problem solving

Student Ideas Matter

Engaging students in math has always been a goal for me.  No… more than a goal…. a passion! And it’s not always easy to do.  For example, I used to hate teaching students how to find the sum of an arithmetic series.  I didn’t hate it because it was difficult to teach or because students had an overwhelming difficulty learning it.  I hated it because I was the only one that saw the beauty in it.  I was the only one who was passionate about it.

This lesson was “fun” (I use the quotes to denote that this was a fun lesson for me – not so much for my students).  But this all changed when I allowed my students the opportunity to think for themselves.

The task was very simple in concept:  Find the sum of the series of the numbers 1-20.

Before going any further, it may be useful to know about the

  • Class norms:
    • Estimate first,
    • the answer is never enough,
    • reasoning, explaining and looking for patterns are all expectations,
    • if you found one way, look again, you may find a more efficient way,
    • get out of your own head and talk about the math with your partner/group while you work

Several started adding 1 + 2 + 3 + 4 + . . .+ 19 + 20.  I noticed this and asked those groups for one word to describe their strategy.  Sample responses:  boring, lame, tedious (actually proud of that one), calculator worthy…

My reply to each of their descriptions:  If your strategy is [insert one: boring, lame, tedious, or just plain calculator worthy] why do you feel the need to use it?

Sometimes students get stuck in their own thinking and just need to be made aware of it. To help nudge students to think in other ways, I had bowls of tiles with the numbers 1-20 written on them available for groups to use.

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It took several minutes before students began to grab tiles and began to notice things like:

  • “Hey, Mr. W., we can make a bunch of 20s.”

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  • “We got a bunch of 21s. 10 of them.  It can’t be that easy, right?”

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  • “We made 10s and 30s.  How did you make 21s?”

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  • “We did the 20s too.  That’s the easiest way for us.”

It was a bit chaotic, and I didn’t know it then, but there was a passion building.  This wasn’t just engaging, these students were ALL IN.  They were more than engaged and wanted to learn more about the strategies they came up with.  They wanted to share. Needed to know.  And the answer was almost irrelevant. The connections between all of their strategies became the focus.

From here, getting to the algebra made sense.  How would you find the sum of the numbers 1-50?  1-90?  1-100?  What about 5-50?  Some saw their ideas with the tiles transfer easily to an algebraic expression and equation.  Others not so much. So, more time to talk and share.  More time to find a strategy that is more convenient to generalize for a series of numbers of any range.  The success of the students’ mathematical ideas gave them power to reach further – to take another chance.

Teaching the lesson this way was a definite improvement on the original. In this version, the students’ ideas matter, so students matter.  In this version, students think for themselves and collaborate with others, and in turn get validation of their thinking, so students matter.  In this version, students built some passion.  They fed off of each other. And the content mattered because of the students’ interaction with it.

Is this lesson the best it can be? I’m not sure.  So, I’ll continue to try to improve on it.

Thoughts and comments welcome.

 

 

 

 

Unlikely Students in Unknown Places

I recently got back from Santa Fe.  I was attending a conference there for a few days last week and afterward, I drove to El Paso to visit my brother’s family (he’s currently stationed in the middle east so I didn’t get to see him – unless you count face time) for a day before flying home.

Let me preface this story by saying that we all probably have a story similar this, but how we handle it can be a possible game changer.

Somewhere on my long drive, I stopped at a fast food restaurant to grab a quick bite.  So, I went inside and got in line.  The following outlines the beginning of our interaction as I stepped to the counter:

Cashier:  May I take your order?

Me:  Yes, please.  I’d like a number 2.

Cashier:  Large or medium.

Me:  Medium, please.

Cashier:  (after pushing more buttons than is conceivably necessary to enter my choice of “medium”): Your total will be $6.05.

I dug through my wallet (receipts from the trip and everything) and found that all I had was a $10 bill, so I handed it to her.  She entered $10.00 correctly and the correct change of $3.95 showed up on the little screen.  At just about that point, I remembered that I had a bunch of change in my pocket and said quite enthusiastically, “Oh, wait, I think I have a nickel.”  Who wants to carry around $0.95 in change in their pocket.

The cashier didn’t miss a beat, and said, “So, your change will be $4.00 even.”  I kind of smiled as I continued to look through my change, proud that she had a mental strategy to adjust to the situation and that she seemed quite confident and comfortable using it in this situation.

