Learning Number Concepts

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This summer is moving by quickly and thoughts of the upcoming school year are in view.  I’m in the process of preparing materials and the first few lessons.  This year my math sections are larger than before and it changes some of the ways in which I prepare. While researching a few ideas I came across a Tweet from Jamie.

 

I’ve heard of Kathy Richardson and her role in the math education world, but haven’t delved too deep into what she’s written.  This book interested me mainly because of the use of Kathy’s critical learning phases.  As students progress in school they visit different stages of mathematical understanding.  It’s not always linear, but these stages tend to follow each other. I wanted to learn more about this process.

So the book arrived yesterday and I was able reading through the first section. Kathy states that children learn number concepts in different learning phrases.  She coined the phrase critical learning phases to describe the stages that kids pass on their way to making meaning of the math they encounter.  The first section of the book focuses on understanding counting.  On first glance I thought this would be very basic.  Advertised and delivered.  It’s basic, but also intriguing and gave me a few takeaways.  After reading this section I started to draw parallels to how my own students make sense of numbers.  I then wanted to reach out to my colleagues to direct them to this section. Understanding counting, although basic, is a foundational skill that is built upon.  My paraphrased version of Kathy’s learning phases are below.


Counting objects

  • Counts with 1:1 correspondence
  • Knows “how many” after counting
  • Counts out a specific amount
  • Spontaneously adjusts estimates while counting to make a better estimate

Knowing one more/one less

  • Knows one more and one less in a sequence without relying on counting out
  • Notices if counting pattern doesn’t make sense

Counting objects by groups

  • Counts by groups
  • Knows quantity stays same when counted by different sized-groups

Using symbols

  • Uses numerals to describe amount counted.  Connects symbols to amount counted.

 

As I read through this I started looking through my school’s teaching materials for grades K-3.  Some of the materials follow a linear progression while others tend to favor spiraling.  I have a few takeaways after reviewing this first section of the book and looking at my district’s materials.

  • If counting is important than students will start to see why keeping track/organizing numbers is important.  If it doesn’t matter students won’t care whether they come up with a different amount when a specific amount exists.
  • Being able to “spontaneously adjust estimates while counting to make a closer estimate” is a skill that’s found as students progress through the elementary grades. This plays a major role in measurement as students are asked to estimate using non-customary  tools.  I also see a connection to Estimation180 as students might use proportional reasoning after initially counting to find an estimate.
  • Moving from 1:1 correspondence counting to group counting can be confusing for students. Understanding that counting by 2s, 5s, or 10s is a strategy that’s more efficient and that may be learned in time.  Students might not initially come to this conclusion and replace 1:1 with 2:1, 5:1, or 10:1 if they don’t understand the reasoning and if they’re taught in the same period of time. For example, I find this evident when there’s a disconnect between counting by 10s and grouping numbers into 10s and then counting the groups.

 

 

 

 

Proportional reasoning

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This summer I’ve had opportunities to review math units that I’ll be teaching in September.  I’ve been beefing up the units with formative assessments and intentional questions that focus on math reasoning.  One skill that seems to need bolstering every year falls in the category of proportional reasoning.  This becomes quite evident when students encounter fractions and rates. Some students may use proportional reasoning,but it’s not necessarily identified as a strategy or communicated using that specific vocabulary.

I also picked up this book over the summer.  I read it about a year ago, but I’m finding so many gems in there a second time around. The authors reveal research that indicates teaching proportional reasoning has benefits.  The authors also showcase that proportional reasoning is difficult to define, but they can categorize what people can do with this type of reasoning. People that use proportional reasoning understand the relationships that numbers have put together and how they relate individually.  They can analyze numbers and look the difference (additive) between them and observe the ratio (multiplicative).

My takeaway from this section of the book comes from the authors’ five reminders.  These reminders come in handy when thinking of how to create learning experiences involving proportional reasoning.

