Friday, November 21, 2014

About Teaching Fractions... How Third Grade Math Links to Sixth Grade Math

I had a conversation with an educator at Freeport Middle School recently about the great number of middle school students who still lack basic understanding of fractions.  While I have a great appreciation for Everyday Math and all the research that goes into it, a common criticism of the program over the years is that it hasn't addressed many math concepts deeply enough.   The spiral format allows students opportunities to experience and re-learn things they might have struggled with in the past, but often those learning experiences have been brief and far between.  Having taught middle school math in multiple districts (some were Everyday Math districts and some were not), my students often displayed learning deficits when it came to fractions.  And for the middle school teacher, that poses a far too common dilemma: Do I take extra time to teach them foundational mastery of fractions, or do I drag them through the unit with the limited understanding they currently possess?

There is good news on two fronts!  Firstly, Everyday Math is rolling out its latest edition of the program (Everyday Math 4), and from what we have seen in the new grade K-2 units we are piloting this year, there seems to be deeper exploration into key concepts.  The new edition of Everyday Math 4 is aligned to Common Core, which brings us to good news item number two:  The third grade Common Core standards emphasize teaching fractions not just pictorially, with fraction bars, rectangles, pizzas and divided quantities, but also representing fractions on a number line, so students can also become familiar with fractions as sequential numbers.

Understanding fractions is crucial for success in middle school, but having a rich learning experience involving hands-on opportunities, part-to-whole relationships, parts-of-quantities relationships, and placing fractions on a number line will also have a positive impact on our students' understanding and mastery of multiplication and division.

Below is a link to a short video that explains about how fractions are introduced in the Common Core math standards.   The video gets a little wonky around 1:30, but it only lasts a few seconds, so you shouldn't (?) have to take any dramamine.  It is worth watching, in my opinion.     :^)

Please contact me if you are looking for more strategies relating to teaching fractions.

Thursday, November 20, 2014

Everyday Math, Cognitive Development, and "Explain..."

"Explain how your number model matches the number story."

At the beginning of the year, my middle school students often sunk into a state of mental anguish and defeat at the mere sight of the word explain on a math assessment.   They often were not nearly as intimidated by mere math problems as they were math problems that involved writing.  It was like a double-whammy of academic anxieties.  First you want me to do math, and now you want me to write... at the same time.

Even more interesting is to see the reactions of first graders when they are asked to "explain" how they solved a problem.  There are so many questions to consider when we ask young students to explain their thinking.  Are they cognitively ready to explain their thinking?  Do they understand what we mean when we ask them to explain their thinking?  Do WE understand what we mean when we ask that question?  Should we give them a low score if they are not yet able to write the words down on the paper?

The increased emphasis on self-explanation, critical thinking, and concept-based instruction in teaching math is more present than ever in Everyday Math and has some parents asking a lot of questions.  They wonder: What ever happened to memorizing facts and learning how to add and subtract?  Math will always be math... Why has teaching math changed so much?

Are they cognitively ready to explain their thinking? 

There is increasing evidence coming from years of research that seems to indicate children benefit greatly from concept-based learning and self-explanation (being able to explain one's thinking to another) as they mature.  In other words, the earlier students become accustomed to showing and sharing their thinking, i.e. being able to not just solve the problem, but also to show a partner exactly how they solved the problem, the better they become at both procedural and conceptual learning as teenagers and adults.  They become better problem solvers.  So some students are more ready than others for this kind of thinking, but the consensus among the pedagogical researchers is that starting early has a positive impact.  Here are some links that help explain this... The first is a video, then the other pieces are articles that cite research.

It is also important for us to remember that as we see a great variety of abilities to explain how they solve problems, that we are not asking them to analyze their thinking in grades 1 and 2 as much as were are simply asking them to show their thinking.  This is the precursor to the kind of critical thinking they will be asked to demonstrate later.  Being able to show their thinking is a very important step in the development of analytical thinking and being able to self-critique and revise their thinking.  Still, it is often difficult for children to put into words their mathematical thinking.  It often amounts to a language and writing lesson.

"What did I do?  First I read the number story. Then I wrote down 5 + 1 = 6 because 5 tickets and 1 more ticket equals 6 tickets in all."

I am amazed at the experience and skills veteran teachers demonstrate when they read and make sense of student answers!  Likely there is inventive spelling and lack of punctuation, digit and letter reversals, and interesting drawings to interpret.  First grade teachers become real experts at making clear sense out of beginning writers' answers.  Sometimes, it is appropriate for the teacher to write the words down as the student dictates his or her thinking.

