Reform! Research on learning shows: the entire standards-based Some - - PowerPoint PPT Presentation

Making Sense of Math Makes Sense

Randall I. Charles

Carmel, CA rcharles155@icloud.com

Brooklyn

March 2020

1

Signs of Change

• New standards
• Focus, Coherence, Rigor
• College and career readiness
• Higher expectations
• New curricula
• New assessments
• etc

Reform!

“…the entire standards-based reform movement [is] one whose focus [is] more depth of understanding.”

(McLaughlin, Shepard, Day, 1995)

Research on learning shows:

Some benefits of understanding mathematics:

• Better retention of knowledge
• Better fluency with procedures
• Better transfer of knowledge
• Better ability to apply/solve problems
• Positive dispositions toward and attitudes about mathematics
• Helpful beliefs about learning mathematics
• More likely to enroll in future mathematics classes
• Better performance on assessments

Many Traditional Expectations

Making Sense of Math Understanding

Session Overview

• Part 1: Essential Understandings & Big Ideas
• Part 2: Teaching for Understanding (and Procedural

Fluency)

• Problem-based learning
• Computation procedures
• Part 3: Quantitative Reasoning (Word Problems)

Instruction/Assessment Task

Each of these fractions refers to the same whole. Place these fractions in order from least to greatest. 4/3 4/8 4/5 4/12 4/4

NY-4.NF.2

Compare two fractions with different numerators and different denominators.

• Recognize that comparisons are valid only when the

two fractions refer to the same whole.

• Record the results of comparisons with symbols >, =,

and <, and justify the conclusions.

Part 1

Essential Understandings and Big Ideas

9

NY-4.NF.2

Compare two fractions with different numerators and different denominators.

• Recognize that comparisons are valid only when the

two fractions refer to the same whole.

• Record the results of comparisons with symbols >, =,

and <, and justify the conclusions.

Research says

Many teachers obviously would like their students to understand the mathematics they study but, when asked to specify the goal for a particular lesson, most U.S. teachers… talked about skill proficiency; few mentioned understanding. (TIMSS, Hiebert and Stigler, 2000)

Skill-Focused Standards

• NY-3.NBT.3 Multiply one-digit whole numbers by

multiples of 10 in the range 10-90 using strategies based

• n place value and properties of operations.
• NY-4.NF.2 Compare two fractions with different

numerators and different denominators.

• NY-5.MD.1 Convert among different-sized standard

measurement units within a given measurement system when the conversion factor is given. Use these conversions in solving multi-step, real world problems.

Standards – “Understand”

• NY-1.OA.4 Understand subtraction as an unknown-

addend problem within 20.

• NY-2.NBT.1 Understand that the three digits of a

three-digit number represent the number of hundreds, tens, and ones.

• NY-7.NS.1.c Understand subtraction of rational

numbers as adding the additive inverse, p – q = p + (–q).

Takeaways
 Pitfalls to Avoid

• Avoid a mindset that, “Today I am teaching a skill;

tomorrow I am teaching understanding.”

• Avoid interpreting every content standard only as a

statement of skills students should acquire.

Takeaway

• For every lesson, make explicit what you want

students to KNOW and, as appropriate, what you want them to be able to DO.

(Do You UNDERSTAND? Do you know HOW?)

“We understand something if we see how it is related or connected to other things we know.”

• J. Hiebert, Signposts for Teaching Mathematics through Problem Solving In F. Lester & R.

Charles, Teaching Mathematics Through Problem Solving, Grades PreK-6. NCTM: Reston, VA, 2003.

A web of ideas! Understanding

• Connections
• Connections
• Connections

What are Essential Understandings and Big Ideas? What do you want students to KNOW. Essential Understandings

• The size of a fraction is relative to the size of the whole.
• On the number line, a fraction to the right of another is the

greater fraction.

• If two fractions have the same numerator, the one with the

greater denominator is the lesser fraction.

• If two fractions have the same denominator, the one with the

greater numerator is the greater fraction.

• Many fractions can be compared using number sense and

how each compares to ½ and 1.

• Etc.

You cannot teach for understanding unless you know what understandings to teach.

A fraction is relative to the size of the whole. NY-4.NBT. 4. Use place value understanding to round multi-digit whole numbers to any place. Round 41,915 to the nearest thousand

Essential Understandings

• Rounding is a process of finding the closest multiple of 10, 100, 1,000, etc.

that a given whole number is closest to.

• Rounding is based on knowing the halfway point between two consecutive

multiples of 10 or100 or 1,000 etc.

