The Teacher Studio: Learning, Thinking, Creating
In my earlier post, I talked about the many factors that make fractions challenging for students.  If you missed it, you can read it by clicking RIGHT HERE.  Before my next few posts where I tackle some "in the trenches" ideas about fractions, I want to talk about an instructional strategy that is true in good math instruction across ALL topics.

### What is "Gradual Release" of Instruction?

So although I want to continue to address some of these foundation fraction concepts that can be so difficult, I want to stress something that is true in math instruction overall--not just for fractions.  When we want to utilize a gradual release of instruction model, we often think of the following:

I show the students.

We do it together.

They try it alone.

Now, I'm not going to lie.  I feel this is a very over-simplified model of what a true gradual release of instruction plan should involve.  A true gradual release is NOT linear; it is recursive and cycles around and around as we layer our instruction.  That's for another day!  But what I see happening oftentimes is that we teach, we practice, and then we assess--and the results of the assessment don't always make us so happy.

I'm going to propose a SECOND type of gradual release that is particularly pertinent in math instruction.  It looks a little something like this.
Now, what this arrow represents is that when we introduce a concept like fractions, it is absolutely critical that we provide our students with the opportunity to spend time--lots of time--in the world of the concrete.  Whether this means paper folding, using pre-made fraction manipulatives, base ten blocks, actual pans of brownies, paper pizzas--WHATEVER--we must immerse students in experiences where they see, feel, and move objects to make discoveries.

As we do this, it is important that we help students start to make connections between these "objects" and the real world.  This is where stories, real-world examples, and meaningful problem solving come in to play.

Here's an example.  A student with very little fractional understanding or experience can perhaps start to make connections when we present a problem like this.

"Sue walked into a bakery to buy some cookies for her family.  She noticed that each tray held 24 cookies.  If half of the cookies on one of the trays had sprinkles, how many would that be?  What if half of the cookies with sprinkles also had chocolate chips--how many would that be?"

The problem we have shared is essentially this:  "What is 1/4 of 24?", a problem that surfaces in most fraction units later on...and usually with a page full of similar problems.  By telling the story in context, helping students visualize ("Can you imagine what that tray of cookies might look like?"), perhaps even sketch or model it--this more advanced skill is very accessible to even the most beginning student.

Only after spending time building, folding, exploring, playing, and "seeing" how fractions make sense in the world should we move to these "bare number" tasks...tasks like:

3/4 + 4/4 = ?

5 x 3/8 = ?

Generate a list of 3 fractions equivalent to 4/5.

These are all the types of questions that we find rampant in our textbooks and teaching resources--and they DO have value.  The question is WHEN.  WHEN does it make sense to do these?  In my view, the correct timing is when the understanding is there and students need simply to work on the fluency and accuracy of the tasks.  Students do not learn best by doing tasks like this; they learn to get more fluent and adept at tasks like this if the foundations have been clearly set.

### I really thought they were understanding...

One thing I have found--especially for the more beginning levels of fractions, is that students may seem to be understanding.  They fill in the answers correctly on their practice sheets.  They can identify a given fraction. They may even be able to do the bare number tasks listed above...but do any of these show a TRUE understanding of fractions and their real-world application?  That takes us, as teachers, getting in trenches with them watching, listening, and asking questions.

Provide students with open-ended tasks and challenges that give them time to explore, discuss, and for YOU to observe.  If your math series is full of computation--that's fine!  Just WAIT to do that work until students have demonstrated readiness.  Unfortunately, many series don't give us enough of these experiences, which is what led to the unit I created to use with my own students and many of the other fraction resources in my store.  I simply didn't feel as though we spent enough time on the left side of the arrow.  If you want to see more, here are the links to a few resources.  See you soon for more fraction thinking!

Want to check out the fraction unit I use?
How about a bundled set of TEN fraction resources that support best practices?

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### More fractions? YES!

