True or False? A Warm-Up That Blows Open Misconceptions About Comparing Fractions

I’ve been filming in Tabitha Eutsler’s classroom all week to capture a progression of student learning.  Every day I have been in her classroom this week there has been moment after moment of excitement as we see the understanding of students unfold through purposeful questioning and student feedback.  

Tabitha is a master facilitator.  She spends lots of time at the beginning of the year (and throughout) building her classroom community because she knows that is what supports students in being able to communicate with each other, disagree, argue, challenge each other and celebrate each other so that every student has not only the opportunity to learn, but the opportunity to belong.  Her students are mathematicians and they know it.

These were the learning goals for each lesson this week:

  • Monday: 
    • Generate equivalent fractions. 
    • Understand two fractions as equivalent if they are the same size.
  • Tuesday: 
    • Identify and generate equivalent fractions.
    • Understand two fractions as equivalent if they are at the same point on a number line.
  • Wednesday: 
    • Represent and compare fractions in a way that makes sense to them.
  • Thursday: 
    • Compare two fractions with the same denominator with the symbols < or >.
  • Friday: 
    • Compare two fractions with the same numerator with the symbols < or >.

Today is Friday and students started the lesson with an instructional routine called True or False?

The first problem posed was:

True or false?

Students were asked to think independently for a minute silently and then Tabitha asked them to share and turn and talk to a partner to see if they agreed or disagree with each other.  She then asked for students to pop their thumbs if they had an answer they were willing to share.  Below are the conversations that followed for each of these tasks:

Student 1: I think it’s false 

Teacher: Why?

S1: Because 2 and 4 are two different numbers and 4 is greater than 2.

T: So you looked at the numbers on the bottom and thought 4 is bigger than 2 so this must be false? Did I get that right?

S1: Uh-huh

T: Okay, thank you for sharing.  Continue. S2 what do you think?

S2: Actually I agree and disagree.  I agree it’s false but actually I would look at the top because it is actually equal, not 4 is greater than 2. 

T: So why do you think it is false then?

S2: Because it says that ½ > than ¼ and actually that is actually equal.

T: So you think it should say ½ = ¼ and they are the same? Ok.  Does anyone have feedback?

S3: I disagree with you because the top are equal, but the bottom numbers are not.  

T: So do you think it is true or false?

S3: False

T: Okay go ahead and give us your reasoning. 

S3: 1 and 1 is equal but the 2 and the 4 aren’t.

S2: I kind of see what you are saying.  The top is equal but the bottom is not. 

T: So are you saying that ½ is the same as ¼ now or you think they aren’t actually the same.

S2: No I don’t think they are the same.  I want to change my answer because of what he said. 

T: I think I’m gonna throw out a different challenge.  How about we get our books out so you have a place to write and I printed some more fractions strips and number lines for you guys.  So how about now you show me what is going on in your brain.  Can you write it down and show me?  So it is asking  ½ > ¼ and we are saying that is false.  So I want you to prove it to me.

Go ahead and talk to your partner.  Is it still false?  Or is it true?

If you want to bring your books up or use the markers you can.  I’m gonna review where we are at. We have 3 of you saying that is false.  You are going to come up and still defend false or if you changed your answers.

S1: I think it’s true because I can see that ½ is bigger than thirds because would you want one of a third or would you want a full half?  The half would be bigger so you would have more.  That’s why I think this is true.

T: Feedback?

S2: I disagree, I think it is false because I drew a number line and on this you said thirds instead of fourths.  Oh yeah fourths. Right here, that would be where ½ happened.  Right here would be ¼. It’s diagonal. 

S1: I agree with how you saying that but you can also split ½ into 1/4s so it would be a smaller piece.  It would be ½ broken up into fourths so you would get a littler piece.  I’ll show you.  We have this whole big piece right here, now you want to get this piece.

T: Can you show him what you mean by pieces using HIS number line?  Can you show him where the pieces are on his number line to compare the sizes?

S1: Yeah. So this is ½ and this is ¼ so this piece is bigger.

S2: Oh, I get it so what you’re saying is this is here.  I agree with S1 now.

T: Everybody we haven’t seen fraction strips yet.  S3 is going to show us fraction strips.  So be looking at your paper.  Does everyone have true written down?

S3: I drew two number lines.  This one is split in halves and this one is spit in fourths.  The half is double the size of the fourth so that’s why I think it is true.

T: Feedback for S3?

S4: I agree because this goes further than this one.  

