Like all other forms of languages, mathematics developed because of the necessity to communicate ideas. In the case of numbers, it was to express and record ‘countables’. Below is a little video that can explain this to pupils in the primary grades. Its title is History of Mathematics but I believe it should be titled History of Numbers. There’s no mention of the Greeks and the mathematics they developed.
Our invention of numbers and number system shows us that we are inherently problem solvers ad that we invent symbols and systems to express and communicate our ideas. Mathematics is the most powerful language humans have invented.
I like these graphs which show how a mathematician and a typical student solve a problem. The first two graphs were from the post “Some research discoveries”. The last one is mine, on teachers time-line graph in doing problem solving.
This is how mathematicians solve problems:
This is how a typical student solves problems:
Here’s my time-line graph of a typical teacher solving a non-standard problem in the class. I asked these teachers how they solve challenging math problems and how long it usually take them to find the solutions. It actually resembles that of the mathematicians. For some reason when they do the problems in their classes they present them like it’s magic, effortless.
I’m sharing in this post some of the “theories” underpinning the lessons, learning tasks, and math teaching ideas that I blog here. This is part of the literature review of my dissertation titled “A Framework of Growth Points in Students Developing Understanding of Function”. If you are a researcher and wants to see the references, you may download the dissertation here.
Good ideas are networks
Understanding as making connection
To understand something is to connect it with other ideas. The stronger the connection, the more powerful the knowledge. The more connected an idea is to other ideas, the easier it is to retrieve from memory and the wider the applications. In mathematics, this implies that one understands when he or she can make connections between ideas, facts or procedures (Hiebert & Wearne, 1991). In making connections, one not only links new mathematical knowledge to prior knowledge but also creates and integrates knowledge structures (Carpenter & Lehrer, 1999). Thus, the process of understanding is like building a network. Networks are built as new information is linked to existing networks or as new relationships are constructed (Hiebert & Carpenter, 1992). If one imagines a weblike structure, the mental representations constructed in the process of understanding can be thought of as nodes. These nodes are themselves “networks”. These smaller networks resemble what is called a schema in cognitive psychology which is a network of well-connected ideas, skills and strategies an individual uses in working with a particular task (Marshall, 1990).
The importance of the acquisition of cognitive structures (schemas) has been shown in studies of people who have developed expertise in areas such as mathematics, physics, chess, etc.
Bransford, Brown & Cocking (1999) summarised the key principles of experts’ knowledge. Some of these are:
Experts notice features and meaningful patterns of information that are not noticed by novices.
Experts have acquired a great deal of content knowledge that is organised in ways that reflect a deep understanding of their subject matter.
Experts’ knowledge cannot be reduced to sets of isolated facts or propositions but, instead, reflect contexts of applicability: that is, the knowledge is “conditionalized” on a set of circumstances.
Experts are able to flexibly retrieve important aspects of their knowledge with little attentional effort (p. 19).
Von Glasersfeld (1987) described understanding as a “never-ending process of consistent organization” (p. 5). It is not an all or none phenomenon hence “it is more appropriate to think of understanding as emerging or developing rather than presuming that someone either does or does not understand a given topic, idea, or process” (Carpenter & Lehrer, 1999, p. 20).
Conceptual vs Procedural Knowledge
Related to the notion of understanding is knowledge of concepts and procedures. Conceptual knowledge in mathematics is “knowledge of those facts and properties of mathematics that are recognized as being related in some way” (Hiebert & Wearne, 1991, p. 200). It is “knowledge that is understood, … a knowledge that is rich in relationships. … A unit of conceptual knowledge is not stored as an isolated piece of information; it is conceptual knowledge only if it is a part of a network” (Hiebert & Carpenter, 1992, p. 78) [italics, mine]. This implies that the quality of conceptual knowledge is a function of the strength of the connection or relationships between the concepts involved.
Hiebert and Carpenter defined procedural knowledge as a sequence of actions and as such, the connection between concepts involved is minimal. An example of procedural knowledge is knowledge of standard computation algorithms, which consist of a step-by-step sequence of procedures of symbol manipulation. Hiebert and Carpenter argued that procedural knowledge could contribute to mathematical expertise only if it is related to conceptual knowledge: “From the expert’s point of view, procedures in mathematics always depend upon principles represented conceptually” (p. 78).
One of the ways to help students to make connections among concepts is to give them problem solving tasks that have many correct solutions or answers. Another way is to make sure that the solutions to the problems involve many previously learned concepts. This is what makes a piece of knowledge powerful. Most important of all, the tasks must give the groundwork for future and more complex concepts and problems the students will be learning. These kinds of task need not be difficult. And may I add before I give an example that equally important to the kind of learning tasks are the ways the teacher facilitates or processes various students’ solutions during the discussion.
I would like to share the problem solving task I made to get the students have a feel of the existence negative numbers.
We tried these tasks to a public school class of 50 Grade 6 pupils of average ability and it was perfect in the sense that I achieved my goals and the pupils enjoyed the lessons. This lesson was given after the lesson on representing situations with numbers using the sorting task which I describe in my post on introducing positive and negative numbers.
Sorting is a simple skill when you already know the basis for sorting which is not case in the task presented here.
Just like all the tasks I share in this blog, it can have many correct answer. The aim of the task is to make the students notice similarities and differences and describe them, analyse the relationship among the numbers involved, be conscious of the structure of the number expressions, and to get them to think about the number expression as an entity or an object in itself and not as a process, that is speaking of 5+3 as a sum and not the process of three added to five. The last two are very important in algebra. Many students in algebra have difficulty applying what they learned in another algebraic expression or equation for failing to recognize similarity in structure.
Here are some of the ways the pupils sorted the numbers:
1. According to operation: + and –
2. According to the number of digits: expressions involving one digit only vs those involving more than 2 digits
3. According to how the first number compared with the second number: first number > second number vs first number < second number.
4. According to whether or not the operation can be performed: “can be” vs “cannot be”.
5. This did not come out but the pupils can also group them according to whether the first/second term is odd or not, prime or not. It is not that difficult to get the students to group them according to this criteria.
Solution #4 is the key to the lesson:
During the processing of the lesson I asked the class to give examples that would belong to each group and how they could easily determine if a number expression involving plus and minus operation belongs to “can be” or “cannot be” group. From this they were able to make the following generalizations: (1) Addition of two numbers can always be done. (2) Subtraction of two numbers can be done if the second number is smaller than the first number otherwise you can’t. You can imagine their delight when they discovered the following day that taking away a bigger number from a smaller number is possible.
One pupil proposed a solution using the result of the operations but calculated for example 3-10 as 10-3. This drew protests from the class. They maintained that 3-10 and similar expressions does not yield a result. Note that class have yet to learn operations on integers. And obviously they could not yet make the connection between the negative numbers they used to represent situation from the lesson they learned the day before to the result of subtracting a bigger number from a smaller. To scaffold this understanding I ask them to arrange the number expressions from the smallest to the biggest value. This turned out to be a challenging task for many of the students. Only a number of them can arrange the expressions for smallest to the biggest value. My next post will show how the task I gave to enable the class to make the connection between the negative number and the subtraction expressions.