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If you have followed me or seen any of my webinars, you know that I wholeheartedly believe that all students can learn math at a high level--and we, as teachers, need to constantly strive to refine our teaching strategies and methods so that we reach ALL students...no matter their starting point.  This is especially true for fractions which can be one of the most challenging things we teach.  This post begins a series about fraction instruction that I hope you find helpful and meaningful.
To begin, I don't think it would be surprising if I told you that a majority of intermediate grade teachers declare that fractions are one of (or THE) most challenging concept for them to teach--and for their students to learn.  So why is this?  And more importantly--what can we do about it?!  These questions will be the foundation for this series of blog posts.  We CAN make a difference in how we reach our students and deepen their fraction knowledge.

## What makes fractions so challenging to teach?

I have done some research (formal and informal) by looking at what the experts have said and by asking countless teachers in the trenches, and I think I have some answers as to what makes this topic so overwhelming for many.  Why is this important?  If we can identify the stumbling blocks, then we can start to chip away at them and begin to learn more about what WILL work and how we can overcome these obstacles.  I am constantly on a quest to find ways to "make sense" of fractions for my students (which is what led to me creating a full fraction unit to use with my students!)

### Fractions are not always "concrete".

Although we may start our fraction instruction with objects--perhaps paper folding or plastic fraction pieces--we too often move very quickly past these real-world models to paper and pencil computation.  I'm going to argue that even drawings of fractions fail to be concrete enough for some students with limited fraction sense.  To many, this is just a circle with lines.  The physical act of cutting, folding, and manipulating is so important for many.

To make matters more complicated--this is just the beginning!  When we multiply whole numbers, we can make "groups of" objects, right?  3 times 6 can be represented by 3 piles of 6 objects.  Do we take the time to really make our more advanced fraction concepts such as multiplying and dividing concrete?  Do students really "see" what happens when we multiply fractions (HINT:  We can still model 3 groups of 1/4 or 6 groups of 2/3...and it can make a huge difference!).  When we move too quickly to "bare numbers", we make a lot of assumptions about what our students truly understand.

### Fractions are numbers--really!

Here's another statement that seems a little obvious, right?  Of course, fractions are numbers!  But wait...try to get yourself into the mindset of a struggling student.  We show them pictures of pizzas and candy bars cut into little pieces.  We talk about the pictures in different ways...and we help students learn to "label" them as representing 1/5 or 3/8 or whatever the picture represents.  Do we really help them understand that fractions allow us to represent real numbers--and even parts of numbers?  That when we have a drawing of 3/8, it means that we have LESS than a whole object or amount?  That fractions allow us to show amounts BETWEEN whole numbers?  This is a critical part of fractional understanding; the drawings we use to show fractions are merely representations of numbers from a number line.  More to come in upcoming posts.
 (These number lines are a part of my fraction number line resource--also part of my 10 resource fraction toolkit)

### The understanding of "unit fraction" is missing

This leads me directly to this concept.  The term "unit fraction" refers to a fraction with a
"1" in the numerator...such as 1/4, 1/7, or even 1/500.  Understanding that this is "unit" of measure or amount is so critical.  I like to interchange the terms "unit fraction" and "counting fraction" with my students because I really want them to understand that, just like whole numbers, unit fractions can be counted, added, subtracted, composed and decomposed--and more.

In the primary years, a great deal of time is spent counting whole numbers in different ways--by ones.  By 5's.  By 10's.  We even practice counting backward which helps develop subtraction understanding.  The same is true for fractions--and we really cannot make assumptions that all our students can do this automatically.

When we count fractional parts ("unit fractions") using manipulatives, on number lines, and even orally, it helps build that understanding that fractions are numbers.  It helps students begin to develop their sense of the relative size of fractions.  It helps lead to a greater understanding of what happens when we count and get the same numerator and denominator (a new "whole") and then count past that to make improper fractions and mixed numbers.  When we can count fractions we start to see the additive nature of them...I have 2/5 and I count up one more fifth and get 3/5--and this is something that can often be missing in student understanding if time isn't spent working on this.  It really forms the foundation for addition and subtraction of fractions--and is truly an essential skill.

In fact, this is one of the first interventions I do with students who are struggling. I get out a manipulative of some type (I linked to some of my favorites...note, these are affiliate links) and we practice counting.  We notice what happens when we reach a "whole" and then beyond.  We essentially "play" with these counting fractions and then begin to record our findings on paper and pencil. I will often pull these back out when reteaching is needed.

