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Teaching Place Value and Kidwatching

Yetta Goodman, way back in 1978, coined the term, kidwatching.  Kidwatching is an observational assessment of children’s performance and responses to instruction throughout the school day. Anecdotal records or more structured teacher checklists often document kidwatching. It’s truly one of the most effective ways of learning about children and how they think…and today, in my 4th grade math class, I saw some really cool behaviors and learned some new things about many of my kids.

I began 2 weeks ago teaching a group of 4th graders (and one 3rd grader who joins that group) about binary numbers. One of our goals for teaching place value is to help kids understand that the PLACE of a digit tells the VALUE of that digit. Kids typically don’t get that, either verbally or intuitively–but they have memorized the decimal place value chart by 4th grade, so the strategy many kids use is to read the whole number, then look back at the question to find which digit they need to name, then try to remember what they said as they read the number, or make a place value chart so they can translate that one digit from what they they just said. Sound confusing?  It is–and there are many, many kids who make silly mistakes on tests, simply because they don’t understand the bold italicized sentence above. They think the digits cannot stand alone in that place, without the whole number.

So I decided to try to teach that sentence to understanding with these kids who come to me for extension in math two days a week.

Here’s what I did:

The first day, I began with a quick preassessment where I asked them to copy into their journal this sentence “There are 10 kinds of people in the world, those who understand binary numbers, and those who don’t.”

I asked them to write a brief comment on what they thought it might mean. Most just repeated or reworded the sentence a bit, not seeing anything wrong with saying ten when they saw the 1 and 0.

Then, we used a website, Math Is Fun,  and scrolled down on that page to the section labeled decimal vs. binary. Next, we discussed what place value meant, then we talked about the word part “bi” and thought of other words that have bi- in them and then I gave a quick intro to how binary numbers worked. I talked about computers, showed them the periods of the binary system and they immediately made connections between the sizes of their iPads and computers, and they brought up the game 2048, recognizing the binary numbers from playing that game in 3rd grade.

I deliberately used the terms decimal number system and binary number system instead of base 10 and base 2, because I wanted them to relate to the word system and look for the system of the numbers we were going to compare. That was the end of that day’s lesson.

For the next several days they first made the binary chart from 1-20, then worked on a couple of worksheets converting binary numbers to their decimal equivalents and decimal numbers to their binary equivalents. They also had some addition and subtraction problems regrouping binary numbers.

This week, in their homerooms, they began a multiplication unit. I had planned this week to have them work in other bases to make sure they understood how the number systems worked, so I decided to continue that. At the beginning of today’s class, one kid raised her hand and asked me what bases had to do with multiplication and why weren’t we working on multiplication. I responded that I had decided to go ahead and finish out the work on bases I had planned, knowing that after today’s work, she would understand.

Today’s work was varied, depending on the kid–some were simply working through all bases 3-9 to write the numbers 1-20. Others were counting in bases at a random decade (from 70-90 in base 8, for example.) Yet others were taking random numbers (49, 76, 114, 162, etc.) to convert them to several different bases. Most were making up their own challenges for themselves, sometimes depending on their interest, and sometimes depending on the partner they had chosen. They were all totally engaged, and working hard the whole hour they were in my room.

So, at the end of class, two boys came up to ask me to check their work on converting a number in the decimal system (that was in the thousands) to their base 4. They were absolutely correct, and I asked them how they had figured it out without a periods chart. They then turned the paper over and showed me their chart of 4 to the 5th power-1, 4, 16, 64, 256, 1024. The number they were trying to convert was 1,028. They said when they saw 256, they had doubled it, and when they saw 512, they remembered from 2048 (the game) that 512 doubled to 1024, so they knew the conversion then- 1 set of 1024, 0 sets of 256, 64, or 16 + 1 set of 4, and no sets of 1s or, in base 4, 100010.

I asked them if they had used multiplication and they said yes–so I then asked the class if they had used multiplication while working in different bases, and they resoundingly answered yes!  They also said they had had fun and they were looking forward to tomorrow’s class to keep going. The girl who had asked what bases had to do with multiplication at the beginning of the lesson asked me at the end if she could work on her charts during lunch–I asked her how much she had used multiplication in the last hour and she said “about a billion times.”

