Thursday, October 30, 2014

Great Things I See In Math Classrooms Part I

I thought it would be useful to share some of the great things I see in classrooms as I am out and about each week.  I get to see math happening at all grade levels, from Kindergarten through Middle School, so I will share reflections from math classes of multiple grade levels.  Keep in mind early elementary content and learning has many connections to middle level learning, and vice-versa.

I will not use any names unless individuals have given me specific permission to do so.

Students Sharing Their Strategies With Confidence... In 2nd Grade!

This is so hugely important I cannot even begin to emphasize it enough.  I witnessed a great sequence of whole class instruction into stations (aka centers) recently.  The whole class instruction portion began with a number story projected on the wall, and each student had individual white-boards with them.  Students were asked to write their solutions and share their strategies for finding each solution. What was special about this was not just that these students were being asked to share their thinking, but how comfortable and confident they were in the process of sharing their thinking:  

"I knew the answer was 7, because there were 13 and we had to take away 6.  I know that 12 minus 6 is 6, because it is a 'double' fact.  So since 13 is one more than 12, the answer must be one more than 6, which is 7."  

Not every student's strategy was correct, but it was clear to me that these students had become comfortable with this kind of sharing-- and that is not easy to facilitate as a teacher, especially with grades 1 and 2.  This section of the lesson, which took about 25 minutes, allowed students to practice their addition and subtraction strategies with four separate number stories.  It also gave them a chance to practice sharing their thinking out loud.  These second graders were not being asked to reflect upon their thinking; they were being asked to show their thinking and they were making great progress!   And just as the kids were showing signs they had had enough of this kind of thinking, they were asked to transition to the floor to hear a very quick set of directions for their centers.  The centers involved rotating through two separate fact activities and an opportunity to work on math boxes/ independent practice.   Students were able to have fun and demonstrate some independent responsibility while the teacher had opportunities to check in with individuals.

The key take-away from this math lesson is that every opportunity students have to practice sharing strategies is greatly beneficial to their learning and understanding of concepts and operations, and also beneficial to their learning how to become effective communicators.  As the years pass, these students will be asked to share their thinking a lot, as well as collaborate to solve problems and transfer their thinking/ strategies into writing.  These students' 3rd grade teachers next year will be appreciative of the work they have done this year in 2nd grade (and their 7th grade math teacher will appreciate it five years from now, too).  

Tuesday, October 28, 2014

Some Thoughts for Open Response Lessons

In recent weeks I have been teaching and observing a variety of Everyday Math lessons.  In RSU 5, K-2 teachers have the advantage of piloting the new EM4 units that have two-day Open Response and Re-engagement lessons imbedded within each unit (in addition to Open Response questions on every other unit assessment).  Grades 3-5 will have this next year when EM4 is released for those grade levels, but in the meantime we are working with the Open Response problems in the old units, and are turning them into two-day lessons.  It is clear to me that turning the old Open Response questions (from the end-of-the-unit assessments) into two-day lessons is not easy and causes anxiety for some.

Here are some things to consider, in the form of a Q+A session, when preparing and carrying out your Open Response lessons, whether you are teaching the new K-2 units or the older 3-5 units.

  • Open Response is all about students showing and sharing their thinking.  One of the hardest things for our students to do is respond to questions that start with the word, "Explain," especially when writing is a challenge for them.  I have created a half-sheet to help with this that contains three guiding questions for showing students' thinking processes.  The questions are:  What do you know?, then What do you need to find out?, and finally, What did you do to find out?   I offered these questions to a third grade group before I even handed out the actual Open Response assignment.  I had introduced the Open Response task to them, writing key bits of information on a white board, but I asked the students to share some possible ideas for solving the problem first.  Then I gave them the half-sheet with the three questions and asked them to write all the information they already know about the problem, followed by exactly what it is they need to find out.  Then, after sharing ideas for solving the problem with a partner, I asked students to write their strategy on the half-sheet, as an answer to the third question.  Only at that point did I pass out the actual Everyday Math Open Response assignment for them to work on.  This way, when they get to the section that asks them to explain their thinking, they already have something written down to refer to.  Otherwise, students often rush to solve the problem and get stuck when it comes to actually explaining what they did to find their solution.  As the year goes on and students become more familiar with Open Response lessons, they may not need to be prompted with the three questions and you can take the half-sheets right out of the sequence.
  • Open Response is one of the most important parts of our students' learning experience.  These lessons give them the chance to communicate their thinking to one another, and analyze each other's thinking.  One of the Common Core Standards for Mathematical Practice is for students to construct viable arguments and critique the reasoning of others.  Everyday Math breaks each the Standard for Mathematical Practice (SMP) into several more specific "Goals for Mathematical Practice," or GMPs.  One of the GMPs for this SMP is:  Make sense of other's mathematical thinking.  This is all part of the overall mathematical goal of helping students to become more confident and capable of working with others to solve problems.  The Smarter Balanced assessments will use problem-solving tasks to assess not only whether or not a student is capable of solving problems, but whether or not a student can show his or her thinking that led to a solution.   
  • Open Response lessons are two-part lessons, so students can begin building and testing their strategies on day one, and revisiting and revising their strategies on day two, as well as refining their explanations (showing their thinking). Expect a lot of difficulty at first; many students are not used to this kind of thinking and problem solving.  You will see progress throughout the year.

Multiplication Algorithms (from October 2014)

Here is a link to some interesting information regarding the term "standard algorithms," and how that applies to what we teach and what will be assessed in the Smarter Balanced Assessments this spring.

This item was also shared with RSU5 grade 3-5 teachers as a Google Doc.

Place value (From September 2014)

Here is an email from early September with some thoughts and observations around teaching place value.  Please feel free to let me know if you have any questions about this, or anything to add.

Regarding Place Value:

I have had a few conversations about place value as it relates to number and numeracy, and I thought I would share a couple basic concepts that may help with your students' mastery of place value.  I have been working on identifying some interventions for number and numeracy, so this is relevant to that also:

The emphasis of simple games is really important in engaging students toward quantitative thinking.  Since a prerequisite for understanding place value is understanding ten and multiples of ten, games that reinforce place value mastery often emphasize and involve the number ten in one way or another.  Using base-ten blocks, for example, is a common and effective introductory method for showing students how groups of ten, when put together, represent other quantities with names like "30," and "70," and "120."  But one thing those base-ten blocks to is leave the training wheels on... Students can count each "one" on every base ten block.  For example, if they count the spaces between the notches in a ten block, they count ten ones.  If they count all the squares on the hundred block, they can count ten sets of ten, or one hundred ones.  This is not a bad thing for introductory lessons, but the next step would be to eliminate the notches so students can see and grasp REPRESENTATIONS of ten without counting, such as with Cuisinaire rods.  That serves as a bridge to grasping the number value in symbolic form.  For example, if a student sees "172," and they can grab a representation of 100 (base ten 100 square), seven tens (cuisinaire rods), and two ones (Cuisinaire), without having all the notches to count, that is one short step away from recognizing 172:  "1" as 100, the "7" as seven tens, and the "2" as two ones.  

Other things besides Cuisinaire rods can be used to represent tens and hundreds (there is a neat Unifix cube trick you can do on your hands; I can show you if you don't already know this).  

The more opportunities children have to use representations of quantity without being allowed to count by ones to find the answer, the more confident (and eventually, fluent) they will be with their learning of place value.  

I hope this is at least a little helpful.   Let's keep these discussions going!