mathematics and the human brain problem posing and problem solving in mathematics

Mathematics and the human brainProblem posing and problem

solving in mathematics

Student concerns: 27 students

3 Students per group. 9 groups 9 15 mins = 135 mins

= 2 hours 15 mins 2 ¼ hours≅Time remaining after today’s session = 11 3 hours = 33 hours

Subtract 3 hours for last session for practical work.30 hours remain.

Student concerns: Time remaining for lectures = 24 7 =

17 hoursNumber of topics remaining = 13 topics Number of topics per session = (13

topics 17 hours) 3 hours = 2.3 3 topics (round up for ≅

emergencies)

Please  go  to  my  website  http://sites.google.com/site/ctmathelp .

The  notes  been  loaded  on  the  teachers  page. B.Ed  notes.Some  have  difficulty  accessing  the  links  fromthe  power  point  file.If  you  are using  the  power  point  programme  and  you  clickon  a  website ,  you  may  need  to  come  out  of  power  pointand  then  click  on internet  browserat  the  botom  of  the  screen.  The  word  files  are  done  in  word  2007. I  have  saved  in  word  2003 to  allow  access   to  those  who  do  not   have  word  2007. You  also  need  adobe  9.

Assessment/ Lesson plans/ 15 minute teaching

Practical activities e.g. 15 minute session simulating with B.Ed class group teaching infants or junior school pupils a concept or reinforcement lesson while developing higher order skills.

A portfolio of 5 uniquely prepared and presented / shared mathematics lessons. Each must be accompanied by

a rationale for the method tried, a description of the students involved, their learning styles, age, class level, who they are; include your reflections on the learning

process, assessment and evaluation.

E-MAILS RECEIVED FROM Lisa LlanosMichelle De GourvilleFerida WelchOlivia GuillenCorann BrowneGermaine Emmanuel-LetrenJulia Andrews  Patricia PrescottJamie Peters

E-MAILS RECEIVED FROM

Sr. Olivia OculienNatasha HardingShellyann GillPatricia BelconPhillis GriffithCoromoto Fernandez – SirjuFrancisca MonsegueHazel Warner-PaulPatricia Joseph-Charles

GROUP 1Olivia GuillenCharmaine Figeroux Shelly GillLisa Llanos GROUP 2Corann BrowneNathalie FariaNatasha Harding GROUP 3Julia AndrewsPatricia Joseph Charles Phyllis  Griffith  

  GROUP 4Patricia PrescottSr. Olivia Marie OculienMichelle De Gourville GROUP 5Coromoto Fernandez - Sirju                           Francisca Monsegue                                     Jamie Peters

E-mail contacts

Review of last lecture : language, concepts, procedures, questions.

Set Induction:

Mathematics and the human brain

Evidence accumulated more recently suggests that the effects of sex hormones on brain organization occur so early in life, that from the start, the environment is acting on differently wired brains in boys and girls.

Such effects make evaluating the role of experience, independent of physiological predisposition, a difficult if not dubious task.

The biological bases of sex differences in brain and behaviour have become much better known through increasing numbers of behavioural, neurological and endocrinological studies.

We know, for instance, from observations of both humans and nonhumans that

males are generally more aggressive than females,

young males engage in more rough-and-tumble play than females

females are more nurturing

in general males are better at a variety of spatial or navigational tasks.

Mathematics and the human brainTall, D. ( 2004). THINKING THROUGH THREE

WORLDS OF MATHEMATICS http://www.emis.ams.org/proceedings/PME28/RR/RR213_Tall.pdf

http://books.google.com/books?hl=en&lr=&id=dWXPJvhco_UC&oi=fnd&pg=PA133&dq=Mathematics+and+the+human+brain&ots=neyWMnCWVh&sig=mKKNmE0veTswxqeUkt3mZfT2S54#v=onepage&q=Mathematics%20and%20the%20human%20brain&f=false

Theorists, such as Piaget (1965), Dienes (1960) and Bruner (1966), have something to say that had particular relevance in mathematics.

At one time, Piagetian theories held sway, with an emphasis on successive stages of development and a particular focus on the transitions between stages.

Piagetian theory was a tripartite theory of abstraction:

empirical abstraction focusing on how the child constructs meaning for the properties of objects,

pseudo-empirical abstraction, focusing on construction of meaning for the properties of actions on objects, and

reflective abstraction focused on the idea of how ‘actions and operations become thematized objects of thought or assimilation’ (Piaget, 1985, p. 49).

Meanwhile, Bruner focused on three distinct ways in which ‘the individual translates experience into a model of the world’, namely,

enactive,iconic and symbolic (Bruner 1966, p.10).