1.) Use unit and multiplicative models.  Double-down on using the idea of a rates, which can be applied to the idea of a proportion.  Specifically, I can think of rate tables to be helpful with this.

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2.) Identify proportional and non-proportional comparisons.

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3.) Include measurement, prices, graphs, and geometry to show proportions. Proportional reasoning can be found in a variety of contexts.

hexagon-014.) Solve proportions using different strategies.  Focus on reasoning.  This may be ignited by planting questions that elicit different ways to solve the problem. Students should be able to compare and discuss what comparisons exist.  This can also be addressed through the use of “what’s my rule” tables.

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5.) Have students recognize that short-cut methods such as cross-multiplication aren’t helpful in developing reasoning.


Being able to identify proportional reasoning can help teachers emphasize its usefulness. Having in-depth conversations about this type of reasoning has benefits.  I realize that this post is heavy on tables and that’s not the only form that proportions take.  I do feel as though the tables help students observe the relationships a bit easier since it’s organized. While exploring this topic I came across a MARS activity that I’m planning on using in September.

Inservice Days

Many district are is in the midst of planning their 2016-17 inservice/institute days.  These days, sometimes called PD days, often include communicating initiatives aligned to district goals.  Sometimes school goals are included in this process.  As far as I can remember inservice days have always been part of my school year.  The content is sometimes applicable to what’s happening in a particular school, other times it’s more aligned with a district goal. Most teachers have experienced successful and unsuccessful sessions.

Last night I came across this Tweet:

David asked an important question.  I’m not an expert in the field of PD, but I’ve experienced some amazing and not-so-amazing sessions in the past.  I’ve also put together plans for PD and other sessions.  Through this experience I’ve been able to evaluate PD sessions a bit better.  Below are four questions to consider before putting together a PD session:

 


 

Are there clear expectations?

Being intentional in communicating expectations is key.  I’m not necessarily talking about listing the objectives of the session. I’m more concerned in what participants should be able to do with the information after it’s been delivered.  How will this impact teaching and learning?  Having a clear understanding of what’s expected and a timeline can help avoid confusion.

Is there an explanation of why?

I think this is sometimes missing from PD sessions.  Why are we learning about guided math, reading workshop models, grading practices, etc.?  Giving the why can help people understand the reason for a particular session.  If it’s not explained than staff may feel as though the reason is directly associated with someone not in the school, which may or may not be a good thing.

Will there be opportunities to revisit this initiative?

Educators aren’t generally fans of participating in a PD session that communicates that what’s being discussed will be fully implemented but it doesn’t happen. If the expectations is that all classrooms need to do x, y z than that should actually happen.  Starting an initiative and abandoning it halfway through the year doesn’t help with rapport or climate.  A successful PD session allow opportunities for additional help and follow up as needed.

Is there a reflection opportunity?

This may be more of a matter of personal opinion.  I tend to learn best by reflecting on what I’m learning and finding ways to practically put it into practice.  That reflection can happen after the session but embedding it in the session can be a valuable.  Sometimes a reflection opportunity can reveal itself through follow up conversations.  It also keeps the conversation going to ensure that consistently is occurring in a school/district.


When creating a PD session I tend to consider the questions above.  The questions aren’t always applicable, but it’s a place to start.  Would you add any other questions?

 

 

 

Reflection

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It’s official.  The 2015-16 school year has concluded.  The final bell rang last week I’m starting to look at my summer book list. My reading takes on different forms during the summer.  I have a few books on hold at the local library just for that purpose.  I’m looking forward to digging into those later this week.  Before reading these I enjoy catching up on blogs that I missed during the last hectic month of the school year.  This year I’m also looking back at my personal goals for this past year.

As I reflect back on the school year I often categorize how classes went that year.  Were the classes successful?  How did students learn?  Did I create an environment that optimized student learning and their curiosity?  Did I leave a lasting impact that students will remember?  How many of these students will invite me to their graduation.  Okay, the last one was a joke.  Kind of.  I tend to reflect back on these questions as well as others.  Last August I wrote a post about the goals that I had for the new school year.  This post is designed to reflect on those goals.


