Early Algebra Math Summit v2 no video - Sas Institute · Early&Algebra&& in&the&Era&of&Common&Core& MariaBlanton ... Result:“Arithmetic-then-algebra” approach leading to abrupt

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Early Algebra in the Era of Common Core

Maria Blanton

The research reported here was supported in part by the Na8onal Science Founda8on under DRK-‐12 Awards #1154355 and #1207945. Any opinions, findings, and conclusions or recommenda8ons expressed in this material are those of the authors and do not necessarily reflect the views of the Na8onal Science Founda8on.

Early Algebra in the Era of Common Core

Maria Blanton Angela Murphy Gardiner

Bárbara Brizuela Katie Sawrey Ashley Newman-‐Owens

Eric Knuth Ana Stephens Isil Isler

The research reported here was supported in part by the Na8onal Science Founda8on under DRK-‐12 Awards #1154355 and #1207945. Any opinions, findings, and conclusions or recommenda8ons expressed in this material are those of the authors and do not necessarily reflect the views of the Na8onal Science Founda8on.

Trivia….

Some questions you might ask…

ì  What is early algebra?

ì  Why do we need to start this in Kindergarten?

ì  Won’t it be enough to teach arithmetic “really well” so that children will be ready for algebra in middle grades?

ì  Does it really matter?

ì  How can elementary students be expected to do the kinds of mathematics with which older students struggle?

Why do we need to start this in Kindergarten?

America’s “Algebra Problem”

ì  Around 1900: “shopkeeper arithmetic” for all; algebra for the elite (3%-5% of school population)

ì  Introduction of“Algebra for all”

ì  Result:“Arithmetic-then-algebra” approach leading to abrupt introduction of high school algebra, resulting in high student failure/dropout

“Algebra has become an academic passport for passage into virtually every avenue of the job market and every street of schooling. With too few excep8ons, students who do not study algebra are therefore relegated to menial jobs and are unable oUen to even undertake training programs for jobs in which they might be interested. They are sorted out of the opportuni8es to become produc8ve ci8zens in our society.” Alan Schoenfeld, 1995

Some (more) reasons we need to do “algebra” well…

ì  Successfully completing Algebra 2 is the best leading indicator of college graduation*

ì  53% of college students take remedial mathematics or English; most of them will not graduate*

ì  Lack of sufficient math preparation is a significant factor in students not pursuing/completing technical majors*

ì  “Children’s ini8al skill levels predict their later learning in many domains, but the rela8on between early math knowledge and future math achievement is…roughly twice as strong as the rela8on between early and later reading achievement”. (Diamond et al., 2013)

* Courtesy of Richard Schaar, Texas Instruments, keynote address at MAA/NSF Algebra Conference (sources: NSF, NCES, Tapping America’s Potential)

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The response to “America’s Algebra Problem”: Algebra as a K-‐12 Strand of Thinking:

(1989)

(2000)

(2006)

COMMON CORE State Standards Ini8a8ve

1989 2000 2006 2010 What is early algebra?

What is early algebra NOT?

ì It is not pre-algebra

ì It is not “algebra early”

What is early algebra?

Four Key Practices

ì  Generalizing mathematical relationships and structure

ì  Representing generalizations in diverse ways (e.g. words, variable notation, tables, graphs, pictures)

ì  Justifying generalizations

ì  Reasoning with generalizations as mathematical objects

Does early algebra ma_er?

Project LEAP (National Science Foundation)

ì  Developed a grades 3-‐5 early algebra curricular progression focusing on core content domains: 1.  generalized arithme8c

2.  equivalence, expressions, equa8ons, inequali8es

3.  func8onal thinking

4.  variable

ì  Developed validated assessments to measure learning along this progression

Project LEAP

ì  300 students in grades 3-‐5 from one school district;

ì  ~20 one-‐hour early algebra lessons (one per week) taught by one member of the research team;

ì  Same (grade 3 level) lesson taught at each grade;

ì  One-‐hour assessment given as pre/post measure.

Project LEAP

ì  No significant differences on overall performance at pre-‐test between experimental and control students at each of grades 3-‐5, across all content domains;

ì  At post-‐test, experimental students significantly outperformed control students at each of grades 3-‐5 (most significant at grade 3);

A closer look at third-‐grade students’ performance and strategy use

!