The foundational symbolic system is language, with two important symbolic systems especially relevant to mathematics:number and logic (ibid. pp. 18, 19).

In his research on the development of children (1966), Bruner proposed three modes of representation:

enactive representation (action-based),

iconic representation (image-based),

symbolic representation (language-based).

Rather than neatly delineated stages, the modes of representation are integrated and only loosely sequential as they "translate" into each other.

Symbolic representation remains the ultimate mode, for it "is clearly the most mysterious of the three."

Bruner's theory suggests it is efficacious when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners.

A true instructional designer, Bruner's work also suggests that a learner (even of a very young age) is capable of learning any material so long as the instruction is organized appropriately, in sharp contrast to the beliefs of Piaget and other stage theorists.

Like Bloom's Taxonomy, Bruner suggests a system of coding in which people form a hierarchical arrangement of related categories.

Each successively higher level of categories becomes more specific, echoing Benjamin Bloom's understanding of knowledge acquisition as well as the related idea of instructional scaffolding.

In accordance with this understanding of learning, Bruner proposed the spiral curriculum, a teaching approach in which each subject or skill area is revisited at intervals, at a more sophisticated level each time.

Efraim Fischbein, was from the very beginning interested in three distinct aspects of mathematical thinking:

fundamental intuitions that he saw as being widely shared,

The algorithms that give us power in computation and symbolic manipulation, and

The formal aspect of axioms, definitions and formal proof (Fischbein, 1987).

Richard Skemp, balanced his professional knowledge of mathematics and psychology with both theory and practice, also developed a general theory of increasingly sophisticated human learning (Skemp, 1971, 1979).

He saw the individual having receptors to receive information from the environment and effectors to act on the environment forming a system he referred to as ‘delta-one’;

a higher level system of mental receptors and effectors (delta-two) reflected on the operations of delta-one.

This two level system incorporates three distinct types of activity:

perception (input), action (output) and reflection,

which itself involves higher levels of perception and action.

The emphases in these three-way interpretations of cognitive growth are very different, but there are underlying resonances that appear throughout.

First there is a concern about how human beings come to construct and make sense of mathematical ideas.

Then there are different ways in which this construction develops, from real world perception and action, real-world enactive and iconic representations, fundamental intuitions that seem to be shared,

via the developing sophistication of language to support more abstract concepts including the symbolism of number (and later developments),

the increasing sophistication of description, definition and deduction that culminates in formal axiomatic theories.

In geometry, van Hiele (1959, 1986) has traced cognitive development through increasingly sophisticated succession of levels.

His theory begins with young children perceiving objects as whole gestalts, noticing various properties that can be described and subsequently used in verbal definitions to give hierarchies of figures, with verbal deductions that designate how, if certain properties hold, then others follow, culminating in more rigorous, formal axiomatic mathematics.

Meanwhile, process-object theories such as Dubinsky’s APOS theory (Czarnocha et al., 1999) and the operational-structural theory of Sfard (1991) gave new impetus in the construction of mathematical objects from thematized processes in the manner of Piaget’s reflective abstraction.

Gray & Tall (1994) brought a new emphasis on the role of symbols, particularly in arithmetic and algebra, that act as a pivot between ado-able process and a think-able concept that is manipulable as a mental object (a procept).

Two further strands also emerged, one encouraged by the American Congress declaring 1990-2000 as ‘the decade of the brain’ in which resources were offered to expand research into brain activity.

The other related to a focus on embodiment in cognitive science where the linguist Lakoff worked with colleagues to declare that all thinking processes are embodied in biological activity.

Brain imaging techniques were used to determine low grain maps of where brain activities are occurring. Such studies focused mainly on elementary arithmetic activities (eg Dehaene 1997, Butterworth 1999).

Other studies revealed how logical thinking, particularly when the negation of logical statements is involved, causes a shift in brain activity from the visual sensory areas at the back of the brain to the more generalised frontal cortex (Houdé et al, 2000).

This reveals a distinct change in brain activity, consistent with a significant shift from sensory information to formal thinking.

At the other end of the scale, studies of young babies (Wynn,1992) revealed a built-in sense of numerosity for distinguishing small configurations of ‘twoness’ and ‘threeness’, long before the child had any language.

The human brain has visual areas that perceive different colours, shades, changes in shade, edges, outlines and objects, which can be followed dynamically as they move.

Implicit in this structure is the ability to recognize small groups of objects (one, two or three), providing the young child with a fundamental intuition for small numbers.

In the second development, Lakoff and colleagues theorized that human embodiment suffused/ inundated all human thinking, culminating in an analysis of Where Mathematics Comes From (Lakoff & Nunez, 2000).

