Mathematical Mindsets: Chpt. 9


This is it. The last chapter of Mathematical Mindsets by Jo Boaler. This chapter recaps key takeaways to help foster a growth mindset classroom. There is so much information to digest from this book to help pave way to mathematical mindsets for all students.


...,it is important to remember that not helping students is often the best help we can give students. ~Jo Boaler (179)

Key Takeaway: Let's encourage students to be mathematicians. Rather than set limits on what students learn and explore in math,it is okay to introduce higher level math in an intellectually honest form to students to pique curiosity and allow them to grapple and wonder (189).

Classroom Connections:
  • It is important to establish classroom norms and expectations that are not only shared verbally, but also shown through teachers' actions.
    • Every student is valued and expected to achieve at the higher levels.
    • Mistakes are valued. Mistakes make the brain grow.
    • Failure and struggle are integral in the process of learning math.
    • Multiple pathways and representations for math are important and valued more than speed.
    • Questions are welcomed and encouraged!
  • When giving feedback, we need to be cognizant of the words we use. Words such as "smart" need to be replaced with feedback that reflects specific student actions and strategies used such as: "It's great that you learned that." or  "I really like how you are thinking about that problem." (179) The word "smart" develops a fixed message that it's great when I know things, but when I make mistakes and fail...I am no longer smart. We want students to struggle in math. Without failure or mistakes, students may not be getting the opportunity to wade into new areas of mathematics or to develop a deeper understanding of complex and rich mathematics. 
  • One reality of mathematics is that in today's work world, computers are available to do calculations. We need to grow students in being comfortable making conjectures, posing questions, and defending their reasoning. Rather than always using problems posed in textbooks that tend to focus on procedural calculations and contrived situations have students use real-world data to generate their own mathematical questions. Modeling with mathematics (SMP 4) and interpreting mathematical results also can help ramp up how we can engage our students in the math classroom. If a student quickly gets an answer ask them to model their thinking...how did they arrive at that answer? This can bring in a whole new level of mathematics to some students.
Thank you for joining me on this journey. Now as the school year unfolds, the goal will be to take some of these ideas by Jo Boaler and embed them into daily practice. youcubed.com has a Week of Inspirational Math that could be used as a springboard to begin to transform the way students view mathematics. Wishing you all an engaging and successful mathematical journey!

Mathematical Mindsets: Chpt. 8


This is one of the chapters I have been waiting to read: Assessment for a Growth Mindset. It seems somewhat contradictory when you think what growth mindset stands for and then the word assessment is associated with it. Let's begin a recap of chapter 8 in Jo Boaler's book, Mathematical Mindsets.


The most important preparation we can give students is a growth mindset, positive beliefs about their own ability, and problem-solving mathematical tools that they are prepared to use in any mathematical situation. ~Jo Boaler (142)

Key Takeaway: Jo Boaler emphasizes the need to transform how we assess students in the math classroom. The goal would be to replace fixed mindset testing with growth mindset assessment that channels mathematics that is broad, creative, and rich in problem solving. In this way, students would be empowered. Rather than students seeing a grade, students should see diagnostic comments/feedback that celebrate their knowledge and guide them to improvement.

Classroom Connections: As teachers, we need to help students become more cognizant of the math they are learning and where they are in the learning process (151).
  • Self-assessment through reflection can help students become more aware of the mathematics they are learning. When asked what are you working on in math, we want students to focus on the content of mathematics and not the chapter, lesson, page, or problem. Students can choose one of the following to respond to at the end of class (158):
    • What was the big idea we worked on today?
    • What did I learn today?
    • In what situations could I use the knowledge I learned today?
    • What questions do I have about today's work?
    • What new ideas do I have that this lesson made me think about? 
  • Exit tickets can be used as a tool to guide instruction. The exit ticket can focus on the key target understanding from the lesson. In the past, I have asked students upon completion to place their exit tickets in a red, yellow, or green basket. In this way, I can get an understanding of student's understanding of content, while allowing students to self-reflect on their ability in completing the skill: green: confident, yellow: getting there, red: help needed. If a student demonstrates understanding yet places his/her exit ticket in the red basket, this may indicate some positive growth mindset messages may be needed. If a student does not demonstrate understanding yet places his/her exit ticket in the green basket, this may indicate a misconception needs to be addressed. 
  • Grading should move beyond checking the accuracy of procedural questions. How might student work be used differently with a shift in focus to whether students ask questions, use multiple representations to show math, build on the thinking of others, and justify their thinking? Math is multidimensional and simply grading if an answer is correct/incorrect does not convey a complete picture of a student's mathematical performance.
Above all else, it is important to view assessment for learning! Assessment should provide information about a student's learning. Rather than focus on grades, the emphasis should be on constructive feedback. Ultimately, the parameters set by a district set the expectations for teachers in terms of grading and assessments, but within those parameters, fostering positive messages and focusing on student learning, and not only achievement, are steps that can be taken.