Unfortunately, I didn’t have a nickel, but I still didn’t want change falling out of my pocket into the depths of the rental car, never to be seen by me again.  So, I told her, “Oh, I’m sorry, I don’t have a nickel, but I do have a dime.”

As I handed her the dime, I saw her face morph from a confident smirk to a confused, almost terrified look of despair .  I had just taken her from a mathematical point of “Yeah, I can do this math stuff.  I may not use the computer for the rest of my shift” to “Holy $#!+, what the #=|| just happened!”

I went into math teacher mode and waited patiently for her to begin breathing again.  And then I waited for her begin thinking.  She adjusted my change with my introduction of the idea of a nickel, why not a dime? After what seemed like 5 minutes (probably more to her), it was painfully obvious to all around that her anxiety in this situation was taking over her ability to tackle this problem. So, I tried to think of a “least helpful question” to ask.  Now I put myself on the spot.  If she only knew that we were both now feeling some of this pressure.

So, I finally asked her my question and she gave me the correct change within a few seconds.  She smiled as she gave me my change and my new “unknown” student and I parted ways.  I know I felt good about helping someone develop a strategy outside of the classroom.  I hope she had a similar feeling about learning to make sense (no pun here) of making change.

Being a math teacher is a 24-7 job sometimes and we can find our students anywhere – even in a fast food restaurant in New Mexico!

What you would have asked the cashier in this situation.  I’d love to hear what your “least helpful” question would have been. No pressure, take as long as you like.  No one is waiting in line behind you!

Feed the hungry!

Oh, here’s my question:  “If you could change the dime into some other coins, what would you change it for?”

Math: A Fun After Homework Activity

All week long I’ve been asking Connor, my 9th grade son, what he has been working on in coordinate algebra.  Here’s a snippet of a recent conversation:

  • Me:  So, Connor, what have you been working on in your coordinate algebra class?
  • Connor:  We’ve been graphing.
  • Me:  Graphing what?
  • Connor:  Graphing different lines.
  • Me:  What kinds of lines are you graphing?
  • Connor:  Ummmm…
  • Me:  Are they linear functions.
  • Connor: Yeah, there are linear functions, but we also do curves…
  • Me:  Like what kind of curves?
  • Connor: Umm… exponents
  • Me:  Ok.  Anything else?
  • Connor: Umm…
  • Me:  Hey, I want to show you something. . .

Versions of this conversation happened several times this week.  Due to soccer practices, games, homework, and Life in general, we never got much past Connor’s last “Umm…”

Until yesterday!  The conversation changed a bit:

  • Connor:  We did something cool in class today.
  • Me:  Oh, yeah?  What was it?
  • Connor:  We had to build a picture using graphs of different lines.  We built a shamrock.
  • Me:  That’s what I’ve been meaning to show you all week.  Go grab my laptop.
  • Connor:  (playing game of war on an ipad) But I finished my homework.
  • Me:  Just take a look at this for a few minutes and see what you think.
  • Connor: (heavy sigh)

Enter Des-Man from Desmos.  Once he had gone through the tutorial, he was hooked. . . for a while!  He engaged in this for about 2 1/2 hours.  When he wanted to make something happen, but didn’t know how, he would come to me and ask.  We’d figure it out together.  The best part of this whole experience was when he realized he knew how to create something on his own and went to his math work from class as a reference.

Fast forward to 2 1/2 hours later, when Connor finished his Desman.

Des-man

 

 

 

 

 

 

 

To see the picture in detail along with the equations Connor used to create this graph, click Connor Face Graph.

It didn’t stop there.  I had some tabs open and clicked on one with the In-N-Out Burger task from Robert Kaplinsky.  He was curious enough to work through it even after all of the Des-man work.  So, I showed him more by clicking on the Open Middle tab (also from Robert Kaplinsky).  I selfishly pulled up the task that I wrote in collaboration with Graham Fletcher called The Greatest Difference of Two Rounded Numbers.  After making sense of the problem, and a lot of eye opening moments that led to phrases like “Oh, I can make it larger!” He got what he thought was the final answer and we validated his reasoning by clicking on the answer.  A slight smile!

