Problem based in learning in the context of math wars. Thoughts are myp.o.v.

I was recently given an article from Suril Shah (@thrilsuril), a colleague of mine in the peel Board, ( and then later on another article from another colleague Aviva Dunsiger, a teacher in the Hamilton School board ( ).  Both articles discuss (or rather reprimand) the notion of “Discovery Math” needless to say I had to respond.

As many of you know from my blog posts, math is a very important passion of mine.  I have in a way devoted my educational career to learning about math education and how it can help transform student learning.  This has gone on for me for the last 9 years of my teaching career and five years before volunteering at an amazing school in Peel.  Over the course of these 14 years the arguments in the above articles have always been happening; so I think it is funny that when Ms Wente mentions that this is a “new faddish fuzzy notion.”  Since mathematics was first introduced into the curriculum in the fifteenth century it has always been a debate over skill versus conceptual understanding.  This debate will always be there all I can give you is fact from experience and from the classroom (which I will say many who write articles in the Newspaper or make policy cannot).

Let me first start of with my own evolution.  Like many of you I was taught with very traditional methods.  My father drilled in me from a young age that fact recall was the most important thing.  I still remember practising for hours on hours flash cards and being randomly asked multiplication questions to see if I knew these facts.  I also remember that my Math class was all in a work book and my teacher sat at the front of the room and wrote many things on the board and then we did questions to practise and show what we learned.  This continued all the way through school and as I got into the high school and eventually University this is what I remember of my Math class.  Did it help?  No, I don’t think it did.  Don’t get me wrong, I did learn math.  In fact Math has never been a hard subject for me (except problem solving).  I was able to work through and memorize what was needed and then when the test came I was able to retell those facts and get an A.  My problems never came until University Calculus.  Here I because I didn’t have a good foundation in Calculus I struggled, in fact I failed. 

Sorry I digress here.  This method of teaching stuck with me, more so because this was all I knew.  During University I changed majors and decided to become a teacher.  I was able to volunteer at an Amazing school in Peel and soon learned Reform Mathematics (what discovery math was called at that time).  I was also fortunate enough to have an Amazing principal who let me question her and learn what reform mathematics was all about.  At first I said the same things that many of these article, and our parents say when they see problem based learning. You have probably heard these before (I know they are in the article):

1)      What is wrong with Rote, it worked for me?
2)      What about facts? There not learning them like I did?
3)      I memorized and got good grades?
4)      They can’t possible learn this on their own?
5)      What do you mean discovery? What is your job then?
6)      You’re the teacher so teach?
7)      This look chaotic, there is no order, how can they learn?
8)      What about the language, seems like more reading than math?

I can go on but they start to sound the same.  During this process I was able to see students truly excel and showcase their learning.  In fact, looking at scores (which is not the end all to be all), the school went from 42% to 93% in that first year in mathematics.  I was also able to reflect on my own learning and how I learn.  This started the ball rolling and has helped me to ask questions back.  Here are a few to think about:

1)      How do you truly learn as an adult learner? 
2)      Do you memorize things and then succeed? Or did you have to make mistakes, go back and relearn or have someone help you through it?
3)      When you are learning do you like to ask questions? Or just sit and receive information?
4)      (my favourite one) As a successful adult how did you become successful? What traits do you like in your employers?

Here are my thoughts to these questions:
I personal learn by doing, struggling, asking questions and then going back to relearn it.  True mastery comes from doing something over and over and over again.  Yes I can see how this backs learning facts, and I am not saying facts are not important, but my learning is in context to the concept not in isolation.  Memorization only works with some things but I still make mistakes no matter what I am doing and then I learn from them.  As for success to me I value students who are free thinkers, creative, adaptable and able to see past just simple direction.  This has been the case even when I was managing people in the private sector in my University jobs.  I don’t think the world can evolve from people who can only follow direction and not think beyond what is on the paper.
With this in mind I began my teaching career.  Here I too continued to question but now I also had to field questions from the general population about my style of teaching.  Here are my responses.
Q: Why is this better than traditional learning?
 A: I hope that I may have answered this above but most students, and adults do not learn through traditional learning.  There are a very few who do and we also have to consider that style but many don’t.  Learning is developmental.  It doesn’t happen in a linear fashion and PBL allows for this to happen.  Learning in PBL also doesn’t happen in isolation from the world, or other subjects.  It is always connected to a context, which helps all students to hold on to something and work with it.  Furthermore, all learners can access PBL, whether gifted or with a learning disability all students can do the problem.  Also, personally, it makes the day go by a lot faster, I enjoy it and so do my students.  Check out this video: for student reflection on what context can do.

