Tag / problem solving

by Beyond Traditional Math

How to Stop Meaningless Problem Solving

Today while working with some third graders in my intervention group, I had that moment that I know you all have as a teacher. When you pose a story to be solved (I’m tired of calling them problems…is it ok to call them stories to be solved?), and the students just start panic crunching numbers. Like throwing numbers around and adding up every number they hear in the story.

Today it was simple story, one that I know these three could probably picture in their minds, or at the very least draw:

“There are two trees, each one has 15 cherries. How many cherries are there altogether?”

I’ve been reading about the benefits of using vertical white board surfaces, to get students to do more thinking. One student wrote 2 + 15 = 17 with her partner. She was CONVINCED (and there was nothing I could say to change her mind), that because there were two trees and 15 cherries, those two numbers must be added together. The other student knew to add 15 and 15, but used an inefficient strategy of drawing each individual cherry on the two trees. I will address the inefficient strategy issue in another post, but today I really want to address the idea of students just crunching numbers without thinking.

I think for too long, students who struggle with math have been led down this road of not have the proper exposure to tools, drawings and then the abstract notation of mathematics. I see tools disappearing as early as first grade in classrooms, which we all KNOW are critical in child development. Not only are tools disappearing, but I see math programs that encourage abstract notation before connecting tools and drawings to that notation.

Take for example, 3 x 14 or 14 x 3. This student was able to build it with tiles (showing an understanding of area), then was able to take a shortcut and show an area model (without the tiles inside), and eventually use partial product notation to solve.

Somewhere along the line, students who struggle to problem solve missed these connections. They look at you with panic in their eyes as they try to figure out what you are asking and what they have to do. Then in pure desperation instead of solving, they just start mindlessly crunching numbers, not making any connections whatsoever.

Here are some ways that I intervene:

  1. Make tools available, have them act it out first, then draw what they made. Don’t worry at this point if the notation is correct, can they make sense of the problem with tools? At first it might be helpful to provide the tool, then the next time have a few different ones to choose from. Start watching if they are choosing an appropriate tool.
  2. Write a hint in a different color. Explain that the hint will lead them in a different direction, and that they need to “unthink” some part of what they did. In the case of the cherry trees, I drew an additional tree next to the one they had already drawn and wrote 15 under it. That little hint made all the difference.
  3. Look for places they are actually making connections and compliment them on that. The student who wrote 2+15 was actually thinking about multiplication! I told her that, but then asked the two of them to back up and think about what that would mean if they multiplied those two numbers. Find SOMETHING that you like about what they did. Those compliments stick, and they remember it again next time and will use it.
  4. Don’t give up on them, and have high expectations. I hear this said over and over again, “My ‘low’ students have a really hard time problem solving.” When you call them low, you will expect nothing but low performance.
  5. Try not to give the answer at the end. Explain that problem solving is about thinking, not the answer. This will enrage them at first, but eventually they will start thinking instead of spitting out an answer. They will also see that they need to persevere while solving instead of waiting for the teacher to give the strategy and answer in the end.
  6. Have them compare two strategies next to each other. Have the students look for similarities and difference.
  7. Choose random partners whenever possible. This allows students to learn different strategies from others, and it keeps your groups fresh.

Students who struggle in mathematics can be excellent problem solvers. It’s so critical that we believe this, and that we continue to challenge them at high levels with the right supports.

by Beyond Traditional Math

An Approach that Works for Struggling Learners EVERY Time

I have been reading about the Concrete-Representational-Abstract Sequence of Instruction for some time now, especially since I began working with our most struggling math students at our school.

I’m hooked and am a firm believer in this approach!

I know you know that moment… where you find students looking at you with the deer in headlights look.  In my intervention groups, I see it several times in 30 minutes! I was desperately searching for more ways to make math meaningful for them when I discovered this approach. And, I will tell you, it works EVERY time. I mean, EVERY SINGLE TIME. There has not been one single concept that I haven’t been able to master with a child when I used this approach.

If you don’t have time to read the article, the approach is summed up quite simply in three steps:

  1. When a student is introduced to a new concept or something unfamiliar, you allow the use of tools. (Concrete)
  2. When the student can perform the task, they move on to representing the concept with drawings or pictures. (Representational)
  3. When the student can master the task with a drawing or a picture they move to using only numbers and symbols. (Abstract)

    * Note it is important to keep all three of these ways visible to promote strong connections and deep conceptual understanding.

I realized that this could be even MORE powerful when students could self assess where they are in this approach. I made this poster with them and we refer to it constantly.

Concrete-Representational-Abstract-Approach-Instruction

They are constantly checking “where their brains are at” when they are struggling through a problem.  When the numbers and symbols don’t make sense, they actually back themselves up to drawings. If that still doesn’t make sense they back up and use concrete tools.

