Category / problem solving

by Beyond Traditional Math

What Does Success Look Like?

Pretend with me for a moment, that you have never seen an apple before in your life. Now pretend that someone has asked you to peel it, but you’ve also never peeled anything before in your life.  How do you know what to do to be successful? How will you know when you have been successful?

That in a nutshell is what “success criteria” is.  It’s all about letting students know what success looks like, and how they will know that they have met the learning target.

In my mission to examine learning targets and communicate them, I learned that it wasn’t enough to simply display them for students. It STILL wasn’t enough for students to write them in their math journals.  My students needed to see the learning target, write it, and then have some sort of interaction with it.  This is where I combine this idea of success criteria (Hattie 2012) with Marzano’s Levels of Understanding.

Here is an example of what this can look like.  You begin your content lesson by reading the learning target out loud, allowing students time to write it down in a math journal (or some other place to take notes). Notice the learning target starts with “I can”.  (It would be just like saying, I can peel an apple.)

I can read and write numbers up to 1,000.

Now some students will have prior knowledge about the learning target, so allowing them a moment to interact and think about this learning target is essential.  You can show them (or you can do this orally) what the different levels of understanding look like. Here is what it could be for this particular learning target:

Using success criteria in the classroom can help students understand the learning they are supposed to do.

Using success criteria in the classroom can help students understand the outcome of their learning.

I use Marzano’s Levels of Understanding to anchor my thinking (which I blew up and posted on the wall-this is a free resource by the way!) because the students connect easily to the language.  After I show them what each level looks like, I have the students rate themselves on the current target. Their goal is always the same every day, get to the next level, get higher and get better.

After this mini intro, I teach the lesson. We practice with tiered examples so that everyone is challenged, we talk it through with each other, we help each other come to an understanding.  We break into independent practice work where I can catch the students who still feel like a 1 or 2.  Then we close the lesson with an exit slip or an assignment, rating ourselves once again to see where we fall on the scale.  I take a look at what they wrote for their final rating and catch those students during the review, intervention block or some recess time the next day.

This seems like a lot of work, and I won’t lie that at first it was for me. It was a different way of thinking.  But soon after I started to do this, I noticed that it was easier and easier to think about what a 0-4 looks like.  If I ever skipped the rating part, my students would actually shout at me “What does a 3 look like?!” They wanted to know what it would take to be successful! It was very powerful.  You may not have time to write it out like this for every lesson, but you can do it orally while referring to the levels on the wall.

This tweak to my instruction was a total game changer.  Thank you John Hattie and Robert Marzano for your inspiration!

 

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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

You are NOT Bad at Math

Did you know that there is not a “math gene” that makes us good at math? If you haven’t read this yet: Why Do Americans Stink at Math? I would highly suggest taking the time to read.  The long and short of it, is that during the industrial revolution, we took the fast track and tried to teach math the fastest way possible.  We taught shortcuts instead of conceptual understanding.  This method of teaching allowed our students to go directly from their education into a factory job, but it did a great disservice to a generation.  This is why people believe that they are “bad at math”.

The world has changed and we can no longer do this to our students, but according to the article above we still are.  Despite many attempts at changing our math practices, we still find the majority of U.S. teachers using traditional methods. Of students attending 2 year colleges, 60% of them are placed in remedial math classes, and only 25% of those students pass those classes! (Silva & White, 2013)

I would strongly recommend that you watch this video by Jo Boaler, of Stanford University. It is 20 minutes long, so if you can’t watch it all, try the first 8 minutes.  In it she talks about how we need to make math a learning subject (exploratory, messy, open ended and challenging), not a performance based subject (math is only about answering questions correctly).

To return math to being a learning subject, we can use rich open ended tasks, inquiry activities, real world projects and problems that encourage math talk and discourse!  Please check out my free Reasoning Puzzle Set to try out an activity that will really get your students thinking and talking and most importantly, learning at high levels.

Reasoning Puzzles to promote student to student math talk.

Reasoning Puzzles to promote student to student math talk.

These are most appropriate for 3 and 4th graders, but even could be beneficial for fifth graders that are not used to thinking this way!  If you end up using them, I’d love to hear how it goes.

by Beyond Traditional Math

Sometimes They Explain It Better Than You

As much as my ego doesn’t like to admit it, often times my student’s explanations of math concepts and processes is WAY more effective than mine.  In the past I would actually make myself a little crazy as I tried to think up 15 different ways to say the same thing to teach a tough concept. Then, one tiny little 8 year old would trot up to the board, say it perfectly, and the light bulbs would go on.