• Most whole numbers can be rounded to more than one place value. The

context can determine the most appropriate place value for rounding.

• On the number line, numbers to the left of the halfway point between two

consecutive multiples of 10 or 100 or 1,000 etc. are closer to the lower multiple; numbers to the right are closer to the greater multiple.

• A convention in mathematics is to agree that numbers exactly halfway

between two consecutive multiples of 10 or 100 or 1,000 are closer to the greater multiple.

What is a “big idea”?

A Big Idea is a statement of an idea central to the learning of mathematics, one that links numerous essential understandings into a coherent whole.

These ALL connect to the same big idea!

“Change the way it looks but don’t change the value.”

“Equivalence”


Big Idea: A given number, measure, or expression can be represented using symbols in more than one way where each has the same value.

Big Idea

• Big Idea: Basic Facts and Algorithms for

Rational Numbers Most basic facts and algorithms for operations with rational numbers, both mental math and paper and pencil, can be carried out using equivalence to transform calculations into simpler ones.

Big Idea

“Change it to a simpler problem or problems”

Research Effective Teachers

✓Have their own mathematics content knowledge

anchored on big ideas and essential understandings.

✓Use big ideas as the glue for teaching, learning, and

assessment (connections).

Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum, 1999. Stigler, James. “The Teaching Gap: Reflections on Mathematics Teaching and How to Improve It.” Paper presented at the Pearson National Educational Leadership Conference, Washington, D.C., March 2004.

Big Ideas and Essential Understandings

• NCTM, “Essential Understandings” Series.
• R. Charles (2005). “Big Ideas and Understandings as the Foundation for Elementary and Middle

School Mathematics.” Journal of Mathematics Education Leadership, 8 (1), pp.9-24.

• D. Clements & J. Sarama (Eds) (2004). Engaging Young Children in

Mathematics: Standards for Early Childhood Mathematics Education. Lawrence Erlbaum: Mahwah, NJ.

• Christopher Cross and others (2009). Mathematics Learning in Early

Childhood: Paths Toward Excellence and Equity. Washington, D.C.: National Research Council.

Key Takeaways

• For every lesson, make explicit what you want

students to KNOW and, as appropriate, what you want them to be able to DO. (Do You UNDERSTAND? Do you know HOW?)

• Take every opportunity to make connections using

“big ideas.”

Part 2 Teaching for Understanding Understanding Student-Made Connections Previously learned math ideas to new math ideas

“Problem-Based Learning” PBL

“Researchers have identified a three-phase classroom activity structure [for high-quality mathematics instruction]. In well executed lessons…the teacher poses a problem and ensures that all students understand the context and expectations, students develop strategies and solutions (typically in collaboration with each other), and, through reflection and sharing, the teacher and students work together to explicate the mathematical concepts underlying the lesson’s problem.” Munter, Charles. “Developing Visions of High-Quality Mathematics Instruction.” Journal for Research in Mathematics Education, Volume 45, Number 5, November 2014, pp. 584-635.

PBL: Three-Phase Instructional Model Phase 1: Teacher poses a problem and ensures that all students understand the context and expectations, Phase 2: Students develop strategies and solutions (typically in collaboration with each other), and, Phase 3: Through reflection and sharing, the teacher and students work together to explicate the mathematical concepts underlying the lesson’s problem.

“First Instruction”

Old Model Step 1: Show and Tell Step 2: Try it Step 3: Practice and Problem Solving New Model Step 1: Problem-Based Learning Phase 1 Phase 2 Phase 3 Step 2: Try it Step 3: Practice and Problem Solving

Problem-Based Learning

Research shows that understanding develops during the process of solving problems in which important math concepts and skills are

• embedded. (Lester & Charles, 2003).

Solve and Share

Suppose you were going to plant a rectangular garden that covered 60 square feet. The side lengths are whole numbers. What might it look like? Draw all possible outlines of your garden. Write and explain why you think you found them all. [Extension: 11 square feet]

Understanding - Connections

Use prior knowledge to solve a problem – connect their prior knowledge to something new

• What prior knowledge was used?
• What was the new idea?

NY-4.OA.4

• Find all factor pairs for a whole number in the range

1-100…Determine whether a given whole number in the range 1-100 is prime or composite. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 11: 1, 11

Phase 3: Enhanced Direct Instruction

….telling the right things at the right time.

Mathematical Practices

• 1. Make sense of problems and persevere in solving them.
• 2. Reason abstractly and quantitatively.
• 3. Construct viable arguments and critique the reasoning of others.
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• 6. Attend to precision.
• 7. Look for and make use of structure.
• 8. Look for an express regularity in repeated reasoning.