T: Oh so the half is further than the fourth?

S4: Uh-huh

T: After listening to our family give me a thumbs up or thumbs down is it true or is it false?  Okay.  Thank you.  This is true.  So you might be thinking I kind of see it but we’re in luck because  we have a couple more problems to practice this.

True or False?

T: Are you defending true or false?

S1: I am defending false. As you can see here when you make fourths you get smaller pieces.  When you make thirds you get big pieces.  I labeled them small and big.  

T: Feedback – what do you think about that with the pieces she was talking about?

S2: I agree because I did the same thing but I didn’t notice that.  I didn’t do the number line but I did the picture and it blowed my mind.

T: Thank you so you had a connection that your picture was similar but doing it this way grew your brain even more? Thank you.  Who would like to defend next?

S3: False because if Ms. Eutsler gave me a candy bar would I want the bigger piece or smaller piece? Since it is a candy bar I would want the bigger piece.  So it’s false because that piece is not bigger than that piece.  

T: You shared something with me about the bottom numbers.  Can you share that with the class?

S3: So on the bottom you always want the lesser number because the less number has the bigger pieces than the bigger number. 

T: So if we look back, a yes or no please.  Based on everything that you heard, is this true? It is false.

True or False?

S1: I think it’s true because um if you draw it on a number line you see that 6 is farther away than 8 so the less away that it is is the one that is true.

T: Feedback?

S2: I agree that it is true but you forgot one. These are five pieces and you need to draw one more. 

T: Look at your paper, do you have true based on what our family just told us?  It is true.

Following the warm up, the students worked on a word problem in which two students disagreed about 5/6 or 5/8 being greater. Several students shared and a student ended with this comment:

S: This kind of goes back to our true or false questions that we did earlier where if the numerator stays the same then the denominator determines which is one is greater.

T: And how would you determine with the denominator which one is bigger?

S: When the denominator is smaller.

T: Because the size of the pieces is what?

S: Bigger.

This is the power of a problem based classroom.  The students uncover misconceptions as they begin the task, but as the teacher supports their productive struggle by allowing them time to grapple with new ideas, share their thinking with others, and see solutions from other classmates they begin to make sense of the mathematics and develop reasoning and proof.  This is possible because of the teacher’s understanding and application of the mathematical teaching practices.  Ms. Eutsler poses purposeful questions, supports productive struggle and facilitates discourse so students are positioned as the experts in the room.  When they need support, she provides it in the means of tools, questions, and the purposeful sharing of student work that provide visual representations from which others can compare, contrast and connect to their own. It is the skillful art of teaching.

Problem based classrooms provide students time and space to develop deep understanding of math concepts. They struggle, they perservere. They make mistakes and they correct them. They agree and they disagree. These are the habits of mathematicians. The world doesn’t behave like a math textbook. Students need to know how to make sense of contexts and scenarios by playing with mathematics in a classroom with a teacher who values mistakes and problem solving, holds the belief that every student can do math to high levels, understands that math is about depth not speed, and knows that students will meet the expectations that are set for them.

Why I Won’t Put My Son in Accelerated Math (Part 2)

I wanted to post an update to my first post about acceleration in middle school.  My son is in 7th grade math this year and I thought it was important to share his progress.

I am using his test results below to demonstrate two points regarding some misinformation and in an attempt to debunk the myths that tend to get hurled by those who believe in back to basics (procedural, wrote, calculation) math.

Myth #1: Inquiry based math will not meet the needs of students who are on or above grade level.
Myth #2: Students who score on or above grade level should have access to an accelerated pathway to get the challenge they need to make growth in mathematics.

Before looking at his assessment results, here are a few facts about this year:

  • Our middle school students lost a quarter of learning last year, spent more than a quarter of this school year only attending classes 2 days a week, and the other part of this year attending 4 days and 1 virtual day (his virtual day he attends a choice program).
  • My son’s teacher is using Illustrative Mathematics (via the Desmos curriculum platform).
  • His teacher uses the 8 Math Practices and the 5 Practices for Orchestrating Discourse to purposefully look at, discuss, and connect student strategies.

This is his assessment from the beginning of the year.  According to this report, to meet typical growth of a student at his level, he should grow by 11 points on this assessment by the end of they year.

Skye diagnostic 1 typical growth

These are his middle of the year assessment results (after 1 quarter of 2 days of instruction and 1 quarter of 4):

Skye diagnostic 2 typical growth

What do you notice?  What do you wonder?