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Project LEAP

ì Rela8onal thinking (children’s understanding of the equal sign)

Project LEAP relational thinking

ì  Item 1a

Fill in the blank with the value that makes the following number sentence true. How did you get your answer?

7 + 3 = ____ + 4 Why?

ì  Item 2b

Circle True or False and explain your choice.

57 + 22 = 58 + 21 True False

How do you know?

Project LEAP relational thinking

ì  Sta8s8cally significant gains in experimental students’ ability to correctly interpret and to think rela8onally about the equal sign.

ì  Control students con8nued to exhibit misconcep8ons (opera8onal thinking) and showed no evidence of rela8onal thinking.

Project LEAP

ì Represen8ng unknown quan88es

Project LEAP representing unknown quantities

ì  Sta8s8cally significant gains in experimental students’ ability to correctly interpret and to think rela8onally about the equal sign.

ì  Control students con8nued to exhibit misconcep8ons (opera8onal thinking) and showed no evidence of rela8onal thinking.


7. Tim and Angela each have a piggy bank. They know that their piggy banks each contain the same number of pennies, but they don’t know how many. Angela also has 8 pennies in her hand.

a)  How would you describe the number of pennies Tim has?

b)  How would you describe the total number of pennies Angela has?

c)  Angela and Tim combine all of their pennies to buy some candy. How would you describe the total number of pennies they have?


ì  Sta8s8cally significant gains in experimental students’ ability to represent unknown quan88es with variable nota8on and to represent related quan88es in meaningful ways.

ì  No controls were able to use variables in any way pre or post.

Project LEAP recognizing & representing arithmetic structure

Marcy’s teacher asks her to figure out “23 + 15.” She adds the two numbers and gets 38. The teacher then asks her to figure out “15 + 23.” Marcy already knows the answer.

a) How does she know?

b) Do you think this will work for all numbers?


recognizing & represen8ng arithme8c structure

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6. Evelyn computes the following:

8 – 8 = ___ 12 – 12 =___

She gets an answer of 0 each 8me. She starts to thinks that any8me you subtract a number from itself, the answer is 0. Which of the following best describes her thinking? Circle your answer.

a)  a + 0 = 0

b)  a = b + a + b

c)  a – a = 0

d)  a × 0 = 0


ì  Experimental students were more likely to recognize the underlying structure of fundamental proper8es and use this as a basis for jus8fying generaliza8ons;

ì  Experimental students were be_er able to interpret equa8ons with variable nota8ons and navigate between representa8ons (natural language and symbolic);

ì  Experimental students were more likely to understand that a generaliza8on might hold over a broad domain of numbers, not just a par8cular instance.

Project LEAP

recognizing and represen8ng co-‐varying rela8onships

!

Project LEAP recognizing and representing co-‐varying relationships

10. Brady is having his friends over for a birthday party. He wants to make sure he has a seat for everyone. He has square tables.

a) If Brady keeps joining square tables in this way, how many people can sit at:

3 tables? 4 tables? 5 tables? Record your responses in the table below and fill in any missing informa8on:

b) Do you see any pa_erns in the table? Describe them.

c) Find a rule that describes the rela8onship between the number of tables and the number of people who can sit at the tables. Describe your rule in words.

d) Describe your rela8onship using variables. What do your variables represent?

e) If Brady has 10 tables, how many people can he seat? Show how you got your answer.

!

He can seat 4 people at one square table in the following way:

If he joins another square table to the first one, he can seat 6 people:

Project LEAP recognizing and representing co-‐varying relationships

ì  Experimental students significantly outperformed control students in their ability to iden8fy and describe func8on rules in words or variables;

ì  Experimental students were more likely to choose variables than words to represent their rule.

!

In one year, in about one day of a students’ life, we can sta8s8cally significantly improve children’s algebraic understanding

!

Children’s Understanding of Functions (National Science Foundation)

Ø  Grades K -‐ 2

Ø  8 weeks instruc8onal interven8on focused on algebraic thinking in the study of func8ons

Ø  Two 30-‐45 minute lessons each week (16 lessons total)

Growing Train

There was a train that ran the same route everyday. As it went along, it picked up two train cars at each stop.

1.  How many train cars did it have at stop 1? How many train cars did it have at stop 2? How many train cars did it have at stop 3?

2.  Organize your informa8on in a func8on table (t-‐chart). Write an equa8on that shows the rela8onship between the values in your table for each set of values.

