HELPING TEACHERS STRUCTURE THEIR INTUITIVE KNOWLEDGE

ABOUT CHILDREN’S UNDERSTANDINGS OF NEGATIVE

NUMBERS

A CGI Approach to Integers

Dr. Randolph Philipp San Diego State UniversityDr. Jessica Pierson Bishop University of Georgia

This work is supported by the National Science Foundation, grant number DRL-0918780

Let’s Start with a Game

Write a number. Any number. Circle it.Add 10Multiply by 2.Add 4.Add 8.Divide by 2.Subtract 17.Subtract your original circled number.Draw a rectangle around your final number.

“What’s in a name? That which we call a rose by any other name would smell as sweet.”

Circle the larger, or write “=“ if they are the equal.

1

More than two thirds of 5th graders circled 1.

negativeToday, let’s try not to think negatively about negative numbers.

Solve each of the following and think about how you reasoned. If you have time, solve

another way.

1) 3 – 5 = ___

2) -6 – -2 = ___

3) - 2 + ___= 4

4) ___+ -2 = -10

Project Z: Mapping Developmental Trajectories of Students’ Conceptions of Integers

Lisa Lamb, Jessica Bishop, & Randolph Philipp, Principal Investigators

Ian Whitacre, Faculty Researcher Spencer Bagley, Casey Hawthorne, Graduate

Students Bonnie Schappelle, Mindy Lewis, Candace

Cabral, Project researchers Kelly Humphrey, Jenn Cumiskey, Danielle

Kessler, Undergraduate Student AssistantsFunded by the National Science Foundation, DRL-0918780

So, Why Negative Numbers?

Even secondary-school students who can successfully operate with negatives have trouble explaining.

Even Successful Students Can Not Explain.

Keep Change Change

So, Why Negative Numbers?

Even secondary-school students who can successfully operate with negatives have trouble explaining.

Many middle-school students do not understand what they are doing with negatives.

Many Middle-School Students Do Not Understand

Valentina and friends—Does 6 – -2 = 6 + + 2?

To a student who answered that the two expressions are not equal, the interviewer said, “That’s kind of crazy that you are allowed to change the problem and it gives you a different answer.”

The student responded, “That’s math.”

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 (in language and mathematics)7 Look for and make use of structure.8 Look for and express regularity in repeated reasoning.

The Common Core State Standards (CCSS) of Mathematical Practice

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 (in language and mathematics)7 Look for and make use of structure.8 Look for and express regularity in repeated reasoning.

The CCSS of Mathematical Practice

What Is Mathematics to Students?

70 - 2353

76- 2353

“Yes, math is like that sometimes.”

So, Why Negative Numbers?

Even secondary-school students who can successfully operate with negatives have trouble explaining.

Many middle-school students do not understand what they are doing with negatives.

Many young children hold informal knowledge about negatives on which instruction might be based.

Reynaldo, 4th grade, -3 + 6 = ___

In a video clip, Reynaldo says that 3 of the 6 go to -3 leaving 3. He then uses a context of money, saying that if he borrowed $3 from a friend, he might give the friend $3 from $6 his mom gave him, leaving $3.

Noland, 4th grade, -5 + -1 = __

“Minus 5 plus minus 1 equals minus something …; 5 plus 1 equals 6, so this is minus 6 …. If you add these two together, it makes it farther from the positive numbers.”

Liberty, 2nd grade, -2 + __ = 4

Liberty solves -2 + __ = 4 using a counting strategy:

“Negative 1, negative 0, 0, 1, 2, 3, 4, so the answer is 7.”

So, Why Negative Numbers?

Even secondary-school students who can successfully operate with negatives have trouble explaining.

Many middle-school students do not understand what they are doing with negatives.

Many young children hold informal knowledge about negatives on which instruction might be based.

Understanding negatives is necessary for algebra success (inverses, even/odd functions, absolute value, rates of change, etc.): Compare – x and x.

