Mathematical Mindsets for Parents
The powerpoint I created is meant to be shown to my students’ parents/guardians at out open house night in August. I would talk them through this presentation and allow them the opportunity to ask questions at the end.
Goal #1 – Implement the 5-phase model of mathematical arguments
Action Step: create open minded math tasks for Kindergarten age students based of off common core state standards and our math curriculum goals. These tasks will replace our typical daily worksheets and activities. I want to implement them similar to how Jo Boaler explained, two to three days a week for twenty to thirty minutes at a time.
Deadline: By the first week of October. That way, I will have given students time to settle into our classroom, become comfortable with classmates, have our schedule set in stone, and have introduced all of our hands on manipulatives and expectations with them.
Resources: Our Kindergarten math curriculum-Tara West KinderMath, common core state standards, colleagues, and the 5-phase model of mathematical arguments.
Potential Challenges: The amount of time that it will take to create the tasks, and the time constraints of our daily schedule. As much as I would love to let my kiddos stay engaged and actively learning, I need to make sure that I stay on top of watching the time.
Goal #2 – Facilitate mathematical discourse
Action Step: Create opportunities, and time each day, for students to share understandings, ideas, and thoughts about mathematical concepts. This will start as basic conversations not about math, to get to know each other, and then slowly become part of our “table time” for all subjects.
Deadline: Almost immediately, at the beginning of the school year. Although we don’t start our curriculums right away, I plan to implement discourse and discussion right away with games, activities, and hands on manipulatives.
Resources: Our Kindergarten math curriculum-Tara West KinderMath, common core state standards, appropriate mathematical games and activities, and hands-on manipulatives.
Potential Challenges: Some students will not feel comfortable talking with each other right away, while others will have a hard time when asked to stop talking (welcome to Kindergarten J). I will need to figure out my student’ personalities, and be willing to rearrange seating, so that not all dominant personalities get put at the same table; and vice versa for my quiet and shy students.
Goal #3 – Purposeful questions
Action Step: while creating any sort of math lesson or math task, be mindful and actively thinking about questions to ask my students, and the benefit of the questions.
Deadline: Almost immediately, at the beginning of the school year. There would be no reason to wait, even though we technically don’t begin our math curriculum right away. However, we still do mathematics tasks, lessons, and activities that can incorporate purposeful questions.
Resources: Our Kindergarten math curriculum-Tara West KinderMath, common core state standards, and the Purpose to Action textbook for this course-which explains and justifies purposeful questions.
Potential Challenges: When asking certain questions, I get nervous about the answers that I could potentially get in return. That also includes not getting answers at all. So, how many questions will it take to get my students to give the answers I want, and what will those questions be that get them there. I will need to be prepared and almost script out my questions, possible answers, and where to go next. I also need to be very mindful that I don’t question my students too much, or ask narrow questions, as it could turn into funneling them to the answer, instead of guiding them along the right path.
This math investigation follows the 5 phase model of mathematical arguments, and is designed for Kindergarten age students. This task is based off of the Common Core State Standard: (CCSS.MATH.CONTENT.K.NBT.A.1) Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
Phase 1 – Noticing Regularity
19=10+9
18=10+8
17=10+7
16=10+6
15=10+5
Teacher: What do you notice? What do you see happening here?
Student: I see that the number 10 is in all of them
Student: I see that the numbers are going down over here, like 9, 8, 7…
Student: I see that it says 19 and then has a 9, and 18 with an 8…
Student: I see the numbers are being taken apart into 10 and then the leftovers. Like 19 has 10 and 9 left over, and 18 has 10 and 8 left over…
Phase 2 – Articulating a claim
Teacher: Now that we see the way these number sentences are changing each time, we need to come up with a statement or a rule that they follow. Does this make sense? This sentence must be true, and we should make sure that it works for all five of the sentence that are on the board. We are first going to work with our triangle partners, and then come back together on the carpet to hear what you and your partners came up with.
Pair 1: All the number sentences have 10 in them
Pair 2: The numbers on this side are going down on that side
Pair 3: Where it says 19, that sentence has 9 on the other side, where it says 18, that one has 8 on the other side, and that happens for all of them
Pair 4: We can take apart the big numbers like 19, into 10 and the ones that are left over which is 9
Phase 3 – Investigating through representations
Students will work together in their groups to create representations of their statements. They will be allowed to use unifix cubes, snap cubes, counting bears, and paper/pencil.
Teacher: Now that you and your group have worked together to come up with a statement, you get to use manipulatives or paper to prove that your statement is true and that it fits all of our number sentences on the board. You may use unifix cubes, snap cubes, counting bears, and paper. I will be walking around during work time to check on each group.
Phase 4 – Constructing arguments
Groups of students will take turns sharing out their ideas and how their representation shows their statements to be true for the original number sentences. This is where the idea of “mathematical argument” comes into play as students will contribute different pieces of information and build upon each others explanations. We will then come together and collaboratively create a complete idea/solution.
Teacher: After hearing everyone’s statements, is there one or maybe even two, that fit our number sentences the best? What do they have in common? Do they both prove to be true? Which sentence could we change, or add to, to make it more specific?
Phase 5 – Comparing operations
Does this work for subtraction too?
Teacher: Do you think that we can take numbers apart in a similar way? Can we start with a large number, and subtract 10 to get leftovers?
19-10=9
18-10=8
17-10=7
16-10=6
15-10=5
Teacher: What do you notice? What do you see happening here?