 

1.  I plan on taking the first few days of school to engage students in community building activities. The class will be completing a “get to know you” survey and set expectations for the class. We’ll also be completing the marshmallow challenge and have some rich conversations around math and mindset. I feel like instructional strategies make little impact if students have a fixed mindset. The same could be said for teachers. Before delving into content I want to ensure that the classroom community is moving in the right direction.

Looking back, I was ambitious with my planning.  At the time I thought this was a realistic goal.  I started off the school year with community builders.  We completed the marshmallow challenge and other activities.  I didn’t actually survey the students.  Instead, students wrote in their journals about math experiences.  I reviewed their journal entries and had brief conversations with each student.  The students felt comfortable in the classroom and seemed to develop rapport with each other.

I didn’t get into the rich discussions about math mindset as much.  Having a growth mindset has been emphasized in my district but the practice of it in individual classrooms vary. This is also a byproduct of the mindsets coming from other students, at home and at school. Honestly, it was challenging to not dive into content immediately.   Regardless, the classroom community was set on a sound foundation.  That foundation played a pivotal role throughout the rest of the school year.

2. I‘d like to make learning more visible in the classroom. I’m planning on having students use math journals to reflect and document their learning journey. I’m also planning on using effect size data to show student growth over time. To do this I’ll need to create additional pre-assessments to analyze pre/post data. I’m also planning on moving away from letter grades on unit assessments. Instead, I’m going to have students reflect more on the skills being learned in class.  This is a change from past practices so a lot of modeling may be needed.

I had students use math journals this year.  I intentionally had students use them to reflect on assignments/projects throughout the year – more so at the beginning of the year.  I also dabbled with students using foldables this school year.  The foldables were used primarily for process-oriented skills involving conversions.  These were glued or taped into the student math journals.  By the end of the school year the math journals were thick and looked like scrapbooks.  I’m looking at changing this format next year.

I used effect-size with one of my classes this year.  Students took  a pre-assessment and explored a particular concept for around three weeks.  After the three weeks, they took the same pre-assessment.  I calculated the effect-size and placed the data in a spreadsheet that was shared with my teaching team and administrator. I felt like this was good practice as my district is moving towards effect-size next school year.  Students received both the pre-assessment and assessment back at the end of the unit to see how much progress was made.

My unit tests didn’t include letter grades on the top of them.  This seemed to bother some students as they wanted to know their exact grade.  By the end of the year, all some students weren’t as concerned about the percent/grade.  I emphasized, as much as I could, that the skills were the focus.  I believe progress was made in this area and I’d like to keep this practice intact for next year.


I tend to agree with the philosophy that deep reflection can lead to growth.  I’m looking forward to the new school year in August and have some new goals that I’d like to put in place.  For now, it’s time to reflect and recharge before the new school year comes around the bend.

Google Forms and Presentations

Google Forms

My 3-5th grade classes are finishing up their math genius hour projects this week.  Fittingly, it’s the last the week of school so we made it just in time!  I have two days to fit in 15 project presentations.  This last round of projects lasted around two months and the final projects will soon be revealed.

Students created questions, found a math connection, researched and are presenting this week.  During the last two years students present their projects and the audience asks questions about the topic. This technique seemed to work but I tended to have the same students ask the presenter questions.  Around five or so of the same students asked the presenters questions.  In a class of 25 that’s not ideal.  It was great that the students were asking questions, but five or fewer was disappointing.  Bottom line – the audience wasn’t as engaged as they should be.  So this year I decided to give the audience more of a voice in the process during genius hour presentations.  This actually stemmed from a class that I took this spring about using Google tools in the classroom.

