A Balanced Math Program

With the endless humdrum of the math wars happening, it is easy to forget that what teachers really need is simple help to understand practical ways to improve or acknowledge their math program.  We have heard both sides for decades. One side is about the context and deeper conceptual understanding while the other side worries about the basics. To be fair there really shouldn’t be any sides. Mathematics is a combination of both concepts and procedures. Even more realistic you will never find a teacher that doesn’t do both.

I love this picture from one of the presentations that Matthew Oldridge and I do on this topic:

Screen Shot 2017-10-16 at 12.07.09 PM.pngWhat it shows is a continuum of teaching. At times, we may be closer to the fully guided while at times we do some unstructured unguided lessons. However, most of the time we are some where near the middle. For myself I lean more towards the 3/4 mark of the line.

A couple of years ago I wrote a post about a balanced math class but since then I’ve had some small tweaks that I thought would be useful to highlight.

When I first thought of this subject I thought of six things that should be in the program (you can read about each section in my post):

  1. Guided Mathematics
  2. Shared Mathematics: Students work together to “Mathematize”
  3. Conferencing/ Monitoring
  4. Congress
  5. Reflection
  6. Math Games and Math Facts

Now my opinion about these things haven’t changed I still think you need to have all of these components but I want to simplify a bit and think more about the practical side.  For this reason I want to steal a little line from the Leaf’s Head Coach Mike Babcock, think of a five day block of time.

Now, before I go into detail I want to preface that this is just my opinion and in no way is this the only way. I think as teachers we need to have professional judgement to choose what is best. I also don’t expect to have these ideas prescribed like a five day must follow. I just want you to reflect on these components.

I broke it into five days because I really felt that it was easy to look a five day segment in time. Some times these components may take more time or less but on average I try hard to stick to this.

Day 1: Problem Solving

I am a firm believer that our math program should be predominately a place where students are problem solving and exploring math concepts. During this time, the teachers role is to explore the concepts with the students. It is a fine balance between a guided approach for some to a more let kids explore. As a teacher I am also conferencing, questioning and monitoring students work. I am checking it to landscapes of learning and thinking about how I will debrief the learning. What misconceptions are students having? How are they tackling the problem? What collective conclusions are they making? are all questions that go through my head.

Day 2: Congress

This to me is one of the most important things we can do in a math class and where that shared, guided and explicit instruction is happening. During this time, I am questioning and explicitly linking the math concepts to their problem solving. Where I may allow students to wander a bit in exploration I am tightly keeping the reigns around the big ideas and misconceptions I observed in the problem.

Day 3: Number Talks

These have been one of the best decisions that I have made as a teacher. Number talks allow me to discuss strategies, talk through misconceptions and help students visually see the mathematics that is happening around them. Number talks is also a 15 to 20 minute exercise so they happen frequently and often in the classroom. Another great aspect is that it allows students to communicate and talk about math in a meaningful way.

Day 4: Reflection

The more I read about this topic the more I believe that this needs to be integrated more in the classroom. We need to explicitly show students how to reflect about their learning and how to set goals in order to improve. This year in my class I have purposefully set time aside for students to regularly talk about their math learning.

Day 5: Purposeful Practise (Math games, Centers and regular practise)

Yes I said it Purposeful practise. This may be in a worksheet but if it is I hope it is geared toward each child’s needs. For me purposeful practise is about seeing where a child is developmentally and finding things that may work for them. This year it has been center work, using board games or math games and digital games like knowledgehook and Mpower.  The important part is understanding that it is purposeful and meaningful.

Overall, I think we need to think less of this war between concept and procedure and meet in the middle. How can we help our students learn and build bridges mathematically.

I would also love to hear your thoughts. If you have any opinions or questions please feel free to leave a comment.

Here is my slide deck on a balanced math approach.

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What is Wrong with an Algorithm?

Nothing! Yes I said nothing! Let me elaborate.

In conversation today I was asked, “What is wrong with an algorithm?”  I get this question a lot. Now this question was asked because like us all we were taught to do an algorithm so we always do what is familiar for us. The question also came about because they were afraid of teaching their child the wrong way.

This inspired me to write this blog post. There is nothing wrong with learning an algorithm; there are some risks in only learning this but there is nothing wrong.

