Making kids doers of math instead of Doing math!

I have been thinking about this topic for quite some time now and then when asked to do the OAME 2015 ignite I thought this would be an amazing topic to push thinking.

My biggest fear in education right now is that we are having our kids go through the paces of doing school. We our turning our students through the drudgery of school.  Before I started to really question this thinking I saw it in my own class. My students were coming to school going through the paces and then leaving. Sure they enjoyed it but was I really making them think? What type of work was I making them do? Why was I teaching them these skills?

I then came across this statement by Fosnot:

The purpose of teaching is to learn, but without learning there is no teaching!

I was shocked. Was she saying that if my students didn’t learn then I wasn’t being a good teacher. The answer was yes! And the more that I reflected on this the more I agreed with this statement. Over time, i realized that even though I was teaching different kids the common denominator was still me. So when I asked questions like, why don’t they get this? The answer was because I am not doing a good enough job. I wasn’t making them understand because I was just making them be there instead of embodying the learning.

I see this a lot in math and this is partially because of my lens. In a math class we traditionally stand I front of students, give a lecture, let them work and then test them to see if they understand. B how many of our students are really learning? How many of them become mathmaticians? 

VanDeWalle suggests that The goal is to let all students believe that they are the authors of mathematical ideas and logical arguments.

So then how do we go about doing this?
I would like to propose three key points to this:  Link back to my thesis always link back

1) Role of the teacher
2) Environment of Learning
3) Accountable Kids

Role of the teacher

I want to first preface that teaching to me is about turning my kids into mathematicians through inquiry and exploration but I start with this point because as a teacher we have the most critical role to play.We are not to sit back and allow our students free reign but to ignite (lol) and actually talk about math. I know really insightful!

Researchers have suggested that children should being engaged in problem versus talk procedures. But our role is to bring out the math not by telling students information and expecting them to regurgitate it but by creating contexts for learning asking critical questions and debriefing the math. In my research I found three types of questions that worked the best for creating these conditions:

1) Interrogation: Just like the title suggests → a lot of why’s and how comes2) Going beyond: Pushing the thinking beyond the schema the student has created. These questions include, have you thought? What about this? Can someone else explain3) Comparing: Often I compare strategies together to see if students can move from one to the next. This includes, what are the differences? Similarities

In order for this process to really work “Teachers must have the [student learning] in mind when they plan activities, when they interact, question and facilitate discussions” ~ Fosnot pg. 24

The key to everyone one of these questions is that it was linked to a big mathematical idea. One that was key to the learning of the student. The same goes to the various talk moves that a teacher can make. These should include: Wait time and revoicing. I cannot stress how important these two items are to the success of building mathematicians. To often we don’t give students enough time.

Creating an environment of learning

In a mathematical environment , students feel comfortable trying out ideas, sharing insights, challenging others, seeking advice from other students and the teacher, explaining their thinking and taking risks. ~ VanDeWalle pg 36. When students do mathematics in an environment that encourages risk and expects participation, it becomes an exciting endeavour. Students talk more, share more ideas, offer suggestions, and challenge or defend the solutions of others. When a context is real and meaningful for children, their conversation relates to the context. They mathematize the situation. ~Fosnot

Making kids accountable:

No one is allowed to be a passive observer ~ VanDeWalle pg 36

I love this quote. I think it is exactly the whole idea around accountable talk. Many teachers may think that just because the student is not talking they are not participating but the key is not to be a passive observer, which doesn’t always involve talking but listening. However, that has not been the case in school. We have been so use to hearing teachers talk that many of our students are use to being told the answer that they are not use to talking. 

In my thesis research I saw that when I asked an Initiate respond Evaluate types of questions (basically questions I already knew the answers) I got no further discussion happening. My kids just sat there. But when I asked going beyond types or comparing questions, basically critical thinking questions, that was linked to big ideas kids talked about math.  They became active users of the information and doers of math not just following the paces.

So I guess I want to ask: How do you make your students into Doers versus just doing? This question doesn’t need to be math as it is a broader problem in education. Love to hear your thoughts and ideas.

Fraction kit and playing games

Fractions have always been a passion of mine. Started researching the concepts in my math part 1 AQ class and have been fascinated ever since.  I even ended up completing my Masters’ of Education thesis in the subject.  Through my studies I came across fractions, Marilyn Burns’ fraction kit and games.  I still haven’t found something anywhere close that helps students understand fraction concepts like this kit.  
For those not familiar with it, let me tell you about it.  The kit in itself is very simple, it is five strips of paper. Each piece is to be cut to a corresponding fraction (halves, quarters, eighths, sixteenths, and a whole). 
Now you may ask yourselves how is this the best thing ever it’s just a bunch of paper. It’s the best thing ever because of the talk that it generates. Since finding this in my research I have done some modifications that really bring out the talk. 
First and foremost, I have them create the kits. It does you no good to create them for your students.  By them creating the strips, the students explore how fractions are division, fair sharing, why fractions are a part of a whole and many more fractional concepts.
Second, I created a context to go with the problem. As many of you know who read this blog, I truly believe in contexts. A good context makes kids think beyond arithmetic and focus on mathematical big ideas.  For this problem I tell my students a story of how I need to clean up my mom’s back yard, she has a huge yard and in payment my mom buys me a large party sub. Now many students now don’t know what a party sub is because they don’t sell them anymore, so you may have to show them a picture: 