Mathematical Mindsets: Chpt. 7


We are winding down with just three more chapters left in Jo Boaler's, Mathematical Mindsets. This week the focus is on tracking to growth mindset grouping. Boaler emphasizes the need to de-track students and group heterogeneously. I know this has been an ongoing debate in some elementary classrooms.

I do not value speed or people racing through math; I value people showing how they think about math, and I like creative representations. ~Jo Boaler (135)

Key Takeaway: The repetitive theme throughout Boaler's book is to move beyond the one-dimensional math class where the practice of executing procedures correctly is valued above others (121). Transforming classrooms into multidimensional classrooms where good questions are asked, ideas are proposed, different methods are explored and connected, multiple representations are encouraged, and reasoning is encouraged through different pathways is a goal to strive for. Dream big!
 
Classroom Connections:
  • When students are grouped heterogeneously, there are different ways to encourage students to look at a problem. These include asking good questions, rephrasing problems explaining, using logic, justifying methods, using manipulatives, connecting ideas, and helping others (122). In this way, there are multiple entry points to a problem for students of different readiness levels. At the start of the year, it would be interesting to ask students how they can approach/look at a problem and record their responses. Then as the year progresses, we can add to that list. Hmm...definitely something to think about.
  • When students are working collaboratively, the role of the teacher is the facilitator. I really liked the idea of how groups were assigned roles such as a reporter. If the teacher needs to share something new to move the task along rather than stop the whole class, the teacher can call the reporters from each group over for a huddle. Love the idea of a "huddle." Once the information is shared with the reporters, they go back to their respective groups and convey the necessary information. Sometimes trying to get everyone's attention when they are immersed in a task to stop can be challenging. I will be adding this idea to my toolbox.
  • Feedback is essential for student learning, especially for those who may not find confidence in the subject. Boaler recommends that feedback to raise the status of those who may not feel like they are as "good" as others in their group needs to be public, intellectual, specific, and relevant (135).
    • public dimension: highlights that a specific student contributed the idea
    • intellectual dimension: addresses the aspect of the mathematical work the student offered
    • specific dimension: showcases exactly what the teacher is praising
Math really is a social subject where students can bring their independent understandings to help accomplish a group goal.

Mathematical Mindsets: Chpt. 6


Well, hello, again! Here we are in Chapter 6 of Jo Boaler's book, Mathematical Mindsets. The focus of this week's chapter was mathematics and the path to equity.

Mathematics is a broad subject that goes beyond computation and procedural speed and involves understanding of ideas. ~Cathy Williams (101)

Key Takeaway: Mathematics needs to be a balance between computational/procedural skills, conceptual understanding, and problem solving.  It is really important that we take steps to move beyond procedural based math classrooms and transform math into a connected, inquiry-based subject (104). As Jo Boaler would say, "Viva la Revolution!"

Classroom Connections:
  •  "Mathematics is not more difficult than other subjects--I would challenge people who think so to produce a powerful poem or work of art. All subjects extend to difficult levels; the reason so many people think math is the most difficult is the inaccessible way it is often taught (Boaler, 96)." This is such a powerful statement that I will need to keep nearby as I continue this journey and support others in helping our students develop a mathematical mindset.
  • Creating hands-on experiences, offering project-based curriculum, providing real-life applications, and fostering collaborative work are ways to help strengthen engagement for students, especially girls (103). 
  • The issue of homework was discussed. Boaler's recommendation is that homework should be given only if the homework task is worthwhile and draws upon the opportunity for reflection or active investigations around the house (109). In her beliefs, homework should be eliminated, reduced, or changed. The idea of procedural practice with repetitive problems for homework needs to be re-evaluated in our quest to foster equity for all students. 
As a teacher, intentional decisions need to be made to ensure equitable opportunities are provided for all students to thrive in mathematics. Mathematics is a gateway to their future!

Mathematical Mindsets: Chpt. 5


August is here! Thank you for stopping by. This week we are talking about rich mathematical tasks from Jo Boaler's, Mathematical Mindsets.


