Adventures in 3rd grade Mathematics
Welcome to Adventures in 3rd grade math, teaching students to love math with a growth mindset.
To give my parents about 3rd grade math at open house I created a video using Prezi and Screencast-o-matic. I will have a document with a QR code available at open house in August for parents to be able to view the video. I have included the link and the QR code for you! Enjoy!
Action Plan
Kelly Johnstad
3rd Grade
Short Term Goals
Goal 1
Goal: Incorporate technology effectively into my classroom and math lessons.
Date to be incorporated: Begin Thursday August 31st at open house with QR code activities for students to introduce them to 3rd grade math (see “parent letter”) and throughout the year.
Resources: Ipads, Chromebooks, Laptops, QR Codes, AR activities, Dreambox math program, 3 Act Math videos, Smart Board
Challenges: Getting access frequently to the school technology carts to allow students to have time with technology. The time it takes to preplan and create effective technology for students to use.
Overcoming challenges: I have already begun conversations with my principal about the necessity to have access to technology on a regular basis. We are working on a schedule to allow each 3rd grade class to have 5-7 technology tools (Ipad, Chromebook, Laptop) everyday during math for student activities. This would be one cart of 30-35 tools split between the 5 classes. This way we could work together to share the technology evenly, or allow one classroom to have the entire cart for a lesson if needed. Time management can, and will always be a struggle for teachers to prepare effective lessons daily with the time allotted, as well as using personal time. I frequently arrive to school 30 minutes each day, I will be trading less assessments for more time to create effective technology integration for my students.
Possible activities: Dreambox math program 2-3 times a week. 3 Act Math once a week. Augmented Reality activities at various times (example during geometry to bring 3D shapes to life).
Justification: Technology is an important tool for meaningful learning of mathematics. Technology can be a wide range of tools, such as those listed above. These tools, when used effectively, help students make sense of mathematics, engage in mathematical reasoning, and have the skills to be able to communicate mathematically. These tools make applications, websites, and activities to available that help students explore math, and help make sense of different concepts and procedure. These tools can also aid in helping students stay engaged and use mathematical reasoning skills. (pg. 78 Principals To Actions)
Goal 2:
Goal: Incorporate 3 Act Math Tasks once a week
Date to be Incorporated: September 2017 School Year
Resources: 3 Act Math tasks found on teacher blogs, twitter feeds, and Dan Meyer’s website, as well as 3 Act Math tasks I will create myself.
Challenges: Finding the time to incorporate 3-Act Tasks, while still completing district mandated curriculum. Finding and creating 3-Act math tasks that support state standards and concepts being taught through district curriculum.
Overcoming Challenges: I can overcome these challenges by seeing the benefits of 3-Act Math and making it a priority for my students to have access too. Also, I need to stay active on twitter and searching for quality 3-Act Math tasks that are already created and finding the ones I need to create myself. Through this class I have already found many that can be implemented in my classroom and saved their links to use later.
Possible Activities: The 3-Act Task I created for class, Robert Kaplinsky 3-Act Math Tasks (http://robertkaplinsky.com/lessons/), Tap Into Teen Minds 3-Act Lessons (https://tapintoteenminds.com/3acts-by-common-core/grade3-ccss/)
Justification: 3-Act Tasks can be used to engage students into mathematical reasoning and understanding, provide students with opportunities to discuss mathematics, encourage flexible thinking and allowing students to use multiple approaches, encourage to students to find an answer and justify it. (pg. 6 Three-Act Tasks: Engaging Students In Math)
Goal 3:
Goal: Incorporate Why Does It Work tasks 3 times a week for 15-20 minutes. The whole procedure may take more than one day/week for students to complete.
Date to be Incorporated: September 2017
Resources: But Why Does It Work text book, WDIW twitter feeds, WDIW blogs/websites
Challenges: Teaching students how to make mathematical arguments and the phases we will go through to create a conjecture, justify our thinking and extend our thinking to other operations.