So, we’re looking at close to 3 hours of after homework math investigation that ranged from rounding numbers to graphing equations, and solving problems.  Sounds like a great evening to me.  Great conversations and fun while learning and reinforcing mathematics understanding!  What could be better?  Talking Math With Your Kids – High School Edition.

Feed the hunger of all ages!

More with Connor:  Real Math Homework and Real Learning

Connecting Percents and Fractions

Not understanding mathematics can be extremely frustrating for students.  As a teacher, figuring out how to help students understand mathematics can be just as frustrating.  My primary go-to resource for these situations is Teaching Student-Centered Mathematics, by John Van de Walle et. al.  because it’s all about focusing on big ideas and helping students make sense of the math they’re learning in a conceptual way.

Recently, I was asked to model a lesson for a 6th grade class who was having difficulty working with percents.  So, I turned to my go-to resource, and during planning, I realized that I didn’t know anything about these students other than that they were struggling with percents.  So, I couldn’t assume anything.  I ended up creating three separate lessons and combined them into 1.

First, I handed groups of students a set of Percent Cards and Circle Graph Cards.  Their task was to match the percent with the corresponding circle graph.  As students were working on this, I heard groups reasoning about how they were matching the cards.  Many started with benchmarks of 25%, 50%, and 75%, while others started with the smallest (10%) and matching it to the graph with the smallest wedge.  As groups finished, they were asked to find pairs, using the matches they made, that totaled 100%.  Once finished, a discussion about their process for completing these tasks revealed a solid understanding of percent as representing a part of a whole.

Now to shake their world up a bit.  I asked them to leave their cards because they would be using them again shortly.  I introduced these Percent circles and asked them what they were.  A brief discussion revealed some misconceptions.  Some students said they were fractions, others said they were wholes because nothing was shaded.  I altered my planned line of questioning to questions that eventually led to a common understanding of what fractions were and how the pictures of the fraction (percent) circles really showed wholes and parts (fractions).

Their next task was to match their cards with the equivalent fraction circle.  This was incredibly eye-opening. Groups began to notice that some percent card matches could fit with multiple fraction circles (50% could be matched with the halves, quarters, eighths, and tenths).   Thirds and eighths were the last to be matched.  But their reasoning didn’t disappoint.  One group noticed that the percents ending in .5 all belonged with the eighths because they were too small to be thirds (the other percents with decimals).

Students were eager to share their thoughts about what they learned about fraction circles and percents:

  • Fractions and percents are the same because the pieces look the same.
  • 1/4 is the same as 25% and 2/8
  • I don’t get why the eighths end in .5.
  • The percents all can be fractions.
  • 1/8 is 12.5% because it’s half of 25%

Finally, I asked students to solve a percent problem (now that they’ve all realized that fractions and percents can be used interchangeably).   I gave them the m & m problem from this set of percent problem cards.  The only direction I gave was that they had to solve the problem using some representation of the percent in the problem before they wrote any numbers.

My bag of M&M’s had 30 candies inside.  40% of the candies were brown.  How many brown candies is that?

While this was problematic at first, students looked at their fraction circles and percent cards and realized they could use four of the tenths since each tenth was the same as 10%.  Most students needed just one “least helpful” question to get on the right track:  Where do the 30 m & m’s belong in your representation?

Most groups were able to make sense and persevere to solve the problem correctly, and explain why they “shared the 30 m & m’s equally among the ten tenths in the fraction circle” and why they “only looked at four of the tenths because that’s the same as 40%.”

My beliefs that were reinforced with this lesson:

  • We can’t assume understanding from correct answers alone.  We need to listen to students reason through problematic situations.
  • Students really want to share their thinking when they realize that someone is really interested in hearing it.
  • Students crave understanding.  They really want to make sense.
  • Procedures are important, but not at the expense of understanding.
  • Empowering students by allowing them to build their own understanding and allowing them to make connections allows students to feel comfortable taking risks in problem solving.

Please take a look at Jenise Sexton’s recent blog about percents with 7th grade students for some fantastic ideas about students using number lines and double number lines to solve percent problems.  It’s SWEET!

 

Re-viewed: Children’s Mathematics. . . It’s a Beautiful Thing

About a month ago, I was asked to preview the new edition of Children’s Mathematics and write about it on this blog.  I was more than happy to oblige!  Children’s Mathematics is one of a select few books that I’ve read in the past decade that have really had an impact on how I teach mathematics.