Teachers are the ones who can create exciting mathematics environments, give students the positive message they need, and take any math task and make it one that piques students' curiosity and interest. ~Jo Boaler (57)

Key Takeaway: Mathematical tasks should be challenging but accessible to students. The tasks should be engaging and require students to think about math visually and numerically.

Classroom Connections:
  • When giving students problems to solve, pique curiosity and prime their brains for learning by giving students the opportunity to explore problems even before the methods to solve the problem have been explained to them. Yes, you heard that right, even before they know the methods to solve. Through exploration, students may come to a point when a method needs to be explained for them to progress further in solving the problem. At this time, the method can be introduced in response to students discovering the need.
  • These are six questions that can help guide in creating and offering rich mathematical tasks (90) to students. These are great questions to have nearby when planning.
    • Can you open the task to encourage multiple methods, pathways, and representations?
    • Can you make it an inquiry task?
    • Can you ask the problem before teaching the method?
      • This is a definite one to try out! Pose the problem first. See where students take the problem. Be sure to ask them to be able to justify their thinking!
    • Can you add a visual component?
      • Drawings can put a new lens on a math problem for students!
    • Can you make it low floor and high ceiling?
      • Tasks can be created so that they are accessible to students of various readiness levels while offering extensions to those you are ready.
    • Can you add the requirement to convince and reason?
  • Flipping questions found right in textbooks is a starting point for creating rich mathematical tasks. Starting slow is key. In this way, you can begin to build a bank of rich tasks that parallel the standards you need to teach.
    • Here are some websites to peruse for additional ideas. It is important to consider your students when choosing problems. Raise the ceiling and see how high your students can soar!

As the new school year is upon us, let's consider ways to transform the types of math problems/tasks students see in the math classroom.

Back to School Activities

It is hard to believe the start of the school year is just around the corner.

Here is a get-to-know-you resource you might like to add to your teacher toolbox for the start of the school year.



Back to School - Get to Know You Activities has two printable games to get students interacting and talking with each other during the first few days of school. These activities encourage students to learn more about each other as they share information about themselves. 

https://www.teacherspayteachers.com/Product/Back-to-School-Get-to-Know-You-Activities-2597089

https://www.teacherspayteachers.com/Product/Back-to-School-Get-to-Know-You-Activities-2597089

Here is another activity you might want to take a peek at. Engage your students with these back to school writing activities in a PRINTABLE and a DIGITAL version. 

School Days will get the creative juices of your students flowing! Roll-a-Cube is a way to engage students in reviewing language skills and applying these skills to short writing tasks. The skills included on the Roll-a-Cube: onomatopoeia, personification, comparison, imagery, exaggeration, and dialogue using quotes. The four writing prompts are sure to pique the interest of your students.


What fun activities do you do with your students to set the tone for the school year? Wishing you a great start to the school year!

Mathematical Mindsets: Chpt. 4


Almost halfway through Mathematical Mindsets by Jo Boaler. So much mathematical information to process and think about. Starting to tuck away ideas to implement next school year, which is not so far away. Let's take a look at Chapter 4, Creating Mathematical Mindsets: The Importance of Flexibility with Numbers.


Mathematics is a conceptual domain. It is not, as many people think, a list of facts and methods to be remembered. ~Jo Boaler (36) 

Key Takeaway: Math facts are best learned through the use of numbers in different ways and different situations. The goal is to help students develop number sense where they can think about numbers flexibly using different combinations. Timed math tests should be replaced with conceptual mathematical activities. Activities where students play with numbers and begin to see relationships and patterns of numbers. Check out "Fluency without Fear" for some ideas to implement in the classroom and grow students understanding of math facts. The point that really resonated with me in this chapter was when Boaler explained that readers do indeed need to memorize the meaning of many words (41), but it is not the fast memorization and recall of words. Math facts is a hot topic that can be a mind shift for some teachers and some parents alike.

Classroom Connection:
  • In the area of math practice, more is not always better. Synapses fire when learning takes place, not when doing the repetitive practice of isolated methods in their simplest form. When having students practice math, reduce the number of problems to practice. A whole page of practice is not necessary.
  • Students need to be introduced to math beyond its simplest form that is often seen in textbooks. For example, when identifying geometric shapes, students need to see many, varied examples, not just regular polygons that are the "perfect" example of each shape. Students should see different examples and non-examples. Jo Boaler mentioned how perfect examples can lead to misconceptions. I recall a time when learning about types of angles. Rather than draw a right angle opening to the right, I drew a right angle that opened to the left. Some students did not think it was a right angle and even questioned, "Why isn't that a left angle?" From that point forward, I now draw angles that open to the right, left, up, and down.
  •  With the integration of technology into more and more classrooms, it is important to think about choosing apps and games that develop a conceptual approach to math. I visited the website, Youcubed, and followed the link for math apps and games. There are a handful of games to view. This is a great starting point for building a bank of more conceptually based apps and games that do not rely on computation and speed.