Overcoming Challenges: In the beginning we will need to “go slow to go fast” while students warm up to the idea of making mathematical arguments. I predict students will come to 3rd grade ready to answer numerical math questions (such as 2+3=5), but not justify and analyze how they got this answer, and share their thinking with their peers. Setting up respectful groups, with high expectations (as explained in Chapter 7 of Mathematical Mindsets) and a value of discussion and sharing ideas will be crucial to the success of these activities.
Justification: Lingering on mathematical arguments requires students to focus on the operations and understand their directions and behaviors, learn about mathematical practices, require students to analyze and justify their thinking, and level the playing field for all students. (But Why Does It Work?).
Long Term Goals
Goal 1:
Goal: Foster a growth mindset in my students, and celebrate mistakes and struggles.
Date to be incorporated: Begin September 2017. Goal is to change mindset for students for years to come in their mathematical futures.
Resources: Mathematical Mindsets, But Why Does It Work?, 3-Act Tasks
Challenges: Students who enter with a fixed mindset and parents who foster a fixed mindset at home. For example, parents who say “It’s ok that math is hard, you’re more of a reading kid”.
Overcoming challenges: I need to educate parents about a growth mindset, and foster growth mindset in my class room through activities, celebrating mistakes and struggle, giving diagnostic comments, changing assessments, and learning to “say the right things”. For example, telling a student “I think it’s great you learned that” instead of “you’re so smart”.
Justification: The research on growth mindset and the positive effects in a classroom are overwhelming. Just Google growth mindset and you can see lists of research that promotes the importance of fostering a growth mindset in students. One great resource for the importance of a growth mindset is Mathematical Mindsets by Jo Boaler.
Goal 2:
Goal: Encouraging the team of teachers, interventionists, Special Education teachers, and ELL teachers to “jump on board” with believing all students can learn and foster a growth mindset.
Date to be incorporated: July 6th 2017. I just called a meeting with my team to discuss some of the changes I would like to see in math this year, as well as attempting to get my principal on board to support this new mindset.
Challenges: Changing the mindset of teachers that have a fixed mindset, are comfortable in their ways of teaching, and do not have the background knowledge on the importance of growth mindsets and mathematical reasoning.
Overcoming Challenges: I can work to overcome these challenges by beginning to teach other staff what I have learned. One of the most powerful ways to teach others the power of growth mindset and mathematical reasoning will be through showing success and changes in my classroom and with my students. I think the most effective way to help my colleagues see the benefits of what we have learned is to “lead by example” and allow them see the positive changes in my students and teaching.
Justification: Students who enter my classroom with a fixed mindset will have an easier time remaining in that mindset if other teachers are allowing them to remain in a fixed mindset, and teaching in practices that foster a fixed mindset. It is my responsibility as an educator to not only encourage growth in my students, but all students I come in contact with, or my colleagues come in contact with. This information we have learned is too important not to share with others and allow their students to benefit from these activities as well!
For my math investigation, I chose to work with one of my former students from last year on some core ideas of addition as shown on page 84. This student will be entering fourth grade, but lacks confidence and a growth mindset in mathematics.
Phase 1: Noticing Regularity
I presented my student with eight number sentences written on a piece of paper. Four of them were already solved and asked him to solve the remaining four number sentences. (Gathering Information, pg 36 Principals to Actions)
When I asked the student to solve the remaining number sentences, he immediately used his fingers to solve them. He did not right away notice the pattern that the sum increased by 1, and the addend increased by one as well.
I prompted the student with: What do you notice? (Probing Thinking, pg 36 Principals to Actions) I think this questioned surprised my student. He was so used to someone asking him to find the answer, but not asking him any follow up questions. He sat and looked at the problems for a minute, and said they are all adding.
I asked again- “What else do you notice?”(Probing Thinking, pg 36 Principals to Actions) . I gave him some time to really look at the work and the problems presented. Finally he said, “they are one more”. I asked him “what is one more?”(Probing Thinking, pg 36 Principals to Actions) . He pointed out the sums in the first problem stating that 13 is one more than 12.
