Tag / math

by Beyond Traditional Math

Small Steps for Differentiation: Tier It!

I’m still searching for ways to differentiate in small ways that take just a second or two. An activity that is tiered is something that is leveled differently. A true tiered activity means that there are two (or sometimes more) options that account for a different level of thinking.  Not everything can be tiered, but some basic math skills can be tiered quite easily. Here is an example.

I saw this post on Pinterest the other day for a primary classroom. So easy and so creative!

Screen Shot 2015-03-10 at 7.47.12 PM

This is an awesome activity for students that are just starting out with numbers and subtilizing.  But what about that small group of kids that the K-1 teacher doesn’t have time to differentiate for? Well, I think that the answer is all about having the right materials, in this case more advanced dominoes. I pulled this together for my kindergartener at home and we had a blast doing it.  (Hello 20 minutes of math play!)

differentiation-tips-easy

I think if we systematically think about what the next “level” of some of those basic math skills are, we can slowly incorporate the correct materials into our centers, our assignments and our games. In this way we ensure all students are making growth!

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by Beyond Traditional Math

Math is a Learning Subject: More Small Steps for Differentiation

My favorite thing about math is that is a messy thinking subject. It is a learning subject. It should be messy and full of questions. We need to teach kids that it can be glorious when it suddenly is no longer messy and the patterns and the discoveries are right in front of our faces!

We have to model this for students, and more importantly we need to give them opportunities to make math a learning subject. So often we want to give all the answers, and tell them all the patterns, and show them how magical it is, that they lose their passion for discovering math at an early age. They begin thinking that math is a performance subject…teacher asks the question, student gives the answer…25 times in a row…on a worksheet.

Instead we need to give students meaningful explorations that can often run in the background of the school day.  These can often be very simple, and they really allow for differentiation. Some students will take these explorations much further than others.

Here is a third grade example:

math-is-a-learning-subject

The keys to making this work are:

  1. Give enough time for the exploration. This one will be 2 weeks.
  2. DO NOT, and I really mean this, DO NOT give them the answers. (This is very difficult, I know.)
  3. Tell them to work with each other! Isn’t that how we learn best? The second we want to know something we email, text or call someone. Let them teach each other.
  4. Make them research it, prove it and let them feel some confusion. This teaches perseverance and also that math is truly a learning subject. Bring in iPads, computers or have them look it up at home. (Hint: Use school tube when searching! Great resource!)
  5. Be sure that they understand that the most important part is not the answer they give you, but rather the method they use to solve it and WHY IT WORKS. That is the number one most important thing that they can get out of this inquiry activity.

Will all of the students be able to do this? Possibly…their level of understanding will vary from student to student. But in the end, when you bring them all together let the students do the talking. They will get there, if not now…they will have some prior knowledge for 4th grade.

by Beyond Traditional Math

Differentiate in Small Steps: Give Them Two Problems

Differentiation is difficult. There is no doubt about it. I’ve been on a mission to find small ways to differentiate without stressing myself out, and without stressing out the teachers I work with.

I often found myself realizing that I was giving one math problem to the whole class when I’d look at my gifted kids faces. You know that look on their face? That boredom in their eyes look…where they’d rather be someplace else than sit and do another problem that they already know how to do. That is what inspired my idea of just putting two problems up, a meets the target (happy) problem and an exceeds the target (stressful) problem. I always explain to the students that if you are able to complete the happy problem correctly, you are meeting the target. It is even more impressive if you can do the stressful problem, but it’s not necessary.

Here is an example. Today in a second grade classroom the learning target was adding 2 digit numbers mentally (without regrouping).  I put up two problems, the happy one was a check for me to see who had it.  (They worked in their notebooks but it’s also great to use dry erase boards.)  The stressful problem is the one that students who need to stretch their thinking just a bit might try after doing the happy problem.

Using two faces makes it both visual and fun for students. Cut them out and reuse them over and over!

Using two faces makes it both visual and fun for students. Cut them out and reuse them over and over!

