Teaching for a Mathematical Mindset Sue Myette and Nona Wright, ISD Instructional Coaches August 18, 2016

Preparation for Facilitator Handout Packet (1/participant): Math mindset questionnaire (see next slide) Mindset messages practice page Sample growth mindset norms Number talk handout Close to 100 Other materials Square paper for tables Large chart paper and markers for number talk

Note for facilitator: Entry task will be for participants to complete a brief questionnaire. Questionnaire will have 5 statements with a 5-point answer scale (strongly agree=0, agree=5, neutral=10, disagree=15, strongly disagree=20) 1.When math gets hard it makes me feel that I’m not very smart. 2.The parts of math I like best are the ones where I make the fewest mistakes. 3.My level of math ability is a trait I inherited from my parents. 4.When I struggle with a hard math problem I feel like giving up. 5.Math is much easier to learn if you are male or come from a certain culture. Participants will find their total score. The scoring will range from 0 (all responses are strongly agree) to 100 (all statements are strongly disagree). Participants will use their score for the slide that follows.

What color are you on the Math Mindset Meter?

Let’s Do Some Math! Work with a partner to complete these paper folding tasks. Explain to your partner how you know each shape you make has the specified area. 1.Construct a square with exactly ¼ the area of the original square. 2.Construct a triangle with exactly ¼ of the area of the original square. 3.Construct a square with exactly ½ the area of the original square. 4.Construct another square, also with ½ the area, but oriented differently from the one you just made.

How did the task feel? For reds and oranges? For yellows? For light greens? For dark greens? Developing a mathematical mindset—a growth mindset applied to approaching mathematics conceptually—can change how you feel about math, how your brain works, how you learn, and how you teach.

Objectives: By the end of this session you will… Understand the impact of mindset on mathematics learning. Begin to develop and encourage a growth mindset in math. Understand, and be prepared to apply, 3 strategies to teach math for conceptual understanding.

Part 1: Encouraging a Growth Mindset in Mathematics The highest- achieving students in the world are those with a growth mindset, and they outrank other students by the equivalent of more than a year of mathematics. (Boaler, 2016)

What messages did you receive about math? Math class is tough! I’m slow so I must be stupid. It’s bad to make mistakes in math. Boys are better than girls at math. I have to find answers the same way my teacher does. Asians are naturally better at math. Math is boring, not for creative people.

More than 50% of Americans between ages 18 and 34 say they can’t do math. Yes, we can!!

Listen for Four Growth Mindset Messages

Video Discussion Discuss with a partner: What were the four main messages for students? Which parts aligned with your previous thinking? Which parts surprised you or made you wonder? What implications do these messages have for teachers?

What do growth messages sound like? I notice you began by drawing a picture. Good thinking! What are you thinking about doing next? I need help. I see you’re using what we know about adding whole numbers to add the numerators and denominators. Now let’s look at… 1/2 + 1/4 = 2/6 I don’t get division. This is too hard.

What do growth messages sound like? Teaching math is so hard!

Teaching Actions to Consider Establish growth mindset classroom norms …the most successful students are those with a growth mindset. Intentionally send growth mindset messages about math …a growth mindset can be developed at any time in life. Teach students to value mistakes as opportunities for learning …struggles make our brains grow more than when the work comes easily. Reconsider the emphasis given to timed math fact tests …speed is not the most important skill in math.

Part 2: Teaching for Conceptual Understanding Giving students growth mindset messages will not help them unless we also show them that math is a growth subject. (Boaler, 2016)

Why is Conceptual Understanding Important? The brain can only compress concepts; it cannot compress rules and methods. (Thurston, 1990 ) Successful math users search for patterns and relationships and think about connections. (Boaler, 2016) Experts see meaningful patterns of information and use them to organize their knowledge in ways that reflect a deep understanding of their subject matter. (Bransford et al., 1999)

Methods vs. Concepts Counting Concept of Number Counting on Concept of Sum Repeated Addition Concept of Product (Boaler, 2016)

How can we help all students successfully learn mathematical concepts? 1.Facilitate number talks 2.Use visual representations 3.Present multidimensional, real world problems

Number talks promote number sense and mathematical mindset Choose problems intentionally to encourage flexible thinking beyond standard algorithms. Focus on having students share multiple pathways to the answer. Use both visual and numerical problems. Number sense and mathematical mindsets develop together. Moreover, number sense is the foundation for all higher level mathematics. (Boaler, 2016; Feikers & Schwingendorf, 2008) =

Let’s try it! 15 x 12 =

Visual representations promote number sense and conceptual understanding Practice multi- plication facts with visual representa- tions to encourage understanding, not just memorization.

Visual representations promote number sense and conceptual understanding Practice multi- plication facts with visual representa- tions to encourage understanding, not just memorization. How do you see this pattern growing? Use cubes to build Case 5. Use colors to show old and new squares. Describe Case 10. How many squares?

Visual representations continued… How do you see this pattern growing? Use cubes to build Case 5. Use colors to show old and new squares.

The Turkey Problem: How would you solve it? A man is on a diet and goes into a shop to buy some turkey slices. He is given 3 slices which together weigh 1/3 of a pound but his diet says that he is allowed to eat only 1/4 of a pound. How much of the 3 slices he bought can he eat while staying true to his diet? Sometimes a visual solution path makes more sense than an algorithm.

Multidimensional, real world problems encourage reasoning and application W ARNING ! Beware of pseudo-contexts: they can make math seem irrelevant. A pizza is divided into fifths for 5 friends at a party. Three friends eat their slices, but then 4 more friends arrive. What fraction should the remaining 2 slices be divided into? This is stupid. Normal people would just order more pizza.

What do mathematicians do? 1.Pose a question about a real world situation. 2.Make a mathematical model to find a way to answer the question. 3.Perform some calculations. 4.Analyze the results to see if the original, real-world question was answered. It used to be that employers needed people to calculate; they no longer need this. What they need is people to think and reason. (Wolfram, 2010)

How many of each kind of chip should go into the box? What kind of “model” or approach should we take to answer this question? What information would we want to have? How could we get that information? What calculations would we do? What would that tell us?

Objectives: By the end of this session you will… Understand the impact of mindset on mathematics learning. Begin to develop and encourage a growth mindset in math. Understand, and be prepared to apply, 3 strategies to teach math for conceptual understanding. Everyone can learn math at high levels. Struggles and mistakes help our brains grow. Understanding is more important than speed. Our math mindset comes from messages we receive. Mindset can become a self-fulfilling prophecy: whether you believe you can or believe you can’t: you’re right. Number talks to promote number sense. Visual representations. Multidimensional real-world problems.

Useful Resources Teaching resources, research, and courses from Jo Boaler Many links to mindset resources for the classroom -of-the-month/download-problems-of-the- month -of-the-month/download-problems-of-the- month Differentiated low floor, high ceiling tasks for conceptual understanding