One Last Reason…

Negative numbers comprise (almost) half of the reals!

So what?

Looking Back

Joan has 4 tennis balls. Her mother gives her some more and now she has 9. How many tennis balls did Joan’s mother give her?What type of problem is this? A) for you?B) for the parents of your students?C) for textbook writers?D) for your students?

CGI Join and Separate ProblemsFrom Carpenter, Fennema, Franke, Levi, and Empson, 1999

Result unknown

Change Unknown Start Unknown

Join Leon has 9 pencils. He gets 5 more pencils. How many pencils does Leon have now?

Joan has 4 tennis balls. Her mother gives her some more and now she has 9. How many tennis balls did Joan’s mother give her?

Mary has some marbles. She went to a party and won 5 more, and now she has 11. How many marbles did she have before the party?

Separate

Lesley baked 9 cookies. Kevin ate 3 of them. How many cookies are left?

Eleven children were playing in the sandbox. Some children went home. There were 3 children left still playing in the sandbox. How many children went home?

Mary had some bookmarks. She gave 3 to Kevin. Now she has 9 left. How many bookmarks did she have to start with?

Some Integer History

Twenty children go on a field trip. If each car can take 4 children, how many cars are needed?

Four children fairly share 6 chocolate chip cookies. How many cookies will each child get?

Children and Mathematicians Agree

Resisting quantities that are “less than nothing” and the lack of tangible or realistic interpretations.“Negative numbers aren’t really numbers because we

don’t count with them. And there’s no negative 1 cube.” (Violet, 2nd grade)

“Zero is nothing and negative is more nothing.” (Rebecca, 2nd grade)

One “must reject the definition … of -a, that it is less than nothing” (De Morgan, 1902, p. 72)

Children and Mathematicians Agree

Resisting removing something from nothing or removing more than one has“Three minus 5 is 0 because you have 3 and you can’t

take away 5, so take away the 3 and it leaves you with 0.” (Sam, 1st grade)

“I know some who cannot understand that to take four from nothing leaves nothing” (Pascal, 1669/1941, p. 25)

So what?

Ways of Reasoning

Students who have negative numbers in their numeric domains typically approach integer tasks using one of the following five ways of reasoning:• Order-based reasoning• Analogically based reasoning• Computational reasoning• Formal mathematical reasoning• Alternative reasoning

Order-based Reasoning

Students use the sequential and ordered nature of numbers to reason about a problem. Strategies include counting strategies and number line/motion.

Are any of the strategies we shared earlier order based?

Did any of the students in the videos shared use an order-based strategy?

Predictions

Rosie, a 2nd grader, will use a number line and/or counting to try to solve each of the following problems. Predict which problems she will successfully solve.• 3 – 5 = __• __ + 5 = 3• -5 – 4 = __• 5 + __ = 2• -9 + 5 = __• 6 - -2 = __

Order-based reasoningRosie, Grade 2

The interviews with Rosie occurred over a period of 3 months.

3-5 = __ Rosie solves 3 – 5 = __ by jumping to zero.-9+5 = __ She said -9 minus 1 first. Rosie is

leveraging the idea that subtraction reduces the size of a number and addition increases the size or magnitude of a number. But with negative numbers, addition is reducing the magnitude of the number but not the number itself. Thus her use of the language minus.

__ + 5 = 3 Note that Rosie solved it by using a number line and trial and error.

Rosie, Grade 2

Think about Rosie who solved, 3 – 5 = __, -9 + 5 = __, __ + 5 = 3, and -5 – 4 = __ but could not solve 5 + __ = 2 and insisted that it was not possible to solve 6 + -2.

What made the last two problems “not possible” for Rosie?

Order-based reasoning

Rosie, at the end of our problem-solving interviews

Analogically (Analogy)-based Reasoning

Students reason by relating negative numbers to another idea, concept, or context. Students reason about negative numbers on the basis of behaviors observed in this other concept or idea. Negative numbers may be related to a countable amount or quantity and tied to ideas about magnitude. They may also be related to contexts.

