1. How Students Learn Mathematics in the Classroom

2. Informal vs. formal mathematical strategies
Brazilian street children could perform mathematics when making sales in the street, but were unable to answer similar problems when presented in a school context.
Carraher, 1986; Carraher et al., 1985
At a very early age, children begin to demonstrate an awareness of number. The rate of development varies at least in part because of environmental influences. BUT it is NOT ONLY the awareness of quantity that develops without formal training. Both children and adults engage in mathematical problem solving, developing untrained strategies to do so successfully when formal experiences are not provided. For Example ---

3. Informal vs. formal mathematical strategies
A study of housewives in California uncovered an ability to solve mathematical problems when comparison shopping, even though the women could not solve problems presented abstractly in a classroom that required the same mathematics.
Lave 1988; Sternberg 1999

4. Informal vs. formal mathematical strategies
A similar result was found in a study of a group of Weight Watchers, who used strategies for solving mathematical measurement problems related to dieting that they could not solve when the problems were presented more abstractly.
De la Rocha, 1986

5. Informal vs. formal mathematical strategies
Men who successfully handicapped horse races could not apply the same skill to securities in the stock market.
Ceci and Liker, 1986; Ceci, 1996
These examples suggest people possess resources in the form of informal strategy development and mathematical reasoning that can serve as a foundation for learning about more abstract mathematics. BUT they also suggest the link is not automatic. If there is no bridge between informal and formal mathematics, the two often remain disconnected.

6. Principle #1: Teachers Must Engage Students’ Preconceptions
Preconception #1 – Mathematics is about learning to compute.
What approximately is the sum of 8/9 and 12/13?
Conceptually
Procedurally
Sense making
The first principle emphasizes both the need to build on existing knowledge and the need to engage students’ preconceptions---particularly when they interfere with the learning.
#1 What, approximately, is the sum of 8/9 and 12/13? Conceptual almost one plus almost one is almost two. Procedural – find common denominator. If one believes that mathematics is about computation, sense making may never take place.

7. Principle #1: Teachers Must Engage Students’ Preconceptions
Preconception #2 – Mathematics is about “following rules” to guarantee correct answers.
Systematic pattern finding and continuing invention
Tower of Hanoi
Units used to quantify fuel efficiency of a vehicle
Miles per gallon
Miles per gallon per passenger
Compare different procedures for their advantages and disadvantages
2. In reality, mathematics is a constantly evolving field that is far from cut and dried. It involves systematic pattern finding and continuing invention. Use Tower of Hanoi problem. Consider the selection of units that are relevant to quantify an idea such as the fuel efficiency of a vehicle. If we choose miles per gallon, a two-seater sports car will be more efficient than a large bus. If we choose passenger miles per gallon, the bus will be more efficient (assuming a large number of passengers). If mathematics procedures are understood as inventions designed to make common problems more easily solvable, and to facilitate communications involving quantity, those procedures take on a new meaning. Different procedures can be compared for their advantages and disadvantages. Such discussions in the classroom can deepen students’ understanding and skill.

8. Principle #1: Teachers Must Engage Students’ Preconceptions
Preconception #3 – Some people have the ability to “do math” and some don’t.
Amount of effort
Some progress further than others
Some have an easier time
Effort is key variable for success
#3In many countries, the ability to “do math” is assumed to be attributable to the amount of effort people put into learning it. Of course, some people in these countries do progress further than others, and some appear to have an easier time learning mathematics than others. But effort is still considered to be the key variable in success.

9. Key Point to Remember
Without a conceptual understanding of the nature of the problems and strategies for solving them, failure to retrieve learned procedures can leave a student completely at a loss.

10. How to best engage students’ preconceptions and build on existing knowledge
Allow students to use their own informal problem-solving strategies
Wrong answer (usually partially correct)
Find part that is wrong
Understand why it is wrong
Aids understanding
Promotes metacognitive competencies
1st bullet - A wrong answer is usually partially correct and reflects some understanding; finding the part that is wrong and understanding why it is wrong can be a powerful aid to understanding and promotes metacognitive competencies.