Q: You know my kids don’t know facts, why aren’t you teaching them?
A: First and foremost, I want this to be said, “FACTS ARE IMPORTANT!” they must be taught and learned; however, how are we learning them.  Let’s go back to my question back to you.  Can you recall something where pure fact learning has help you be successful?  If yes, no think was it just fact memorization or was it in a context?  Fact knowledge is important and needs to be done.  I prefer to do this through games and mini-lessons.  This allows me to talk about a strategy and have students discuss the pros and cons of the strategies.  The talk focuses the learning.  Check on my previous blog post on it.

Q: “Teachers and Students are learning together” Great so now we have the blind leading the blind!
A: This is the one that bothers me the most.  It bothers me because PBL actually takes more understanding, more planning and a lot more patience then traditionally teaching.  I have almost completed my thesis, in where I researched the impact of my questions on students learning of fractions.  It was interesting to see where I had moments of direct teaching that my students stopped talking.  In fact, they just sat there.  Which is exactly what traditional teaching does, students sit and listen then do.  PBL takes planning.  In another of my posts I talk about five practises that teachers should be following for PBL implementation ( ).  Teachers actually need to learn the mathematics and it is through critically placed questions that the learning is brought out.  Students develop at a faster rate through this proper questioning style and can achieve a higher level of understanding.  I have grade twos right now who are learning about equivalent fractions, ratios, division and adding three digit numbers in their head.  It is truly amazing to see what they can do.  But this takes planning on my part.  It takes understanding of learning trajectories and  understanding what students are doing (so you can redirect or push beyond) and understanding the math to be effective in PBL.

Q: Test scores are falling?
A: this might be so but I would caution you on this.  First of all tests are a snap shot of learning at a particular moment in time.  They have a place in assessment.  In my personal opinion a very far place but a place nonetheless.  There are many factors to low test scores: 1) poverty, parents education, home life, social problems that day, being sick, stress, reading level, context, etc. The list is endless.  When we put all emphasis on test we are taking away so many other factors of learning.  I know more about a student from a problem that they solve then by what they can retell me on a test, just a matter of fact.
 I am going to stop here for now as I think I have written more than I ever have in a blog.  This topic is very dear to me and I have heard a lot of the questions in this “Math War.” It will not go away but please don’t take this as a discouragement to stop PBL or even start.  To me PBL is the best way for ALL students to learn.  It gives the teacher the most time for true assessment and understanding of their students needs and next steps.  It allows you to meet all levels of students and be able to get to all of them.  I have and will continue to only teach through PBL (problem based learning).  Love to hear your personal stories, questions or answers to this lovely debate. 

Accountable Talk in the Classroom: Practical Advice for the Classroom

I have recently finished one great book and one great article on Accountable Talk and Classroom Discussions. 

Stein, M. K., Engle, R., Smith, M. & Hughes, E,  Orchestrating productive mathematical discussion: Five practices for  helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10313-340. 

Chapin, Suzanne, O’Connor, Catherine, & Anderson, Nancy. Classroom Discussions: Using         math talk to help students learn. California: Scholastics. 2009.

Accountable talk is one of my passions as I have spent the last four year studying the impact it has on my classroom.  I highly reccomend these two readings for anyone interested in learning more about accountable talk.  However, I also know that in teaching we really don’t have time to sit down and read.  For this reason I thought I would summarize them for you and include them in my blog (I appologize in advance as this will create a rather long post).  These ideas come from the two resources above and my own thesis work.  I hope they are practical advice for anyone in their teaching practice.

