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.  

Being Reflective Practitioners

"If we teach today as we taught yesterday, we rob our children of tomorrow..." - John Dewey

I'm currently reading "Instructional Rounds in Education - A Network Approach to Improve Teaching and Learning" by Elizabeth City, Richard Elmore, Sarah Fiarman, and Lee Teitel.

 I actually picked the book up over a year ago because I was intrigued by the title.  As you may know, before going into Education, I was a nurse, and nursing rounds were a huge part of our practice.  So I wanted to know how the authors proposed to use the "rounds" process in the world of Education.

I just wished that I had read the book BEFORE this past school year. Why?  Well, this year, for the first time, I experienced our School Effectiveness District Review process.  This process is mandated by the Ministry of Ontario.  I actually learned so much going through this process, not just about the practice of teaching and learning, but also about human nature, relationships, and the pressure associated with feeling like you are "under the microscope."  I also witnessed the stress teachers experience with the advent of change.

Going through the review process gave the majority of the us the impetus to move our practice forward at a faster pace than we might have otherwise.  That was a good thing.  It also helped us to be much more reflective in what we do, why we do it, and the impact we have on student learning.  I just wish we didn't have to go through a "Review Process" to behave this way!

The first thing that we had to do for the Review Process, was determine a Problem of Practice.  I'd have to say that this was the most difficult part of the whole process. We used our classroom assessment data along with our EQAO data to determine what our Problem of Practice is.  What was interesting was how many people took issue with the term "Problem of Practice".  Many didn't like the insinuation that there WAS a "problem".

This is where Instructional Rounds in Education would have come in handy.  It paints a clear picture of what a "problem of practice" is.  Had I read it before going through the Review Process, perhaps I could have helped alleviate some of the tension.  From my current understanding, a Problem of Practice does not reflect bad teaching.  It simply reflects a need in the school.  For example, we realized that, in general, the students in our school have a very limited vocabulary.  We started there.  But as we continued to reflect, we also realized that the students in our school have difficulty comprehending texts. We wondered, was the reason they had trouble comprehending because their vocabulary was so limited?  Or was it a bigger problem?  Was it that they couldn't make inferences?  Were they having trouble visualizing?  Or is it that overall, they lack background knowledge?  We also noticed that in our Junior Division, our students had trouble solving rich, multi-step math problems.  Should we focus on Math?  Or was the issue with the Math problems actually related to a reading comprehension issue, they couldn't understand the questions?

It was really great to notice the change in the conversations in the hallway.  Teachers were having discussions about teaching metacognition, whether they should do it explicitly in the beginning of the school year, or towards the end and whether or not we were using a common language for Math instruction.

After determining our problem of practice, (we decided to go with reading comprehension), we had to develop an "If... then" statement about something we were going to change in our practice to meet the need we wanted to address.  Again, it would have been helpful to read Instructional Rounds in Education first because it helps outline how to do this.  It defines the "If... then" statement as a "Theory of Action".  Eventually, (it wasn't until JANUARY!) we finally came up with our statement.

The next step was for everyone to participate in professional development so that we could successfully implement the teaching strategies we outlined in our Theory of Action.  The interesting thing I learned from reading Instructional Rounds in Education is that, according to the authors, just because teachers plan together and receive the same PD, it doesn't necessarily follow that what will happen in their respective classrooms will be the same.  After observing classrooms, it became evident to the authors that teachers interpret and implement what they have learned in different ways.

Enter "Intentional Interruption: Breaking Down Learning Barriers to Transform Instruction" by Stephen Katz.  Katz describes the barriers we have as teachers to learning.

I could see all of these barriers in action this year at my school.  It is so hard to get past them.  We tend to look at something new, and say "oh, I'm already doing that, I just call it something else", and not really take the new learning seriously. The other thing I hear most often "that wouldn't work with the students I have this year".

Why is change so threatening?  Why are we so reluctant to teach differently?  These are the questions I hope to tackle this year.

Since we only came up with our Theory of Action in January, and then we had to have PD, we were still novices at implementing the new strategies by June.  We haven't even had a chance to reflect on whether or not they have had an impact on student learning.

As a school, we won't be reviewed again for another five years.  That is just too far away.   It is so important that as educators, we reflect on our practice regularly.  We need to look at our teaching practices critically.  What impact did we have on student learning?  Not just most of the students, but ALL of the students.  Which group of students were we not successful with?  What adjustments do we need to make next?

This seems natural to me.  Maybe that is because of my nursing background.  As nurses, we were each the Primary Care Taker for 10 - 13 patients on the ward (depending on the unit).  As the Primary Care Taker, we were responsible to make decisions about the care for our patients.  Although I had 10 patients, and they all just had some sort of surgery, their needs were very different, and they healed at different rates.  We met weekly to discuss each patient's progress.  If someone wasn't progressing at the rate we thought they should, we worked together as a team to come up with different strategies of care. No one took it personally, no one thought it was a reflection of their nursing practice if someone didn't recover at the predicted rate.  We weren't reluctant to try new strategies - new dressings, new medication pumps, new protocols for healing; we were willing to try anything to be successful, and we documented DAILY the impact of what we did on patient recovery.  We charted the patient's "complaint", our observations, our plan of action, and an evaluation of the impact of our plan.

I think our students deserve that same quality of care, that same reflective practice.  I hope in September, at the first staff meeting, my school will begin by looking at their Theory of Action again and recognize that as a theory, we need to determine what evidence exists, if any, to validate it.  What adjustments need to be made to the theory? How will we dig deeper in the coming school year?

I leave you with the following video that I find so inspirational.  It is time to move from an Industrial Model of Education to a 21st Century Model of Teaching and Learning.  That means we have to be reflective practitioners, practitioners that can define our practice and the impact that we have on student learning.  In the video, Mr. Lichtman says we need to be self-evolving learners and in doing so, we teach our students to also become self-evolving learners.  We can't be afraid to let our practice evolve along with the world as it continues to change at such a rapid rate.  Watch the video - be inspired!