Meaning and Algorithms

My third and fourth grade classes are exploring computation algorithms this week. Taking apart and putting together numbers is one of the skills needed to tackle higher level math concepts. I find it interesting when this topic is brought up as I sometimes hear students commenting that they already know how to add/subtract. Many of my students have been instructed to use the traditional algorithm to add large numbers.  After digging a bit deeper I started to see students falter with their explanation of the traditional method.  I asked them to explain their thinking. I found that their reasoning started to turn into comments about the process. Students explained the steps involved to find the product.  One student told me about the example below.

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Traditional

The process, although important, doesn’t reflect mathematical understanding. Students didn’t mention place value at all.  Place value within the traditional method is evident, it’s just that the students weren’t able to explain it with confidence.  It reflects more of an understanding of the procedures required to find a solution. Most of the students that I see are used to adding/subtracting numbers on a number line. They can hop up and down the number line with ease.  Once they feel comfortable with that, parents or others start to show students the traditional algorithm. Students become familiar with the terms carry, borrow, cross-out, add a zero, and others without necessarily knowing the reasoning behind the process.

This past week I introduced the partial-sums algorithm. This is one of my favorites as students start to see the reasoning behind the process that they’ve been following for years.  Although at first it can seem clunky, students started to see how place value can be a key indicator in determining the sum. That’s an #eduwin in my book.

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Partial-sums

Later that hour students were given an assignment where they had to find the sum using the partial-sums method. Some students struggled a bit to move out of the traditional algorithm process and move to a different algorithm. Afterwards I gave student the option to use whatever method best meets their needs. It’ll be interesting to see the progression that students make as they decide on an algorithm. When asked about how and why the algorithm works I’m hoping they’re able to confidentially create their own proof.

We’re making progress!

Math Puzzles

My students enjoy puzzles. It often doesn’t depend on the type of puzzle. They like the trial-and-error of attempting to find patterns and eventually solutions. In math class these puzzles take on many different forms. I believe that patterns and puzzles play an important role in the math classroom. Some skills are more aligned with using puzzles than others.

Puzzles offer learners multiple entry points. Students have the option to look at a puzzle and decide to start in one section while another student decides to start in a different section. One puzzle that’s been thrown around the Internet is below.

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I gave this puzzle to my fourth grade students. Students started to calculate how much each horse was worth and worked from there. I had other students that immediately looked at the horseshoes and found that the total worth would have to be divided by two. After working with each other the students then conversed in groups. The discussion was fabulous. Students started to identify where a mistake was made and corrected their papers as needed. During this time students were engaged, using math vocabulary and practicing skills that they will see again throughout the year. What I found interesting was that zero students had the correct answer the first time. It took perseverance as well as a thorough amount of collaboration to get to a consensus. The class had a conversation afterwards.

The key I find is connecting the puzzle to the skill/standard. Afterwards, students understand that this was a fun problem, but was the puzzle connected to a certain skill? Connecting the puzzle to prior skills not only shows how this fits into a continuum, but also gives students a picture of what skills are being addressed. Maybe the skills can be introduced after the lesson. I know that the exploration that students participate in is a valuable piece in the learning process. What are students exploring as they unpack the problem? The objective isn’t to just solve the problem. The bulk of the student learning experience is using the substitution process to find a solution.

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After discussing the solution I drew the picture above on the whiteboard. Through this I attempted to bridge the puzzle to more of an abstract model. This made sense to some students while others were debating on whether this matched the earlier puzzle. Regardless, the transition seemed beneficial in having students use substitution to find a solution.

Next week the class is tackling a different problem.

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I’m already looking forward to using this with my students.

Evolving Teaching Strategies

 

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This past week I was traveling and was taken off guard when I saw a phone station. At first glance I thought this was a modern day phone booth. I looked for the phone, credit card reader, directions and buttons but couldn’t find anything. I actually looked a bit closer at the parts. There were only two.  There was a small partition and place to put your phone as you speak with people.  After a few weird looks from commuters I concluded this was a modern day phone station/booth.

It had me thinking of how the phone booth has evolved over time.  It had a purpose back in the past and seems it still has one now.

Examples of how products have evolved over time can be found just about everywhere.  This can be true for products as well as strategies/processes.

I believe the same can be said about instruction in schools. New research can impact the strategies that teachers use in the classroom. These strategies have also evolved over time. Marzano and Hattie are just two names out of many that have impacted the field of instruction and teaching strategies.  Some strategies have been proven to be more efficient than others.  Books, articles, administrators, coaches and other professionals often impact what new techniques educators utilize. How students respond to those strategies is important. Some of the strategies I used when I first started teaching continue to work and others were cut after the first year. Educators reevaluate tools and techniques in their classroom. I believe this reevaluation is a form of evolution that comes with experience and betters teachers and their students over time. Some of the strategies that I use stay the same from year to year while others change. I question some of the tools and strategies that are used or given to me. Are they efficient?  Do they provide opportunities for students to make meaning? Is this the best strategy for my students? Educators often adapt and evolve their teaching strategies to meet their students’ needs. Teachers evolve over time and this is a driving force that can impact students for the better.

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.

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.