During a response day, when I noticed that students were struggling a bit with perimeter, I made tiered/leveled posters around the room of perimeter tasks.  Leveled posters allow for differentiation, and they are also formative so that you can see who is meeting your learning target vs. who is not.

For example, this learning target was: I can calculate perimeter when a side is missing.

  • Level 2 (basic): all the measurements were labeled on all of the sides of the figure.
  • Level 3 (proficient): one side of the figure had a missing measurement.
  • Level 4 (advanced): more than one side was missing their measurements.
Perimeter Task

Level 3 Poster: The students had to add the two sides that were known, to find the unknown side. (The two in the corner signifies that this is the second problem as they numbered their papers.)

Their goal was to try to get through them all. The problems were numbered, so they simply numbered their half sheet of paper and tried each one.  I tell them ahead of time what it means when they successfully answer each problem.  They knew the levels and consequently tried hard to get all three. (No one got the Level 4 problem, very close, but not quite!)

Perimeter Task

Level 4 Poster: This one perplexed even my accelerated gifted and talented student. He missed the insides of the figure.

My favorite part is the math talk session where they discuss their answers with each other. We move from poster to poster, and students speak up and share their thinking. I don’t have to say a word.  You hear a lot of shouts of understanding and mistake finding as they explain.  Again, it is okay for me to take a backseat. I don’t have to blabber my mouth (for double the time) when very quickly they speak to each other.  When everyone is stuck, that is when I step in.

When students do the talking, they do the learning!

 

by Beyond Traditional Math

Classifying Shapes Using Tangrams

Since our geometry unit is so full of vocabulary, I am always looking for ways to have students apply the vocabulary to new situations. After learning how to make our own tangrams yesterday, we took it to another level today. We decided to sort and classify the shapes that make up the tangrams. Because we’ve spent a few days describing two types of polygons: triangles and quadrilaterals, we had some really nice anchor charts up on the wall.

Their task today was to classify and describe the shapes from their tangrams using sides and angles. This meant that some of their shapes could be described in more than one way. This was a really cool opportunity for students to participate in math talk, and it was even better to see them using the anchor charts we made together:

Classifying Triangles

The student on the right is trying to classify his triangle by looking at angles.  The student on the left quickly realized that he was holding a shape with four sides.

Classifying Quadrilaterals

This student is checking to see if all four of her sides are equal, as well as if there are four right angles in her shape.

There is nothing better in a math class than hearing students argue over whether something is a right angle or not. Hearing words being used authentically was really cool, especially since they could actually touch and manipulate the shapes! I’ve really taken geometry a long way since I started teaching (ahem…I used to use worksheets only…please don’t judge me!), and the terms and vocabulary has started to stick because of it.

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

Fraction Tile Strategy: Hide the Labels

I used to tell students WAY too much when introducing new concepts in math, and I neglected to let them discover things on their own.  When students are told what to think, I believe that it limits understanding. I used to stand up in front of my students and talk about fractions by telling them which ones were halves, thirds, quarters, etc. It is embarrassing to admit that I would teach that way, but now at least I can share what I’ve replaced that method with…lots of inquiry based hands on learning.

Today the learning target was: I can use benchmark fractions to estimate. After spending lots of days in a row with tangible real world concreteness, it was a big leap to move to representational models of fractions. We pulled out the fraction tiles (basically harder versions of fraction strips) that came with our new math series.

Fraction Tile Strategy

You’ve seen these, you may even have them, If not there are likely a dozen free ones out there that you can print and cut. Just try a web search for “fraction strips” or “fraction tiles”.

After we took them apart, I had them flip over the entire thing, putting the red whole number 1 at the top.  There were a LOT of confused faces, and even one student said “how will we know what they all are?” I asked them to trust me and luckily they do, so they flipped them over. Then, I asked the same question that the student had just asked. How do you know which one is which? Which one is 1/2, 1/4 or 1/3? A few students right away started to lay them on top of the red whole piece like this:

Fraction Tile Strategy

Flip over the tiles so that the labels are NOT showing.

After a little buzzing with their partner, they all pointed to the pink one immediately for 1/2.  It took them only a few more minutes to identify the 1/3 and 1/4 sections.  When we talked about how they discovered this, there were a couple of cool things they noticed:

  1. “I just counted, if there were two of one color, I knew it was two halves! If there was three, I knew it was thirds.”
  2. “I lined them all up below the whole tile to see how many could “fit” in one whole. Sometimes I didn’t need to fill up the whole thing, because I could just tell.” (Boom! Learning target met!)
  3. “I counted them and stacked them, as I they got smaller, they got taller.” (I really like the size thinking there.)
Fraction Tile Strategy

These students counted and stacked each one to figure out the fraction piece label. Then they noticed the sizes of their fraction tile stacks began to grow!