Also, consider when you look for manipulatives that some have the fractional part written on them and others are blank--both are great, but make your decisions based on what you want to accomplish.  Similarly, if you ONLY use circles or ONLY use fraction bars, students may struggle to generalize the learning.  Even using tools like pattern blocks can be really helpful as you study basic fraction concepts.

### Fraction notation and terminology can be challenging

So, when talking about playing with unit fractions and counting fractions, I mentioned that I work to connect the physical fractions (fraction circles, folded paper, etc) to the written symbols--and this leads me to yet another challenge associated with fractions--notation and terminology.  We throw around words like "numerator" and "denominator" and sometimes forget that students may not have internalized those terms and have most likely never heard them.  Words like "equivalence" and "improper" and "reduce" can also add to the confusion.  If you see my other posts from the past (or other lessons from my fraction unit)

To make it worse?  We can write fractions in different ways!  Check out the image below--and then think through the lens of a struggling mathematician.  Both of the green fractions represent the same thing--but they are written in two different ways.  One, if written carelessly, could end up looking like 2,110--the other is easy to read but difficult to make using technology.  How about the purple ones? Students need to understand that both of those are the same value--written in two different ways.  Even understanding that the "1" is a "whole" and the 7/8 represents part of another whole can be confusing for many.  We need to be constantly assessing for this type of misunderstanding.  Finally, with a tie back to unit fractions, students need to understand that one "whole" is the same as when it is notated with the same numerator and denominator.  We cannot make assumptions that this understanding is in place.  Dig in and find out--and fix any misconceptions along the way.

### Fractions appear in multiple contexts

I'm almost finished--I promise!  I know so many of these things seem trivial--but if we don't catch them in our strugglers, we just keep moving them forward with an unstable foundation--and we KNOW that really understanding fractions is the key to success in algebra and other upper math classes.  We can't fail our students!

Another frustration for students is that fractions appear in so many contexts.  We can divide a pizza in fourths.  We can divide SETS of things (like bags of jelly beans or baseballs) into groups.  We need to learn how to use fractions of more abstract things like units of measure--like fractions of an inch or a pound.  It's no wonder students get confused--1/2 can mean so many different things!  Even thinking about 1/2 of a pizza...is it a 7 inch pizza?  A 16 inch pizza?  Are those halves the same?  What about 1/2 of the pizzas in the restaurant?  Or half of a slice?

Because 1/2 can VARY, this is extremely confusing to many students.  In fact, this challenge forms the foundation of a great deal of the fraction work I do in my class--to really help students grasp how fluid fractions are.  You can see lots of examples of this in my complete fraction unit that has been such a game changer for me and literally thousands of teachers who have used it.

### A growth mindset is missing!

This is perhaps the hardest to overcome.  Students aren't born thinking fractions are hard--somewhere along the way, they have gotten this message. Whether it be from their families ("I was terrible at fractions, too!") or the way we, as teachers, present the information--somehow we are sending messages that fractions are our nemesis, something only to be understood by the lucky ones.  We have to think about our own biases and make sure we are NOT sending those fixed mindset messages to our students.  Fractions ARE accessible to everyone...but we do need to make sure that we build and nurture a strong foundation of understanding and a mindset of growth and perseverance.

I hope I gave you some things to think about today--and stay tuned for more fractions posts coming soon!

Want to check out the unit I was referring to?
Want to check out a bundle of 10 fraction resources I use to help build this understanding?

### More fractions? YES!

Let me send you a freebie and more ideas to try in YOUR classroom!

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It's my day to blog over at Upper Elementary Snapshots!  Check out today's post all about teaching COMPARISON PROBLEMS...and how important it is to help students really dig deeply into their problem solving.  I hope you get some helpful tips!  Whether you call them comparison problems, tape diagrams, or strip diagrams--these problems can really helps students "make sense" of problems.

Just click the image to take you there...

It's coming for many of our upper elementary teachers and students.  They have thought about it.  Worried about it.