Hopefully you noticed a few of things:

  • The curriculum is not holding him back. 
  • The course pathway is not holding him back. 
  • His teacher is not holding him back. 
  • In fact, the opposite is true.  He is being challenged and he STILL LOVES MATH!

He met the criteria to be placed in 8th Grade Algebra next year (at the beginning of this year – see a problem here?).  An email went out and I promptly responded asking that he be placed in 8th grade math next year.  Why?  Because I believe that 8th grade math is the foundation on which Algebra 1 and Geometry are built and I want him to enter high school equipped to succeed in both of these gateway courses.  AND because I trust educational research done by experts in their fields.  And the research is clear;  There is no place for accelerated math pathways in middle school.

Some common arguments (and my responses) for keeping accelerated pathways are:

  • My child scored advanced on the state assessment/my child is gifted/my child is special.
    • Great!  Let’s keep that going by providing them a strong foundation for future math success.
    • If they love math and want extra opportunities, have them join a math club or explore (awesomely fun math)!  
  • I want my student to graduate with Honors.
    • Maybe the current honors system is set up to disadvantage the many and needs to undergo change.
    • If the system cannot or will not be changed, I suggest opt in honors – let kids decide if they want to take honors courses and agree to additional or modified requirements.  (This keeps the classrooms diverse and allows students a stake in their educational decisions while still offering the opportunity for honors credit).
  • My child has to take Algebra I in middle school in order to take Calculus in college.
    • Do they?  Do they really?  Have you talked to the colleges that your child (is planning to attend 5-6 years from now) and do they require Calculus?  What must your child demonstrate in order to be placed beyond Calculus in college (MANY students end up having to retake Calculus in college even though they took it in high school).   This is sometimes determined by ACT/SAT score or entrance exam.
    • And also, is Calculus the course your child should take in high school or are they entering a career that would be better supported by Statistics?
    • And also also, there are ways to take Calculus without being accelerated in middle school.  That road is still open to my son (if he chooses that is the course that will set him up for success in college).

If you find yourself wondering if your child should take Calculus in high school, I encourage you to read these articles.

The most common argument for taking Calculus in high school is that students need it to get into great colleges.  I decided to see if that was true, since I had just attended a lecture from Stanford professor, Jo Boaler, who mentioned Stanford does not require it and that colleges would rather have students take Statistics their senior year.  Here is a table I created after looking at the websites of each of these universities and their math requirements:

If you want to learn more about the research from the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics, check out these articles and books:

(2/21) UPDATE!!! Kate Nowak shared this resource in which IM synthesizes the research above and suggests some additional action steps for districts.

Show Your Work

This post has been one of many sitting in my drafts, but I decided to post it even though it is a few years old because it offers an opportunity for self reflection.

My son was doing math homework a few weeks ago, he came across a problem that went something like this:

Mr. Jones had 3.75 ounces of Kool-Aid and he put them into cups of 0.25 ounces.  How many cups of Kool-Aid was he able to pour?  

I was checking email while he was completing his homework, but looked up when I heard him mumble under his breath. “Ugh, it’s 15, but I have to show my work.” I asked him why he was frustrated and he said because he knew the answer but he had to show all of the steps.  Knowing his teacher puts a strong emphasis on Number Talks, I told him I’m sure that’s not the case, I’ll just write a little note that says that you solved it mentally.

The next day he came home and brought me his homework.  He said well I got 21/22 because I didn’t show my work on the problem.  I was perplexed so I emailed his teacher to discuss.  I thought this must be a misunderstanding.

The reason I share this story is because I think sometimes as educators, we lose sight of what our goals are for students in mathematics and we can send a message that we ourselves do not believe in.

Let me first say that I greatly respect my son’s teacher and that I know that she wants what is best for him.  She uses a number talks routine in her classroom and has students work on mental math to build their number sense strategies on a regular basis.  However, I think my son is confused as to when he is “allowed” to use mental math and when he has to calculate on paper.

As I reflect on this experience, I think back to when I was a teacher and I made graphic organizers for my students to “draw a picture” to solve.  Although I think this is a good way to get students to visualize problems they haven’t encountered, I now feel that I overused this method and I can recall some students who were frustrated by this practice.

It is important that we not lose focus of our purpose and that we constantly reflect on our practice to ensure our philosophy matches up with our actions in the classroom.  It is easy to get bogged down by state testing, mandates, homework practices, etc. but we owe it to our students to check ourselves and question our practices when it leads to frustration.