An algorithm was first invented in order to make math easier to do. We have to remember that calculators were not invented at this time and may people were using abacuses and basically counting by ones or a base 16 system. With the use of an algorithm counting became procedural and easier to do.  However, those using it understood mathematical foundations of why the algorithm worked.

This is the problem with introducing the algorithm to early. Let’s just reflect on our own understanding of basic addition. We honestly did not learn how to add by using an algorithm first and to be fair our first insistence of remembering an algorithm (at least in my guess) was probably around grade three maybe four. This means that for four-five years, depending if you were in the pre-school or JK error you had a lot of experimentation and exploring in counting, saying numbers, ordering numbers and subitizing.  The problem is that we often forget all of the steps it took us in learning an algorithm.

For the past ten years I have been looking closely at how students learn mathematics. One of the foundation research pieces have been Fosnot and Dolk’s landscape of Learning (2000). (full file)

I know that the picture is small but you can see all of the learning that a student has to go through in order to get to an understanding of algorithms.

Now here is my problem with starting with algorithms to early:

1) It doesn’t teach proper number sense:

Now you may question me on this but in my honest opinion it doesn’t teach proper number sense. The problem is that when a person does an algorithm it forces you to only look at numbers from 1-9. There is no true understanding of our base ten system. Students do not really understand why or how 1 group of ten can be ten things yet 1 ten. This is often seen when you ask a student tell you the value of a two digit number (let’s say 24). When you point to the ten’s column they will say the value is two. They will even go as far as counting 24 objects then pulling two of them to signify the value. This can also be seen when students use base ten blocks and they count the rods as two instead of twenty.

2) Basic procedural errors:

Now these errors may be dismissed as, they just have to learn the proper procedure. However, these are troubling errors because students not only are procedurally doing the operations wrong they are also struggling with reason-ability of answers or what I like to call no number sense.  
What I propose instead:
In my math class we work hard on developing a number line.  Learning this strategy does a lot of things: 
1) It allows all students to grasp a strategy whether they are counters by one or able to skip count proficiently. 
2) It teachers all of the rudimentary learning of number sense. Students learn one to one tagging, cardinality, subitizing, magnitude and many more options all in one strategy.
3) Eventually they develop more proficient mental strategies because they can conceptually understand what is happening to the numbers.
4) There is no need to regroup and or borrow with a number line, just the use of number sense.  I find this to be the biggest problem with algorithms. For example, let’s take 56-49=7. Now for many of us we may not need to use an algorithm but ask your child or a younger student they will start by borrowing from the 5 (not fifty) to make 16 because you cannot take 6 from 9 (which we also no is not true– ask anyone in debt or who has a mortgage) and then they will realize the answer is five. However, if students are taught proper number sense, they realize that they can just count up to the answer.
A number line also develops efficient mental math strategies that are far faster then whipping out a piece of paper and borrowing or regrouping.  Take a look at some of my grade twos thinking (all done 

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By allowing my students to explore, question and develop their number sense they are better mathematicians and yes we have discussed an algorithm, we have developed an understanding of how to use it but my students make less mistakes with a number line then with an algorithm. And to be honest they much prefer the number line then the algorithm, most of the time they do this because that is what they think they need to do.

One of the biggest problems in our math programs today is that we often jump to far into the abstract without thinking about the concrete work that people need to develop in order to understand these concepts.

What do you think? When and where should algorithms be taught? How do you develop number sense? Love to hear your thoughts.

For further reading I recommend reading:

1) Fosnot, C. T., & Dolk, M. L. A. M. (2001). Young mathematicians at work. Portsmouth, NH: Heinemann.

2) Anghileri, J., Beishuizen, M., & Van Putten, K. (2002). From informal strategies to structured procedures: mind the gap!. Educational Studies in Mathematics,49(2), 149-170. 

3) Kami and Dominck: The Harmful effects of Algorithms in 4-9

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Why teach through inquiry? A real testimonial

Now I know that I have posted on this subject before but with the day I had I just had to write about it again.  Inquiry: WOW!  Man I love it.