Let me begin by saying that no matter which edition you read, it’s worth it.  If you’re a teacher (or parent) and have an unread edition of Children’s Mathematics sitting on your shelf (for whatever reason), do yourself and your students (or children) a huge favor and read it.  Then do exactly what it says to do!

The research-based approaches to teaching mathematics you’ll learn from the contents in this book are invaluable.  Cognitively Guided Instruction.  That’s what it’s called.  And it’s a beautiful thing to see in action.  And it’s even easier to see in action as you read the book (more on that later).  Empowering students to think, make sense of, and solve problems based on their own understanding.  Why don’t we all teach this way?  It makes so much sense.

To be clear, Cognitively Guided Instruction (CGI) is not a program or a curriculum.  It’s an approach to teaching and it’s based on research on children’s mathematical thinking and how it develops.  The idea that’s most intimidating to teachers (and parents) is that no direct instruction is used before giving students a problem.  Many would argue that students won’t know how to solve the problem unless they are shown how first.  This is so NOT TRUE.  The contexts of the problems give the students all they need to jump in to the problem.  Their pathways to solutions are defined by their own understandings.  For example, students may be given a problem such as:

Luke had 7 toy cars.  His friend gave him some more cars for his birthday.  Now Luke has 12 cars.  How many cars did his friend give him?

Students given this problem may solve it by counting down from 12 or up from seven,  They may begin by choosing 7 objects to represent the cars, then counting some more to get to 12.  They may even make two sets (one set of 12 and one set of 7). There are multiple ways students can represent the problem.  All of them valid.  Some are more efficient than others, but regardless of the strategies used, it’s a beautiful thing.  It’s especially beautiful when students share their strategies and learn from each other.  When we listen to students’ thinking we best know how to work with them in order to move them along their own mathematical journey.

Now this is all great, but you can get this and more from any of the earlier editions of this book.

Here’s some of the new goodness you get from the latest edition (out later this month):

  • A chapter dedicated to Base-Ten number concepts – this was nice to see, since base-10 understanding is a huge part of elementary mathematics.
  • Quotes from real teachers using CGI in the classroom.  These can be found at the beginning of each chapter.  Its a small part of the new edition, but really it’s one of the things I really enjoyed!
  • Video clips that you can watch as you read! No more CDs to have to load, or lose or break.  When you’re reading and want to see the accompanying video, just scan the QR code in the book with your phone.  It just pops right up!

Overall, this new edition has some updated content and makes it easier (thanks to technology) to see in action.  As one teacher from the book put it:

The better I get at listening to children, the clearer I hear them tell me how to teach them.

Have I said this already?  Beautiful.  Absolutely beautiful!

What are you waiting for?  Go out and get Children’s Mathematics and read it.

And then, go out and get the “sequels”:

Extending Children’s Mathematics: Fractions and Decimals

and

Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School

Check out the new edition here:  http://heinemann.com/ChildrensMath

The Penny Cube

I just finished a 5th grade 3-Act task called Penny Cube that I created last spring.  I tried it then, but just to get some feedback from students and see what I might need to change about how the task should be presented.  Now, after completing this task with two groups of students (at two different points in the year), I’ve learned three things:

  1. Students see a video and notice a bunch of things that teachers don’t even realize are there.
  2. The curious questions students ask first are often “why” questions.
  3. There’s no way to predict everything a group of students might wonder.

I’ll take this reflection from the beginning.  First, I let students know that I was going to show them a video clip.  I also told them that I was going to ask them what they noticed when it was finished playing.  I gave students a chance to brainstorm ideas about what they could do to make sure they would be able to share what they noticed once the clip had finished playing.  Their ideas were amazing:

  • We could look for expressions (on faces – I found out later that this student was thinking about context.  A person’s facial expression can tell a lot)  Unfortunately, there were no facial expressions in this video.
  • Listen carefully (they might be able to hear something that might give them a clue about what was going on – these students were already expecting a problem situation!)
  • Stay focused on the clip.
  • Take notes.
  • Try to remember as much as you can.

I had never done this before, but after hearing their ideas, I will be using this again.

