Posted in Number Sense

Introducing negative numbers

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.

Posted in Lesson Study, Number Sense

Patterns in the tables of integers

Mathematics is said to be the science of patterns. Activities that involve pattern searching is a great way to engage students in mathematical thinking. Here are some of my favorites for teaching positive and negative integers. If you are wondering why most of my posts are about integers it’s because I’m doing a Lesson Study with a group of Mathematics I (Year 7) teachers about this topic. Last week we concluded the first cycle of our research lessons on teaching subtraction of integers.

The first task students need to do with the tables is to list 3-5 observations. From there you can start asking the ‘whys’ for each observation. Sample questions are provided for each table below:

1. Adding integers

Sample questions for discussion:

a) Under what conditions will the sum be positive? negative? zero?

b) Why are there the same numbers in a diagonal?

c) How come that the sum is increasing from left to right, from bottom to top?

2. Taking away integers

Sample questions for discussion:

a) Under what conditions will the difference be positive? negative? zero?

b) Why are there the same number in a diagonal?

c) How come that for each row/column, the difference is decreasing?

3. Multiplying integers

Sample question for discussion:

a) Under what conditions will the sum be positive? negative? zero?

4. Dividing Integers

Sample question for discussion:

a) Under what conditions will the difference be positive? negative? zero?

       b) Does dividing integers still results to an integer? What do we call these new numbers?

Feel free to share your ideas/questions for discussion.

You may also want to share other  math concepts that students can learn with these tables.

Posted in High school mathematics, Lesson Study

Pedagogical Content Knowledge Map for Integers

I’m working with a group of Year 7 mathematics teachers doing Lesson Study for the first time. The teachers chose to do a lesson study for what they believe to be the most difficult topic in this year level – integers. Part of my preparation as facilitator is to draw a map of what I know about teaching the topic. The map is more than a concept map because it includes not just related big ideas or concepts but also how  these are taught and learned. Hence, I call this pedagogical content knowledge map (PCK map).

The PCK map I present here is a product of my own readings and my own experiences of teaching the topic. This means that it may not be the same as other teachers especially the ‘teaching part’ of the map, the ones in orange colors. For example, experience and research results back my claim that the number line is a very good way of representing the set of integers but not in teaching operations. Click here for my post about this. Notice that I gave emphasis on students knowing when a negative, a positive or a zero result rather than the rules for operation. I believe that without this, a conceptual understanding of the operation involving integers will be weak. Also, experience has taught me that although integers are numbers, the teaching of it must be algebraic. The instructions should be so designed so that students are learning algebraic thinking as well. I have noted this in the PCK map.

The map is not yet complete. I intend to include descriptions of effective activities and students’ learning trajectory of the concept after my research with the teachers. Please feel free to give your comments and share experiences for teaching integers that I could look into in my study.

pedagogical content knowledge
PCK Map for Integers

Please click the link to see my PCK map for Algebraic Expressions.

Posted in Elementary School Math, High school mathematics, Number Sense

Teaching positive and negative numbers

A popular approach for teaching numbers is to use it to describe a property of an object or a set of object. For example, numbers are used to describe the amount or quantity of fruits in a basket.

In introducing integers, teachers and textbooks presents integers as a set of numbers that can be used to describe both the quantity and quality of an object or idea. Contexts involving opposites are very popular situations to show the uses and importance of positive and negative numbers and the meaning of its symbols. For example, a teacher can tell the class that +5 represents going 5 floors up and -5 represents going five floors down from an initial position.

Mathematics is a language and a way of thinking and should therefore be experienced by students as such. As a language, math is presented as having its own set of symbols and “grammar” much like our spoken and written languages that we use to describe a thing, an experience or an idea.But apart from being a language, mathematics is also a way of thinking. The only way for students to learn how to think is for them to engage them in it!  Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking as well.

Sort the following situations according to some categories

  1. 3o below zero
  2. 52 m below sea level
  3. $1000 net gain
  4. $5000 withdrawal from ATM machine
  5. $1000 deposit in savings account
  6. 3 kg weight loss
  7. 2 kg weight gain
  8. 80 m above sea level
  9. 37o above zero
  10. $2000 net loss

The task may seem like an ordinary sorting task but notice that the categories are not given. Students have to make their own way of grouping the situations. They can only do this after analyzing each situation, noting commonalities and differences.

Possible solutions:

1.  Distance vs money (some students may consider the reading the thermometer under distance since its about the “length” of mercury from the “base”)

2. Based on type of quantities: amount of money, temperature, mass, length

3. Based on contrasting sense: weight gain vs weight loss, above zero vs below zero, etc.

The last solution is what you want. With very little help you can guide students to come-up with the solution below.

Of course, one may wonder why make the students go through all these. Why not just tell them? Why not give the categories? Well,  mathematics is not in the curriculum because we want students to just learn mathematics. More importantly, we want our students to think critically and creatively hence we need to give them learning experiences that develops good thinking habits. Mathematics is a very good context for learning these.

Here are my other posts about integers: