Mental Math Strings

Wow…I haven’t posted in a long time.  It’s absolutely a busy time of year, but then again, what time of year isn’t a busy time?  Having said that, I had to post because I had such an exciting lesson on my mental math strings today.  Once a week, I dedicate a forty minute period to “whiteboard math,” focusing on mental math strings.  This is an approach inspired by the work of Catherine Fosnot in her book, Young Mathematicians at Work.  Essentially, a “string” is a set of computations that builds understanding towards utilizing a particular strategy.  I have been focusing on the strategy of “jumping by tens” to add two numbers together.  For example, I started my string with 63+10 and I asked my students “How many jumps of ten do I need to make?”  Using an open number line, students begin with the bigger number and make one jump of ten to add.  This particular activity hits several strands:  addition, patterning, and place value.

I have attached pictures of my students’ work.  Please note that it has taken me two months to get students to this point (“training” them on how to use an open number line, how the jumps are made, including the computations and numbers on the line, etc.).  However, perseverance has paid off and I managed to get through an entire string in 40 minutes, with much success!  Here is the string I used today:

63 + 10

30 + 63 (I put the larger number second sometimes to reinforce the idea that they begin with the larger #)

63 + 13

45 + 25

65 + 34

132 + 41

If anyone is interested in learning more about strings, check out this blog:  http://mathcoachondemand.blogspot.ca/2011/03/mental-math-strings.html

Problem Solving

Over the past week, I have been including a variety of games and problem-solving questions for students to work on.  Personally, I see a lot of value in the math games, as it provides a highly engaging way of helping students understand concepts of number.  I am hoping to include some updates and pictures of these games in my next post. 

In conjunction with our work with estimation, I asked students to solve a problem posed by the parent council.  The question:  They want an estimate of how much food they will need to order a special lunch for primary students in grades 1-3.  They need to order enough food for each student: 1 hot dog, 1 salad, 1 milk, and 2 cookies.  We discussed how many primary classes there are (10), and that there are approximately 20 students in each class.  We also had a conversation about the fact that not all students will want a hot dog since some students cannot eat meat.  I paired up the students, and gave them chart paper and markers to work on solving the problem.

As you can see in this group, they spent the whole first day drawing pictures of food.  Clearly, they were having difficulty on where to start (even with some open-ended questions to get started).  On the second day, more direct (closed) questions were asked to get them thinking about how to solve the problem.

This next group estimated how much food they would need in each grade (not as a whole division).  When they finally finished, they realized that it would be quite difficult to add it all up (I offered a calculator at this point to avoid frustration).

This group organized their thinking with a chart with each food item at the top.  They accurately added how many students there were altogether, but did not take into consideration that they might need to adjust the numbers according to food preferences/choices.

Finally, this group calculated how many students there were altogether, and made a one-to-one calculation for all the food items (the most efficient way to solve the problem).  They also accounted for students who may not choose to eat/drink the items.  They did forget to double the number of cookies (2 per student), but this strategy was employed by 1 or 2 other groups who did remember to double the amount (I didn’t have a picture of this example).

Since some groups solved the problem after the first day, I prepared a more “challenging” question for day two.  They needed to figure out the same food items for the junior classes with an average of 25 students per class.  I put this question on the board last minute – much to my surprise, these students were finished in a couple of minutes.  How did they figure it out so fast?  Well….we only have 8 junior classes, which brings the total number of students to 200.  Yikes!  Lesson learned for me – I need to plan extension questions ahead of time!  It gave me a good laugh for the day. 

Inquiry Into Estimation

Yes, my second post of the day.  I intended to post my diagnostic yesterday, but was too tired (September exhaustion!).  But I felt like I needed to post my inquiry into estimation because it really didn’t go the way I had planned, and the whole point of this blog is to document the “messy” parts of my math inquiry, too.  Armed with my lesson plan and materials (chart paper and centimetre cubes in cups), I asked my students to get together in groups to estimate how many cubes I had in each container.  Based on the diagnostic, I grouped students who were already unitizing in their own groups, hoping that the students who counted by 1’s might be able to discuss strategies to estimate and count more efficiently.  I had my clipboard ready with anticipated strategies – those who used a benchmark to estimate, and those who did not.  When I went around the table groups, not one group was using a benchmark!  Oh oh…not at all what I was expecting.  I was hoping that at least one group would be able to use that strategy, and select them to present during math congress.  Most students estimated about 80 cubes, but each container contained about 200!  So, I brought them back to the carpet for a brainstorming session.  I knew some scaffolding was needed, so I raised the container, and asked “How might looking at the bottom of the container help you with your estimates?”  Alas, one student said “You could count the cubes on the bottom, and add up each layer.”  Employing good accountable talk habits, I asked the students to repeat this idea.  I suggested that maybe they could all go back and change their estimates.  Going back around the groups, many did change their estimates, but they actually lowered them.  Could it get any worse?  Ha ha.  This is the point where every teacher who embarks on inquiry based learning wants to rein them all in and just directly teach them how to do it!