I asked “What is happening here?” and pointed to the number sentence (Probing Thinking, pg 36 Principals to Actions) . At this point, he shrunk in his seat wondering why I was asking so many questions. It was as if he was thinking, “I gave you the answer, now let’s move on”. It dawned on me, he has escaped explaining how and why he does things, and noticing patterns in his work. Finally, after some good “wait time”. I got out of him, “This one is one more than that one”. I asked him if he could say that another way. Finally, I heard some math talk! He restated his statement into, “the sum goes up by 1”! Fantastic! A math observation! A notice of patterns! I followed this up with, “what else do you notice?”.(Probing Thinking, pg 36 Principals to Actions) With some more wait time, he began to explain. He explained that the sum is going up by one because one number stays the same, and one number goes up by 1.
Phase 2: Articulating a Claim
At this point, I explained we would write a conjecture, stating our observations. The student wrote- The sum gets bigger by one. One number stays the same and the other number gets to be one number bigger.
As a side note, I asked him what the sum was. He was quick to respond with the answer to an addition problem. I then asked if he knew what the other numbers in an addition number sentence were called, and he did not have the vocabulary addend.
Phase 3: Investigating Through Representations
I used the example on page 35 of But Why Does It Work with my student and told him the story problem but changed the names to his brother’s name and his name:
“Nomi found 4 rocks and Alec found 6. How many rocks did they find?” (pg 35, But Why Does It Work)
Using fingers, the student told me they had 10 rocks altogether.
“On the way home, Alec found 1 more rock. Now Nomi has 4 rocks, and Alec has 7. How many rocks did they find?” (pg. 35 But Why Does It Work).
After, recounting he told me 11. To help keep this information in our head, we wrote down the number sentences on paper.
4+6=10
4+7=11
I asked him to create a representation to show why the sum must increase by 1. (Making The Mathematics Visible, pg 37 Principals to Actions) Immediately, he drew a set of dots.
First, he drew 6 dots, plus 4 more dots.
I began to ask more questions.
“Where do you see 6?”
“Where do you see 4?”
“Where do you see 10?”
“How is addition represented?”
Where do you see the 1 that was added to the 6?”(Encouraging Reflection and Justification, pg 37 Principals to Actions)
This is where he was unable to answer, because he had forgotten to show adding the 1 to the 6. So far, he had given me 6+4. To show that he added one more, he drew a dot off to the side, and an arrow to the set of 6, and then added that dot to the set of 6 to have a set of 7.
Phase 4: Constructing Arguments
Just looking at his drawing could be confusing to some students, so I asked him to explain it. He explained that he found one more rock, so he added another circle to his drawing to turn 6 into 7 and make the sum 11 now. (Making The Mathematics Visible, pg 37 Principals to Actions)
I asked him to think of another way to show his thinking, but he was unable to produce another form to explain how and why this conjecture worked. This is when it would have been a great opportunity to have more minds to show other way of thinking and other proof.
At this time, I pretended to be a student as explain to him; I was thinking in the form of a number line so he could see more than one way to represent and explain why the conjecture must be true. As soon as I began working, we had that moment all teachers hope for, when the lightbulb turns on! Before I could finish explaining my thinking, he was finishing it for me. I began with my number line and showed; 4 rocks and 6 more rocks show 10 rocks. He chose to take the pen and add in the jump of 7 to show the sum increases by 1 to 11.
After seeing it represented a different way, I asked him again if he could show me another way. At this point, he was able to show me 6 cubes, plus 4 cubes colored in is 10 cubes. Add another cube (represented with lines drawn on it), and the sum changes to 11.
Phase 5: Comparing Operations
To consider if this works for other operations, I next gave him a sheet with some subtraction problems on it.
I asked if the same conjecture we wrote would be true for these number sentences as well.(Exploring Reflection and Justification, pg 37 Principals to Actions) He was able to tell me it did not, because as one number went up in the number sentence, the difference went down by 1.
In Conclusion: I noticed my student became very uncomfortable when I asked him to really look at what we were doing, locate patterns, and explain his thinking. “We want students to feel free as they work on math, free to try different ideas, not fearing that they might be wrong. We want students to be open to approaching mathematics differently, being willing to play with mathematics tasks, trying “seemingly wild ideas” (pg 14, Math Mindsets). Through these phases, my student was able to struggle, look for patterns, and try different approaches to solve and explain his thinking. We were able to go through the four stages of mathematics as proposed by Wolfram (2010).