Suddenly all mathematics is claimed to be embodied. This is a powerful idea on the one hand, but a classification with only one category is not helpful in making distinctions.

If one takes ‘embodiment’ in its everyday meaning, then it relates more to the use of physical senses and actions and to visuo-spatial ideas in Bruner’s two categories of enactive and iconic representations.

Following through van Hiele’s development, the visual embodiment of physical objects becomes more sophisticated and concepts such as ‘straight line’ take on a conceptual meaning of being perfectly straight, and having no thickness, in a way that cannot occur in the real world.

THREE WORLDS OF MATHEMATICS.docx

JOURNEYS THROUGH THE THREE WORLDS

JOURNEYS THROUGH THE THREE WORLDS.docx

Process and Product

Process and Product.docx

Ministry of education SEA guidelines [mathematics]http://www.moe.gov.tt/parent_guides/SEA-GUIDELINES.pdf

Van de Walle, J.A. (2004). Elementary and Middle School Mathematics: Teaching

Developmentally. Boston: Pearson Education Inc.Of doing mathematics pp. 13-14.

Problem posing and problem solving in mathematics

http://jwilson.coe.uga.edu/emt725/PSsyn/PSsyn.html

http://www.emis.de/proceedings/PME28/RR/RR117_Lin.pdf

The reformed curriculum suggested that every instructional activity is an assessment opportunity for teachers and a learning opportunity for students (NCTM, 2000).

The movement emphasized classroom assessment in gathering information on which teachers can inform their further instruction (NCTM, 1995).

Assessment integral to instruction contributes significantly to all students’ mathematics learning.

The new vision of assessment suggested that knowing how these assessment processes take place should become a focus of teacher education programs.

Problem-posing involves generating new problems and reformatting a givenproblems (Silver, 1994).

The quality of problems in which students generated depends on the given tasks (Leung & Silver, 1997).

Research on problem posing hasincreased attention to the effect of problem posing on students’ mathematical ability and the effect of task formats on problem posing (Leung & Silver, 1997).

Problem-posing task was that the task teachers designed that requires students to generate one or more word problems.

The professional standards suggested that teachers could use task selection and analysis as foci for thinking about instruction and assessment.

According to De Lange (1995), a task that is open for students’ process and solution is a way of stimulating students’ high quality thinking.

Training teachers in designing and using assessment tasks has also been proposed as a means of improving the quality of assessments (Clarke, 1996).

Brown and Walter provide a wide variety of situations implementing this strategy including a discussion of the development of non-Euclidean geometry.

After many years of attempting to prove the parallel postulate as a theorem, mathematicians began to ask"What if it were not the case that through a given external point there was exactly one line parallel to the given line? What if there were two? None? What would that do to the structure of geometry?" (p.47).

The reformed curriculum calls for an increased emphasis on teachers’ responsibility for the quality of the tasks in which students engaged.

The high quality of tasks should help students clarifying thinking and developing deeper understanding through the process of formulating problem, communicating, and reasoning.

Problem-posing is recognized as an important component in the nature of mathematical thinking (Kilpatric, 1987).

More recently, there is an increasedemphasis on giving students opportunities with problem posing in mathematics classroom (English & Hoalford, 1995; Stoyanova, 1998).

Research papers have shown that instructional activities that engage students in generating problems as a means of improving ability in problem solving also improve their attitude toward mathematics(Winograd, 1991).

Nevertheless, such reform requires first a commitment to creating an environment in which problem posing is a natural process of mathematics learning.

Second, it requires teachers to figure out the strategies for helping students posingmeaningful and enticing problems.

Thus, there is a need to support teachers with a collaborative team whose students engage in problem-posing activities.

This can only be achieved by establishing an assessment team who support each other mutually by providing dialogues on critical assessment issues related to instruction.

For teachers, the problem-posing tasks allows them to gain insight into the way students construct mathematical understanding and this served to be a usefulassessment tool.

As an assessing tool, the tasks incorporated into everyday instruction, decisions about task appropriateness were often related to students’communication of their thinking, or the students’ problem-solving strategiesdisplayed in classroom.

The mathematics concepts to be taught at a grade level became as an elementary element of designing assessment tasks integrated into instruction.

Other decisions concerning the appropriateness of a task were relevant with teaching events students encountered in everyday lesson.

LINK

Comparing to the assessment tasks generated by individual, sharing multiple perspectives of appropriateness of task in a school-based assessment team was likely to achieve the purpose of task and the variety of task.

Next

•Steps in the problem-solving process•The use of mathematics in our daily lives

The End.