Implementing Classroom Discussions
Establishing and Maintaining a Respectful, Supportive Environment:
·         LAY DOWN THE LAW (in a collaborative manner):
o   that every student is listening to what others say
o   that every student can hear what others say
o   that every student may participate by speaking out at some point
o   all have an obligation to listen
·         neither student or teacher will participate in bad environment.  Everyone needs to feel comfortable.
·         Emphasize the positive and forestall the negative
·         Establish classroom norms around talk, partner work, and discussions (what does it look like, sound like and what should we be doing)
·         everyone has the right to participate and an obligation to listen
Focusing Talk on the Mathematics:
·         During the discussion time you need to focus the talk on math:
o   plan your questions carefully
o   Have good formative assessment happening at all times
o   Make a plan as to what big ideas you want to cover
o   Anticipate problems and possible solutions
Providing for Equitable Participation in the Classroom Talk:
·         Here are some strategies that will assist you in making it all equitable:
o   Think-pair-share
o   Wait time
o   Group Talk
o   Partner Talk
o   Debates
o   Random  Choice on who Talks


Types of Talk Moves:
Talk Moves That Help Students Clarify and Share Their Own Thoughts
·         Say More:
o   Here you literally ask the student to explain more.  “Can you tell me more?”, “Tell us more about your thinking.  Can you expand on that?”; or “Can you give us an example?”
o   This sends the message that the teacher wants to understand the students’ thinking.
·         Revoicing:
o   It is sometimes hard for students to clearly articulate what they are trying to say by revoicing or having a student do this it allows the original student to check and make sure what they said is true or to hear it in a new way
o   It is not just repeating but more of paraphrasing the students ideas
·         Model students thinking:
o   This is not so much a talk move as it is a way to help talk
o   As students talk record what they are saying without comment.  When they are done ask them , is this what you meant?
o   This allows students to reflect and think about what they said in comparison to what was written
·         Wait Time:
o   Wait time is so important.  I cannot stress this enough.  The longer you wait the better responses you will get.  It allows students to process what you or another student asked and be able to formulate their thinking
Talk Moves That Help Students Orient to Others’ Thinking
·         Who can Repeat?
o   I would classify this under the first category but it also helps students with understanding what their peers are saying
Talk Moves that Help Students Deepen Their Reasoning
·         Press for reasoning
o   Here you are basically asking students to think about why they did this.  This can be done by asking:
§  Why do you think that?
§  What convinced you that was the answer?
§  Why did you think that strategy would work?
§  Where in the text is their support for that claim?
§  What is your evidence?
§  What makes you think that?
§  How did you get that answer?
§  Can you prove that to us?
o   Not only are these excellent talk moves but excellent questions that push students beyond their thinking and make excellent mathematical connections.
Talk Moves That Help Students Engage with Others’ Thinking
·         These are excellent questions that help students build upon their own thinking and the thinking of the community
·         Do you agree or disagree…and why?
o   This really brings students into direct contact with the reasoning of their peers
o   You can do this by:
§  Thumbs up or thumbs down
§  Why do you agree or disagree?
·         Who can add on?
o   When you ask this question make sure that you wait for answers as this may need time to develop connections.
1: Anticipation (P.322)
The first thing is for the teacher to look and see how students might mathematically solve these types of problems.  In addition, teachers should also solve them for themselves.  Anticipating students’ work involves not only what students may do, but what they may not do.  Teachers must be prepared for incorrect responses as well.
2: Monitoring students’ work (P. 326)
While the students are working, it is the responsibility of the teacher to pay close attention to the mathematical thinking that is happening in the classroom.  The goal of monitoring is to identify the mathematical potential of particular strategies and figure out what big ideas are happening in the classroom.  As the teacher is monitoring the students work, they are also selecting who is to present based on the observations that are unfolding in the classroom.
3: Selecting student work (P.327-328)
            Having monitored the students, it is now the role of the teacher to pick strategies that will benefit the class as a whole.  This process is not any different than what most teachers do; however, the emphasis is not on the sharing, but on what the mathematics is that is happening in the strategies that were chosen. 
4: Purposefully sequencing them in discussion (P. 329)
With  the students chosen, it is now up to the teacher to pick the sequence in which the students will present.  What big ideas are unfolding, and how can you sequence them for all to understand?  This sequencing can happen in a couple of ways: 1) most common strategy, 2) stage 1 of a big idea towards a more complex version or 3) contrasting ideas and strategies.
5: Helping students make mathematical sense (P.330-331)
As the students share their strategies, it is the role of the teacher to question and help  them draw connections between the mathematical processes and ideas that are reflected in those strategies.  Stein et. al. suggest that teachers can help students make judgments about the consequences of different approaches. They can also help students see how the strategies are the same even if they are represented differently.  Overall, it is the role of the teacher to bridge the gap between presentations so that students do not see them as separate strategies, but rather as working towards a common understanding or goal of the teacher.