Giving them time to explore has made fractions way less intimidating. The models came much easier to them when they didn’t have confusing labels staring them in the face.  I am definitely a fan of using fraction tiles/strips with the labels hiding.  What an awesome way to get them to estimate using benchmark fractions.

Tomorrow, we’ll explore equivalent fractions with these same strips, and try to connect it to the real world.  I can’t wait to see their thinking!

by Beyond Traditional Math

3 Hands On Earth Day Activities that Integrate Math

Here are three really amazing Earth Day activities for your elementary classroom that all include math. I’ve done them all and they’ve been memorable, educational, and fun! The best part is they always lead to deep moral and ethical conversations.

1.  Hold a Trash Free Lunch Picnic:  This is a two day project.  The first day, you ask the students to keep track of how many pieces of trash they have as they eat lunch.  For the second day, you send home a note asking parents to pack a trash free lunch, (as trash free as possible) to see if you can cut down on the amount of trash.  On the day of the trash free lunch, you ask the students to count how many pieces.

Earth Day Activity Trash Free Lunch

Here is an example of a student’s trash free lunch.

 

We kept track on a tally chart and realized the impact we can have if we change one simple thing, how we pack our lunches!  Then, the students draw a bar graph or a pie chart to show the results of the tally chart.

Earth Day Activity: Trash Free Lunch

Record the data of a trashy lunch vs. a trash free lunch!

2.  Build a Solar Oven and Bake S’Mores:  Show a tutorial for how to make a solar oven a few days before Earth Day.  Here is one that could be made from a pizza box!

Tell the students to bring their own supplies from home (cardboard boxes, plastic wrap, aluminum foil, tape) and give them time to make them when they first get to school.  It took our class about 2 hours. The math involved is awesome, measurement, measurement and more measurement! I brought the graham crackers, chocolate, and marshmallows and they melted like crazy in the sun. It was super fun! Here are a few photos.

Earth Day Activity: Build a Solar Oven

Students bring in their own materials, but you may want to have some extras on hand.

Earth Day Activity: Build a Solar Oven

This student even put a skewer in the middle of the oven, too bad they decided not to use it in the end.

Earth Day Activity: Build a Solar Oven

Some students learned the hard way that you need to CLOSE the solar oven!

3.  Study an Important Environmental Issue and Act on it: Perhaps the best Earth Day activity we’ve done is something that felt meaningful, like we could make change happen!  We studied the Great Pacific Garbage Patch by watching videos, reading about it, doing some math problems surrounding conservation, and by writing persuasive letters.  We ended the project by doing a water pollution science experiment.

Here is a video about the Great Pacific Garbage Patch if you haven’t heard about it (Depending on the age of your students, you may be able to show it to your class.):

Here are some photos of us trying to “clean” water, so students could find out how truly difficult it was.  As they work, each tool they borrow from me costs them money. They have to keep track of the cost of their clean up.

Earth Day Activity Water Pollution Experiment

Students are trying to clean out a polluted basin of water using different tools (all of which cost different amounts of money). They keep track of their successes and the cost of cleaning out their basin of water.

Earth Day Activity Water Pollution Experiment

A student is trying to remove vegetable oil from their basin of polluted water. NOT easy!

The bottom line, is there are so many things that students can do to learn about alternative energy, and to study current environmental issues. Instead of encouraging them to recycle with a coloring sheet or a worksheet, engaging them in these issues will help them feel an authentic push to do it!

I love Earth Day and the awareness it brings to young people! What kinds of things do you do with your class on Earth Day? Share below in the comments. 🙂

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

Things That Come in Halves and Quarters: The Real World Connection

Whenever I have connected math to the real world, I’ve seen a boost in achievement in my classroom. Fractions are REALLY important concepts that must be connected to student’s lives. When I first started teaching, I would just plod along in the book. I would hand out worksheets with rectangle boxes that students would just fill in.  They’d write the numbers without really connecting it to much of anything. It was kind of a disaster!

Now, I love to collect real world examples, and put them on an anchor chart.  Because fractions are so abstract, we put this anchor chart together after brainstorming with a partner first:

IMG_8500

 

On the student planner that day, I put an assignment to look for things that come in halves or quarters at home as well. We can always add more! Now, each time we talk about a fraction, we try to picture something from this list.

I am hoping that these concrete examples will really help them understand when I move them into representational symbols and numbers.