It has been simply amazing, and you must try it!

by Beyond Traditional Math

Help! What’s My Error?

Every so often I have one of those moments where I want to stop the entire class to show everyone something a student did. Today was one of those moments when I watched how a student tackled a word problem. He had asked me for help and I was guiding him through it.  He was doing some mental math, realized that something didn’t seem right and checked his thinking. When he realized that he was 10 away from the target number, he very quickly realized that he could adjust his thinking and his answer. (You can see he changed 44 to 34.)

helping-students-persevere

This is perseverance and precision!  This is what we are constantly hoping that kids will do without us having to remind them. The problem is we are running around asking students to do this on an 1-on-1 individual conference. Imagine how powerful it would be if students shared examples like these and learned from them, how much more time would be free up in our classrooms to really dig deep with kids!

Here are some simple ways to share:

1.  Stop the entire class and have the student show their error and how they fixed it.

2.  Build in share time at the end of your lesson for students to tell a story of how they found and fixed an error in their thinking.

3.  Here is my favorite idea…make a “What’s my Error?” chart!  This is a simple chart where students (while they are working on an assignment) could put up problems that they are stuck on.  We’ve all been there before where we keep on getting the same answer, but we know that something isn’t right.  Other students during a share time could help figure out the error and write their thinking on the chart.  So often adults turn to others for help when we need it (for technology, for many things), but often in math class we leave students to figure out these things alone.  A “What’s My Error?” chart could help students explain their thinking AND help them to be more interested in finding the error in their ways in the future.  Like all things, you have to manage it by making it a routine and having general expectations (imagine the students fighting over the markers, crowding around the chart), but isn’t that a good problem to have?

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Let me know if you try it I’d love to hear how it goes!

by Beyond Traditional Math

A Math Book to Change Your Teaching

blog banner august hop

This is a funny title for a post because I have to say that this math book hasn’t “changed” my teaching exactly, but it has opened my eyes to how we can simplify our teaching.

I’m reading Building Mathematical Comprehension: Using Literacy Strategies to Make Meaning by Laney Sammons. I chose to read this book because with all the demands in our teaching profession, I was seeking a way to simplify things.  We cannot get everything done in a day, a week or even in a school year, so I keep thinking that there has to be a way to integrate things so that we aren’t going crazy every day.

Now literacy and math are not the same. But they have similarities, check this out:

Similarities-between-math-and-reading-strategies

 

The entire book focuses on ways to use literacy strategies in math.  And when you read it, you’ll be thinking “Oh my gosh! This just makes sense!”

One thing I’m going to try this year as a math coach, is to model some problem solving lessons in classrooms. In this book she talks about using comprehension strategies before, during and after solving a problem. Since math truly is all about problem solving, I’m envisioning an anchor chart adapted from my reading (pg. 35-38). I think as we work through problems, it may help students get “unstuck”.  It’ll look something like this:

Problem-solving-strategies

I say that it’ll look “something” like this, because I want to come up with the anchor chart WITH the students. I think doing a problem as a whole class (and using a think-aloud strategy) would help them see the kind of thinking that should be going on in your mind while problem solving.

This book has even more literacy strategies that you can use in your math classroom.  Using some of the same vocabulary in your math and literacy blocks can help students make great connections!

To win a copy of this book, you will have to enter by clicking the link below.

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There are a bunch of us talking about great books that have changed our teaching, so don’t forget to check out Mr. Elementary Math’s book, he is next in this blog hop.

 

by Beyond Traditional Math

Build A Strong Math Culture With The Standards For Math Practice

With all of the current criticism about the CCSS and “Common Core Math” (I put that in quotes because that phrase has been driving me crazy, now that is another blog post in my mind), I’ve been happy to see that the Standards for Mathematical Practice have been left alone.

I’m glad that they’ve been left alone in the criticism because the Math Practice Standards are all encompassing thinking habits, more than they are standards to be met.  They encourage us to teach mathematics more as a learning subject than a performance based subject. Math absolutely should require lots of messy critical thinking, deduction, discussion and reasoning.

Someone last year said/asked me, “I just don’t understand what these standards are for.  What are they?” I explained them the best way I could, since I had just spent a month researching them to know them better:

  1. The math practice standards are a set of math habits, ways in which we should think about math. (That sounds so simple, but the way the standards are worded, it has driven many of us crazy while trying to understand them.  Reason abstractly and quantitatively? Huh?  It sounds like a completely different language!) The standards are all about developing positive habits and attitudes about math.
  2. They allow students to explore math as a learning subject.  They begin to understand that math is not about the teaching asking a question, and the student must answer it correctly. Most importantly they begin to see the connection to their lives.  Math connects so beautifully to real life, but because the U.S. has such a worksheet culture, we’ve lost that connection.
  3. Math from K-12 has the same underlying theme with these standards. As the years tick by the content standards become more complex, but the practice standards remain the same.  With the Standards for Mathematical Practice, we can develop a very positive culture surrounding mathematics.  A culture of persevering when encountering a problem, making sense of the world with math, using prior knowledge to solve new problems, being precise and reflective, patterning to find faster ways of working, explaining our thinking, understanding others thinking, knowing what tools will best help to solve problems, and connecting the world with abstract numbers and symbols. This all makes us excellent THINKERS.