Scoring questions such as these is not often a fast and simple process.  Careful analysis is required, and sometimes challenging interpretation.  I know some teachers get together to score assessments so they can consult with one another on such matters, and when it is possible, that is a terrific idea.

In closing, I should emphasize that this kind of learning experience is coming from scientific research, which influences the mathematicians, teachers and pedagogical experts who wrote the Common Core Content Standards for Mathematics and the Standards for Mathematical Practice, and those standards have influenced the writers and publishers of every major math curriculum in America right now.  The simple answer to the befuddled parent is this:  The trend today in mathematics instruction is teaching more for understanding; we are not abandoning fact practice and procedural learning, but we are introducing more ways to teach for greater understanding of math concepts.  Research shows that greater understanding of math concepts comes from giving children more opportunities to think for themselves, to construct and deconstruct mathematical problems, and to be able to explain their problem solving strategies to others.

Monday, November 17, 2014

Online Links to Extension Activities and Projects for grades 3-6

I am in the process of seeking out valuable and meaningful extension activities for grades 3 and up that offer high-achieving students a chance to be challenged and engaged in the regular classroom.   I spent some time looking into online resources, knowing there is a lot of not-so-great stuff out there to sort through in order to find the occasional goldmine.  The good news is that with Common Core now spread through most U.S. states, teachers are sharing more and more resources around the country and I was able to find a number of collections of rich problem-solving tasks, projects and activities aimed at higher-achieving learners.  As you browse, you may notice that some of these links are more easy to navigate than others, but all have some activities that are accessible and might be useful to you.

Don't be afraid to look beyond the grade level you teach.  You might find an activity in a grade above that fits perfectly with the unit you are teaching now, or something in another grade level you think might really match the interests of a particular student of yours.

Note... The previous post (below this one) discusses classroom differentiation and opportunities to introduce tasks like these during math lessons.

I will continue to look online and in print for more engaging extensions for your fast-paced and/or easily bored mathematicians!   Stay tuned.  Here is the list so far:

3rd Grade:

4th Grade:


5th Grade:

6th Grade:

Differentiating for High and Low in the Workshop Model

Over the past few weeks there have been a number of similar requests for help with differentiating instruction.  I have put together a short list of online extensions (in the next post), but I wanted to make sure and address this with respect to the Math Workshop model of instruction.  This can really make a difference, whether you are trying to engage the higher achiever in your room, or you are trying not to neglect the slower learners among the group.

Every math class you teach has different dynamics, and sometimes the diversity of learners among a class of 20 or so students can be a major challenge.  Maybe you have more students than usual who are really struggling and are demonstrating great gaps in foundational concepts.  Or it might be that you have one or two students who seem to be miles ahead of the others and are getting bored.  Or you could be experiencing both of these conundrums.

Will Pidden has let me borrow a book from a professional development session that happened before I joined RSU 5 that offers a way of clustering groups for learning.  This model of grouping students in a way that honors a diversity of abilities and learning styles while attempting to create a manageable range is attractive.  But I am going to assume that is not possible, and you simply have a wildly diverse group of learners.

There are different teaching styles and different learning styles; every district in which I have taught has had its range of each!  Some teach math in the more traditional style, with mostly whole-group instruction, and do it very well.  New concepts are demonstrated in front of the class, often with teacher play-by-play narration, followed by an opportunity for students to attempt the new work together as a whole group, perhaps with guidance from the teacher.  Then, triumphantly, students complete the new tasks all on their own,  and practice more for homework.   The next day, homework is gone over in class and a new concept is introduced in the same manner.

This is how not most, but all of my math classes were taught when I was in school.  It worked for me at first. I didn't love math as a child, but I was reasonably good at it and found most concepts easy to understand.  That is, until I became a very distracted teenager.  Advanced math was agony.  I sat in the back and zoned out, completely unengaged, handed in dreadful work, and "got by" with mediocre grades and test scores.  I was neither gifted nor exceptionally challenged, but I was definitely not engaged.

The most effective workshop-style classrooms I see allow opportunities for independent work to be done by a small group of students while others play math games or work with the teacher to complete problems they are having trouble with.  It involves a certain mastery of classroom management; students must be very aware of the routines and respectful of the learning environment, which takes practice and time.  But the benefits can be pretty amazing.