Plan B….ask the students, in groups, to count the cubes.  Maybe this will help them see that their estimates were not really reasonable?  To my surprise, the students who I grouped because of their ability to unitize based on the diagnostic were counting by 1’s.  Yes, it could and just did get worse.  Calmly, I asked “Hmmmm…..is this how you counted the kernels yesterday?”  “No, I counted by 5’s yesterday.”  I replied, “Well, do you think you might want to do that again today?”  Right or wrong, I thought this was an appropriate suggestion to salvage some of what my learning objectives were for the lesson.  Eventually, everyone counted their cubes, and recorded the total on the chart paper.

After lunch, we held a brief math congress to evaluate our learning from the day.  The students concluded that their estimates were low.  I asked two groups to present – these were the groups who unitized to get their totals.  Much discussion and repeating of ideas ensued, and I can only hope that they internalized the big ideas for the next activity.  One saving grace from the day – I asked them to play a game of “Stars” where one partner draws as many stars as he/she can in one minute.  After a count, his/her partner does the same, and determines whether or not he/she could draw more stars in the same amount of time.  With much glee, virtually all students circled stars into groups of 5 or 10 to determine the quantity.  Whew!  Maybe it wasn’t as bad as I thought!

Place Value Diagnostic

A couple of days ago, I conducted a diagnostic assessment to determine whether or not students would be able to unitize when counting large quantities of objects, which is one of the big ideas of place value.  I gave each student a small cup containing popcorn kernels and asked them to count how many were in their cup.  I recorded students who counted by 1’s, and students who grouped the kernels together (and by what quantity).  I was surprised to learn that 68% of my students counted by 1’s, and none of the students who unitized their kernels grouped in quantities of 10 (some even grouped in 4’s rather than 5’s).  This is not to say that they can’t unitize, but rather did not automatically employ this strategy when counting a large quantity of objects.  Attached are pictures of how some students recorded their count, along with the diagnostic recording sheet I used.  This will help me determine next steps for instruction and inquiry.  🙂

First Week

We’ve been busy during the first week, mostly with “getting to know you” activities.  One of the activities that we discussed as a group during the summer was a “Math About Me” diagnostic.  The idea was to get students to think about how they could describe themselves using the different strands of math.  We tossed around the idea of asking students to do it without a model to see exactly what they knew about each strand.  While this might be a good idea for older students who are more familiar with each strand, we thought that a very simple teacher model might be more appropriate for younger students.  Therefore, I provided limited guidance with my own “Math About Me” poster containing one piece of information in each strand.  The activity was a real eye-opener, not only for which students could add information, but also on whether or not they could include operations to describe important numbers in their daily life.  I attached a photo of one student (prior to his picture and name added for privacy).  As you can see, he was able to provide (albeit a little simplistic) his house number and the number of letters in his name using addition.  He also created a repeating pattern and accurately recorded the pattern rule (very few students wrote the rule on their posters).  I thought it was clever that he created a map for geometry, but it would have been more related to math if he included math vocabulary to describe it.  Finally, he utilized the computer to access information on exactly how old he was in months, weeks, hours, minutes, and seconds.  Overall, the students had a lot of fun using measuring tapes, metre sticks, and the computer to help describe themselves in a mathematical way.  I would definitely consider this activity during the first week of school next year, too!

Linking Math Games to Curriculum Expectations

Here is a chart that shows how the games I planned are linked to the Ontario curriculum.  I hesitated to “check off” the two expectations that included “real life situations”.  Are games real life?  I tend to think that the intention of these expectations lean more towards activities like shopping, cooking, and reading signs/billboards.  I wasn’t sure, so I left those expectations blank.  Some of the games spill over to expectations in other strands, but my chart only includes the expectations for place value.  This was definitely a worth-while activity.  I think that as educators, we always need to be prepared to explain what we’re doing, and why we’re doing it.

Setting Up

Over the past week, I spent many hours in the classroom setting up for the new school year.  During a meeting in August, I had a discussion with colleagues of mine on how to best set up the classroom to facilitate inquiry-based learning.  Part of that discussion was the idea of having the students help in the arrangement of desks and books.  While I see the benefit of doing that, I opted to set up the desks and resources myself (for now).  Personally, I like the idea of students coming in the classroom on the first day of school with a sense of comfort and readiness.  During the first month, I may ask students for their input on whether or not the arrangement works for them.  