After showing the video clip for Act 1, I immediately had them talk about what they noticed with their groups.  Then, they were asked to share with the whole group.  Here is what they noticed:

Penny Cube Notices

What’s missing from this picture is the wonderful reasoning given for some of these.  For the last bullet, “container is open in the front,” the student told the class that it was open in front so the pennies could be placed in the container more easily (I never thought they’d see or think about that).  They even began to wonder a bit here – “it might be an expression or it might be counting.”  My favorite, though, is the estimation by the girl who said “it looks like 100 pennies in the stack ($1.00).”  This was particularly interesting to me because of what happened when they were asked to estimate for the focus question.

The wonders were typical from what I usually get from students new to 3-Act tasks, but I handled it a bit differently this time.  Here are their wonders (click here for a typed version of Penny Cube Notices&Wonders):

Penny Cube Wonders

In my limited (yet growing) experience with teaching using 3-Act tasks, I’ve noticed that the wonders are initially “why” questions (as stated in number 2 above).  I told the class that I noticed that the questions they were asking were mostly “why” questions.  I asked them what other words could be used to begin questions.  Rather than trying to steer students to a particular question, I decided to focus the students’ attention on the kinds of questions they were already asking, and guide them to other types of questions.  It didn’t take long!  Within about 5 minutes, students had gone from “why” questions to “how many . . .” and “how much . . .” questions which are much easier to answer mathematically.

The students were then asked to figure out what they needed to solve the problem.  From experience with this task, I knew that most students would want pennies, so I had some ready.  I didn’t give them the Coin Specifications sheet, because no one asked for it.  I did have it ready, just in case.  Every group asked for pennies and rulers.  I wasn’t sure how they would use them, but I was pleasantly surprised.

Here’s what they did:

How many pennies in 2 inches

How many pennies in 2 inches

How many pennies in an inch

How many pennies in an inch

How many pennies fit on a 6 inch edge of the base

How many pennies fit on a 6 inch edge of the base

How many pennies cover base

How many pennies cover base

How many stacked pennies in 2 inches?

How many stacked pennies in 2 inches?

 

 

 

 

 

 

 

 

 

 

 

The students all started in a place that made sense to them.  Some wanted to figure out how many in the stack, so they stacked pennies and quickly realized (as I did when filling the cube) that you can’t stack pennies very high before they start to wobble and fall.  So, they measured smaller stacks and used that info to solve the problem.  Others wanted to find number of pennies along an edge to find how many cover the base, then work on the stacks.  Students were thoroughly engaged.

After three 1 hour classes, students were wrapping up their solutions.  Some groups were still grappling with the number of pennies in a stack.  Others were finished.  A few were unsure about what to do with some of the numbers they generated.  All of this told the classroom teacher and me that there were some misconceptions out there that needed to be addressed.  Many of the misconceptions had to do with students disengaging from the context, rather than integrating their numbers into the context:

  • One group was unsure of whether to multiply  the number of pennies in a stack by 12 (6 inches + 6 inches) or to use 64.
  • Another group found the number of pennies to cover the base and multiplied it by itself to get their solution.
  • A third group found 37 pennies in 2 1/2 inches and was having a difficult time handling that information.
  • A fourth group had come up with two different solutions and both thought they were correct.  Only one could defend her solution.

Eventually, several groups arrived a solution that made sense to them.

Time to share!

I chose one group to share.  This group had a reasonable solution, but their method and numbers were different from many of the other groups, so this is where we were hoping for some light bulbs to begin to glow a bit.

This group shared their work:

20140919145046-3351215

 

 

 

 

 

 

 

 

 

 

I asked the class what they liked about the work.  The responses:

  • The math (computations) are written neatly and they’re easy to follow.
  • I know what their answer is because it has a bubble around it.
  • The question is on it.
  • It’s colorful.

All good.  Now, for the best part:

What questions do you have for this group?  The responses:

  • Where did you get 34?
  • What does the 102 mean?
  • How about the 64?

Any suggestions for this group to help them clarify their work to answer some of your questions?

  • Maybe they could label their numbers so we know what the numbers mean.
  • Maybe they could tell what the answer means too.  Like put it in a sentence so it says something like “6,528 pennies will fit in the container.”
  • Maybe they could have a diagram to show how they got a number like 64 or 34.  I know that would help me (this student had a diagram on his work and thought it was useful).