One of the assignments required students to create a Google Form that could be used in class.  My first thought was to create a student rubric for presentations.  I decided to create my own after dabbling around the Internet for a few examples. Initially the form was going to be used by the teacher to evaluate presentations.  After starting the form I changed my thinking.  I thought about possibly having all of my students use the same form to evaluate the presenter.  The genius hour feedback form was built from that idea. Click the image below for the form.

Feedback Form

This week my students have been using the form to evaluate their peers.  Students are asked to present their projects while the audience listens.  At the very end (not at the beginning as some students want to get a head start) students take an iPad and scan a QR code to access the Google Form.  Individual students evaluate the speaker and submit their response.  It’s not confidential as students have to pick themselves (the evaluator) and the presenter.  I tell the students that this information will not be revealed to the presenter.  So far it’s been working well.  The last presentation took less than 2 minute to collect 21 feedback submissions.  Another bonus is that you can have a class conversation about the overall quality of the presentations.

Data

I then export the file to Excel, hide the evaluator column and then print out the sheet for the student.  The student is then able to reflect on the data at a later time.

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The form needs some work as I’m thinking of making some of the questions more clear.  I’d also like to add a section on the form where students can ask the presenter questions.

Rates and Pizza

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My third grade class is studying unit prices this week.  They investigated in/out tables on Monday and determined how to find missing values.  During this process students started to explore strategies to solve function machines.  The class transitioned to unit prices on Wednesday.  This was more challenging, especially when students needed to find the price per ounce.  Being able to round answers and place the decimal in the appropriate place seemed to cause some issues.  Students knew what operation to use but had trouble placing the decimal.

On Friday students started off the day with a brief activator involving price per ounce.  Students were given three Starbuck cup sizes.  They had to find the price per ounce.  Students worked in groups and experimented with different strategies to find the right place to put the decimal.  The groups checked their reasonableness by multiplying their unit price answer to find the total price.  This took around 10-15 minutes.  Click here to find the template.

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After this activity the class formalized a process to find the unit price of an item.  We then moved to the main project for today’s lesson.

Students were given iPads and asked to visit a pizza site.  The site was up to the student.  Students visited Papa Johns, Pizza Hut, Giordano’s, Dominos and Lou Malnati’s websites.  They were asked to customize one large pizza.  Students put together their dream pizza and found the final price.  In order to be consistent students were told that each large pizza has around nine pieces.

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Students took a screen shot of their fake order and saved it to their camera roll.  They submitted their screen shots and added a caption that included a number model of the unit price.

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I’m reviewing the student screenshots tonight and am finding that students are becoming better at understanding unit rates.  Next week we’ll be looking at better buy problems.

 

Reasonable Answers

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This week my third grade class started to learn about measurement.  Measurement is one of those topics that doesn’t see much light of day.  Number lines and computation are very prevalent in elementary classrooms, not so much with measurement.  This is especially the case when it comes to metric measurement.  Every grade level teaches a form of measurement, but conversions aren’t discussed on a regular basis.

As I was looking for ways to review I came across a gram measurement set in my school’s storage room.  I was looking for a digital scale but couldn’t fine one.  I dusted off the tiny weights and also found a balance.

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Winner!  So I brought them back to class.  The next day my third grade class had a discussion about how to calibrate a balance and use small weights to find measurements.

While reviewing I started to find that students struggled with the reasonableness of their answers.  They had trouble identifying what a gram, 100, or even 100 grams really looked like.  They had an easier time with conversions, but that is more of a process and not an understanding.  So I brought the class to the back table and found an empty container.

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It measured exactly 510 grams.  It was empty so students knew that it couldn’t be that exact amount Students took turns and added the weights to see if they could balance the equation items.

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Not balanced

Students were excited to add/deduct the weights to find balance.  It seemed like a puzzle to them.  The balance was even at 41 grams.

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Even steven

It took a small amount of time before students calculated that the oats would weight approximately 510 – 41 = 469 grams.  This also led to a quality discussion about the weight of products vs. the container that it fits in.

This was decent activity to bring out a better understanding of what a gram really is.  Next week we’ll be exploring kilograms.