Trouble Shooting Talk in the Classroom

My Students won’t Talk:

v  First ask yourself: our my students silent because they have not understood a particular question? –> sometimes they need to hear the question a few times and have time to think
§  if this is the case then give students time to think  (wait time is very important)
§  also revoice it or have another student revoice the question
v  Second they may be shy or unsure of their abilities:
§  If this is the case you may need to revisit strategies for talking
§  Think-pair-share is an excellent way to get kids comfortable to talk
§  it will also take time to get kids comfortable.  Wait time again is important as it holds students accountable.  Also making them feel comfortable and that mistakes are okay will assist with this difficulties
The same few students do all the talking:
v  Wait-Time:
§  I know that I say this a lot but it allows the other students to think and then participate while making the ones who always participate  (it will feel awkward at first but wait as long as you can)
v  Have students Revoice:
§  This is good strategy to bring validity to students answers and encourage others to talk
v  Conferencing with the ones who talk a lot:
§  You also don’t want to ignore the ones who talk  all the time.  You can talk to them and let them know that you are not ignoring them but are just trying to allow others to participate.
v  Turn-Taking/ Random presenters/ group discussions:
§  These are all roughly the same strategy.  It allows you to have certain presenters share their thinking without offending or allowing others to take over the conversation
Should I call on students who do not raise their hands?
v  there is research to suggest that students will learn by listening but you will also hinder the class progress in discussion.  To help try creating a positive space that allows all students to feel comfortable and willing to participate.
v  “right to pass”: 
§  allow students at the beginning of the year the right to pass.  You’ll notice that they may do this at first but as you build the community they do this less and less
v  Call on reluctant to students after partner talk:
§  Often when you give them a chance to share first they are more willing to share or at least have a response from their partner
My students will talk, but they won’t listen
v  Set the classroom Norms:
§  remind each students that they have the right to be heard but that this also means an obligation to listen
v  Students Revoice:
§  When students need to revoice then they have to listen
Huh?” How do I respond to incomprehensible contributions?
v  The temptation is to simply say, “Oh, I see.  How interesting….” and quickly move on to another student.
v  Try Revoicing or repeating what they have said.  After you have done this ask them is this what you meant?
v  Record their strategy on the board and ask them is this what you meant?
Brilliant, but did anyone understand?
v  Repeat what they said, then have another student repeat what they have said (if really important have many students repeat)
v  Break the explanation up into small chunks and revoice or have the students
I have students at very different levels
v  Pair students in ability groups:
§   Similar abilities with similar abilities.  This allows students to contribute at their level and to also struggle at their level.  In addition, it allows you as the teacher to differentiate as needed.  When you scaffold you can do so by group not by individuals
v  Parallel Tasks:
§  Give students similar tasks but with varying degrees of difficulty (still around the same big idea)
What should I do when students are wrong?
v  First ask yourself is there anything wrong with having the wrong answer?  Sometimes wrong answers provide rich and meaningful discussions
v  Need to establish Norms around respectful discourse and discussion with wrong answers
v  Mistakes are always an opportunity for learning to happen

This discussion is not going anywhere or Students’ answers are so superficial!
v  This may be happening because you are asking to many students to share or revoice the ideas that are happening in the classroom or in the case of superficial classroom  norms have not been established or the types of questions have been simple and direct
v  Use the working on phase as an opportunity to direct your bigger discussion:
§  As you are walking around and looking at work, look for the progression your students are taking.  This will lead you to a group discussions.  What questions are the students asking themselves?  What problems are occurring?  What big ideas are they trying to work out, have worked out or are struggling with?
v  Look at the type of questions that you are asking:
§  As teachers we are comfortable asking questions but do our questions already have responses?  Are we leading the kids to OUR thinking or our we allowing the students talk to LEADthe thinking.  Yes you are very much in control of the discuss and have to lead but it is not YOUR thinking but THEIRS that should be articulated.
§  Higher order questions build-upon or go beyond the thinking that is being presented.  As a teacher we need to help with the connections in mathematics.  Compare student work?  Compare strategies, Pros and Cons, naming and identifying.  We need to go beyond just show and tell