Next week's chapter is a good one. Chapter 5 talks about rich mathematical tasks. See you then!

Mathematical Mindsets: Chpt. 3


So glad you stopped by for the third week of the book study for Mathematical Mindsets by Jo Boaler. If you asked your students, "What is math?" What kind of responses would you get? Would students say answering questions by adding, subtracting, multiplying, and/or dividing? Would they say problem solving and reasoning? In Chapter 3 the focus is on math...the creativity and beauty in mathematics.


"In order to understand the universe you must know the language in which it is written and that language is mathematics.” -Galileo Galilei

Key Takeaway: Mathematics is all around us. Mathematics needs to move beyond a textbook with right and wrong answers. If we look closely, we can see the mathematics of nature. We can investigate how animals, yes animals, use math. We can see the beauty in art. There is even the National Museum of Mathematics in New York.

Classroom Connections:
How do we help our students to see the beauty and creativity in math?
  • To start the year and to engage learners, I read Math Curse by Jon Scieszka and Lane Smith. It does a great job to help students realize that math is around them EVERYWHERE! The book starts out, "On Monday in math class, Mrs. Fibonacci says, 'YOU KNOW, you can think of almost everything as a math problem.'" Before I read on, I ask students what they think the teacher means. They are quite surprised as I begin to read the story, reading faster and faster as I turn the pages. The text and the illustrations bombard students with how the main character in the story gets tangled up in math problems with every turn of the page. It makes the point that math is everywhere!
  • Another picture book that piques students' interest is Mathterpieces: The Art of Problem-Solving by Greg Tang. In this book, twelve artists are featured. Each piece of art is paired with a problem that focuses on addition for younger students and problem solving in multiple ways for older students. The art piece is featured on one page, and then groups of objects on the adjoining page to find a sum in different ways. The connection relates to the art piece subtlety but the novelty hooks learners. This is just one way to bring art into the math classroom.
  • Rather than pose questions that require a calculation and an answer (29), math needs to be more. It does not mean "faster is better." Math needs to be about collaboration and deep thinking. Using open math problems (29) helps students develop creative ways to problem solve. Flipping questions traditionally found in a textbook to open math problems is one way to move students to think about math differently. Sometimes students feel uncomfortable with these types of problems because there is no one right answer.
In a bag, there are coins (quarters, dimes, nickels, and pennies) that add up to $1.00.
Penny pulls out 33 cents.
What coins did Penny pull out?
How much would be left in the bag?
What coins would be left in the bag?

Math really is a beautiful thing! Thanks for stopping by Pam's Place. Next week the chapter is on flexibility with numbers. See you then!

Mathematical Mindsets: Chapter 2


Welcome to Week 2, Chapter 2 of Mathematical Mindsets by Jo Boaler. This week the chapter was on mistakes. Thank you to Kathie from Tried and True Teaching Tools for hosting this book study.


In Chapter 2, The Power of Mistakes and Struggle, Boaler puts the spotlight on the importance of mistakes. For some, this is a whole mind shift on how parents, students, and teachers should perceive mistakes. Mistakes are evidence of learning.

Key Takeaway: According to Carol Dweck, "Every time a student makes a mistake in math, they grow a synapse."  Yes, mistakes cause the brain to spark and grow. When the brain is challenged, the brain grows the most (11-12).

Classroom Connections:
  • Mistakes happen to the best of us. It is important for students to understand this. Sharing famous people and athletes who have made mistakes/failed and how they overcame those obstacles can be tangible evidence that it takes mistakes/failure to be the best. Two quotes I often share with my students are those associated with Thomas Edison and Michael Jordan. 
    • Thomas Edison: I've not failed. I have found 10,000 ways that won't work. 
    • Michael Jordan: I have missed more than 9,000 shots in my career. I've lost almost 300 games. 26 times I've been and trusted to take the game winning shot and missed. I've failed over and over and over again in my life, and that is why I succeed. (Here is a YouTube video for this quote: Michael Jordan video.)
  • Creating posters of Peter Sims's Habits of Successful People (14) can help to remind students to play with math and approach math differently. It would be important to explicitly talk about each habit so students know what each "looks" like and "sounds" like.
    • Feel comfortable being wrong
    • Try seemingly wild ideas
    • Are open to different experiences
    • Play with ideas without judging them
    • Are willing to go against traditional ideas
    • Keep going through difficulties 
  • Mistakes happen. Having students analyze their mistakes can be helpful. Was the mistake a silly mistake, a math mistake, or a process mistake? Here is an image of an anchor chart similar to one students keep as a reference: Math Mistakes.
Ultimately, we want our students to feel comfortable with making mistakes and learning from them. Mistakes are opportunities for learning! Come back next week for Chapter 3, The Creativity and Beauty in Mathematics.