Not only does this model allow those students who need extra help to receive more attention from their teacher, but it also allows some flexibility for differentiating independent work.  If Johnny is breezing through this unit, maybe he can be working on a more challenging and engaging task instead of plowing through his math boxes at lightening speed.

Ed Techs in the room can help facilitate this model, but it is not impossible to conduct if you are the only adult.  It takes practice, both for you and for your students.  Maybe experiment with one transition at a time.  "Today we are going to try something different for twenty minutes.  I'd like group A to work with me over at the table, group B to start your math boxes on page 78 at your seats, and group C may partner up and play Multiplication Top-It on the floor.  Let's practice transitioning to this new format today, and tomorrow we will switch groups."

Practicing is essential.  Students should know exactly what is acceptable for working on the floor with partners, independently at their desks, and in groups at the table.

This workshop model, sometimes referred to as stations or centers, does not have to be the rule every day, but it can offer opportunities for students to learn in different environments, take brief breaks as they transition, and receive the learning opportunities they thrive most from.

Students at the table with the teacher are receiving in-class (or "tier 1") interventions for their difficulty with the math of the current unit, while students doing independent work have the opportunity to experience "tier 1" interventions for their accelerated learning needs.  On a different day, or after a rotation of groups, the teacher might have the opportunity to work with a small group of faster paced learners and could even experiment with some extensions related to the unit.

Trying new things in the classroom involves some uncomfortable risk-taking, and can be quite challenging to carry out.  Your trusty math strategist can be a resource for this kind of thing!  Also, your colleagues may have some great tips for classroom management and workshop-style strategies for teaching a diverse group of learners.

I am hoping to do some math learning walks this winter, and I also hope to find ways for teachers to watch other teachers teach whole lessons in math.  I see an impressive variety of great teaching strategies everywhere I go, so I can't wait to share notes.  

Wednesday, November 12, 2014

Thinking Time

"Billy, can you tell us how many sixes there are in forty-four?.. OK, make it forty-two.   How many sixes are there in forty-two?  Billy, what is forty-two divided by six?.. How many times does six go into forty-two? Billy??  If I have forty-two chocolate brownies, and there are six people who want them, how many brownies does each person get?? Would it be seven?  And if we started with forty-four, there would be two left over, right?  Seven with a remainder of two.."

Most of us have been there.  I know he can get it... If I just re-phrase it in a way he can understand...

An important item to consider when teaching math in any format is thinking time.  Often, if we don't hear a response from a student, our instinct is to rephrase the question, or offer a quick hint or scaffold, or sometimes even answer the question for the students, with the hope they will be able to figure it out on their own the next time.

I have recently been in a fifth grade classroom and a Kindergarten classroom where I saw thinking time being offered to students in really effective ways.

The fifth graders, when given a question to ponder, were asked to put their pencils and papers away for a moment and discuss with a partner what strategies they might use to tackle the problem.  When the discussion/ think-aloud time was up, students were asked to volunteer some of their strategies.

The Kindergarten students were told, simply, to stop and quietly think before raising their hand to offer a solution/ answer to a question.  The teacher gave them about 10 seconds to think, and then when he signaled to them that the thinking time was over, the hands shot up in great numbers.

For me personally, I need thinking time in almost every situation.  Not everyone does, but I do. Meetings are particularly difficult for me, because I am often two or three agenda items behind, stuck on something that was said when discussing an earlier topic.  I then set myself up for embarrassment when I am called upon to voice my input on the current item.

"Umm... Could you repeat the question, please?"

For me, when other people are talking I have a hard time focusing on the appropriate topic.  And for some of our students, it is even more challenging.  Re-phrasing becomes more like another question added on top of the first one.  Then the third re-phrasing is added, and now the student has three questions to consider.  Indeed, some children have difficulty understanding that their teacher is trying to make it easier for them.  Instead it feels like an interruption to an already challenging thought process.

Many students require time to process their thoughts.  Pair-shares, quiet thinking time, and simply allowing some silence after questioning might make a significant difference in how confident your students' participation is in whole class or even small group discussions.   This way, the worst case scenario is that there is an extended period of silence with no solution.  And that is valuable for formative assessing.   That extra little bit of time might yield a response from a student, and if not, you now know that student needs some help.

"Billy, it looks like you could use some time for this one.  OK, let's all take 30 seconds to talk with a partner about how many sixes can fit into forty-four..."