I was surprised by today’s class for several reasons…the ability they had, as a group, to manipulate numbers in their heads was so different from a 5th grade class I worked with another day, their remembering the numbers from the 2048 game and their use of them in the bases, a kid saying he needed to really understand bases so he could do better programming since he was going to be a computer programmer when he grows up, the engagement of all 17 kids, even a few who were new to the class today, and the total immersion in setting up challenges for themselves as they worked to understand the patterns and ways each base worked.

The fact that most of them were totally working independently, working with a partner, (so there were checks and balances) and having fun was great. But the conversations and explanations I heard clearly said they understand the systems better now that when we began last week. The fact they were looking for combinations of factors of a number showed their deep understanding of decomposing numbers. And, the way they were manipulating digits in various places to both show that factoring and the total number said they have grasped some major understandings of place value–my original goal. Today was a day for kidwatching. Now to help them verbalize those understandings and consciously recognize what they know and have learned…that’s tomorrow’s work.

Why I Don’t Give Grades

My 4th grade math group is a bunch of geeky math kids.  They love puzzles, trying to figure out problems by themselves, and they do math just for the fun of it.  It’s really an amazing hour three times a week.

Our fourth grade is where kids in our county encounter report card grades for the first time.  Up to this point, they’ve gotten behavioral and work habit scoring and a satisfactory (or not) ranking on subject areas–but no A, B, C, D, F to this point.  Here, though, they begin to encounter that grading system we all know and love.

I have nothing to do with their grade–the classroom teachers do that. I also don’t have the same kids all the time.  Our 4th grade teachers pre-assess at the beginning of a unit and I work with the kids who need extension.  Kids stay in their math class Monday and Fridays so the teachers can make sure they get exposed to and work on all pieces of the curriculum. I work with them the other three days of the week.

My current 4th grade math group is working on analogies and patterns in math right now as an extension to their place value work. For the past week, they’ve been working independently on a series of worksheets and problems that stretch them in all kinds of ways, and they’ve been loving it.

They love feedback, so when they finish a page, they find a buddy who has finished the same page and compare answers–when answers are different, they work the problem together to find the correct one. As they discuss and solve problems and question each other, I glance over their work, but I don’t ever sit down and go through their papers problem by problem, to score it in any way.  Our work is collaborative enough and we talk enough about  the work that I know who’s still a little iffy on certain things, who has it solid and who needs lots of support. We end class lots of times by going over the problems someone found hard that day.

Today was hilarious–I was teasing some kid–I honestly don’t even remember about what, but I said something stupid like “if you do so and so, it’ll be an “F”.”  The kid I was talking to looked at me and her eyes starting widening, getting round as saucers.  I looked at her and was thinking–but not saying– “Really?  You really think I’ll give you an F?” She hesitated, and then she said,

“Do you actually  grade our papers?”

I laughed.

It was such a foreign notion to her–she had no clue what that might look like.  Me judging her? Me putting red marks on her paper?  Me crossing out ones she missed and counting them up? (This is the same kid who earlier today had looked at me and said, “I don’t see how you do it–so many kids are asking for help and you help all of us.  Any other teacher would be yelling at us, telling us not to call her name anymore!” )

(I have to say I don’t think that’s really true in my school, but I know we’ve probably all had days we wished we could change our names–even if just for a little while!)

So, why don’t I grade their papers?

Because I think kids learn more from reflective feedback and deep questions and studying and finding and talking through their own mistakes.

Because what we learn from grades is to compare ourselves to others around us–and I’d rather set them up to look for their own growth in relation to themselves, instead of their performance in relation to someone else.

Because I know them–from our class discussions and our quiet one-on-one talks and the questions they ask, and the comments they make and the strategies they share– I feel no need to give them a letter grade to tell them what I think.

Because I get to know their thinking every day as I challenge their sharing, ask them hard questions and honor their responses as a learner–right or wrong.

Because we share strategies and thoughts every day–and they trust themselves to ask questions about stuff they don’t understand–and their questions help me know what to teach and help them learn.