Problem-Based Learning

Teaching through problem solving Inquiry teaching

Phase 1: Teacher poses a problem and ensures that all students understand the context and expectations,

Establish goals for new content to be learned Tasks …Reasoning and Problem Solving Cognitive Complexity

Phase 2: Students develop strategies and solutions (typically in collaboration with each other), and,

Support Productive Struggle; Do not lower the cognitive complexity Facilitate Meaningful Discourse Use and Connect Representations Pose Purposeful Questions

Phase 3: Through reflection and sharing, the teacher and students work together to explicate the mathematical concepts underlying the lesson’s problem.

Elicit and use evidence of student thinking; Students share thinking and solutions. Construct viable arguments, and critique the reasoning of others. Pose Purposeful Questions Make the important mathematics explicit

Non-Commercial Plug!

• F. Lester & R. Charles (EDs). (2004) Teaching

Mathematics Through Problem Solving: Grades PreK-6. NCTM.

• H. Schoen & R. Charles (EDs). (2004) Teaching

Mathematics Through Problem Solving: Grades 6-12. NCTM.

Takeaway

This should be the instructional model for all core lessons. Step 1: Problem-Based Learning

Phase 1 Phase 2 Phase 3

Step 2: Try it Step 3: Practice and Problem Solving

“Students think when they are learning and they learn

when they are thinking.” (stolen from somebody

important) “Give kids a chance to think!”

(Van de Walle, personal communication)

M M

Principles to Actions

Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

NY State Next Generation Learning Standards

Note on Fluency with Procedures: Fluency with procedures (procedural fluency) means students are accurate, efficient, flexible, and know when and how to use them

• appropriately. Developing fluency requires understanding why and

how a procedure works. Understanding makes learning procedures easier, less susceptible to common errors, less prone to forgetting, and easier to apply in new situations. Students also need

• pportunities to practice on a moderate number of carefully

selected problems after they have established a strong conceptual foundation of the mathematical basis for the procedure. (12) (13) For more on developing procedural fluency, see Adding it Up, pp. 121-124.

Procedural Fluency Strategies then Fluency

NY-1.NBT.4 Add within 100 (2d + 1d, 2d multiple 10)…using concrete models or drawings and strategies based on place value, properties…, and/or the relationship between addition and subtraction. NY-2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties…, and/or the relationship between addition and subtraction. NY-3.NBT.2 Fluently add and subtract within 1,000 using strategies based on place value, properties…, and/or the relationship between addition and subtraction. NY-4.NBT.4 Fluently add and subtract multi-digit whole numbers using a standard algorithm.

Mental Math

1.38 + 62 2.185 - 57 3.42 x 3 4.248 ÷ 4 5.168 + 80

Change it to a simpler problem!

6 4

+ 2 8

T O 6

+ 2 4 8

Understanding Subtraction

6 2

• 4 8

Takeaway

• Don’t rush to standard algorithms. Start with students

building and using a variety of strategies based on place value, properties of operations, and number sense.

• Do not expect or require every student to learn each

strategy.

• Allow each student to use the strategy or strategies

that make sense to her.

...understanding is not something that one can teach

• directly. No matter how kindly, clearly, patiently, or

slowly teachers explain, they cannot make students understand.... Teachers can help and guide their students, but understanding occurs as a by-product

• f solving problems and reflecting on the thinking

that went into those problem solutions. (Lambdin 2003, p. 11)

Benefits (from NY)

• Understanding makes learning procedures easier, less

susceptible to common errors, less prone to forgetting, and easier to apply in new situations.

• Students are more accurate and flexible; they know

when and how to use procedural strategies appropriately.

Part 3 Quantitative Reasoning

(Solving word problems)


 
 .

Write an equation that shows the relationship between feet and yards. Let f = the number of feet Let y = the number of yards

Multiple Choice

a)

3 y = f

b)

3 f = y

c)

None of the above

Solve

Willie’s total came to $11.90. He paid with a $20 bill. How much change should Willie get altogether?

Key Words –Noooooo!


• More
• Less
• In all
• Left
• Altogether
• Times
• Gave away

Lessons Learned

Just because a child can read a word problem, knows all vocabulary in the problem, and can identify the relationships stated in the problem it does not mean that he or she can solve it (Knifong & Holton, 1976, 1977).

Problem Complexity

Byron has 45

• pigeons. He

keeps them in 5 pens with the same number of pigeons in each. How many pigeons are in each pen? Pam had 5 bags and put the same number of apples in each

• bag. She ended

up with 45 apples in bags. How many did she put in each bag?