We’ve had math coaches, administrators and other teachers pass out posters to put up in our classrooms. We’ve seen freebies and posters that we are meant to download and print. We’ve put them up on our walls with very few of us digging in to what they actually mean. I WAS one of those people. I had a poster of the kid friendly standards up for two years, and it wasn’t until last year that I realized one of the standards was completely inaccurate on the poster.  I had never bothered to check, and I assumed that the source knew the standards.  Can you blame me? I didn’t have the TIME to dissect what each one means.  It felt like another thing…another plate to spin…another added responsibility. I truly didn’t understand the importance of the standards to create a culture.

When I decided to figure out what they really mean, introduce them to my students from the start of the school year, work through the problems with them, and embed the language in the classroom, the culture really changed. We became mathematicians who could work through anything. It was remarkable. We put our work on the walls to help remind us that these were to be a part of our classroom daily.  We truly became vicious problem solvers, we worked together and math was about learning.  Math became FUN.

After a month of research I created posters, problems and activities to help myself understand them, but also to help teachers understand them, too.  (Feel free to check out the preview which walks you through the first standard.)

Build a culture by introducing, working with and revisiting the Standards for Math Practice.

Build a culture by introducing, working with and revisiting the Standards for Math Practice.

Even if the Common Core goes away (which it most certainly will, and already is in many states), I will always keep the Standards for Mathematical Practice.  It is a foundation in which we can all build upon, year after year!

 

by Beyond Traditional Math

Build Your Own Restaurant

Today was a totally delightful 20 minutes of math play. It ended up being almost 40 minutes as we created our play situation.

I have to admit that sometimes pretending can be exhausting.  Maybe as I’ve gotten older I’ve lost that spontaneous creativity. So I was happy to find a way that I could “pretend” a real life situation…ordering food at a restaurant. Today, it was almost lunch time and my 5 year old and I decided we were going to make our own restaurant…PB & J. We built the menu, with her telling me the items and prices.  It was fun to think of the different categories and to put together the menu. We grabbed a notepad, marker and some cash and we were ready to go. (Notice some of her choices, “Soda is not very healthy so we can make it very expensive.”)

Our menu, cash and the ordering pad for the server.

Our menu, cash and the ordering pad for the server.

Right away she wanted to order a Hawaiian Punch juice box, and an appetizer of crackers.  That was when I asked her how much money she had.  She counted her cash, “I have 8 dollars.”  I asked her if she had enough money to buy a lunch.  This was one of those moments where I wish she would think out loud, because she immediately changed her drink order for milk. I bet all kinds of good mathematical thinking was going on there! Now, if I know my daughter, it’s because she wanted enough money for dessert!

Here she is counting her money after ordering her milk and crackers, to be sure she would have enough.

Here she is counting her money after ordering her milk and crackers, to be sure she would have enough.

Sure enough, I took her order and it came out something like this:

real-life-math-playShe quickly realized that she was NOT going to have enough money for dessert, so she sprinted off to go and get her piggy bank, coming back with coins.  She asked me so innocently, “How many of these do I need for a dollar… one?” That was the perfect moment to tell her that a dollar is ten dimes, or four quarters…the perfect intro as to why she needs to know about coins and their values.

Which will lead to many more fun money play sessions!  We can use this same menu, but change up the ways to pay, the amounts and combinations of money.  All of which she will have a strong reason to want to know how to do it.

The best part? I told her that she needed to make sure to leave the server (me) a tip. As I left the room with her dirty plates, she secretly wrote this on a piece of paper and presented my “tip” to me when I came back:

real-life-math-play4

 

I’ll take that tip over 20% any day!

 

by Beyond Traditional Math

Crank It Up a Notch: Add Something They Can Touch When Problem Solving

One of my favorite ways to amp up problem solving is to throw something into the mix that they can touch.  This makes the project or problem so much more interesting to students in one instant.  We are working on the Design a Dream Bedroom project, so I picked up some free samples from the hardware store:

Hans On Real World Math

Give them stuff to touch when they are working on real world math activities.

