What's Wrong With How We Are Teaching Math?

Ontario needs to improve teacher training in math, the province’s Education Minister has said in response to standardized test results that revealed students are losing ground in the subject for the fifth year in a row.

CAROLINE ALPHONSO AND ADRIAN MORROW
The Globe and Mail
Published Wednesday, Aug. 28 2013, 11:31 AM EDT

It seems all eyes are focused on Junior Math in our schools this Fall.  Everyone is asking, why are our scores dropping in Math?

For the last several years, the Education Ministry has been encouraging teachers to teach Math through problem-solving using a three-part lesson.  This mandate is grounded in research.  In our board, like many others, our Elementary teachers have been involved in the CIL-M (Collaborative Inquiry and Learning in Math) meaning they have been learning together, through a co-planning, co-teaching, debriefing model to learn how to teach Math through-problem solving using a 3-part lesson.

So why have the scores been dropping?  Over and over again I hear teachers saying "So, when are we going to admit that the 3-part lesson doesn't work?" And I hear parents saying "Teachers need to go back to teaching the basics.  They need to teach Math the way I was taught."

Are they right?  I've been thinking about this question a lot lately, and I have to say "NO!"  The 3-part lesson and teaching through problem-solving WORKS and it is the best method for helping students to develop deep conceptual understanding of mathematical ideas.  So, if I am right, why haven't we seen the scores improve in Junior Math on our EQAO Assessments?

I have several theories that might explain why the scores are not improving.  I believe that our lack of improvement is actually the result of several factors.

1.  Teachers have not fully bought into the 3-part lesson and teaching through problem-solving

• What I have noticed during many of our co-teaching sessions is that teachers have a really difficult time "giving up the floor". The first part of the 3-part lesson, which should be a 15 minute introduction or "Minds On" tends to go on for closer to a half hour of straight teacher talk. This is because old habits are difficult to break and teachers don't fully trust that the learning should and will come out during the third part of the lesson, the Consolidation.  They feel that they have to provide students with enough information at the beginning of the lesson for them to be successful in solving the problem during the "Action" portion of the lesson.  They run out of time in their lesson, and the Consolidation ends up being only 5 minutes long, if it takes place at all.  So while on paper, the plan looks like a 3-part lesson, what is happening in the classroom is much closer to traditional teaching.
• Some teachers see the 3-part lesson and teaching through problem-solving as a "Friday" activity and teach traditional methods during the rest of the week.
• Teachers did not choose to be a part of the CIL-M, they were told to be a part of it.  I've asked many people to explain what their "inquiry" in Math is about and they don't even seem to know that they are inquiring about anything. I've also asked "What is the goal of your PD session today?" I've yet to meet a teacher who is a part of the CIL-M who can answer that question.  They are attending as a form of compliance not because they are hoping to learn something new.  Just like our students can be compliant with a learning task without knowing or understanding the learning goal, so can our teachers!

2. In order for teaching through problem-solving to work effectively, teachers have to be excellent at formative assessment

• In order for teaching through-problem solving to work, the problems have to be in a student's zone of proximal development.  This can only happen if the teacher has good assessment for learning practices and knows exactly what his/her students are ready to learn next.  The teacher has to differentiate for students at different places on the Math learning continuum.  
• Teachers need to be able to determine, through assessment and making learning visible, what students misconceptions are so that they can address them.  This leads to very tailored, explicit instruction, but again, requires that the teacher is an expert at formative assessment practices. 

3. Teachers have to themselves have an excellent understanding of the Math Curriculum 

• Many of our teachers lack confidence in teaching Math.  They don't consider themselves experts in Math, and struggle to identify the "Big Ideas".  Teachers today were taught Math by traditional methods; many of them only have rote learning of mathematical algorithms, and don't have deep understanding of mathematical concepts.  Here is a small example of what I mean: how would you compare 6/7 to 7/8?  When I ask teachers this question, they automatically begin to create equivalent fractions using common denominators.  That's because they don't truly understand fractions.  They should be able to see that each fraction is one piece away from a whole, and the eighths are smaller pieces than the sevenths so 7/8 is the larger fraction.  

So what do we need to do about this?  Well we need to acknowledge that we have a problem of practice.  And saying that we are going to teach Math through problem solving isn't going to truly address that problem.  We need to dig deeper, be more specific.  What do we have to change about our teaching practice to address this learning deficit in Math?

