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.  

Why Are We Looking at This Data?!?

Percentage of Students at Level 3 and 4
EQAO School Report September 18, 2013

Last year was my first year back in the classroom after working as an Instructional Coach for two years.  I really tried to put into practice all of the strategies I had read and learned about.  It was a wonderful year, the best year of my teaching career thus far.  My students and I had a blast.  In comparing assessments from September to June, I truly believed that my teaching had had a positive impact on my students' learning. 

Then the EQAO scores came out.  I was crushed! I fully recognized that the Junior EQAO Literacy and Numeracy Assessments really are just one type of assessment and can't possibly validly assess all that my students had learned over the past year.  My students had learned to work collaboratively to solve problems, share their thinking, and connect with others outside of their classroom to learn authentically from the world around them.  How can EQAO possibly assess that?  But still, I believed that all of those experiences would have had a positive impact on their ability to be successful on the paper/pencil EQAO test. 

That's probably because one of my very favourite books is "Hooray for Diffendoofer Day" by Dr. Seuss (with the help of Jack Prelutsky and Lane Smith).  In fact, I have always read it to my Grade Six students on the first day of school.  In case you are unfamiliar with it, I've included a YouTube video of the story.  Basically, at Diffendoofer School, the students learn differently, and while they never prepare for the high-stakes test, they are more than ready for it because of their unconventional learning.  Oh - you've just got to watch it - it's still one of my very favourites!

So in my new role as a Curriculum Consultant I was recently at one of the schools I support conducting an Item Analysis of the EQAO data for the Primary and Junior Divisions.  Together with the School Improvement Team, we analyze the EQAO data to determine our students' needs because our student needs tell us where we, as teachers, need to focus our learning.  It is very public knowledge that the scores for Math in the Junior Division across the province are dismally low.  Why?  One of the teachers conducting the Item Analysis with me said "We've been asked to use the 3-Part Lesson in Math and to teach through Problem Solving.  When are we going to accept defeat and acknowledge that it doesn't work?"

And for the first time I had doubts.  I doubted the efficacy of teaching Math through collaborative problem-solving because of the low scores in my own school in Junior Math.  Had I been wrong?  Could everything I had been espousing been incorrect?  I couldn't answer that teacher's question; I couldn't blog - I felt like a fraud.

I needed to take a hard look at the data from the school in which I had taught. What could that data tell me?  What couldn't it tell me?  When we look at the data, we don't only look at the Achievement or Outcome data.  We also have to look at the Contextual or Demographic Data as well as the Perceptual Data.  That is why we shouldn't ever rank schools.  We have a Community Living Classroom in our school and those students are not physically able to participate in EQAO but they are still counted in the overall scores.  We also have a very high ELL population and sometimes have to exempt students who arrive from non-English speaking countries only weeks before testing.  When we looked at the percentages of participating students, our results were a bit better.  When I looked at the scores for my class alone the data looked even better still.  78% of the students in my class achieved benchmark levels in Math. Of course, when you are talking about only 22 students, it is difficult to talk in percentages, but I was relieved none the less. 

I am tired of hearing "these kids can't" and want to prove that these kids most definitely CAN! Last year I became convinced that Carol Dweck was correct in her theory on mindsets.  Basically, she says we can have a growth mindset, where we believe accomplishments are a product of hard work and dedication, or we can have a fixed mindset where we believe intelligence is a fixed entity that can't be changed.  Dweck's research demonstrated that teachers who have a growth mindset are better able to motivate and engage their students.   

Why do I pour over EQAO data?  Because it is one of the best tools we have in this province to reflect on our impact on student learning.  I think my class' EQAO data proves Dweck is correct.  Here's a concrete example.  I had one student who struggled all year in Math.  Early on in the year, she told me she didn't like Math (Perceptual Data).  When I asked her why not, she explained that when she had been in Grade One her teacher had told her she didn't have a brain for Math.  I told her her Grade One teacher was wrong, and I would prove it.  A couple of days ago, I called her at home to celebrate her level 3 on the Math EQAO scores.  She said, "So does that mean I am good at Math?"  I answered, "It means you can be good at anything you want to be!"

So why are the Junior Math scores so low in our province?

I have a theory about the Math.  Traditionally, teachers have always taught the Math lesson to the whole class, then assign a set of questions from the text, and then take up the questions with the whole class.  This teaching strategy has a certain level of effectiveness.  I believe teaching Math through collaborative Problem-Solving is more effective BUT ONLY IF THE TEACHER IS GOOD AT FORMATIVE ASSESSMENT.  If the teacher is teaching through Problem-Solving but does not begin with Assessment FOR Learning, the impact on achievement is lower than the impact of teaching with the text book.  I was able to positively impact my students' learning because I started by finding out what they already knew and what misconceptions they had; then I worked toward closing gaps and correcting misconceptions.  This style of teaching does not work if you don't first teach your students how to communicate their thinking in Math. You have to ask the right questions to elicit their understandings.  You have to know the significance of what they are saying.  If you only ask for an answer, you have no idea how they got there!

During one of the Item Analysis meetings at one of my schools, two of the primary teachers were discussing the following exemplar from the released EQAO Math scoring guides.  

They were having an excellent discussion about whether or not this should be considered a level 3 response.  (By the EQAO scoring system, is was considered a level 30 response).  We ended up discussing what the work told us about the child's understanding of the Math concepts.  Did the child in fact use critical and creative thinking to solve this problem?  What does this child's solution tell us about his/her understanding of Math concepts?  Are we just looking for the right answer?  Or are we assessing the child's level of understanding?

We need to know the kids that we are teaching.  We need to know what they know and what they can do.  We have to give them multiple opportunities to explore Math concepts until they develop deep understanding of these concepts.  

As long as EQAO is out there, I will be pouring over that data, analyzing it, trying to determine what pieces we are missing and how we can do it better.  The current scores in Junior Math tell me that we still need to learn a lot more about how students learn Math.  

I have a fantasy that one day we will live in a world where no one ever says "Oh, I'm not a Math person, I've never been good at Math."

Are They Really Understanding What They are Reading Update - Using Descriptive Feedback

If you read my earlier post on teaching my students how to respond to what they read, you will know that this past week during our Guided Reading sessions I was providing support to my students as they responded to what they read.

We have been reading some great articles on Space about space rocks, extraterrestrial life, and frozen carbon dioxide on Mars.  On Friday, my students had to independently respond to an article they had read.  I told them that I would be using the Success Criteria we had created together to provide them with a grade or "level".

One of my students put up her hand and asked, "Instead of giving us a mark, could you just give us some descriptive feedback so that we know how to improve our work?  And then, could you give us a chance to re-write it?"

Sometimes we are so stuck in our old ways of doing things.  Thank goodness our students are not so encumbered by old habits!  My students have completely embraced the use of descriptive feedback to improve their work.  Now I just have to do the same!