Tag: making connections

Posted in Mathematics education

What kind of mathematical knowledge should teachers have?

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As a result of her research, Liping Ma developed the notion of profound understanding of fundamental mathematics (PUFM) as the kind of mathematical knowledge teachers should possess. She discusses this kind of knowledge in her book Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series). This book is now considered a classic by many mathematics educators. The ‘elementary’ in the title does not mean the book will be valuable to elementary teachers only or those engage in the training of prospective elementary teachers. The book is for all mathematics teachers, trainers, and educators. This book is a must-read to all that has to do with the teaching of mathematics.

Here’s what Liping Ma says in the introduction:

Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers seem far less mathematically educated than U.S. teachers. Most Chinese teachers have had 11 to 12 years of schooling – they complete ninth grade and attend normal school for two or three years. In contrast, most U.S. teachers have received between 16 and 18 years of formal schooling-a bachelor’s degree in college and often one or two years of further study.

In this book I suggest an explanation for the paradox, at least at the elementary school level. My data suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than that of most U.S. elementary teachers. Their understanding of the mathematics they teach and -equally important – of the ways elementary mathematics can be presented to students continues to grow throughout their professional lives. Indeed about 10% of those Chinese teachers, despite their lack of forma education, display a depth of understanding which is extraordinarily rare in the United States….

Why the word ‘profound’? Profound has three related meanings – deep, vast and thorough – and profound understanding reflects all three. From the paper delivered by Liping Ma and Cathy Kessel in the Proceedings of the Workshop on Knowing and Learning Mathematics for Teaching conference, Liping and Cathy offered the following explanation:

  • A deep understanding of fundamental mathematics is defined to be one that connects topics with ideas of greater conceptual power.
  • A vast or broad understanding connects topic of similar conceptual power.
  • Thoroughness is the capacity to weave all parts of the subject into a coherent whole.

A teacher should see a ‘knowledge package’ when they are teaching a piece of knowledge. They should know the role of the current concept they are teaching in that package and how that concept is supported by which ideas or procedures.

To further explain the kind of mathematics knowledge a teachers should possess, Liping and Cathy used the analogy of a taxi driver  who knows the road system well. The teachers should know many connections so that they are able to guide students from their current understandings to further learning.

I think this is how designers of curriculum, writers of curriculum materials, and teachers should interpret the standard “Making connections”.  It is not simply linking.

 

Posted in Elementary School Math, Number Sense

Teaching negative numbers via the numberline with a twist

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One popular way of introducing the negative numbers is through the number line. Most textbooks start with the whole number on the number line and then show to the students that the number is decreasing by 1. From there, the negative numbers are introduced. This seems to be something easy for students to understand but I found out that even if students already know about the existence of negative numbers having used them to represent situations like 3 degrees below zero as -3, they would not think of -1 as the next number at the left of zero when it is presented in the number line. They would suggest another negative number and some will even suggest the number 1, then 2, then so on, thinking that maybe the numbers are mirror images.

Here is an alternative activity that I found effective in introducing the number line and the existence of negative numbers.  The purpose of the activity is to introduce the number line, provide students another context where negative numbers can be produced (the first is in the activity on Sorting Situations and the second is in the task Sorting Number Expressions), and get them to reason and make connections. The task looks simple but for students who have not been taught integers or the number line the task was a problem solving activity.

Question: Arrange from lowest to highest value

When I asked the class to show their answers on the board, two arrangements were presented. Half of the class presented the first solution and the other half of the students, the second solution. Continue reading “Teaching negative numbers via the numberline with a twist” →

Posted in Teaching mathematics

To understand mathematics is to make connection

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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:

  1. Experts notice features and meaningful patterns of information that are not noticed by novices.
  2. Experts have acquired a great deal of content knowledge that is organised in ways that reflect a deep understanding of their subject matter.
  3. 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.
  4. 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).

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Credits

The image is from the post Good Ideas are Networks in  Slow Muse by Deborah Barlow.

A Framework of Growth Points in Students’ Developing Understanding of Function – PhD Thesis by the Author

Posted in Number Sense

Introducing negative numbers

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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.