I think we need to do the following:

• We need to assess our students' mathematical understandings BEFORE we begin a unit of study, but we also need to assess them along the way to determine if we are having an impact.  We need to be responsive to our students' needs.  In order to do this effectively, teachers need to develop their formative assessment strategies.  I like Dylan Wiliam's Embedded Formative Assessment and our Ministry's AER Gains site for professional development on formative assessment.
• We need to develop our questioning strategies to provoke thinking, elicit misconceptions, and scaffold student learning to address misconceptions
• We need to use rich multi-step problems that are open-ended or have multiple paths to a solution; that way students have multiple entry points. Dr. Marian Small's resources help teachers to develop rich problems that address the Big Ideas and common misunderstandings that students have.  Here's a PDF that shares Marian's teaching strategies for focusing on the Big Ideas in Math through open tasks. 
• We need to explicitly teach accountable talk in the Math classroom using prompts to promote student-to-student discourse and discussion. Cathy Marks Krpan's Math Expressions is a great resource to help teachers develop student thinking and problem solving through communication. 
• We need to make Math concrete.  Students can't understand the abstract until they understand the concrete.  We have to stop saying,"Here are the manipulatives if you need them." No student wants to say they need something others don't.  We need to say, "Use these manipulatives to prove your solution." Students who are using an algorithm but don't have conceptual understanding will be forced to explore what the concepts really mean. 

Teachers can learn along with their students, but they can also learn with their colleagues through a co-plan, co-teach and debrief model.  Teachers need to get together to look at curriculum expectations along side their student needs before they develop problems.  Once they know what their students need to learn, together they can develop rich problems or activities for their students.  They need to moderate student work together to establish what they now know about what each student knows and can do, and where each student needs to go next in order to achieve the learning goals.   We need to be intentional in everything that we do!

But most importantly, teachers need to recognize that every classroom is a laboratory, and every lesson is an experiment.  We formulate a hypothesis, "if I teach this concept this way, the children will be able to ...." and like any good scientist, we need to observe the reaction; what are students doing?  What are students saying?  Then we judge the evidence.  Were we effective?  What impact did we have on student learning? If we're not effective, we have to create a new hypothesis and determine how to fine tune our teaching to meet the needs of the students in front of us.  

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!

A Digital Math Story

On Monday, at a 21st Century workshop, I learned about the SAMR, a model for the use of technology in the classroom.   But even before I saw this model, I had already decided that if I am going to be teaching with technology, I want to use the technology to teach differently.  I don't want to use technology as a substitute for our old Blackline Masters.  I started teaching with the SMART board five years ago, I want to do more with it than use it as a fancy whiteboard.

Since I started the etMOOC (Massive Online Open Course about educational technology), I've been  contemplating how to make students creators of knowledge.  21st Century learners are collaborators, creators, and sharers.  I knew that before, but the etMOOC has been helping me to make that into a reality in my classroom. 

During that same Monday workshop, I was introduced to a new version of Bloom's Taxonomy made for iPads: 

The point, of course, is that the highest level of thinking is creating or synthesizing, so I have been spending my week exploring how I can have my students create their own knowledge. 

I liked the way +Michael Buist uploaded a video of his students' completing a math task.  You can see in the video how he scaffolds their learning and understanding.  I was truly inspired by that video.  To me, this was a Digital Math Story.

I decided to have my students create videos of their math solutions in a similar way.  I used the ShowMe app because it is free and so easy to learn and use.  My students quickly caught on and in their collaborative groups, they used the ShowMe app to explain how they used the volume of rectangular prisms to solve math problems.  As always, I was walking around the class while the students were doing their work.  I was thrilled because they all seemed to be understanding the math concepts.  I asked them questions, they gave appropriate answers.  During the consolidation of our math class, we watched one of the videos.  We made comments and asked more questions.  Everything seemed great.  But then when I got home, I watched the rest of the videos, and I discovered so many misconceptions as I listened to the conversations they were having while they were solving the problem.  It was incredible learning for me.  

The next day we watched the videos and discussed everyone's misconceptions.  We also reviewed the first part of the 4-Part Problem Solving Model, because I realized that while my students understood the math, they weren't always understanding or answering the question.  

Today, we solved another math problem; this one required that they use their knowledge and understanding of the surface area of rectangular prisms.   I asked only two groups to create videos.  I chose one of the groups because they had previously demonstrated that they made certain to understand the problem before moving on.  Then I showed their video to the class as a model for how to go through the first step of the Problem Solving model.  

One of my students posted this on Twitter.  I asked her to explain how making the ShowMe videos was helpful and she said, "When I explain it, I understand it better afterward".  And THAT is why it is so important to be a knowledge creator, collaborator and sharer in the 21st Century!