The light bulbs really started to glow as students began making suggestions.  As soon as a suggestion was made, students began to check their own work to see if it was on their work.  If it wasn’t, they added it.  All of the suggestions were written on the board so they could modify their work one final time.  The best part about this whole exchange was that students were suggesting to their peers to be more precise in their mathematics (SMP 6 – Attend to precision).  And, they really wanted to know what 34 was because they didn’t have that number on their boards (which is why I chose this group).

Now for the reveal!  When I asked the class if they wanted to know how many pennies were in the cube, they were surprised when I pulled up the reveal the video.  I guess they thought I’d just tell them (that’s so 1980’s).  They watched to see how close they were and when the total came up on the screen, many cheered because they were so close!

The students in this class were engaged in multiple content standards over the course of 3 days.  They reasoned, critiqued, made sense, and persevered.  It’s almost difficult to believe that this class was a “remedial” class!

Below, I’ve included a picture of each group’s final work.

20140916143126-1706895 20140916143218-1880868 20140916144859-1652302 20140916143241-1966871 20140916143157-1917874

 

 

 

 

 

 

 

Finally, one of the conversations witnessed in a group was between a girl and a boy and should have been caught on video, but wasn’t.  This group had an incorrect solution, but they were convinced they were correct, so to keep them thinking about the problem, I asked them how many dollars would be equal to the number of pennies in their answer (3,616).

  • Girl:  There are 100 pennies in a dollar.  So 600 pennies is . . .
  • Boy in group: $6.00
  • Girl (after a long pause):  1,000 pennies equals $10.00
  • Boy:  So that’s . . . um. . .
  • Me:  How does knowing 1,000 pennies = $10.00 help you.
  • Girl:  We have 3,000 pennies, so that’s $30.00.
  • Boy:  $36.00
  • Me:  Share with your group how you know it’s $36.00
  • Boy:  Because $30.00 and $6.00 is $36.00
  • Girl:  And the rest (16) are cents.  $36.16!

And they didn’t even need a calculator!

Math really does make sense!

How Old is The Shepherd? Revisited in 3rd Grade

Over the past few weeks, I’ve shown the How old is the shepherd? problem to both of my kids and then shown them the video  from Robert Kaplinsky’s blog. Both were shocked at how many students don’t pay attention to what is happening in the problem.  Connor even said, “I guess I’m not one of the 24.”

Here is my son, Connor, with his response to the problem:

Unfortunately, his first statement, “That’s stupid!” was not caught on video!

My daughter, Lura, with her response:

Last Saturday, after ambushing one of my daughter’s friends with the problem while she was visiting, Kim (my wife) became more curious about the problem, so I showed the video to her and shared some of the data on Kaplinsky’s blog. She was also shocked at the results.  We had a brief conversation that went something like this:

Me: This is why we need to teach math content through patient problem solving and sense making!

Kim: Ok. (with a look that says, I know you’re passionate about this, and that it’s important. We’ll talk later. Go make a 3-act video and post it to your blog.)

Me: Ok.

It was left alone until this morning. It’s just me, but I like to think we would’ve talked sooner if I hadn’t been fighting a cold. She texted me and asked me to send her the Shepherd problem. I did, but only with the requirement that she share what she does with it.

Kim (and her co-teacher) gave the problem to each of their students and I just received the results:

  • 3 out of 19 students made sense of the problem (15.8%)
  • One student added 125 five times.
  • One student reasoned that by the time you had 5 dogs and 125 sheep, you have to be in your fifties.
  • One student divided 125 by 5.
  • 6 students added 125 and 5 to get 130.
  • 3 computed an operation with the two numbers incorrectly
  • The other students guessed or showed no reasoning.

Now the good stuff:

  • One student (an autistic child) shared his reasoning about the problem with his classmates:

“The shepherd has no-o-o-othing (said as a sheep might say it) to do with the sheep and the dogs.”

  • Both teachers lost it!

Take aways from this:

  • It’s best that we start teaching math content through problem solving early and consistently K-12 and beyond.
  • Making sense of mathematics needs to be a priority for all students. (SMP 1)
  • All students bring something of value to a classroom.
  • Stories like the student who shared his reasoning sometimes get us through days that are not so much like this.

Below, you will find some of the students’ reasoning.

How old is the shepherd_