1. Posing a question- What is the pattern here? What do we notice? If we increase an addend by 1, does the sum always increase by 1?
2.Going from the real world to a mathematical model- Creating a visual representation of our thinking to prove our conjecture.
3. Performing a calculation- solving our models, to see if they answer our number sentences.
4. Going from the model back to the real world, to see if the original question was answered – Going back to see if our models prove our conjecture, and apply to other operations.
I would like to see how this can work in my classroom with more minds, ideas, approaches working together to explain thinking and identify what we notice and why it works. I noticed some lack of vocabulary in my student, as well as a lack of a growth mindset. It made me wonder how many of my other students are also lacking in those areas, and what I can do to help them have a growth mindset in the upcoming school year. I believe, using this approach to mathematics will help all my students increase their mathematical reasoning, and move toward a growth mindset to have the skills necessary to not only succeed in third grade mathematics, but in life.
Lesson Title: How Many Reese’s Peanut Butter Cups?
Prescribed Learning Outcome(s): Students will pose questions and find solutions to answer the question: How many Reese’s Peanut Butter Cups?
Instructional Objective(s): 3.1.2.3 Represent multiplication facts using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups. Recognize the relationship between multiplication and division.
Students will show multiplication facts in more than one way – including repeated addition, equal sized groups, arrays, area models, and equal jumps on a number line and skip counting.
Act 1:
Click this link to see the Act 1 Video:
Show the students a video of a opening a bag of Reese’s Peanut Butter Cups and placing one on a white card.
Ask the students- What do you notice? What do you wonder?
Students may respond with:
How many fit on the card?
What is the perimeter of the card?
How many rows and columns are there?
Act 2:
Ask the students: what information do you need to solve this question?
Give the students the information that they need: 7 rows and 5 columns.
Allow students to work in pairs to solve: How many dots are there altogether?
Closure: Show students the answer.
Ask students to share their answers. Facilitate a discussion on the different strategies to solve how many Reese’s Peanut Butter Cups fit? Did students draw a diagram? Use repeated addition? Make an array? Skip count? Allow students to model all of their strategies for students.
This week I learned a lot of things to help all students be successful in my classroom and in their future as mathematicians. A few topics that I would like to discuss are the knowledge that every student can be successful, and using growth mindset language in the classroom, and celebrating mistakes.
The research shared in Mathematical Mindsets by Jo Boaler changes the way we have been thinking. For years people have stated that our brains have a certain capacity, some students just have “math minds”, and that fast equals smart. It changes our thinking to understand that all students are capable of learning, succeeding, and growing with the correct materials and teaching. It is our job as educators to help all students make deep meaningful connections through learning in a variety of situations and experiences to help all students grow their mathematical knowledge.
Through learning about growth mindset, I have discovered how important the feedback I give my students is. I plan to enter future school years ready to celebrate mistakes and teaching my students that mistakes are an opportunity to make our brains grow. It can be as simple as restating, “you’re so smart” into “I’m really glad your brain learned that”.
Finally, another eye opening topic Jo Boaler discusses is the fact that brains grow through mistakes. A students who gets all of the problems correct on a test did not have a chance for their brain to grow and create new connections. When we make a mistake there are two outcomes that can happen; our brain can increase electrical activity of conflict between error and a correct answer even if we are unaware there was an error made, or the brain signals us that an error has been made and we can pay attention to that error. This was discovered in a study by psychologist Jason Moser (Moser et al., 2011). From this we learn that mistakes create activity in our brain, causing our brain to grow. During times of struggle, of deep thinking is when the brain grows the most. From this we can learn to celebrate mistakes. This tells us that mistakes are an opportunity for learning and for brain growth. I plan to take this into my teaching and celebrate the mistakes, and give all my students the opportunity to celebrate mistakes, and look at them as a time to discuss and learn from each other. Through mistakes and discussion we can grow together and learn from one another.
With my newfound understanding of growth mindset, I feel confident that I can help my students celebrate learning, mistakes and grow as students and people. I look forward to digging deeper into the growth mindset so I can aid all students into deeply thinking about math, and feeling successful as mathematicians.
Adventures in 3rd grade Mathematics 