Mathematical Mindsets: Chapter 1


So excited to begin this book study hosted by Kathie at Tried and True Teaching Tools. So hope you enjoy the posts on Mathematical Mindsets by Jo Boaler.

I know there has been a lot of buzz about growth mindset so the title of this book intrigued me. How can we help all students develop a mathematical mindset? Let the discussion begin...


In chapter 1, The Brain and Mathematics Learning, Boaler begins with research. I have taken one of her online courses, and I know supporting ideas with research is her thing. Boaler begins with her claim that the brain can CHANGE, ADAPT, and GROW (4)! This is great news for teachers. If the right math materials are used and if students receive positive messages about their potential/ability, students can journey towards a mathematical mindset.

Key Takeaway: One recommendation by Boaler is to praise students for what they DO rather than who they are as a person (8). Flipping how teachers praise students can make a difference.

 "That is an amazing piece of work."
"You have thought deeply about this piece."
"It is wonderful to see how you learned this."

Classroom Connections:
  • Growth mindset. Messages students receive can impact their self-perception. Intentionally teaching students about the difference between a fixed mindset and a growth mindset is important. Explicitly teaching students about the different mindsets can help make what growth mindset "looks" like and "sounds" like more tangible. Creating a classroom environment that surrounds students with growth mindset messages is key. Pinterest is full of different ideas to help support this in the classroom. Here are some ideas I posted to one of my Pinterest boards: Growth Mindset vs. Fixed Mindset.
  • Mistakes. How important they are! Rather than see mistakes as failure, students need to realize they are a pathway to learning. Helping students to see the importance and relevance of mistakes is important and has to be intentional. One strategy I like to use is, "My Favorite No." Check out the video here: My Favorite No: Learning from Mistakes from the Teaching Channel. Although it is recommended for grades 6-8, it can be modified and used with other grades in a manner that fits the readiness levels of the students.
  • Regardless of where our comfort level is with math, it is beneficial for students when we, their teachers, approach math with confidence and enthusiasm (8). Bypassing this onto students can help students view math as reachable and enjoyable (9). Isn't that what we want for all our students? 
Thanks for stopping by Pam's Place. I look forward to sharing with you strategies Jo Boaler outlines in her book in the coming weeks. Stop by next Thursday as I talk about "The Power of Mistakes and Struggle," chapter 2. 

Summer Activities for Kids

Let the summer fun begin. Warmer weather. Longer days of sunlight. Summer brings its own traditions and fun! Here are a few summer activities for kids.

Think summer. Think beach. Think sand. Have you ever made sand slime? Check out this recipe for SAND SLIME. Great summer time fun. As always, supervision with young children is important.

Sand pail with text: Ingredients for summer sand slime
For more details visit: Growing a Jeweled Rose

Summer is a time for family fun and outdoor activities. Have children fill out this Fun in the Sun Bucket List form to get them thinking about summer activities. Parents also can spend time with their child and fill this out together to make a summer activities list. In this way, children have something at their fingertips to keep them engaged. After children have completed one of the items on their bucket list, they can check it off. Children can create a new bucket list for each month of summer vacation. Click on the image below to grab your FREEBIE.

Kids with sunglasses with text: My Fun in the Sun Bucket List - FREE printable

Get this FREE summer reading bingo card and get students excited about reading during the summer! Summer is the perfect time to jazz up reading with some of these fun in the sun summer activities for kids. Share this bingo card with students before sending them home for summer vacation. Parents can encourage reading at home during the summer months. CLICK on the image below and ignite the excitement for reading this summer.

Bingo Card with text: Fun in the Sun - FREE printable

Hope these activities add a splash of excitement to the warm days of summer!

Looking for some summer activities for kids? Check out this blog post for a few ideas and a couple of FREEBIES. Have children complete the FUN in the SUN Bucket List so they have a list of things to do throughout the summer. The summer reading bingo card offers fun ideas and activities to get children excited about reading. Add some summer fun with these FREE printables. #summer #pamsplace

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