Math lessons can be stressful for your students.  Offering some quiet time to think gives them greater confidence when it comes time to present.  It also gives them some ownership when sharing out, even if their answer is wrong.  Thinking time helps them to own their mistakes, and to see those mistakes as an important step in the learning process.  When there is not enough thinking time, students are more likely to panic and toss out a guess.  And while a stab in the dark might get them a correct answer and a high-five from a neighbor, an incorrect guess has not meaning to them and is forgotten.
Thinking time helps reduce math anxiety and it helps kids see that making mistakes is an important part of problem solving.

Monday, November 10, 2014

Scoring the EM4 Assessments

This fall one of the tasks Lisa Demick and I have taken on is  to create a scoring guide and new cover sheet for the new Everyday Math units in grades K-2 (Next fall for grades 3-6) that is compatible with each school's report card format.

I have taken the answer guide Everyday Math provides for the Progress Check assessment at the end of each unit (I will be working on Unit 4 shortly), and have added details about how to score each item.  Here are some details you should know:

  • Many items have been divided into two parts for scoring purposes.  For example:  "How many apples are left if Tate eats two apples from his bag of six apples? Explain how you found your answer."  This question would be divided into two separate questions that address two separate standards.  The "How many apples" part might address standard 1.OA.4, while the "Explain.." part might address a standard for mathematical practice (EM4 calls them "goals for mathematical practice, or GMP's) like GMP6.  If this was item #1, I would call the first part #1A and the second part #1B.  Item #1A would likely be worth 1 point for the correct number of apples, four.  Item #1B would be worth at least 2 points for a complete and accurate explanation of the how the student found the answer.
  • Sometimes a student's response might not exactly match up with the scoring guide.  For example, the sample answers given for a score of 2 and 3 might not match your student's response which appears to fall somewhere in between.  In this case, you can use teacher discretion to determine which it should be, or consult with a colleague or even myself.  
  • These are brand new and there may be errors.  For unit 1, I made a number of errors that teachers alerted me to, and I was able to fix them.  If I make changes to a scoring guide after you have used it to score the assessments for that unit, do not feel like you need to go and re-score all your students' assessments.  If you see errors or typos, please let me know in an email (or in person!) and I will address the issue as fast as I can.    
  • The score guides are somewhat time-consuming, especially at first, but once the assessment is scored the transfer to the cover sheet should get easier and faster in time.   
  • The cover sheets show goals based primarily on clusters of standards, which are groups of Common Core content standards that fall into a sub-category (between standard and domain).  We are using clusters on the cover sheet to simplify the scoring/reporting process and make it more straight forward to calculate a grade on report cards at the end of the trimester.
  • From this point on, these score guides should be ready and shared with all 1st and 2nd grade teachers as Google Presentations in the grade 1 and 2 folders well before you assess your students on that unit.  Up to now, we have been racing time a bit to complete the guides before the unit has been taught.  I think I am catching up, though.
  • Please invite me to PLC meetings if there are questions regarding the scoring guides, and bring sample student assessments so we can address specific issues with scoring assessments as they arise.
  • Open Response questions, and some others, are difficult to score.  First graders in particular have trouble showing their thinking in words.  This has already prompted some parent feedback suggesting we are asking too much of a student in the first grade.  Please keep in mind that most first graders and many second graders have never answered this type of question on an assessment before and will have difficulty doing so.  I am trying to be gentle in the scoring guides when it comes to showing partial understanding (a "2" on the cover sheet).  But some students will score a "1" on an open response question and have higher scores on the rest of the assessment.  The goal is to work with these students throughout the year to help them show their thinking on paper more proficiently and earn higher scores on Open Response questions.
  • Challenge questions are optional and are often not attempted by all students.  They offer opportunities for students to go above and beyond what is expected.  If a student leaves a "challenge" question blank or does not attempt any challenge questions, that should be reflected on the cover sheet as a "NA" for "not assessed."  If challenge problems are incorrectly answered, a "partial" score of "2" is usually given.
  • Most assessment items allow a maximum of 3 points scored on the cover sheet.  This is because the majority of assessment items do not allow a student to "exceed," or go above and beyond mastery.  When an item offers the possibility of a response that "exceeds" expectations and/or the standards, it will be possible to score a "4" on the cover sheet for that item's standard/cluster.  

No doubt there will be more questions and further revisions to the guides.  Thank you for your patience with these scoring guides and also for your feedback.