Another day in patternville

It’s been awhile since I last blogged but we have been on our two week fall break.  It is quite nice to be apart of a balanced calendar.  The kids have been great getting right back into the swing of things Today in math class we are talking about patterns. From my previous blog post students were doing identifying non patterns and patterns. Now they are creating their own patterns for another group to solve. Take a look at the patterns below Once the groups finished they had to tell if it was a pattern, then tell the rule of the pattern and finally extend the pattern by three terms. Here are some of their work: 


This had some very interesting results. I had some students unable to make patterns, and some students who only made repeating patterns.  The interesting part is when it came to explaining all students recognized if it was a pattern or not but some struggled to make them.  I found this interest because of they can recognize patterns they should be able to make them.  However, that is not the case, I wonder why? More work might be needed in explaining patterns and identifying key attributes of patterns.  Overall, this was a great exercise to help the students solidify their understanding with patterns.  Our next big problem is looking at how two patterns are a like and not alike (e.g. 2,4,6,8,10,… And 3,5,7,8,11,…..).  Stay tuned for more on patterns

Patterning in grade two

Today was our first formal lesson in patterning.  What I mean by that is we have been discussing patterns but more in the context of number sense, where we have been learning to count by twos, fives and tens, as well as, doubling numbers. This type of talk has been focused on the magnitude of numbers and associated with place value not so much on growing and shrinking of patterns.

So today we started with a problem that was asking the students to sort eight sets of (patterns and none-patterns) into two categories, a yes it is a pattern and a no it is not a pattern. I have attached the patterns down below.
It was very interesting conversation around this. At first the students put only the repeating patterns in the yes category stating that for it tone a pattern then it had to repeat. I had to remind them about the book we read called patterns big and small. In this book they had a set of nesting dolls, I’m asked the kids did that pattern repeat? After that discussion the kids where better off explaining their reasons for their groupings.
I really liked this question because it made the students really think about what a pattern is and what attributes are needed to make a pattern. They obviously had worked with patterns in grade one but mainly with repeating patterns, which is why they at first they only made piles with repeating numbers.  The non-patterns are also helpful because they can assist us with thinking about what attributes a pattern doesn’t have and therefore in the end has.
Today we are going to be working on the justifying of their answers and then coming up with a definition of what a pattern is and is not.  If your grade two or any grade for that matter I highly recommend this type of problem for your class.

For the patterns we used click on this link:



Madeline Math Problem

This was a really cool problem.  The kids read the story of Madeline and we discussed all of the math problems in it.  After closely looking at the story the kids saw that the girls always walked in two rows.  We had a discussion about how many kids that was and how they knew.  I then had them go back to their seats and work on this problem:

1) Suppose that there was 7 kids in a row, how many would their be altogether?

2) What if there was 8 kids in a row, how many would their be altogether?

3) What is there was 23 kids in a row, how many would their be althogether?

Here is what the kids said:

Dot Plates

It has been an amazing experience working with Dot Plates.  Such a simple exercise but what rich discussion we had.  In this simple exercise my students learned about subitizing, counting on and one to one tagging.  They also learned that numbers are made up of other numbers and that there are parts to numbers.  This is the foundation for addition and subtraction.  If you want to create your own dot plates all you need are simple 35 paper plates and bingo dabbers.  The patterns are simple here is a youtube video to follow:   Our next move is to play a game called part-whole Bingo. I’ll fill you in on how that goes in my next blog.

Inquiry doesn’t promote fact growth, or does it?

I was out with a friend of mine last night when we got into a heated debate about inquiry based learning. I was fine with his opinion until he got to the part that inquiry learning does not promote facts. His argument was that teachers spend too much time letting the kids explore that they forget about the actual computation that is needed for them to do the math.

My argument, was that though I can’t speak for ever teacher who teaches inquiry, only myself and those that I have seen, it does. Not to be frank with my argument but it does. As a teacher it is my job to make sure that my kids are learning mathematics. This means that they aren’t just figuring out amazing ways of solving the problem but are actually talking about the mathematics.  Facts are amazing but without a context they are just facts, meaningless and useless facts.   In an inquiry based learning environment students do learn their facts, maybe not as quickly as if I used flash cards and mad minutes but they don’t forget them once they have learned them.  In addition, my students also learn them in engaging ways, through games and contextual problems.

But this is only my opinion, would love to hear what you all think?