I keep those little face headings handy, they go up on dry erase boards, chalkboards, and easels…wherever we are doing math.  If I forget to put two problems, the students definitely remind me. They love to see the stressful problem face, especially the first time when I draw it in front of them.  You can use this method during quick checks, problem solving, mini lessons, practice, mental math…the possibilities are limitless.  This is something that can be done quickly, and doesn’t require hours and hours of work.

 

by Beyond Traditional Math

Student Talk Leads to Deeper Thinking

I witnessed this cool thing the other day, the thing that I keep on blogging about because I keep on seeing it over and over. I was in a second grade classroom where students were adding two digit numbers.  The lesson was to add the ones first, then the tens by decomposing numbers. The well meaning adult in the room (me) kept on teaching it according to the lesson.  Students were making mistakes and errors like crazy.  Then, I gave them the freedom to try whatever way they pleased.

I was astounded by their thinking, they came up to share one by one with different strategies that made WAY more sense to each other than what I was preaching. It was pretty amazing to see what they were coming up with. Not only did their ways make sense, but they were also accurate. Another reminder that I need to SHUT UP!

student-talk-in-math-class

This student made tens, and then added the ones and tens in the order that made sense to him.

 

So let them talk! The deep thinking and learning that will come from it will be amazing.

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?

helping-students-persevere2

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

by Beyond Traditional Math

What Exactly IS Subitizing?

For a few years, I saw the word “subitizing” and I had no idea what it meant. It is one of those things where you don’t know the word very well, but once it enters your consciousness, you start to see it more and more.

Last summer I learned what subitizing is (I pronounce it soob-i-tizing, some people say sub-i-tizing), and how important it is for K-2 students to be able to do. Quite simply, subitizing is the ability to count a collection of items without having to count each item individually. For example, when you roll a dice you automatically know the number without having to count each dot individually.

Subitizing is important because it means that children can hold an image in their minds. They are able to group collections effectively so that they no longer need to count by ones. We want them to do this with regular patterns and irregular patterns. You know that children are subitizing if they can tell you the number in the collection when it is flashed at them quickly.

Here is an example of an irregular pattern, and what we want the student to think in their minds:

subitizing-irregular-patterns

We want students to group items together to count efficiently.

There are several videos (if you google it) that people have created to help children practice this skill.  Here is just one (warning: it flashes fast!):

How do you have students practice this skill in the classroom?

by Beyond Traditional Math

1 and 2 is NOT 12!

place-value-misconceptions

I know you see it all the time, when you ask a student what a number is made of, and they instantly throw place value out the window. That is why so many of us are doing this “Squashing Misconceptions” blog hop about place value.

I was working with a second grader that was struggling to add some two digit numbers (12 and 24).  I was thinking that if we could split the 12 she could add a ten and 2 more to 24.  So I asked her: “What two numbers go together to make 12?”

“1 and 2!”

Now you might be thinking that I tricked her with the question or that it was a matter of not understanding what I was asking, so I posed the question in a few different ways (“What is 12 made of? What two numbers can you add to get 12?”).  Each time, she maintained that 12 is 1 and 2 put together.

So, what do you do when there is a struggle with place value? Get out the MATERIALS.  ALWAYS. Don’t waste time or think that this will make it more difficult or confuse them. Just get out manipulatives, counters or cubes.  I love cubes because you can group and ungroup them at any time.

I gave her some unifix cubes and told her to prove to me that 1 and 2 make twelve. The second she pulled out those cubes her face lit up.  “That’s only 3!”

So I asked her again what makes 12.  Do you know what that little cutie did? She pulled out a ten and 2 cubes without even having to count the ten.  She was so used to seeing those concrete tools that it was a no brainer.  We had to have those materials out to make it concrete, she was not ready for bare numbers yet.

place-value-misconceptions

We constantly make those leaps too soon, and then time and time again (myself included!) it doesn’t occur to us to pull out tools, manipulatives or materials.  Our young learners need this, as they are very concrete and it can help them make the connection to number sentences.

So let them have those materials! Have them out for ALL students, so that no student ever feels “babyish” having to use them.  This is essential for students to understand place value conceptually.

Want to read some more about misconceptions in place value? Check out the next stop in the blog hop:

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!

 

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.