Exploring Fractions with a Growth Mindset

I love summer. I love it for so many reasons. One of the reasons I love summer is because it gives me the opportunity to tutor students at my own pace with no "curriculum" other than what I believe to be good instructional practice to follow.

This summer I'm tutoring a 10 year old who has just finished Grade Five. I have worked with this student before, (I will call her Grace because she's an incredibly graceful young lady), and I know that she has struggled with Math for the last few years. Grace is pretty typical of many students that I have taught. Traditional math instruction is not that effective for her. She is quiet, and if something doesn't make sense to her she won't ask questions, preferring not to single herself out. 

Grace also has some issues with her short term memory. Many of her teachers have expected her to learn her addition and multiplication facts by rote memory. That is just not a realistic expectation for Grace. She has difficulty memorizing facts. Last summer, Grace and I explored repeated addition, skip counting, and making groups. Although she doesn't know her facts with lightening speed, she can figure out any multiplication question using strategies that makes sense to her. She understands what multiplication means. But some of her teachers don't value this, and as a result, at the tender age of 10, she does not really see herself as a "math person". 

I also love the summer because it gives me time to work on my own professional learning. In past summers I have taken Marilyn Burns' Math Solutions course and attended our Ministry's Math Camppp. This summer is no different and I am currently taking Jo Boaler's MOOC How to Learn Math. (You can watch the Youtube videos here). There is a common thread through all of the learning - Math is not a textbook subject! You don't develop deep understandings of mathematical concepts by completing worksheets or workbooks. In order to develop conceptual understanding in math, in order to see connections between concepts, in order to love math and think creatively in math, students need multiple opportunities to explore math. I love the summer because I can work one on one with students, give them those opportunities to explore and construct their own understandings, and watch and listen to learn how they learn. 

I knew that Grace had had trouble with fractions this year, so that is where we started this summer. 

This was an SOS I got from Grace in the Spring

More than ever, I've been trying to use Growth Mindset language. I've been using "traffic light comprehension" with Grace, asking her frequently if she is red, yellow or green light in her understanding, and asking her to really pay attention to her own learning. I'm being careful about the language I'm using, drawing attention to how successful she has been with her persistence and hard work. I'm also making sure that if she doesn't get something, we add the word "yet". I give her lots of time to explore her understandings and allow her misconceptions to "float" out there, merely asking questions that allow her to re-evaluate her own beliefs and re-adjust her understandings. 

We've spent three 90 minute lessons just on representing proper fractions using fraction circles, fraction strips, sets, area models and number lines. We compared these representations looking at the connections between them. I've been encouraging Grace to name her learning so that she can see her own growth and take ownership for her understandings. Grace has concluded that:

- the numerator counts how many parts you have (or are discussing)

- the denominator tells how parts make up the whole

- a proper fraction is always less than one

- in a proper fraction the numerator is always less than the denominator

- different fractions can represent equivalent amounts e.g. 1/2 is the same as 5/10 which is the same as 0.5

- the equal sign (=) means "the same as"or "is equivalent to" and not "the answer is..."

Grace doesn't have to memorize these things because she came to these understandings on her own. 

During one lesson while exploring proper fractions Grace said "My teacher kept giving me questions like this: 2/3 = ?/6.  I didn't know how to answer those". Those questions had absolutely no meaning for Grace at all.  She did not know what the teacher was asking or looking for. I pointed to the number line we had created and how we had divided it up many different ways. I pointed to the half and asked "How many ways could we name this fraction?" She said we could call it "1/2 or 5/10". Then she went to the fraction circles we had on the fridge and said "It's like 1/2 is the same as 2/4 and 3/6". I explained that is what the teacher was asking, that she was asking what fraction with a denominator of 6 was the same size as a fraction of 2/3. All of a sudden the question made complete sense to Grace but she needed to connect the question to a visual representation. 

 Today we began working on improper fractions. I could not believe how quickly she picked it up.  I guess it was easy after all the work we had done on proper fractions. Grace had such a deep understanding about the role of the numerator and the denominator she quickly deduced that if the numerator was greater than the denominator we were talking about a fraction greater than one. She had no trouble representing them in any format. Next week I will show her how to write an improper fraction as a mixed number. This should be easy for her to understand since she's already been naming them out loud as "two and a quarter" for example because she can see them pictorially as a combination of wholes and fractions.

Along the way we've been comparing fractions. As we put the fractions on the number line, as we use the fraction circles on the fridge, and as we draw our area models, I'm always asking "What do you notice about these two fractions". Eventually, I will teach Grace about common denominators, but not for a while, not until she has a really solid understanding of fractions, and has an idea of benchmark fractions on a number line so that she can estimate the relative size of a fraction. I want Grace to have many ways to compare fractions. She has already noticed that the larger the denominator, the smaller the fractional piece. I want her to realize that 7/8 is less than 9/10 because each fraction is missing only one piece but the tenths are smaller pieces, so 9/10 represents more. I want her to know that 9/20 is closer to half than 4/10 is, and I want her to know this without having to use a common denominator because she understands fractional parts.