Student: They still all have the number 10
Student: The numbers on the side are still going backwards, like 9, 8, 7…
Student: It still has a big number like 19 and then has a 9, and 18 with an 8…
Student: The numbers are still being taken apart into 10 and then the leftovers. Because it still has a large number, you take away 10, and there are leftovers that match the leftovers from when we added numbers.
NCTM questions:
Gathering information: helping students make sense of the information that is prior knowledge
Probing thinking: questions that help students clarify and explain their thinking (What do you notice? What do you see happening here?)
Making math visible: students are active in learning, while teacher is available to assess levels of understanding and comprehension
Reflection and justification: deep levels of thinking, reasoning, and understanding are occurring in order to produce mathematical arguments
NCTM Effective teaching practices:
Implement tasks that promote reasoning and problem solving (by engaging students in problem solving and discussing the task)
Use and connect mathematical representations (by making connections using mathematical representations, which deepen understanding of mathematics concepts)
Facilitate meaningful mathematical discourse (by creating discussion opportunities among students for sharing ideas, approaches, and arguments to mathematical tasks)
Pose purposeful questions (to assess and understanding students’ reasoning and sense making about mathematical concepts)
Support productive struggle in learning mathematics (by supporting students individually and collaboratively, as they engage in mathematical struggles)
Before reading, But Why Does It Work, I had never heard of the 5 phase model of mathematical arguments. I think this model is a great way for students to become active in their learning, and to develop a mathematical mindset. It gives them responsibility, and I am a firm believer in working collaboratively with peers. It may sound cheesy but I cannot wait to try this with my Kindergarteners, when we begin decomposing numbers into tens and ones!
Lesson objective: Students will enhance their geometry skills, by creating shapes using small pattern shape blocks.
Kindergarten math standard: CCSS.MATH.CONTENT.K.G.B.6
Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”
The task: As part of our geometry unit on shapes, kindergarteners will have the opportunity to explore and create shapes by using pattern blocks. Prior to this lesson, we will have completed lessons on shape names and identifying shapes. For this activity, students will be given a pile of square pattern blocks, and a pile of triangle pattern blocks. The questions posed to students will be, “What shapes can you make, only using square pattern blocks?” and “What shapes can you make, only using triangle pattern blocks?” Students will have the opportunity to work independently, or with the three other peers sitting at their table. Not only will students have fun being hands-on and working with shape blocks, but they will be responsible for their own problem solving and creativity.
Purposeful questions: What shape are you trying to make? What shape did you make? How many triangle blocks did you use? How many square blocks did you use? Tell me about what you’ve created with these blocks.
Whole class discussion: After gathering students back at the carpet, I would start by asking, “Tell me what we just did.” I know that my role as the educator is to encourage the discussion, so based on their answers to the first question, following questions could be, “What did the two questions I wrote on the board want you to do?” “How did you know that?” Tell your turn and talk partner, “What shapes were you able to create using triangle blocks?” Turn back around to face me, who would like to share what they told their partner? “What shapes were you able to create using square blocks?” “Did you count how many small ones it took to make your shape?” “What do you think would happen if you kept adding blocks?” As students share out their ideas, the goal would be that someone is able to point out that today we learned that we can use smaller shapes to create bigger shapes.
Since starting this class a week ago, I’ve had several flashbacks come flooding through my mind while reading the first four chapters of Mathematical Mindsets by Jo Boaler (2016). These flashbacks are of this last school year, and they include my students, myself, my co-workers, and their students; of things we did wrong, and of things we did right.
As of June 2nd 2017, I have officially finished my first year of teaching! There were a lot of emotions that came with that day. I was so ready for summer break, like the rest of you, but I was so NOT ready to let “my kids” go. These little humans that I knew nothing about back in September, completely changed my world in a way that I didn’t know was possible. I never expected to become so attached so quickly, or to know every little detail about their lives, their families, their friends, their likes and dislikes. I feel as if they became a part of my life, and I became a part of theirs, not just as their teacher.
One of the most prevalent feelings that I’ve had throughout reading is that of guilt. I feel guilty that I didn’t know this information until now. I feel as if I taught my students all wrong; as if I was a failure of a math teacher for them, because it all makes perfect sense to me now. As a first year teacher, I wasn’t as prepared for all the activities and games that come with some of our lessons, and I definitely hadn’t prepared for the time it took to teach them and play them. However, when teaching my kiddos new concepts that came with a game or activity, those are the skills and tasks that they were very good at. They understood those math tasks the best, the picked up on those skills the quickest, and most importantly those were the days of math time that they enjoyed.
One of my favorite lines that I’ve come across so far is this, “mistakes make your brain grow.” I will absolutely be talking about this with my incoming kinders in the fall. I feel as if I connect with this concept so well because, math has never been a favorite subject of mine. It was something that I always struggled with through my entire K-12 journey, and even in college. My first two math courses as a freshman and sophomore, which were part of generals, were really hard for me. I know that as a teacher, I made several mistakes this last year. However, I learned a lot from them; I grew as a teacher by making the mistakes I did. I don’t want to be that teacher that portrays her weaknesses onto her students. My uncomfortability with math taught me that I need to learn to be comfortable with it. Not always making time, or being prepared for the games and activities taught me that I need to make time for them, and make them this summer so that they’re made and ready to go for the school year.
The four chapters that I have read so far, in Mathematical Mindsets by Jo Boaler (2016), have already changed my outlook on Math. I’m now excited to spend time digging into our math curriculum this summer, to make the games and activities that come with the lessons, and to redo my classroom schedule to accommodate the amount of time math will now take. I feel like I’ve grown and changed as a teacher already, and this class just started! I can’t wait to see whats to come and how I can better myself as me, and as an educator!
My favorite take aways from the book so far are this:
Balzer Blog 