Because I expect them to be learners–and people who care about their own learning don’t much care about outside evaluations of their learning–they know when they know it and when they don’t. They don’t need a grade to tell them that.

So, yeah, when I was asked if I actually grade papers, I laughed…and we do that a lot in my class.

So, Reagan, this blog’s for you–keep asking those hard questions, thinking, looking to make meaning and sense of your world  and most of all, keep laughing with those sparkly blue eyes!

Great Questions For Learning

“Interesting questions to engage kids as thinkers” means different things to different people.  My math collab teacher and I recently had a conference with a parent and we were describing our math class. When we said her child was really engaged with our problems, her response was ,”Oh, yeah, the word problems.”  I was initially confused by that–but thank goodness for my partner–she got it–the parent meant the handshake problem.

Okay, so that started me thinking…my initial thought was that no, we hadn’t been doing word problems. But, I guess you could categorize our shaking hands work as a word problem.  Then, my next thought was that all math problems should involve words. Then, I thought, no, when I see a problem on a piece of paper, it’s not necessarily about words.  Then I started thinking about what constitutes a word problem and what makes a good question?

We read Counting on Frank earlier in our math class and talked about “Henry questions” which are questions modeled after those in the book.  The kids wrote some they thought of here.

The third column is where we’ll go through them and see if we know how to solve such questions–then we’ll do that again at the end of the school year.

But, where do we get good questions to explore?  Here’s one source, a book called Good Questions for Math.  I especially love the ones that have multiple responses–that’s sometimes a great source of easy differentiation!  Here’s another, a pdf about asking effective questions.  And, yet another, Good Questions: Great Ways to Differentiate Mathematics Instruction, which is an awesome resource K-8 and would be great for a faculty book study! But I have to say, my absolute favorite source is the kids themselves…when they’re engaged, and trying honestly to figure something out, they ask the best questions. They also cause me to make connections and ask great questions. And what’s better than getting provoked to think deeply?

So here’s a couple of what I think are cool questions that just play around with numbers (and number sense) for you, my reader…

You multiply two integers.  The result is about 50 less than one of them. What might the two integers be?

Oh–don’t like that one?  Okay, try this–

A shape has some perpendicular sides and some parallel sides.  What might the shape be?

Or, try this:

Using the divisibility rule of three, make a five digit number that is a multiple of both three and five.

Do  you have any good ones to share, or another source of interesting questions?

Why Can’t Kids Be Responsible For Their Own Learning?

“Wow, young lady, you’ve really put me in a hard place!”  I told one of my kids this morning in math class.  You see, we’re trying hard to get kids to do their homework (and all work) in their math journals, so we asked them to practice a few factor trees in their journals for homework last night.  That, of course, meant they had to remember to take them home, do it, then bring them back.  And, up to this point, we’ve been very forgiving of kids who haven’t come to class quite prepared, so today was our day of “cracking down”–we had made a big deal of how important it was to be ready so we could all move on and that we were giving points for procedures as well as work today.

So, I asked them to open their journals to their factor trees from last night and show it to a buddy and get some feedback.  (This is a strategy my collab teacher and I often use for checking to make sure something was done and each kid getting quick feedback. We watch as kids share with one another and listen for big discussions where disagreement may be occurring or a long explanation may be needed, so we can step in if necessary.)  But,  this morning K turned to me immediately and said, “I don’t have a factor tree–I was too busy last night working on my multiplication tables.”  I said, “What?  Why were you working on those instead of doing the homework?”  (I figured maybe someone at home had given her a different task.)  Her response? “Well, I knew I needed to work on my facts, so I thought I should probably spend some time doing that so I’d be better at factor trees today.”

That’s when I told her she’d put me in a hard place. The kids heard me and it got pretty quiet as everyone watched to see what would happen. This was an amazing opportunity to both give a life lesson and teach them we were not ogre teachers.