I'm really enjoying the Jo Boaler MOOC. One thing that Jo said was that intuition is an extremely important part of math competency. I've often thought that having mathematical intuition was a genetic gift - I guess you could say I had a Fixed Mindset about math ability. But Jo has conducted research to show that it is mathematical understanding that helps a person to develop mathematical intuition. And having mathematical intuition, in turn, helps a person to develop their mathematical understandings. Jo Boaler explains it as an iterative process. This makes total sense to me. I'm helping Grace to understand fractions. I'm hoping this will help her develop an intuitive sense about proportional reasoning. Once she has honed this intuition, it will help her solve problems and make sense of problems involving fractions, decimals, percentages, rates, and ratios. It is my goal that Grace sees the connections between all of these beautiful math concepts.

Graces always texts me before she comes over. I told her I had bought her some fraction circles she could take home. She wrote "Do the fractions stick on the fridge?"She wants some like mine so she can play school at home and use them for math homework. Then she wrote "I can't wait". She clearly loves math, she just doesn't love it at school. Isn't that a shame?

Learning Goals in Math

I had a very interesting conversation with my friends Janice and Celiza the other night.  Janice and Celiza are currently Curriculum Consultants in two other school boards in Ontario.

We were discussing the use of Learning Goals in Math.  Janice was of the mind that we shouldn't be naming the Learning Goals up front at the beginning of the Math lesson if we are truly teaching through problem solving.  I understand her point.  If you are teaching Math through problem-solving - which I hope you are - then you are using a constructivist approach, and you want the children to construct their own learning.  You don't want to give away the punch line.  Janice was adamant that the Learning Goal should come out during the Consolidation (the third part of the three part Math lesson).

<For a brief review of the Three-Part Math Lesson click HERE.>

I was very fortunate to go and hear Dr. Marian Small speak last May (I reference her often, I am quite a fan of her work).  She addressed this very question.  She said that we have to tell the students something about what we want them to do.  She asked (and I quote) "Can we say: 'We are going to develop strategies to compare two fractions'?"

What I liked about Marian's talk was that she never told us what we should be doing.  She only ASKED us what we might want to consider.  The fact is, there is no right answer here.  It is trial and error.

I have tried it both ways, not telling my students anything, and also giving them a simple statement about what we are hoping to learn about.  Not what we are going to learn specifically, but what we are learning about.  I have found that giving them a Learning Goal up front helps give them more direction.  It sets a purpose up front for why I am giving them the task that I am giving them.  And, I admit, it speeds up the problem-solving and consolidation.  The fact is, I don't have unlimited amount of time to teach them everything I want to teach them.  I only have 9 months to teach them 10 months of curriculum (because I have EQAO testing in the beginning of June).

What I don't do is give them Success Criteria for the content expectations up front.  That is what I want them to come up with in the consolidation.  I DO however, give them Success Criteria for the process expectations up front. (The process expectations include communication, representing, selecting tools and computational strategies, problem-solving, reflecting, reasoning and proving). Confused yet?

For example, I might write:

• We can reflect on the reasonableness of a solution 
• If we determine our solution doesn't make sense we can look for a different solution
• We can share our solution with others in a way that makes sense to them
• We can explain our thinking so that someone else "gets it"
• We can connect our solution to someone else's solution
• We can restate the problem in our own words

Here was our Math Learning Goal from Thursday and Friday:

For our Minds On, I gave the students fractions to add to a number line.  This allowed me to differentiate by giving different students different fractions.  As the students placed their fractions on the line, I asked them how they knew where to put it.  "Tell me what you are thinking in your head as you are deciding where to put that fraction."

Here are some strategies we came up with for comparing fractions in our Consolidation:

We are still using the "ShowMe" app and "Explain Everything" app to show how we solve a Math problem.
Here is a video from one of the problems my students solved on Friday.  I was only going to give them one problem, but some of the students solved it so quickly and easily, we went to the text book to find a second problem.


I think it is important at this point to mention the obvious.  This lesson could not have happened if it weren't for the previous lessons.  We spent Monday, Tuesday and Wednesday reviewing what fractions ARE.  Before students could begin comparing fractions, they had to fully understand what a numerator and denominator are, and they had to understand what mixed and improper fractions are.  For example, many students knew that an improper fraction was a fraction with a numerator larger than the denominator, but very few students knew that this meant we were talking about a number greater than one.

I also had one student who could compare fractions by using a common denominator.  He had learned how to do this in a previous school in grade five.  But he did not know how to represent those two fractions, and he did not know how to compare those two fractions without using a common denominator.  For example:  students should know that 5/6 is less than 7/8 because sixths are larger pieces than eighths. Just because they can convert them into 20/24 and 21/24 doesn't mean that they understand that concept.

I guess my point here is this:  You have to be extremely deliberate in everything you do when you are teaching.  Before you post a Learning Goal, you have to think, "How should I word this?" "How will knowing this help my students?" "What do my students already know and think?" "What skills do they need?" We have to be reflective practitioners; I am very fortunate to have friends that are interested in having these conversations with me!