“Solving Word Problems”

Quantitative Reasoning

Understanding the quantities involved and how

they are related, independent of the specific numbers.

Fourth Grade

Carrie has 135 U.S. stamps. She has 3 times as many foreign stamps as U.S. stamps. How many stamps does she have all together?

Research Finding

“If students are encouraged to understand and meaningfully represent mathematical word problems rather than directly translate the elements of the problems into corresponding mathematical

• perations, they may more successfully solve these

problems and better comprehend the mathematical concepts embedded within them [quantitative reasoning].”

Pape, S.J. (2004). Middle school children’s problem-solving behavior: A cognitive analysis from a reading comprehension perspective. Journal for Research in Mathematics Education, 35:3, pp. 187-219.

“understand and meaningfully represent…”

• Training children in the process of using diagrams to solve

problems results in more improved problem-solving performance than training students in any other strategy.

(Yancey, Thompson, and Yancey, 1989).

Quantitative Reasoning

Concrete (real-world situation) Bar Diagram (visual representation) Abstract (number sentence)

Quantitative Reasoning

Carrie has 135 U.S. stamps. She has 3 times as many foreign stamps as U.S. stamps. How many stamps does she have all together? (3 x 135) + 135 or 4 x 135 = ?

U.S. stamps

135 135 135 135

Foreign stamps

Grade 1

#1. Jill walks 9 blocks. She walks 5 of the blocks with a

• friend. How many blocks did Jill walk by herself?

5 9 ?

Grade 1

#2. Rita has 3 yellow balloons. The rest of her balloons are pink. She has 7 balloons in all. How many pink balloons does Rita have?

3

? 7

Grade 1

#3. Ashley has 4 flowers. Some are red and some are

• white. How many of each color flower does Ashley

have?

4

Draw Bar Diagrams

Jon is 8 years old. His brother Tom is 2 years older than Jon. Their brother Henry is twice as old as

• Tom. Which number sentence can be used to find

h, Henry’s age? A 8 x 2 = h B (8 + 2) x 2 = h C (8 + 2) ÷ 2 = h D 8 x 2 ÷ 2 = h

Hidden Question?

Tom is 2 years older than Jon. How old is Tom? Draw the bar diagram.

Hidden Question?

Tom is 2 years older than Jon. How old is Tom? Draw the bar diagram.

Can you “see” the correct choice?

A 8 x 2 = h B (8 + 2) x 2 = h C (8 + 2) ÷ 2 = h D 8 x 2 ÷ 2 = h

Draw a Bar Diagram Some adults and 12 children were on a bus. There are 31 people in all on the bus. How many adults were on the bus?

Draw a Bar Diagram

Byron has 45 pigeons. He keeps them in 5 pens with the same number of pigeons in each. How many pigeons are in each pen?

Alice has 24 pens. She puts 4 in a box. How many boxes are needed for all of the pens? Bob has 24 pens. This is 4 times what Carol has. How many does Carol have?

Some Quantitative Relationships

• Addition, Subtraction, Multiplication, Division
• Proportional
• Linear
• Quadratic
• Function

Quantitative Reasoning There are 80,000 tons of garbage in a landfill that can hold 100,000 tons. Each month the landfill increases by 570 tons in spite of the fact that some trash decomposes. In how many months will the landfill have to close unless changes are made? Let m = the number of months Write an equation and solve the problem.

Quantitative Reasoning

Amount in the landfill now Amount added

• ver the

months. Total amount in the landfill

Quantitative Reasoning

100,000 = 80,000 + 570m

80,000 100,000 570 m months

“What Einstein was able to do was to think visually.” (Einstein: His Life and Universe, Walter Isaacson)

Some References

Charles, Randall I. Undated. “Solving word problems: Developing quantitative reasoning.” Research Into Practice. Pearson Education. Karen S. Karp, Karen S., Sarah B. Bush, and Barbara J.

• Dougherty. 2019. “Avoiding the ineffective keyword

strategy.” Teaching Children Mathematics 25, no 7 (May): 428-435.

Takeaway

• Do not teach key words.
• Rich problem structures need to be used with all students to

develop quantitative reasoning.

• Explicit instruction needs to be given to the development of

quantitative reasoning.

• Bar diagrams are an effective tool for developing students’

quantitative reasoning abilities.

Research Fact We must believe that ALL students can be successful in mathematics; all can be successful problem solvers.

Success in math will be determined not by how many answers you know, but by what you do when you don’t know the answer.