I know that recently there has been a lot of discussion about inquiry in the classroom and if it is really making students learn.  There has also been a huge push to go “back to basics” all I have to say is wish you were in my class (even school) today.  Today’s math problem was quite simple: 
“Mrs. Standring, our proud principal, needs help.  Our school has been open for two years now and we got more kids this year, because of that the fire Marshal has asked her to make a new fire plan.  I was telling her that we were studying measurement and she thought you could help.  How far is our door to the nearest fire door?”
The kids went nuts. It took them a while to get over the fact that they were helping Mrs.Standring.  Well they just started with the questions: what tools can we use? How are we starting? Which door is closer?
Most of them saw that a meter stick would be the best measurement tool, we had been talking about measurements for some time and been measuring in non-standard too and knew that it was inconsistent. So they all grabbed meter sticks and off they went.
We got a bunch of numbers and came to the carpet to discuss. They were all in confusion, why do we have different numbers. We used a standard measurement? We then asked the students to demonstrate how they measured.  Some saw that when you lift the ruler up, you sometimes, overlap the space or leave a gap.  I then asked them how can we prevent that?  This brought up the discussion of leaving marks, or placing fingers.  They went back at it.
Students then came up with an answer but when I asked them to tell our principal they didn’t know what to say.  This of course then led us into a discussion about explanation texts, which we then made some success criteria and off they went to write.  When the bell rang half way through the students were very upset that they didn’t have enough time to finish there work.
Not only did this problem happen in my classroom but my teaching partner did it too.  Her kids thought string was the best and then bring it back to measure against a meter stick.
Now you may read this and say so what? So what! The best part of this is that all this discussion was student driven. All collaboration, student driven, all learning student driven.  Yes as a teacher I am incharge.  I have planned this problem, I have thought of the big ideas and questions but it is the passion, and learning of my students that drive this problem.  Also, when looking back (though I will say to make it worth while this should be done first) my students met over 37 expectations from the curriculum and all of the learning skills that are in the report card.  In addition, the talk was amazing and the learning even more. Not only this but when it comes to assessment I have it all, with no tests.  I know my students skills, next steps and a mark of work.
Inquiry for me is the only way to teach.  Yes, students do need facts and knowledge but that fact and knowledge is gained through the inquiry process.  Also, if a student doesn’t have that to start with as a teacher it is my job to scaffold the question so that they do learn; however, it should still be done in a way that the student is discovering the learning.
Now in the end, there is no wrong way to teach, all learning is valid and good. But through inquiry students do grasb and understand concepts faster and with a deeper understanding. It’s been amazing to see our students development as our school adopts this approach. There is less review needed from year to year and the students are talking more and communicating their thoughts.  For me there are a couple of key reasons to teach through inquiry:
1) Students learn and enjoy the lessons more then traditional teaching styles
2) It covers more curriculum and deeper knowledge
3) Students retain information
4) Learning is integrated in real life, why separate at school
5) It validates the students and makes them buy into their learning. If they are invested you have less behaviours
6) students easily tune a teachers voice out but not their peers
7) It’s fun for me too! Shh don’t tell my students
What are some potential problems: (though to me they are not problems)
1) Problems take time: learning is not easily divided into 30, 40 minute time blocks
2) Can be and should be noisy but productive
3) Takes more planning: yes it takes more planning. You cannot wing inquiry. Even though it may appear as if it is winged or that the teacher is doing nothing it is an art form and requires a lot more planning (will tough on that in a minute)
4) Parents: you will get parents complaining and questioning your practice.  This is new for many and with new comes questions and fears. Stand up and proudly defend your practice because when they hear and see their kids they will love you.
5) you may not have all of the answers
What do I need to do to teach through inquiry?
1) know your content and curriculum: when you know your students learning it is easier to formulate questions and scaffold students learning.
2) plan: I wrote a previous blog post about planning but essentially you need to plan.  Inquiry does not happen by the seat of your pants.  You need to anticipate students questions, problems, and ideas.  You need to know what the big ideas are and where you want the lesson to go.  You need to understand learning trajectories and see where your class is and should go next and you need to do the problem first.
3) inquiry should be contextual and related to the kids life.  The best inquiries are ones in which the students really wonder or can invest in.
4) have fun and don’t be afraid to make a mistake.
Overall, I feel inquiry has been one of the best things I could have done. It really benefits the students and it makes my assessment easier.  I would love to hear your thoughts on inquiry? Have you tried it? Struggles? Pointers? Thanks for reading.