I told the kids that most educators’ basic goal in working with them was to help them to be lifelong learners, because we–teachers or parents–wouldn’t always be there and our hope was that they would be independent learners, taking responsibility for their own learning.  My collab teacher added that we wanted to see self-directed learning, questioning and sharing, and that personal decisions had to be based on one’s own knowledge of themselves as learners. “So”, I said, “Let’s talk about what K did last night.”  Was she an independent learner? Yes!  Did she take responsibility for her own learning? Yes! Was it self-directed learning?  Yes!  Was it based on her own knowledge of herself and what she needed as a learner?  Yes!

So how can I be upset she didn’t do what I had given her to do?  She’s meeting the goals I said educators have for her.

I ask again, in a slightly different way, with you, the reader, looking through a slightly different lens perhaps–

Why can’t we allow students to be responsible for their own learning?

Well, we did, for K today.  She absolutely got that check for homework!

Why do we have to be the ones dictating homework?  Why do we set the tasks?  Why don’t kids get a chance to say what they think they need to practice or reinforce?

Well, they will in our class.  How about yours?



Imaginary Numbers?

Okay, a second time I’m doing 2 posts in one day–but really, it won’t happen often!

Those of you who are mathematicians will know what I am referring to with that title–those of you who aren’t may not.
The bottom line is that my kids can’t wait for math tomorrow because they are going to learn about them. We are working on their sense of number, and I happen to believe that the more kids have the big picture, the more they are able to manipulate things within subsets of that big picture. Today’s agenda include talking about real and imaginary numbers–but we didn’t get to it.

I wish I’d had a video of the kids face as I finished one sentence (about consecutive numbers, which is what we had been working on) and then said, “Well, we’ll have to do real and imaginary numbers tomorrow, cause it’s past time–you guys have to go.” They were so disappointed we didn’t get to talk about imaginary numbers. I can’t imagine THAT! (as a kid, I mean…)

So what do you know about imaginary numbers?  Any good resources out there for a bunch of fifth grade geeks? If we don’t get to it tomorrow, I’d like to have some resources they could pore over at home. Thanks!




Shaking Hands-Math, part 2

OMG was the way the imaginary Tweet I began when I left the 5th grade math class. (I sent it later, promising a reflection, and so, here it is.)

There’s never enough time.  I always overplan, but  because this was the first math class, and I really wanted to include all parts, I really rushed the kids. I own that.  They didn’t finish the problem, the time for reflection was minimal, and they mostly complied with the various activities, without deep conversation. They didn’t get to share strategies or talk about the problem in a class discussion, so there is a lot left for the next class.

There are several things I really want these kids to understand–

a.) While the answer, in some cases, is extremely important, we’re going to be using messy questions to think about process and strategies, and sometimes we won’t center on the answer. Today was one of those days.

b.) There are many ways to solve most problems, and there usually just isn’t a right or wrong one.  There will be more efficient or less efficient ways, and we’ll talk about that.

c.)  Complicated problems are what they’ll encounter in life. We’ll practice those, often.  And, math is NOT simply arithmetic.

So today, they got involved in something that encompassed all three of those things, and when many of them figured out we were NOT coming to closure on the right answer, there was an amazing amount of frustration, even though they hadn’t had enough time to really get to the end of their own mathematical processes.  Yet, the comments on our Today’s Meet were mostly thoughtful and honest, not frustrated. I’ll be interested to see what they have to say after we look at some strategies and final answers.

So the next class will be sorting some of their responses to the “what is math?” question, looking at some of their work under the document camera, with them sharing strategies they used, and then groups finishing the work to get to a resolution. When they finish that, we’ll ask them to make up their own handshake questions.  It’s a pretty intense beginning to what looks to be an interesting year!

Shaking Hands

So in the post, First Day Plans, I was brainstorming possible ideas for the first day of an advanced math class.  I want it to be fun for the kids, to be active, to include mathematical thinking and to allow my collaborating teacher and I to observe their problem solving and collaborative skills.

Here’s what we came up with:

The Do Now:

Please open your journal and respond briefly to the question, “What is math?”  Share what you think.  (We’ll be doing this throughout the year so your definition will change and grow and as you do.)


1. Pick a number from 1-3. Write it on the last page of your journal for accountability.

2.  Without talking or giving away your number, you will go around the room shaking hands with your classmates.  Each time you shake a person’s hand, you shake their hand the number of times to match the number you chose.  If you chose 3, for example, you shake their hand three times.  If they chose 1, they will shake your hand once.

