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In his paper The Design of Multiple Representation Tasks To Foster Conceptual Development, Professor Malcolm Swan of University of Nottingham presented five types of tasks that foster conceptual understanding of mathematical concepts. This was developed through their work with teachers. This classification of tasks is a very good framework to use in designing instruction. I have used this framework in one of our lesson study projects.
Types of tasks for teaching with conceptual understanding.
Classifying mathematical objects
Students devise their own classifications for mathematical objects, and/or apply classification devised by others. In doing this, they learn to discriminate carefully and recognize the properties of objects. They also develop mathematical language and definitions. The objects might be anything from shapes to quadratic equations.
Interpreting multiple representations
Students work together matching cards that show different representations of the same mathematical idea. They draw links between representations and develop new mental images for concepts.
Evaluating mathematical statements
Students decide whether given statements are always, sometimes or never true. They are encouraged to develop mathematical arguments and justifications, and devise examples and counterexamples to defend their reasoning.
Creating problems
Students are asked to create problems for other students to solve. When the solver become stuck, the problem ‘creators’ take on the role of teacher and explainer. In these activities, the ‘doing’ and ‘undoing’ processes of mathematics are exemplified.
Analyzing reasoning solutions
Students compare different methods for doing a problem, organize solutions and/or diagnose the causes of errors in solutions. They begin to recognize that there are alternative pathways through a problem, and develop their own chains of reasoning.
Kids are taught math as pets are taught tricks. A dog has no idea why its master wants it to perform. With careful training many dogs can be taught to perform complex sequences of actions in response to various commands and cues. When a dog is taught to perform a trick it has no need or use for any “understanding” beyond which sequence of movements its trainer desires. The dog is taught a sequence of simple physical movements in a specific order to create an overall effect. In the same way, we teach children to perform a sequence of simple computations in a specific order to achieve an overall effect. The dog uses its feet to move about a space and manipulate objects; the student uses a pencil to move about a page and manipulate numbers. In most cases, the student doesn’t know any more than the dog about the effect he creates. Neither has any intrinsic motivation to perform nor any idea why the performance is demanded. Practice, practice, practice, and eventually the dog can perform reliably on command. This is exactly how kids are trained to perform math: do a hundred meaningless practice problems, and then try to do the same trick on the test.
Mr.Brenner’s observation is as true in America as it is here in the Philippines. This is a painful truth but something that we all must take seriously. I strongly encourage our teachers, those writing our new curriculum framework (I think this is our third within the decade), textbook publishers and our DepEd officials to read the entire essay. The author outlined the reasons why math education is failing but he also offers solutions which I believe are doable even if our average class size here is 60! Let me list the 10 point solutions:
Understanding Must be Central in Math Education
Textbooks Must Not be Allowed to Undermine Math Education
Teachers Must Stop Teaching Math as They Learned It
Curricula Must be Coherent and Cumulative
Worked Examples Must be Emphasized for New Material
Curricula Must Include Examples of Excellent Performance
Assignments Must Draw on the Old and the New
Content Must be Meaningful and Contexts Must be Rich
Metacognitive Activity Must Pervade Mathematical Activity
Language Must be Taught, Used and Evaluated Fairly
I do not agree with #5 proposal because I believe that mathematics should be taught in the context of solving problems but I think this is a very good list. Find time to read it. Mr. Brenner also offers very good sample lessons. You may also want to read 10 signs there’s something not right in school maths and let us know your thoughts.
I believe in early algebraization. I have posted a few articles in this blog on ways it can be taught in the early grades. Check out for example Teaching Algebraic Thinking Without the x’s. All the lessons in fact that I post here whether it is a number or geometry or pre-algebra lesson always aim at developing students’ algebraic thinking. What do research say about early algebraization? How do can we integrate it in the grades without necessarily adding new mathematics content?
“Traditionally, most school mathematics curricula separate the study of arithmetic and algebra—arithmetic being the primary focus of elementary school mathematics and algebra the primary focus of middle and high school mathematics. There is a growing consensus, however, that this separation makes it more difficult for students to learn algebra in the later grades (Kieran 2007). Moreover, based on recent research on learning, there are many obvious and widely accepted reasons for developing algebraic ideas in the earlier grades (Cai and Knuth 2005). The field has gradually reached consensus that students can learn and should be exposed to algebraic ideas as they develop the computational proficiency emphasized in arithmetic. In addition, it is agreed that the means for developing algebraic ideas in earlier grades is not to simply push the traditional secondary school algebra curriculum down into the elementary school mathematics curriculum. Rather, developing algebraic ideas in the earlier grades requires fundamentally reforming how arithmetic should be viewed and taught as well as a better understanding of the various factors that make the transition from arithmetic to algebra difficult for students.
The transition from arithmetic to algebra is difficult for many students, even for those students who are quite proficient in arithmetic, as it often requires them to think in very different ways (Kieran 2007; Kilpatrick et al. 2001). Kieran, for example, suggested the following shifts from thinking arithmetically to thinking algebraically:
A focus on relations and not merely on the calculation of a numerical answer;
A focus on operations as well as their inverses, and on the related idea of doing/undoing;
A focus on both representing and solving a problem rather than on merely solving it;
A focus on both numbers and letters, rather than on numbers alone; and
A refocusing of the meaning of the equal sign from a signifier to calculate to a symbol that denotes an equivalence relationship between quantities.
These five shifts certainly fall within the domain of arithmetic, yet, they also represent a movement toward developing ideas fundamental to the study of algebra. Thus, in this view, the boundary between arithmetic and algebra is not as distinct as often is believed to be the case.
What is algebraic thinking in earlier grades then? Algebraic thinking in earlier grades should go beyond mastery of arithmetic and computational fluency to attend to the deeper underlying structure of mathematics. The development of algebraic thinking in the earlier grades requires the development of particular ways of thinking, including analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting. That is, early algebra learning develops not only new tools to understand mathematical relationships, but also new habits of mind.”
The foregoing paragraphs were from the book Early Algebraization edited by Jinfa Cai and Eric Knuth. The book is a must read for all those doing or intending to do research about teaching algebra in the elementary grades. Educators and textbook writers should also find a wealth of ideas on how algebra can be taught and integrated in the early years. Of course it would be a great read for teachers. The book is rather expensive but if you have the money, why not? Here are some section titles:
Functional thinking as a route in algebra in the elementary grades
Developing algebraic thinking in the early grades: Lessons from China and Singapore
Developing algebraic thinking in the context of arithmetic
Algebraic thinking with and without algebraic representation: A pathway to learning
Year 2 to 6 students’ ability to generalize: Models, representations, and theory for teaching and learning
Middle school students’ understanding of core algebraic concepts: equivalence & variable”
Check out the table of contents for more.
The following books also provide excellent materials for developing algebraic thinking.
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