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



Place Value

These last few months we have been focusing on place value. Place value is such an important beginning for any primary student in mathematics. We have started the unit with basic counting. Now this may seem too basic and you may think, “what kid doesn’t know how to count by grade two?” This may seem an obvious skill to many but it is something that many (not some) still struggle with.

 Counting goes beyond being ale to tag each object and say its corresponding number. By grade two students should be seeing groups of objects, especially twos, fives and tens, and be able to count by them efficiently and effectively. Students are still grasping with recognizing fives and tens as they count often still counting by ones till they get to five and then putting that aside. Students should start to see 5′s as 2+3 or 4+1 or even better 10′s as 9+1. 5+5, 2+8, 4+6, 7+3, without having to count.

 To help with this we have been collecting and organizing objects in our classroom. Students have been counting bins, pencils, books, etc. in order to tell me how many is in each basic. We then moved to figure out how many bundles of tens there was in each basket and if there was any patterns we noticed in the numbers. Students soon realized that the number (or numbers) to the left became the amount of groups of tens. I told them that this was because that is called the tens column in the place value system and really it is saying 1 group of 10 or 1 x 10.

 We are now trying to see how many groups of fives and tens there are in the bins. Now again, I thought to myself this should be an easier concept. Obviously if they see the fives then they will see how many tens. I also thought that since we worked on doubling so much in patterning that they would see that there was two fives in one ten. However, I was wrong again. Like many students, we are struggling to see how one group of objects can be called a 1 group but still be 5 or 10 things. Another mistake that my students are making is assuming that the ones place value tells us how many tens we have. They assume that if the left column told us the tens then the right must tell us the ones. We are currently working on this concept by looking at numbers and asking how many tens and how many fives? The follow up questions are simple: What patterns do you notice? Why does this occur? My hope is that students will see that there are two fives for every ten and if the leftovers (after making a group of ten) is greater then five it is just one more group. Example: 76: The number 76 has 7 groups of tens because there is 7 tens in 70 (10+10+10+10+10+10+10=70). We also have 15 fives because there are two fives in one ten and we have 7 tens so you double it; however, we also have 6 leftover which can make another group of five; making the total 15 fives, with one leftover. 

 To help out at home, keep practising the subutizing plates (dot plates) or counting objects in the house and looking for patterns.

Making conjectures and proving them

In grade two we have been exploring the concept of doubles and what is a double. It’s an interesting concept because we probably assume that by grade two students should know what a double is and why it is called a double, but that was not so. Oh of course, all of the kids could count by twos, but when asked what makes a double all I got was blank stares.  With this in mind we went through some problems exploring whAt a double is.  We started with the story of Madeline, see previous post, and then moved to a discussion about where we have seen a double before.

Today, the class looked at pairs of shoes.  The problem was if each person in your house had one pair of shoes in the front hallway, how many pairs would you have and how many individual shoes would you have?

Most of the students drew out the people and then the shoes, they then counted by ones or twos to get the individual shoes. This alone is a good math problem but I decided to take it a step forward. I asked them to look at their results and make theory, so that I could figure out how many shoes I would have for any number of people?  Once they made a theory they had to test t out to make sure it was true.

I was amazed at how many of the students looked puzzled. It was almost like I asked them to fly to the moon. I am amazed every year at how students struggle with thinking. We often say that’s we are teaching 21st century skills but are we really?

As I look at what my students eventually did I think how they are starting to become real mathematicians. Sure I could have told them the rule was the amount of people doubled would give you the individual shoes because each person has two shoes or mode the rule with pictures, t-charts and then follow up with a question like, “what pattern do you see?” or i could count the shoes with the kids, but then would my students have learned?

By doing this, this way, I have allowed my students to make their own theories and thentestthem out and prove them to the mathematical community. They have thought about the process, they have looked at the numbers in relation to the context and the math became real.

So I ask you, what are you doing to make your students think?

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:  http://www.youtube.com/watch?v=POSgVl07Go0.   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.