The students are so impressed and they can’t believe that I would eat this much. Now I tell them that just before I was about to eat lunch one of my friends popped over. Now what?  This continues all the way to eights, the door bell ringing every time we figure out portion we need to cut.  For sixteenths I tell them this is what we are going to do as I really don’t have sixteen friends; however by now we have really constructed a good understanding of the pattern that is happening.  Now why this context. I like this context because it is a linear model like the strips. Having the sub also means students have to think about measurement and division because technically you cannot fold a sub, as all the pieces fall out. The other part is students often will try cutting the their strips horizontally instead of vertically. Now this also brings up interesting discussions about equivalency versus congruency but this context stops that because if students cut a sub horizontally they don’t really get all of the sub.
Third I don’t have the students label their fractions.  When I have done this with my fours it was mainly because I didn’t want them to associate a particular fraction with the strips whole. Basically, 1/2 strip is 1/2 of the kits whole not 1/2 somewhere else.  A big misconception with students thinking is that a what they learn is he only representation of a particular fraction. When you label the students don’t understand that the size of the whole matters.  That 1/4 can be bigger than 1/2 depending on the size.  However, now that I am in primary I see a whole new benifit, it makes students understand what a fraction is. Why is 1/4, 1/4? While my students where playing cover-up, one of Mariyln burns fraction kit games, they asked me which fraction is 1/4? I turned it around and asked them. They then just picked a random strip up. I the. Asked them why that one? This discussion continued as students explored that the amount of pieces that we break our sub into is our denominator and the amount we use is our numerator.   If I had them label the fractions they never would have explored this concept and I would never have realized that they struggled with it.
The final change is the questions that I ask around this particular problem.  It’s not just to make the stud ets create the kit but to think about the big ideas around fractions. Have a listen to my grade two class discussion on fractions:

Day 1 of our Fraction Talk

It is quite interesting the talk that can come from building these kits and the big ideas that come from it. I have played this game in junior and primary and personally I would do this for middle school as well.  In junior I start to add fifths, tenths, thirds, sixths, ninths, and twelves.  By adding these other fractions you also start to see other misconceptions of students halving strategies but for primary halving is still okay.  I hope you really try the kits and see the benefits of it in your classroom.

You can find all of my fraction research and resources on my site:  Feel free to use anything you want.

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

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.

Reflection on Classroom Practise and the types of Talk moves/ Questions I ask

I am in the process of analysing my research for my thesis.  My thesis is on the impact my questioning had on student learning of fractions.  I was quite surprised at the amount of questions I asked and the types of questions I asked.

 Have a look at the chart below:

Types of Questions
Amount of Times Asked
Talk Move
Big Idea
Doesn’t initiate any discussion
T- Building on
49 (16.9%)
T- Introduce new strategy that has not been developed
14 (4.8%)
T- direct teaching
27 (9.3%)
T- Go Beyond
75 (25.8%)
2 (.68%)
T- Initiation- response- evaluation
7 (2.4%)
T- Interrogation
73 (25.2%)
23 (31.5%)
50 (68.5%)
T- question unclear
3 (1%)
T- Scafolding
32 (11%)
T- shares strategy
8 (2.7%)
Total of Questions:
290 (49.3%)
31 (10%)
222 (74%)
37 (12.8%)
T- Air Misconceptions
T- answering with another question
T- Echo’s students words
T- Letting students just talk
T- Monitoring students
T- no confirmation/ in order to push beyond
T- relate back to context
T- relate to other problems
T- Revoicing
T- Student revoicing
T- Think, Pair, Share
T-Wait Time
T- Checking for understanding
Total of Talk Moves
298 (50.7%)
24 (8%)
40 (13%)
234 (78.5%)
Totals altogether
55 (9.4%)
262 (44.5%)
234 (39.8%)
37 (6.3%)

The chart is split into two different groups Questions (in black) and Talk moves (in red).  I tallied all of them together and in a three week unit I ask or did a total of 588 talk moves/questions.  This first of all surprised my that I ask or did so much.  Most of the time we often think of teaching as just standing there and lecturing, not getting the student involved.  however, that wasn’t the most surprising stat.  What really got me going was that even though I may have done more talk moves then asked questions the majority of these actions were related to a big idea.  I wasn’t just trying to get the kids to talk about the subject, I wanted them to articulate a big idea of point in mathematics.