Of course you can be sneaky about introducing the materials.  Before I even went over the problem during math, I spent the morning organizing the materials on a common table when they were arriving for the day. I got about a million questions, and hands were reaching out to touch the carpet and flooring samples before I could even get them in the bucket.

That is all I needed to do to get them interested in the problem.  After I read through the introduction with them during math problem solving time, the students literally leaped out of their carpet spots to run up and grab the problem from me.

That is what we want problem solving to be like…exciting, engaging, rigorous and motivating! Putting things in their hands to make it real world has worked every time.

by Beyond Traditional Math

Third Graders are Googling the Answers to Math Assignments?

My jaw literally dropped when I received this email the other day:

Assign Rigorous Problems

Let me explain two things to give you a little background:

First, I have a “contact me” section on this blog, for anyone who may need to get in touch. I received this email through that form.

Second, I recently put out a free resource called Doggy Dilemma for teachers.  It is an open ended problem that requires a lot of reading, writing and thinking.  There is no immediate answer, and all students would have a different answer in the end.

So…by the powers of observation and inferencing I can only conclude:

  1. This person who contacted me is a child that has received the assignment in class. (I am thinking this due to the lack of punctuation, capitals and misspellings. The “voice” of the writer seems very young, also.)
  2. This person is likely a 3-5th grader (since that is the target age group of the problem).
  3. This person is incredibly resourceful and bold. Not only does she google the problem, but she thinks to contact the author of the problem for an answer!

After I got over the shock of receiving this message I thought to myself, THIS is why I create what I create. No child should be able to google the answers to a great math problem.

What do you think about this?! I would love to know your thoughts.

 

by Beyond Traditional Math

Open Ended Math Problems Promote Reading, Writing AND Math

Last spring I had the opportunity to take a practice version of our new state assessment (the Smarter Balanced Assessment). In some states in the U.S. the PARCC is the new assessment which is similar in nature.

Talk about a jaw dropping, sweat on my forehead, instant anxiety through my whole body moment.

What the students are being asked to do is way more than a few math problems. They are expected to read, write and use appropriate grade level math in VERY complex ways. I realized that I needed to add some deep problem solving to my math instruction.  So I began to make open ended problem solving problems to introduce regularly into the classroom.

I decided to create Doggy Dilemma, a free problem for anyone to try out.  It is a highly motivating, real world problem in which students must read through information to decide what dog they must adopt. They draw a diagram of the dog pen, calculate the cost of the fencing, and write a letter to their parents explaining why they made the choices they did.

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My third graders have gone crazy over it.  They love it!  There are two full pages of reading involved which mimics the new assessments.  I have enjoyed creating it and want to make it available to anyone who teaches elementary math so that you can give your students the experience they need before the real assessments begin. You can get it by clicking on the picture below:

Open Ended Problem: Rigorous Problem Solving for Elementary Students

I’d love to hear how other teachers are encouraging this type of thinking in their classrooms. Please feel free to share in the comments!

I am happy to link up here:

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by Beyond Traditional Math

Concrete Learners: Hands On and Real Life…Every Day

In teaching division this year, I’ve never before used so many counters for so many consecutive days in a row.  I’ve got a core group of students who feel really great about division, some of them have even been memorizing multiplication at a very fast rate, which allows them to make better sense of division.  But I also have the exact opposite end of the spectrum as well.  As soon as they see that division symbol, their eyes glaze over and they become fearful of the problem. They worry about what to do and they think they cannot divide (even through they’ve been dividing all of their lives, they just haven’t seen the number sentence for it).

To help struggling learners, I’ve been trying to make it more concrete. Our youngest learners often need to see visual representations of numbers so that the concept is not so abstract.

In this set of problems, fruit was being divided equally into bags. I decided to lay out paper bags for this student.  The problem was 14 divided by 2, and he could easily solve it. He was both proud and excited to write his answer.  Win!

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Another student though, had a little more trouble. The problem was 8 divided by 4.  Because she had done 10 divided by 2 right before this one, she forgot to put away the 2 counters to start. She hit a wall very quickly and gave up. Instantly, she had her hand up for more help.

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To help her, we re-calibrated a bit by checking the problem again.  She very quickly realized that her counters started out wrong, and she was able to fix it. That check back to the problem is what I want her to do in the first place, great mathematicians do that without any prompting.  It was clear to see that she wasn’t connecting the number sentence with the manipulatives at all. It was a quick 1 minute conference on the importance of paying attention to detail/being precise, a math practice standard that many students struggle with. I told her that the next time she should try that strategy before asking for help.

Giving these quick teaching tips while conferencing with students makes WAY more sense when those tools are right there in front of them.  I used to try to draw on student’s papers, help them extend patterns etc… but the most struggling students need those tools in their hands to actually act out and see the problem. That is when I’ve noticed teaching tips given to them have become ultra powerful!