On the teacher professional side, I purged many files from units I built years ago.  It’s so easy to become a “packrat” … “Oh, I might need this in the coming year (or years)”.  It makes me laugh, because I haven’t used the resources for the past 3-4 years.  I also spent a lot of time transferring current units from file folders to binders.  I have learned that files, although easy, became a mishmash of duplicate copies of sheets and disorganized chaos.  I now have units in binders with lesson plans in page protectors and in the order that I will need them.  Wow, I feel so much better!

I’m excited about the upcoming year…only four days to go!

Unit Planning

I spent the better part of today shopping for back-to-school items for my two daughters.  It’s nice to feel like you’re accomplishing the little goals every day prior to going back to work.

Yesterday, I spent quite a bit of time on my unit plan.  Although I don’t have a single lesson plan completed yet, I feel like I truly have a vision of how it will unfold.  I realize that inquiry-based learning is all about discovery, but I think you still need a roadmap to help with the direction.  The detours are the fun part of the trip, and are yet to be discovered!  I shared my unit plan here (first crack at it).  I spent a lot of time looking for appropriate literature to accompany the topic, and many games that fulfill the expectations of the curriculum.  Come to think of it, I probably should have included which expectations are covered in each game, but at this point, I think I’ll save it for version 2.  My assessment is still a little bit vague (“common assessment”).  I have ideas, but I would like to share them with my grade partners before I commit.  The assessment piece is something I really like to co-create with my colleagues.

In my research into place value last night, I found this great video by Alfie Kohn (link: http://www.youtube.com/watch?v=9gkplk3uEW4).  He spoke about how deep learning is really messy (vs. standards that tend to be nothing but “orderly”).  At 10:51, he quoted Linda McNeil at Rice University as saying “Measureable outcomes may be the least significant results of learning.”  To explain this, he said it’s a lot easier to measure the number of semicolons used correctly in an essay than it is to measure the number of wonderful ideas in an essay.  My interpretation really goes back to conceptual understanding.  A student who can add using the standard algorithm but does not possess the conceptual understanding of the process may meet the expectation, but is he really learning?  Kohn adds: “The more focused you are on measureable outcomes, the more trivial your teaching tends to become”.  My goal is to foster deep learning through inquiry and student engagement.

Procrastination 101

Ok….here I am with my laptop and several resources out on my dining room table.  I’m ready to begin my unit plan for place value.  Let’s see, I have my curriculum document, Fosnot books, my Guide to Effective Instruction in Mathematics, my Van de Walle resource, Marian Small books, Puddles books, and individual lesson plan templates I designed during my Math Part 1 course.  I’m all set.  Or am I?  I think I’ve been procrastinating since I laid them out.  I emptied most of my board email inbox (52 pages!), checked on my newly created Twitter account (no, David Booth does not appear to be on Twitter).  Oh, and I can’t neglect my Words With Friends games on Facebook.  In between all this fluttering around, I have read quite a bit of the background to place value concepts in my Van de Walle book, so not all hope is lost.

I learned about early concepts of counting – that my students’ understanding may be much different than my own.  Even though young children may be able to count to 100, this is based on a one-more-than or count-by-ones approach.  Place-value understanding requires children to integrate new concepts of grouping by tens.  Recognizing that a group of 27 tiles arranged haphazardly is the same quantity as two groups of ten and 7 more may be difficult for some children to recognize.  Van de Walle states: “It is inadequate to tell children that these counts will all be the same.  That is a relationship they must construct themselves through reflective thought, not because the teacher says it works that way.”

Yes, I suppose procrastinating may be part of the planning process (or maybe just my part of the planning process).  I think I’m just a little overwhelmed by the sheer number of resources I have, and changing the way I will teach these concepts.  I’m not afraid to change, just a little unsure of the approach.  I will keep chipping away, and I will hopefully make some great progress over the next day or so.  It’s time for me to get back to work….

Focus on Inquiry

I’ve always thought of a blog as a diary, but the idea that there is no lock and key is somewhat frightening.  This is my very first blog that is intended to document my foray into the world of inquiry-based learning.  I have dabbled with it over the past two years, and I have to say that “dipping my toes” versus “diving in” has worked for me.   Each year, I try to think of how I can expand this approach into other areas of my teaching.   Since I just finished my Math Part 1 course at Queen’s University, my logical next step would be inquiry-based learning in math.  My goal is to blog my experiences from the start.  Often, we see the “end result” in videos that appear to be picture-perfect classrooms.  I want to document the mess, sweat, tears and successes to help me reflect upon what works, what doesn’t work, and how I can move forward to improve my practice. I’m looking forward to sharing and reflecting – September isn’t too far off!