3. You can sort yourselves into three groups if you pay attention and find like handshakers.  That is your job today.  Once you think you have your full group, shake hands with your group members once more to check.  Once you have done that, raise the correct number of fingers as a final check.  Move groups if you need to do so.

4.  Class discussion: We’ll look at the groups and talk about their sizes and see what patterns or relationships we can infer. (We’re anticipating the term probability to come up and we’ll be ready to talk theoretical or empirical if necessary.)

5.  Then, we’ll count off in each group, A, B C.  When that is done, we’ll have smaller groups  to work the next problem.

6.  Small group work:  If every person in this room shakes every other person’s hands, how many handshakes will that be total?  (We will not specifically address whether the adults count or not, but simply refer the kids back to the question. Our intent is to observe the small groups and see who includes what, how thorough they are,  how they organize their work, if they understand the problem and have strategies to attack it, who disengages, who steps up as a leader, who can be a follower or leader, etc.)  We, the collaborating teachers, will simply be observing during this time.

7.  Whole class reflection time: We’ll share strategies, examples of work, and answers and ask them to think about whether this was a good problem to work or not.  The last five minutes will be responding on a Today’s Meet to share what they think of the problem and what their work/behavior told us about them.

8. We’ll pull the work they do Tuesday, Wednesday and Thursday of this week together on Friday by asking them to help make a problem solving rubric against which we’ll look at our work this year.

First Day Plans

So our kids have been in school for 8 days.  Teachers have been building classroom community and pre-assessing for some achievement groupings, and getting to know the kids.  We’ve been doing some pre-work on our new devices (an infusion of 40 Chromebooks and 18 iPad minis this year) and talking about our Design 2015 projects in each grade level.  So far the highlight of the year for kids (and definitely our principal!) has been the Book Brigade, I think.

But the highlight for me is that the first week of September I get to start working with kids more than one on one testing them! So here’s what I’m thinking about math.

I’ll be working with 25 or so kids in 5th grade–who will be heading towards a middle school honors class that is a compacted 6th, 7th, and 8th grade curriculum.  There are two likely paths for these kids when they get to middle school.  Those who succeed in the 3 year compacted class will go into Algebra in 7th grade. Some, who are less confident and secure in the concepts taught next year, will go into a compacted year of 7th and 8th and go into Algebra in 8th grade.

My goal, to amplify their minds this year, is to help to my kids look at the world through a mathematical lens, rather than see math as algorithms to be learned. So, we’ll be working with real life examples whenever possible, they’ll be learning math within real contexts and we’ll do units like the “Artful Engineering” one developed to use the fabricators we have. I’ll be collaborating with a fifth grade teacher for this class, and we’ll mostly do it in her room.

But this first week, I can’t decide quite how to begin…

I want to have discussions about what is math and what does a mathematician do. I’m considering asking them to do/make something to show what they know/think/believe about math. I’m not sure that wouldn’t be too open-ended for the first day, though.

I want to take in some new manipulatives that will end up being snap cubes, but right now are nets of cubes.  I want to ask kids to figure out what they are and design something with them, to see what they build and how they think.

I want to read Counting on Frank and talk about “Henry Questions.”

I want to show the figure from Lockhart’s Lament and ask them how much room the triangle takes up and hear their thinking–and watch how it changes as others share theirs.

Screen Shot 2013-09-01 at 9.50.37 AM

I want them to fiddle with questions such as:

“A piece of wood is 15 feet long. How many 3/4-foot sections can be cut from it?” so I can see how they think.

I want them to list and think about questions that arise from looking at things like:


I’ve thought about building a slideshow of some kind and asking them to pick a picture and describe the math in it. I’d include pictures like those from here and here and here.

I want them to see math as art, as fun, as manageable and achievable and most of all something to invest in and enjoy!

So I’m trying to figure out how to involve and engage them in something cool…something that shares what math means to them… something that helps me begin to know more deeply how they think and what they think doing math is.

What would you do?

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