Thursday, December 18, 2014

The Language Of Math, The Language in Math, and Math as a Foreign Language

The math anxieties and math phobias so many people experience result from a lot of misconceptions about mathematics.  Many people simply consider themselves not mathematically inclined, as if there is a dreaded gene responsible for being bad at math, or some sort of Mathematics Deficiency Syndrome (probably caused by drinking blue Kool-Aid as a kid, or eating those Pop Tarts with the red sprinkles).

Sadly, many people have convinced themselves they are bad at math because when they were a child, an adult told them so.  Sometimes it was a teacher that bore the bad news:

"Some people just aren't math people, and based on your grade in my class, that some people is YOU."

Other times it was a parent:

"Don't worry, son/ daughter, you just happen to come from a long, proud line of good people who are very bad at math.  Just remember that we love you anyway."

What compounds the problem of anyone mistakenly believing he or she is hopeless at math is that it is commonly accepted and shrugged off, or even worn like a badge!  It has become cool in some middle school and high school circles to be "mathematically challenged."

There is no doubt that we all learn in different ways, and some of us excel in areas where others struggle.  But math is such a large and vague category of something to be good at or bad at; while about 7% of the population are born with dyscalculia, the math equivalent of dyslexia, this does not mean such individuals cannot excel in problem solving, or geometry, or even advanced mathematics. Math is a great many subjects in one, and almost all barriers to success in each can be overcome.

There are all sorts of learning disabilities out there, including dyslexias and dyscalculias, but we do not dismiss them as conditions we can do nothing about.  When a child has a reading disability, we work diligently with the child to make sure he or she can read.  Likewise, if a child has a learning disability related to numeracy, we work just as hard.  These learning challenges affect a small percent of our students, yet far more adults and teenagers gleefully proclaim they are no good at math.

What we are experiencing in our classrooms is not a great video game induced increase of Mathematics Deficiency Syndrome, or an increase in the number of children borne with the bad-math-gene.  What is far more likely is that our students today are coming from a wider variety of math backgrounds.  Important learning experiences may or may not have impacted their brains at that crucial stage of child development where kids have huge capacities for learning language and mathematics.

That's right!  Language and mathematics.  Many parents remember when their children began soaking up words and phrases like a sponge, and those of us who are not parents get to see examples of this every day posted on facebook.  "Watch this video of our Junior reciting the French alphabet backwards!"  Brain research has determined we gain huge amounts of language knowledge between the ages of 3 and 9.  And that also happens to be when children learn a lot of number sense and become fluent with their facts.  That is, unless they don't.

Just like a foreign language, math is a language that is best introduced at an early age, and not just with flashcards, but with actual verbal intercommunicating.  We don't learn to speak a foreign language by memorizing long lists of vocabulary words alone, and children do not learn to become fluent in math by memorizing long lists of facts.   We learn new languages by speaking them.  Beginning Spanish students learn how to respond to "Como estas?" before they knew how to spell it, so they can gain an understanding of how the language works and what it sounds like.  Likewise, we often teach the word "half" before we teach what a denominator and a numerator are, and how to even write fractions, so we can introduce the concept for our students to understand.

A criticism of the earlier editions of Everyday Math and other broad math curriculum programs is that they tried to squeeze too many topics into one school year, and sometimes did not give children deep enough understandings of the concepts being introduced to them.  Some students, especially those who for one reason or another may have missed out on important early childhood mathematical learning experiences, advanced through the grade levels with less than stellar understandings of basic operations and concepts.

What we are seeing more and more of with modern mathematics instructional resources and curriculum materials is an effort to teach mathematics more for understanding, and less for procedural memorization.  Even the Common Core features fewer content standards per grade level, with the expectation that math curricula will be constructed to allow for deeper exploration into math concepts.  In addition to the content standards, the Common Core also includes Standards for Mathematical Practice, which are guidelines not for what math should be learned, but how that mathematics learning should occur.  There is a tremendous effort to incorporate a wide variety of learning experiences into mathematics curricula nationwide.

While this is seen by many as a big step toward increasing math proficiency, it also presents challenges.  Students are being asked to problem solve, express their thinking on paper and share strategies with peers not only more often, but at a younger age (see previous post in this blog from November 20, Everyday Math, Cognitive Development, and "Explain..").

Learning mathematics relates to language learning, and today students are also being asked to use language more often in mathematics.  This is hard for students who struggle with language learning, and students who might not be learning in their first language to begin with.

In Lewiston, Maine, nearly 20% of all students are learning English as their second language.  When I taught in Lewiston, I often found it difficult to determine whether my ELL (English Language Learner) students were struggling with the mathematics of a particular lesson or unit, or with the language itself.  I quickly learned to slow my delivery in the classroom, and to check for understanding with greater frequency, but I still found it to be one of the most difficult challenges of my teaching career. I wanted to be able to speak in their first language.  Progress in math can come more slowly for students who have difficulties with reading, and for students who have had fewer experiences playing with numbers and quantities in pre-school.  Now consider a student who is just beginning to master the English language after being raised to speak one or more other languages.

Jessica Sturges shared with me some important pieces for teachers to incorporate into their instruction when there is an ELL student on board (in fact, I would recommend these practices even if there are no ELL students in the class):

#1:  Speak slowly and clearly

#2:  Provide students with vocabulary lists that pertain to the current unit to take home and study before the unit is taught.

Here is a sample Everyday Mathematics graphic organizer Ms. Sturges shared with me that shows how she presents unit vocabulary to some of her students:

In this example, there are some words listed in the student's first language, but that is not always an option, and it is not always necessary.  A word bank used in this manner could be helpful to any student learning new concepts.

Finally, an important thing to remember both with ELL students and other students who have trouble showing their thinking on paper, is that the language of mathematics can be confusing.  Upper elementary, middle school, and even high school students can still be heard using words like "times" or "timesed" when they really mean multiply or multiplied.  Early elementary is a time to encourage all students to use "add" and "plus" appropriately, as well as "subtract" and "minus."  Add is the verb and plus is a conjunction.  Subtract is a verb and minus is the conjunction ("take away" is often used in kindergarten and first grade in place of "minus," but by second grade, we should be encouraging the use of minus as the proper conjunction).  And likewise, multiply is the verb and times is the conjunction.  Division has some challenges as well... Remember guzinta?  6 guzinta 42 how many times??  Division all too often represents a great void of essential understanding.  I think we'll save that for a separate post.

It is hard for struggling students.  Minus, subtract, take-away, what is the difference, how many more than, how many less than, how many are left... The list goes on.  It is ok to use and encourage all of these representations of subtraction, but it is also important to check for understanding frequently.

There have been entire books written on the language connection to mathematics, but for now let's just remember that as we present new ways of learning mathematics, we are trying to tap the natural hunger for learning that occurs in a child's developing brain, in order to build a greater foundation for understanding as they tackle more complex mathematics concepts in the future.  This is why we use the word fluency in mathematics (more on that topic in a future post!).  The more fluent we are in a language, the less it actually hurts our brains to use that language.   The same is true of math.  With greater "fact-fluency" and overall mathematical fluency, complex problem solving becomes less painful, and potentially enjoyable and engaging.

For further reading on math and the brain, and the links between mathematics learning and language learning, here are a couple online resources:

And here is a highly informative series of links from PBS' "Child Development Tracker," for several age levels relating to learning mathematics:

And relating to language learning:

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.  

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!