The. Test.
This is something that I have thought about a great deal.  I hear so many stories about teachers, schools and districts who set aside real teaching and learning to prepare for tests that are simply supposed to be a "dipstick" to measure the state of affairs in our classrooms.  I am a believer (to a degree) in some forms of standardized testing.  Districts need to get some feedback on how their students and programs are performing.  That being said, the evolution of testing into high stakes, pressure-riddled experiences for teachers and students about sends me over the edge.  Because I think this is so important, I have revisited a post I wrote last year about this time to make sure that we continue to think about what is important about testing--and the number one thing we need to remember is our students and what best practices in education really are.

Teachers around the country are worried about if they are preparing their students well enough.  If they have given them enough practice opportunities. If they have spent their instructional minutes providing them with EXACTLY the right amount of exposure to what they will see on the test.

I don't.

I don't make pages of practice questions. I don't do a "real" test preparation unit.  I don't provide ongoing practice on key skills I know will be on the test.  It's not worth my time.  I'm not preparing a group of students to be test takers.  I am teaching them how to think and how to learn and how to tackle ANY problem they encounter--with energy, with perseverance, and with an "I can do this!" attitude.

In my heart of hearts, I truly believe that students who can read, who can think, who are willing to try will do as well or BETTER than students who are given hours of fill in the blank practice.  I want students to learn how to do well on these tests without me telling them what to do and spending hours of their precious time drilling.  I want them to DISCOVER how to be successful by putting them in situations where they can learn this genre in a meaningful way.  Now--before you accuse me of doing my students a disservice, let me tell you what I DO do!  Hopefully you might find a little morsel of information or inspiration below!

1.  I do teach my students about multiple choice questions.  In fact, I try to get them in the minds of a test writer by teaching them about distractors and even having them try writing questions with a right answer, a distractor, and two other relevant answers. We even talk about the art of "coloring the bubble".

2.  I do teach my students about healthy testing behaviors like getting sleep, eating well, and relaxing for best performance.

4.  I do teach my students about staying focused and checking over their work.

5.  I do teach my students about answering questions fully and providing evidence found in the texts.

6.  I do teach my students about what to do when they encounter a challenging problem.  We learn all sorts of strategies that gives us POWER...how to reread directions. How to find key words.  How to "give it a try" on scratch paper.  Even how to SKIP it if it is interfering--and then we come back later. I use resources like my perseverance problems and open ended challenges for this.  Students love the tasks--and don't have any clue that they are really "readying" themselves to do test taking.

7.  I teach my students about problem solving and looking for patterns.

8.  I teach my students to read all sorts of materials...stories...poems...articles...graphs...infographics. I have really increased the amount of informational reading that I do throughout the year, and I have been more deliberate about asking students to read the directions and other information BEFORE I explain it so that they are learning to be more proactive and not wait for me to help them understand what to do.

9.  I teach my students how to work with stamina so they can sit and complete a task that might take them an hour or so--without losing focus.  We talk about this almost daily.

10.  I teach my students how to be ok with doing their best and having an "I can do it!" attitude.  I want them to treat everything they do with that spirit...and to walk away knowing that they did their best--and that's all they can do.  I want my students to walk out after the test feeling great--that they did their job...even when the questions were tough.  A growth mindset is key--and we start that from the very first day of school.  Want to read a post with more information about that?  Just CLICK HERE.

Do I do this with packets?  Nope.  Do I do this for 3 weeks straight?  Nope.  I do this all year long, when it's relevant...and BECAUSE it's relevant.

Now--don't get me wrong--we DO a practice test or two.  In fact, we take it, study it, and break it apart.  I have my students hunt for terms they think are tricky like "passage" or "synonym".  We make anchor charts and lists of "things to know" about taking tests.  We practice this in a quiet room to mimic testing situations.  We talk about filling in the bubbles neatly and checking over our work so we don't miss questions.  If I taught third grade, I would have to do even more of this because the test is so new.  That being said, if we can teach our students to have a great attitude about trying, if they can stay focused and apply what they know, and if they can be successful at whatever task they are handed!