Are any of the strategies we shared earlier analogically based?

Did any of the students we shared video of use an analogically based strategy?

Analogically based reasoning

Randy, 1st grade, -8 – -1 = __Mathematically speaking, Randy is using

the distributive property to factor out the negative 1, though of course he does not state it using those terms!

Visual Representation of Randy’s Strategy

-1 -1 -1 -1 -1

-1 -1 -1

8 negative onesTake away 1 negative oneLeaves 7 negative ones

So what?

Analogically based reasoning

Treating Negatives like Positives. Multiple students solve

__ + -2 = -10 and -5 + -1 = __.

Formal mathematical reasoning

Negative numbers are treated as formal objects that exist in a system and are subject to fundamental mathematical principles that govern behavior. Formal strategies often involve comparisons to other, known, problems so that the logic of the approach remains consistent and underlying structural principles are not violated.

In the last of Rosie’s clips, she used formal mathematical reasoning.

Formal mathematical reasoning

Noland, Grade 4, -5 – -3 = __ compared to -5 + -3 = __Rosie held the first number and operation constant,

varying the second number—she compared 2 – 4 to 2 – - 4 and conjectured that subtracting [or adding] a negative does the opposite than what it normally does. She reasoned that subtracting -4 would move to the right on the number line. Noland, instead, held both numbers constant and varied the operation, leveraging the inverse relationship between addition and subtraction. He used his earlier observation that the addition of two negative numbers moves one farther from the positives to conjecture that subtracting two negative numbers moves one closer to the positives

Teachers’ Knowledge About Integers

To begin to make sense of how teachers think about integers, we interviewed 10 seventh-grade teachers to determine their understanding of integers and their perspectives about teaching integers and about students’ thinking. We posed integer tasks, we asked them about their teaching, and we showed them video clips of children solving open number sentences.

Teachers’ Approaches to Integer Tasks

Teachers possess these rich ways of reasoning (WoR) about integers. They are

Able to flexibly apply WoR, Able to analyze problems and strategically

combine WoR.

Kalani: -3 – ☐ = 2

First, Kalani thinks of moving from -3 to 2, as if the number sentence were -3 + ☐ = 2.“So I am thinking about the number line …. So I am starting somewhere, … and what do I do to end up at positive 2? I am moving 1, 2, 3, 4, 5—five units to the right.” (Order based)

Second, Kalani accounts for the subtraction sign.“I am moving the opposite direction [because of the subtraction sign], so I would write down negative 5 here.” (Formal)

Interpreting Student Thinking

Difficulty understanding student reasoning

Kalani: “I think he got confused. There’s no context involved .… So I don’t see a clear understanding on his part at all.” (referring to Noland’s solving -5 + -1 = ☐)

“I don’t quite understand him when he used the opposite. Opposite of what? …There is no context.” (referring to Noland’s solving -5 – -3 = ☐)

Pedagogical Goals

Did we expect teachers to mention ways of reasoning as goals for instruction?

No! Did we expect teachers to apply ways of

reasoning?We did not know.Did they?Yes!

One RoadA Road Through

Children’s Mathematical Thinking

The Land of Knowledge for Teaching Mathematics

So what?

Children struggle with the same issues about negative numbers that mathematicians struggled with historically, but integers provide a rich domain for reasoning. Furthermore, children come to us already applying these rich ways, and teachers use these ways, too.

A Final Thought:The best approach for teaching integers is….

…..there is no best approach.

We conjecture that flexibility, the ability to select approaches that are efficient and effective, is the hallmark of rich integer reasoning.

We suggest that children be given opportunities to tap into their rich, intuitive understandings of negative numbers, understandings grounded in their ordered-based and analogical reasoning, and then we stand back and let them enjoy the magic of making sense of wacky but wonderful negative numbers!

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