understanding.  You can read all about it here!   

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Minds on Mathematics: Math Workshop Book Study

How is your professional reading coming?  Mine has taken a big hit over the past few months.  I find I am most inspired when I am reading a really good teaching book and have been putting it on the back burner for way to long.  I have been keeping up with my monthly issues of Teaching Children Mathematics and a few of my favorite teaching blogs but haven't had my nose in a really good math book in almost 3 months!  Last year, I read 14 professional books that I can remember and this year I am at 0.  With that in mind, I am putting my professional reading on the top of my priority list.  I think the end part of the school year is a great time to read a new book and try new things.  It is the time of year I sometime struggle with finding the joy in teaching and it is easy to get bogged down in testing and administrative tasks.  Reading a good book keeps me motivated!  Chatting with all of you about what I am reading really keeps me motivated and pushes me to try what I am reading in the classroom.

I am happy to announce that I will be hosting a new book study starting in 2 weeks!  I am going to be taking a look at Minds on Mathematics: Using Math Workshop to Develop Deep Understanding in Grades 4-8.  

You can read more about the big ideas in this book here or read the introduction and chapter 1 here.  

I will be posting on Saturday mornings.  

Here is the posting schedule:

April 2: Chapters 1&2

April 9: Chapters 3&4

April 16: Chapters 5&6

April 23: Chapters 7&8

April 30: Chapters 9&10

Hope you can join me on my journey to learn more about math workshop!  

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Jelly Bean Math

The countdown to Easter (and the spring testing season) has begun!  Today, I planned a quick little project with a group of 10 second graders that was super engaging and a great way to practice estimating, counting, comparing numbers and early division concepts.

It all started with little Dixie Cups of jelly beans (I used these ones).  I put the kids in 3 groups.  I gave each group a little cup of jelly beans and posted these directions on the SMART board.

Want to try this with your students? Grab these directions from Google Drive.

Two of my groups had 3 kids in them and one group had 4 kids.  I gave the groups with 3 kids 45 jelly beans and the group with 4 kids 60 jelly beans.  

I have really been working on independence with these kids and tried my best to step back and observe and not take over!  

I saw so many things that made my math heart happy!

Kids organizing and recording estimates

One group organized all their beans by color and then decided it would be much more efficient to put them in groups of 10.  The other groups went for groups of 10 from the get go. 

When each group had figured out how much they had, it was quite easy for them to figure out how much they were off by.  These kids are so flexible and fluent with double digit subtraction! 

Dividing the jelly beans equally between each group member was fun to watch.  Each group started by giving everyone 10.  Two of the groups were convinced that there would be a leftover bean that would have to be cut up.  The rationale was that because 15 is odd, there will be a leftover.  They were surprised to find that there was none left!  The main strategy for sharing out the beans was to give each person 10 and then share them out 1 by 1 until they were gone.

This part took a total of 14 minutes (including a few minutes of eating the beans!)

Then I pulled the group together and had each group share out how many beans their group got and how many each person ended up with.  There was some outcry over one group getting 60, but it provoked an interesting discussion about how each kid ended up with 15.

I then presented them with this problem."There are 10 second graders in this room and I gave you each 15 jelly beans.  How many jelly beans did I give out?"  I gave kids a minute to think at the circle then sent them off to grab a white board and show me what they knew.  I chose this problem because last winter when I was reading Children's Mathematics I became very intentional about making sure kids have the chance to solve Base 10 Story Problems.  Since 10 kids were in this group, I thought it would be interesting to see who used base 10 knowledge to solve this problem.  I gave out 10 groups of 15 but some kids used the idea of that being the same as 15 groups of 10.  

This student used base 10 knowledge to solve the problem (and the distributive property!)

Another student who used base 10 knowledge

This student added 45+45+60.  They used the number of jelly beans each group had.

Total time for this was 25 minutes.  We had a lot of fun, the kids got some good practice and I got to learn some new things about what strategies my students have.  How are you working on estimating and counting in your classroom?  Want to try this out?  Grab the directions from Google Drive! 

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100th Day Fun!

Today was our 100th day of school!  It was a busy day and I am exhausted!  Here are few of the things we did:

100th Day Bulletin Board

I saw this idea in my son's school and instantly loved it!  Our 100th day is always around Valentines day so having kids think of 100 reasons we love our school was perfect.  I cut 100 hearts out of pink paper and glued them to a piece of chart paper.  We passed it around the school and had kids and adults write reasons they love our school.  I will definitely be doing this again next year but I will be using heart shaped post-it notes because I won't have to cut anything out and they will be a bit bigger.  The hearts were a bit on the smaller size for something I wanted to make a bulletin board out of and kids could really only fit one word.   Here were a few of my favorites math, feeling of community, caring teachers, amazing students, respect, safety, creativity and honesty.

 100th Day Swag

The primary teachers in my building are great about making 100th day hats and glasses and such with their students but I find kids of all ages love 100th day swag.  I made these fun 100th day ties and bracelets with my older students.  I myself supported a 100th day tie, several bracelets, my 100th day pin a 100th day crown and a pair of 100th day glasses that my son made with his class and let me borrow.  There is nothing like a bit of swag to make it feel like a holiday. 

100 Equations

Over the years with my grade 3-6 students we have celebrated the 100th day by writing 100 equations in whatever area we have been working on.  This year I made it more official and dressed it up with some cute clip art.  My grade 4 kids just learned the traditional algorithm for subtraction so they rolled dice to create 100 three and four digit subtraction problems.  The fifth graders just finished a unit on adding and subtracting fractions so they did some dice rolling to generate and solve 100 fraction addition and subtraction problems.  We took all 100 problems and organized them on chart paper to make a display to share in the gym. I love this idea because you can do it with any grade and with any topic.  A great way to celelbrate the 100th day and still work on your current unit.  You can grab my printables from Google drive here. 

100th Day of School Workout

I found this little routine on You Tube when I was looking for a 100th day book.  We have been working on adding more movement breaks into our day and it was fun to have a special one for the 100th day.  If you want to jump right into the workout, skip to the 2:10 mark.  We watched the whole thing through once and then did it a few more times throughout the day starting from 2:10.

Literature

I pulled out a lot of my favorite 100th day books to read to various grades today.  Here are a few of the books that were being read around our building today.

What a fun day!  What is your favorite way to celebrate the 100th day?  

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