How are my test scores, you might ask?  My principal called me in several years ago to ask what I do...because my scores were SO much higher than the average.  It was hard for me to explain.  I told her, "I teach students how to learn, how to work, and how to try."

One resource that has been super helpful to me is the book "Test Talk" by Amy Greene and Glennon Melton.  It gives some GREAT suggestions for how to incorporate test taking strategies into your reading workshop.  Check out the link below for more details.

One final thing I do is ask my students to talk and write about all the ideas mentioned above.  It needs to be more than me TELLING them these things...they need to be able to process them and construct their own meaning.  I have put a lot of this together in an unusual test prep resource--in case you are interested!  Thanks for stopping by--and good luck on the tests.  Make sure you keep it positive and give your students the power to do well AND feel good about it!

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Writing about math thinking is REALLY challenging.  Students in the intermediate grades aren't used to taking their ideas and transferring them to paper when it comes to complex ideas!  If you have asked students to "explain their thinking" about a solution, you may have noticed them writing things like:

"First I took the 64 and the 49 and I added them.  Then I took the rest away." or

"I could tell it looked like about a half so I wrote that." OR my personal favorite...

"I just knew it in my head." (You've heard this, right?  It's not just me?)

### What do the Standards for Mathematical Practice Say?

The Standards for Mathematical Practice and other rigorous math standards have made it clear that we need our students to get better at explaining their thinking and critiquing the reasoning of others (and themselves!).  The standards mention things like, "make conjectures and build a logical progression of statements" and "justify their conclusions" and "communicate them to others" and more.  So how do we get our students to dig deeper and explain their thinking more clearly? (Want to see how I help my students understand these standards?  Just CLICK HERE.)

Today I presented my students with one of the fraction challenges in my fraction unit to see a few things--one, if they had internalized an important fraction concept we had been working on and, two, how they were doing with their "explaining their thinking".

I learned a few things!  First of all--almost ALL students got the right answer!  #boom

Secondly, our written explanations were in dire need of some work!

I thought I'd share with you my next steps--because I REALLY want to see my students make gains in this area...and stay tuned over the next weeks for an update!

### Coaching Students Toward Better Math Explanations

As I was "walking the room" as students were working, I noticed that very few students were writing what I considered to be a quality explanation.  I began to wonder if they REALLY knew what I meant when I said, "Write a clear explanation."

I have noticed that students tend to write procedures rather than thinking.  Instead of writing, "I read that I needed to find the difference between the two amounts so I needed to subtract.", students write, "I subtracted 53 from 82."  I try to tell them to let their math computation speak for itself and let their explanations explain the WHY...but it's hard!

I started by having students work in trios to share their explanations.  They needed to read it aloud and listen to see if they heard WHY and HOW in the explanations.  After a few minutes, I asked if any groups had heard any explanations that they thought did a good job.  As students nominated other students, I asked their permission to share under the document camera.  I reminded them that we weren't looking for perfection--just for ideas on how to improve our work.  I got six samples that we then looked at together.  We collected words and phrases like "proved" and math words like "equivalent" as we went, talking about how important it is to be specific with our explanations.

So...we generated a list of fraction words we might expect in this problem...and made it clear that each type of problem will have math vocabulary specific to it that a reader would expect to see.

I also shared with them an anchor chart I whipped up quickly with some sentence stems...phrases to get them "unstuck" when writing about math.  There are so many more--but I wanted them to see that there are different types of writing about math, and you have to choose what makes sense.  We tested these with our fraction problem and realized that the third one might be really useful.  "I knew that the larger shapes would divide the rectangle into fourths so..." or something like that.

### Revising Their Thinking

So the real learning comes in when students take in these student models, the vocabulary list we generated, and these writing stems to set out to revise and improve their own work.  It's not enough to see other people do it--students need a chance to "give it a go" on their own--so off they went to make improvements.  When they finished, the met back with their original trio to see if the group agreed with the improvements!  Mission accomplished!
We will continue to refine our anchor chart by adding new phrases we find as we work on different problems and will continue our discussions about how to improve our math writing.

Coming up next?
If you are interested in seeing more of the fraction work we do in our class, just click the image below.

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