Math Responses and Discussions

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Last year I experimented with a couple different ways to encourage students to discuss mathematics.  I used a form of a number talk last year and found some success.  Students were engaged the conversations were more productive than in the past.  I also noticed that not all students participated in the conversation.  Even with manipulatives, some students participated minimally and shied away from being called on.  I found that some students dominated the discussion more than others.  This was taking place in most of my classes and I kept on reinforcing the importance of having a positive classroom climate where mistakes were honored.  I thought emphasizing the climate and providing support would help encourage participation from everyone involved. For some that worked, others not so much.

This year is a bit different.  I’m still using a form of number talks with success.  I’m still looking for ways to help improve this process.  I also introduced a more organized way to incorporate math discussion prompts with students.  I first organized students into groups using a randomizing student spreadsheet.

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Students are put into groups and a destination in the classroom.  I put a new slide on the whiteboard once everyone finds their assigned location.

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Students get into their groups and identify themselves as partner A or B.  Usually I use the spreadsheet to indicate the partners.  Partner A starts with the first prompt and I display it on the whiteboard.

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I click the timer and partner A has 40 seconds to respond to the prompt while partner B listens.  After the 40 seconds I pick a few different people in class and ask them about their thoughts about the prompt and their answer.  Partner B then gets a different prompt.

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Partner B gets to respond to the prompt while partner A listens.  I’ve toyed around with 20 – 40 seconds and have landed on 40 because it gives students an ample amount of time, but also the limit encourages them to be concise.  Students usually go through 2-3 questions each and then we have a whole class debrief session.  So far students have been receptive to this medium and I’m hoping to expand it to other classes that I teach.

Files referenced in this post:

 

Finding the Difference

My second grade math group started this week.  I gave a pre-test on Monday and found that students had some trouble with the word difference.  Many of the second graders saw the word difference and immediately thought subtraction. I could see why students would see this as a quick search reveals difference as being “The result of subtracting one number from another” and “How much one number differs from another.”  I think most students in my class focused on the first definition rather than the latter.  While discussing the word more than a few students brought up that they knew how to find the difference using a method.  It ended up being the standard subtraction algorithm.

On  Wednesday students were introduced to part of a 100 grid and asked to use it to find the difference between two numbers.  Some of the students started to see that difference could be interpreted as distance between.

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Students used two different colors to locate and identify the numbers.  Students then counted the space between the two numbers.  They used hops while moving to the right and then down.

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30 + 3 = 33

Another student used the grid to show a different way to find the difference.

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3 + 30 = 33

I showed both methods under the document camera and the class discussed how both could work.  Students were then asked to place their strategy on a number line.

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Another student raised their hand and wanted to show the class something that they created.

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Next week the class will be investigating regrouping strategies.

Categorizing Numbers and Number Lines

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This week my students explored how to categorize numbers. By then end of the week students were expected identify integers and rational numbers and apply them to real-world contexts. The class reviewed what and where to place numbers on a number line and how to classify them as whole, counting, integers, rational, and/or irrational numbers.  This was an introductory lesson and the term rational and irrational were new to them.  After a brief class conversation about the differences between rational and irrational numbers the class took a deeper dive into how to identify the characteristics of each classification.  The class looked at a few true/false statements:

  • Is 1,000,000 a counting number?
  • Is 1,000,000 an integer?
  • Is every rational number in an integer?
  • Is zero is a counting number?

The class went through these types of questions and were able to respond and justify their answers.  The questions started to get more challenging as students needed to circle  multiples answers.

  • Circle all of the numbers that belong to each set.

Integers:   4.5       2/3     102     -6       8       0

 

This was more challenging and took some time to categorize each number to see if it fit accordingly.  Students were then asked to place numbers on vertical and horizontal number lines.  I was glad to see how well the students responded to the vertical number line as I don’t believe they get enough practice with those.