3.  Find a rela8onship between the number of stops and the total number of cars on the train. Represent your rule in words and le_ers (variable nota8on).

4.  If we count the engine, how would this affect your func8on table? How would this affect your rule?

1st-‐Grader and the Growing Train

ì  (video)

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1. Make sense of problems and persevere in solving them.

ì  “Monitor and evaluate [her] progress and change course” when she ini8ally incorrectly represented an arbitrary number of train cars;

ì  Make sense of regularity in problem data by no8cing structure in number sentences;

ì  Explain the meaning of parts of an equa8on (func8on rule), including variables, in terms of the problem context;

ì  Make sense of data in func8on tables (t-‐charts), including interpre8ng these data to find problem solu8ons, such as how to represent the new rela8onship where the train engine is counted.

COMMON CORE MATHEMATICAL PRACTICES

2. Reason abstractly and quantitatively.

ì  Make sense of co-‐varying “quan88es and their rela8onships in problem situa8ons” ì  Decontextualizing

Ø  abstract the problem situa8on about a train that grows in a specific, numerical way (at each stop, the train adds 2 more cars) and represent the func8onal rela8onship in a generalized symbolic form;

Ø  manipula8ng symbols as objects themselves, “as if they have a life of their own” (Meagan’s use of her original func8on rule to represent the rule coun8ng the engine)

ì  Contextualizing – explain at any point the meaning behind the symbols used to represent the rela8onship (‘v’ represents the number of stops the train makes)


3. Construct viable arguments and critique the reasoning of others.

ì  Reason induc8vely about data (here, a set of number sentences) in order to find a generalized rela8onship (func8on rule);

ì  Build arguments that explain the correctness of a generalized rule based on the problem context from which data arose (e.g., “Whatever number, how [sic] many stops it made, if you doubled it, that’s how many cars it would have”. ).


4. Model with mathematics.

ì  Write number sentences (“addi8on equa8ons”) to describe rela8onships between specific numbers in co-‐varying data;

ì  Iden8fy important data in a situa8on and map their rela8onship using tables and formulas;

ì  Analyze the rela8onship in data about co-‐varying quan88es and draw conclusions about a generalized rela8onship;

ì  Interpret results (func8on rule) within the problem context as a way to make sense of their model (e.g., does the model accurately reflect the situa8on?).


5. Use appropriate tools strategically.

ì  Use different representa8ons such as natural language, algebraic nota8on, and tables to reason about co-‐varying rela8onship and navigate between representa8ons;

ì  Use of tables as a strategic choice to organize co-‐varying data.


6. Attend to precision.

ì  Understand that variable nota8on represents a quan8ty, not an object and accurately state the meaning of symbols they choose (‘r’ represents the number of stops, not the actual stops);

ì  Represent func8on rule with precision (completeness and consistency of representa8on), that is, as an equa8on (‘R + R = V’), not an expression (‘R + R’) or in syncopated language (“the number of cars is R + R).


7. Look for and make use of structure.

ì  Able to discern a pa_ern or structure in co-‐varying quan88es;

ì  Able to make use of an exis8ng structure (i.e., func8on rule) to generate and represent a new rela8onship.


8. Look for and express regularity in repeated reasoning.

ì  No8ce that calcula8ons are repeated in number sentences expressing a rela8onship between two co-‐varying values and, from this, abstract the func8on rule and express this regularity in symbolic nota8on (“R + R = V” or “R + R + 1 = V”).


Closing Thoughts

Of the mathema8cal content in grades K-‐5, early algebra is most able to comprehensively address the 8 Mathema8cal Prac8ces of the Common Core.

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Closing Thoughts

If we don’t adequately invest in early algebra…

…we will likely have “algebra early”

Closing Thoughts

L + 2 = 1!

Closing Thoughts

We need to teach children to be mathema8cally fearless

Closing Thoughts

ì  Child cannot learn arithme8c apart from early algebra

ì  Developing children’s algebraic reasoning requires significant professional development for elementary teachers

ì  We cannot con8nue to priori8ze literacy in the elementary grades at the expense of mathema8cs

Ques-ons?

[email protected]

Documents

Early Algebra Math Summit v2 no video - Sas Institute · Early&Algebra&& in&the&Era&of&Common&Core& MariaBlanton ... Result:“Arithmetic-then-algebra” approach leading to abrupt