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.
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
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.
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.
2.) Identify proportional and non-proportional comparisons.
3.) Include measurement, prices, graphs, and geometry to show proportions. Proportional reasoning can be found in a variety of contexts.
4.) 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
My fourth grade crew started to explore circles this week. They looked at the circumference, diameter, radius and area of circles. They’ve had a bit of experience with them earlier in the year during Pi Day. That day was long forgotten this week. The class started with a large dose of reviewing the concept of area and circles. I noticed that some of the students wanted to immediately use the formula to find the area of a circle. For a few, the process of giving the formula to the kids has caused a headache down the road. They know the formula and some even have it memorized. When asked to define the area of a circle they revert back to formula. When asked what the formula means some of the students repeat the formula back to me. This is a red-alert in my mind – indicating a lack of conceptual understanding.
The class went back a few steps to the area of rectangles and triangles on a grid. After a brief review I observed that students were able to 1) describe what area is 2) Use a grid to count squares to determine the area. Students then said it would be much easier to use the formula. So we used the formula and then went back to circles.
Instead of jumping to the formula like I said earlier, the class took out centimeter grid paper. Students traced circular items onto the grid paper and counted the squares.
This seems boring looking back and as I write this but it was a worthwhile activity in retrospect. Students then found the radius of the traced item. It was fairly easy as students could count the centimeter squares. They used the radius, multiplied it by itself and found the product. They then multiplied that by 3.14, which is our abbreviated version of Pi.
The goal was to get our answers between three and four centimeters. Many of the students had trouble at first but refined their counting methods to be more accurate. I could see some of the students making the connections between using the formula and an understanding of the area of a circle. Eventually students started checking their work by using the area of a circle formula.
Near the end of class students started asking what happens when we can’t actually count the squares of an object. Another student mentioned that we use the formula. That brought out a quality discussion about understanding the formula and why it works. We’re making progress.
My own formative assessments have changed over the years. Being able to quickly assess where a student is at in relation to the standard helps me make decisions. These decisions impact classroom lessons and activities. I’ve always been a fan of exit slips and quick review checkpoints that see if students understand what was discussed during a particular lesson. Most of these take on a paper form. In fact, one year my district’s math committee spent almost an entire year working on these types of formative assessments. I still use some of these and find them valuable. These exit slips are usually split into a half-sheet with a small amount of questions related to the skill for that day. Students complete it in around 5-10 minutes.
This year I’ve been relying a bit more on digital formative assessments. These projects involve having students explain their math reasoning through some type of digital recording. Lately I’ve been using Recap (Thanks Kirk!) and Explain Everything for this. Students generally follow a rubric and submit their file to me via SeeSaw or Showbie. In my experience, audio or video formative checkpoints are much more valuable than other formative assessments. Not only can I view these checkpoints at a later time, I can also check for vocabulary use and hidden misconceptions that a student may have adopted along the way. I generally can’t uncover that as easily with an exit card. Moreover, students and parents can review their recordings over time.
I believe all formative assessments have some type of value. The extent of the value depends on how it the checkpoint is used. Blending formative assessment strategies has its advantages. I want to be able to gauge student understanding and give feedback that helps connect understanding. This isn’t always possible with standard curriculum guides. Looking beyond the traditional blackline masters may yield better results.