Students had about 20 minutes left and one project to complete.  I introduced students to a number line project.  I ended up going with Google Draw for this project because I don’t have enough access to iPads at the time and I was able to checkout a Chromebook cart for this particular lesson.  Students were given a prompt to use dice to create numbers and fractions to place on a number line.  They rolled and found their numbers.  Students used their Chrombooks to access bit.ly/mrcoaty.

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Students make a copy of the Google Drawing and added their numbers to the number line.  It took some work to manage the tools involved in this platform.

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I explained what each icon meant and how they could use it to make the number line their own.  It wasn’t as smooth of a transition as I thought it’d be, but students persisted and were eventually able to place the numbers they created on the number line and dragged the label to each number.  Students were then expected to take their drawing, save it as an image and place it in their individual SeeSaw account.

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Not all students finished this in class and I sent it home as optional homework for students to complete.  The above example is from one student that took it home and completed it before putting it into their SeeSaw account.

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Data Landmarks and Context

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One of my classes is working on a unit related to data displays and number systems. Around a week ago the class was putting together sets of numbers to match data landmarks.  This was a challenge as students had to think differently.  The class was also asked which data landmark better represents a student’s performance.   I was meaning to write a post then, but a number of things came up and it never happened.  Fast forward a week and here we are.   

Students were given two sets of scores from two different students. 

Jack’s scores:  85, 81, 78, 100, 84, 89 

Sonja’s scores:  55, 87, 91, 92, 68, 93 

Students were asked to find the median and mean for each student.  For the most part, students were able to identify both of these landmarks.  Here comes the kicker … now students needed to determine which landmark better represents each student’s performance, mean or median?  This was a challenging prompt for a couple reasons.   

  • Students weren’t accustomed to using the word represent in this context.  Students were taking the view that the students should get the higher grade and that would be the mean or median. They explained that the student should receive the higher grade because they (the person) is a hard worker and deserves to be rewarded with the highest score. 
  • Students thought of the word represents as the typical score.  When discussing the mean earlier in the year the word typical would often come up as a synonym. 
  • Students looked at the last score as the most recent and thought that should be the final representation.  My school is heading in the direction of standards-based grading so that’s maybe why students took that approach.  I don’t know. 
  • Students looked at the lowest and highest score of each set of data and reviewed the range to help them pick the median or mean 

 

After struggling a bit, the class came together and we discussed a few possible solutions.  The class agreed that the question allows a lot of room for interpretation and context certainly matters.  The fruitful conversation brought about a change in perspective for some as students started to see this type of math differently than just numbers sprawled across a page.  The numbers had meaning and the context drives the answer.   

A little later in the week students were asked the following prompt: 

If you were the teacher in Jack and Sonja’s class, would you use the median or the mean to calculate students’ grades?  Explain. 

This was a bit confusing at first, but students made progress in understanding the context and how it helped determine which landmark to use.  Again, I had answers related to the teacher wanting to give the higher score to help students with confidence.  Other students used the data landmarks to find the average.  I felt like students were more comfortable using the average as they could say that they used every data point, therefore making sure all assignments counted for something.   

I’m looking forward to next week as we dive into histograms. 

Reflections on 2018-19

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It has been about a month since school let out and I’ve been enjoying the summer so far.  I’ve been reading, working on the lawn, painting and took a vacation.  During the last month I decided to focus my time on things not related to school work.  This balance of time tends to give me a better perspective when I do come back to working on items related to school.  Now I’m starting to see school supplies (already!) in stores and am looking back at how the school year went last year.  Every year I attempt to gather information about my students and how they perceived the school year as a whole.  I give a survey and use that information moving forward for the next year.  I decided to wait a bit over the summer to look over the results.

Back in June I gave a survey to all of my students in 3-5th grade.  The survey was related to instruction models and preferences.  This year I intentionally varied my models throughout the year and didn’t stick with one particular tool for activities.  I started off the survey with a brief question about their favorite math topic this year.

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Before giving the survey I went into detail about each topic.  The purple is measurement and I’m not sure why it didn’t show up with my advanced table Gform add-on. The next question was related to why they felt this was their favorite topic. Here are a few responses:

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Next time I’m going to put a minimum character limit to extract more information.