So I ask you to think about your practise.  What types of questions are you asking?  What are you doing to make your students talk?  What is the majority of your time in a unit spent on?  Just some things to reflect on.

Creating Accountable Talk in the Classroom

Accountable Talk is a big passion of mine.  Seeing the results of the students talking is truly amazing.  Here are just some small tid-bits that I compiled to help create accountable talk in the classroom.
Accountable talk just doesn’t happen, no matter what age group you are teaching, you have to create conditions for it.       

1) Students have to feel like they are welcomed (which I know we all do as educators)

2) All voices are heard à this is the hardest part.  We sometimes only chose certain kids to talk

3) At the beginning of the year like many teachers I spend a lot of time on training my students to work in partners, what talk looks like, and sounds like.  We go over rules for partner talking and what my expectations areThis is rough at the beginning of the year.  I often do this through games, not only is this great in primary but it works well in junior.  As you are playing games you are also teaching many of the math concepts and having small conferencing moments with the students. You get great diagnostic assessment and provide formative assessment right on the spot.  For junior I tend not to spend as much time and introduce games every Friday because of how short on time I am and how dense the curriculum is in junior (spend a week if not two though).

There are also talk moves that you can be constantly do:

1)      Wait time: à when students have enough wait time they will participate (this takes time)
§  At the beginning of the year this wait time feels like hours but if you don’t give it then they won’t talk later
§  When you wait the accountability is on them
§  Kids need time to process
§  Add in think pair share here: à great teaching tool to promote talk
2)      Revoice: 
          When you revoice what the students have said then they feel accountable to the work.  It validates their opinion but at the same time makes them think about what they are talking about
          You can also have the other students revoice: à this holds other students accountable to contribute to the community and that they have to listen
3)      Just don’t talk:
          I think that sometimes as teachers (me included) we talk too much
          I sometimes don’t say anything and then a student jumps in (let it)
Finally talk will not happen if you don’t plan for it to happen.  You must think about what big ideas you are going to be discussing.  They sometimes don’t happen but if you have things planned out you can create questions to lead students back to these ideas or be prepared to discuss what the students are talking about or ready for. 

Asking Good Questions

Asking questions has always been an important aspect of any teachers job but understanding what makes a great question is the hardest part of the job.  As a teacher we have watched those Professional Development (PD) videos on the classes that seem to be in rich discussion, always learning and having students say such wonderful and impacting statements.  I know I often sat in those said PD sessions and said, ” How in the world did that happen?” or “Those students must be the best of the best?”  It wasn’t until I watched my own videos up in a PD session that I realized there was more to this then meets the eye.

Part of my research has been to look at how my questions impact the learning of my students understanding in mathematics.  As a secondary question I also wanted to understand how teachers plan in order to ask good questions.  I have noticed three important aspects that may help in asking good questions.

The first is that as teachers we need to plan for good questions and good talk. Discussion just doesn’t happen.  We may think that they do but real discussion takes time, just like real learning takes time.  If we want to impact our students learning, we, as teachers, need to plan for it to happen.  The first step is planning meaningful, rich tasks that allow students to explore the concepts.  These tasks need to be open ended and have a real context for all students to access the problem/tasks.  The second step is anticipating students problems, responses and learning.  I often have these mapped out based on my experiences, learning and research into the subject matter.  As a teacher we MUST understand my students learning and we MUST understand the curriculum and concepts being taught.  It is more then just opening a textbook and learning steps or procedures to solve the problem.  When you can identify the problems students may have you are better prepared to give a question instead of directly teaching the concept.

The second aspect of asking good questions is the type of questions that we ask as a teacher.  Often, (and I am included in this) we as students questions that only have one answer, or we just want to check for understanding and move on.  If we want our students to develop deeper understanding our questions have to be focused on learning objects and linked to further explanation of concepts.  For this to happen our questions should: 1) push our students beyond the basic procedural output and into connecting it to conceptual big ideas; and 2) introduce connections to other concepts or subjects.

The final aspect is giving our students wait time to respond.  Often, we expect an answer to a question right away.  This is due to the fact we already have an answer that we are looking for or the question only has one answer to respond too.  When we give our students the time to think they have time to develop an true understanding.  When we rescue our students or go right to direct teaching we rob our students of their understanding and thinking.  In addition, the wait time also allows our quieter students to feel a part of the community and wanting to participate.  It honours their learning.

Asking good questions just doesn’t happen.  It takes practise, it takes time and it takes patience.  You will make mistakes but that is okay.  Some times it may seem that you are going backwards in learning but when you sit back and reflect your student’s learning, you may just be surprised; I know I have been.

For more information read the following articles:

Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell.  Mary Kay Stein, Randi A. Engle, Margaret S. Smith, Elizabeth K. Hughes.