11. How to best engage students’ preconceptions and build on existing knowledge
Encourage math talk
Actively discuss various approaches
Learner focused
Draw out and work with preconceptions
Making student thinking visible
Model the language
2nd bullet – Different from simply giving lectures or assigning textbook readings and then having students work in isolation on problem sets or homework problems. Instead students and teachers actively discuss how they approached various problems and why.
Math talk is also important for helping teachers become more learner focused and make stronger connections with each of their students. Math talk allows teachers to draw out and work with the preconceptions students bring with them to the classroom and then helps students learn how to do this sort of work for themselves and for others. – Making their thinking public – visible. Provides an opportunity for the teacher to model the language and help students use it in their descriptions.

12. How to best engage students’ preconceptions and build on existing knowledge
Design instructional activities that can effectively bridge commonly held conceptions and targeted mathematical understandings
More proactive
Research common preconceptions and points of difficulty
Carefully designed instructional activities
3rd bullet – The first two bullets discussed provide opportunities for students to use their own strategies and to make their thinking visible so it can be built on, revised, and made more formal. This third strategy is more proactive. Research has uncovered common students preconceptions and points of difficulty with learning new mathematical concepts than can be addressed preemptively with carefully designed instructional activities.
Share piece on misconceptions.

13. Key Point to Remember
Identifying real-world contexts whose features help direct students’ attention and thinking in mathematically productive ways is particularly helpful in building conceptual bridges between students’ informal experiences and the new formal mathematics they are learning.
Identifying real-world contexts whose features help direct students’ attention and thinking in mathematically productive ways is particularly helpful in building conceptual bridges between students’ informal experiences and the new formal mathematics they are learning.
Talk about the benefits of Lesson Study. Share piece on misconceptions.

14. Principle #2: Understanding Requires Factual Knowledge and Conceptual Frameworks
MDE HSCE for Mathematics p. 4
Conceptual Understanding
Procedural Fluency
Effective Organization of Knowledge
Strategy Development
Adaptive Reasoning
How Students Learn Mathematics in the Classroom
A Developmental Model for Learning Functions
The second principle suggests the importance of both conceptual understanding and procedural fluency, as well as an effective organization of knowledge—in this case one that facilitates strategy development and adaptive reasoning. Page 4 of the MDE Mathematics HSCE.
To teach in a way that supports both conceptual understanding and procedural fluency requires that the primary concepts underlying an area of mathematics be clear to the teacher or become clear during the process of teaching for mathematical proficiency Good teaching requires not only a solid understanding of the content domain, but also specific knowledge of student development of these conceptual understandings and procedural competencies.
Share Table 8-1 page 365.

16. A Developmental Model for Learning Functions.
Levels of Understanding
0 – Recognize and Extend Pattern
1 – Generalize the Pattern and Express it as a function (y = 2x)
2 – Look at graph and decide if a particular function could model it
3 – Using Structures from Level 2 to create and understand more complex functions

17. Principle #3: A Metacognitive Approach Enables Student Self-Monitoring
Learning about oneself as a
Learner
Thinker
Problem solver
Learning about oneself as a learner, thinker, and problem solver is an important aspect of metacognition. Many people who take mathematics courses “learn” that “they are not mathematical”. This is an unintended, highly unfortunate, consequence to some approaches to teaching mathematics. It is a consequence that can influence people for a lifetime because they continue to avoid anything mathematical, which in turn ensures that their belief about being “nonmathematical” is true.

18. Instruction That Supports Metacognition
An emphasis on debugging
Finding where the error is
Why it is an error
Correcting it
Internal and external dialogue as support for metacognition
Help students learn to interact
Model clear descriptions, supportive questioning, helping techniques
Seeking and giving help
1st bullet - Metacognition functioning is also facilitated by shifting from a focus on answers as just right or wrong to a more detailed focus on debugging a wrong answer, that is, finding where the error is, why it is an error, and correcting it.
2nd bullet – Students can learn to reflect on and describe their mathematical thinking. They can learn to compare methods of solving a problem and identify the advantages and disadvantages of each. Peers can learn to ask thoughtful questions about other students’ thinking or help edit such statements to clarify them. Students can learn to help each other, sometimes in informal, spontaneous ways and sometimes in more organized, coaching-partner situations. Of course, teachers must help student learn to interact. Teachers can model clear descriptions and supportive questioning or helping techniques.
3rd bullet – Students must have enough confidence not only to engage with problems and try to solve them, but also to seek help when they are stuck.
Ideas from Reading Apprenticeship. Surveys from DCA II.

19. How Students Learn Mathematics in the Classroom
Careful consideration was taken to the points presented in this powerpoint to insure that the development of our units was guided by the known research