The next section, which was the largest, was related to instruction models/activities. Students rated them (1-5) 1 being the least effective for learning and 5 being the most.  A brief explanation of the items is in each caption.

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Teacher gives prompt and students discuss as a group and share out responses to the class

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Used for retrieval practice – students used this individually and in groups of two

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Similar to Kahoot, but more self-paced – used to reinforce concepts

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Similar to Number Talks – prompt is displayed and class discusses concept

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Used  for notation on slides and showing student’s work to the entire class

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Used K-5 math for whole class or independent exploration of math concepts

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Students are digitally randomized into groups and then they work on a particular task with that partner.  Students report out their responses afterwards.

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I display work (sometimes student work) on the document camera and the class discusses strategies


I had 59 students take the survey, but I have 65 + students in 3-5th grade.  Some of them were out for other activities during the time the survey took place.  Something to consider … some of these activities were used more frequently so students had a larger sample size.  Overall though, it seems students enjoyed most of the tools/activities for learning about mathematics.  I think it should be mentioned that there’s a difference between a tool and strategy and I might be blending the lines a bit in this post.

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The tools in the box were used independently or with a partner.  They also required some type of technology (iPad, Chromebook), while the other four didn’t.  I think having a blend between the tools/strategies is helpful and students aren’t dependent on using one medium to show their learning.  I’m looking forward to diving more into this data as the summer progresses.

 

Scale Factor – Part Two

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During the last week of school one of my classes explored dilatations.  It was a rather short lesson since there were only a couple days of school left.  After some review, I pulled out a project from last year and thought might be applicable since it addressed the same standard for that particular day. I looked it over and made a few changes so this year it would run smoother.  Here’s what changed:

  • I had the students create an exact 4cm by 6cm grid using rulers.  This was different than my initial project.  I made sure to check each grid before students moved on to the next step.  I’m not a fan of having a simple mistake or unclear directions derail an entire project (which it did for some last year) – so I decided to check each students initial grid.

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  • I also created a random piece to the amount of dilation this time around.  This picture is from last year’s post.update.pngLast year students already knew the grid to use and basically used a “paint by number” approach to fill in each square.  Although that was fun, it didn’t really hit the objectives as much as I’d like.  I had students roll a die to determine the dilation this time.  This gave four different options for students.

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  • I put together a criteria for success component where students could check-off items when completed.  I set up the different dimension papers on one of the tables so students could easily grab them depending on their dilation.  I also added a short debrief piece near the end of the project where students discussed how they increased the size of the image.

These changes helped improve this particular project and I believe it created a better learning experience for the students.   There are times where I completely scrap a project and other times I make tweaks in order to make it better.  I opted for the second option this time around.

* Next year I’m planning on updating the project to include dilations that involve reducing the size of an image.

Measurement and Reasonable Solutions

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My fourth grade students finished up a project involving area last week.  Students were asked to find the area of different playing areas for certain sports.  They first calculated the areas of the playing field by multiplying fractions and then found the product.

The next step involved creating a visual model on anchor chart paper.  Students worked in groups to put together their athletic park involving the field areas.  They were given the area of the park and then had to place the fields where they wanted according to the team’s decision.  Students also added additional facilities for their athletic field and then presented their projects to the class.

While presenting, students in the audience were required to either 1) ask a question or 2) provide a constructive comment.  Most of the questions that were asked related to why certain fields were placed in specific areas on the field.  One question stood out more than the others … does the distance make sense?

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The locker rooms had to be adjusted (see whiteout) as one student said, “It doesn’t make sense so I changed it to match the 120 yd.”

 

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Students were looking at the length of the fields and observing whether it was reasonable or not compared to the total length.  The class then had a conversation about the terms reasonableness and proportions.  The discussion involved how a double-number line and a grid could’ve helped visualize how the distances match.

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I’m hoping to revisit this idea during the next few weeks as the school year finishes up.