Posted in Assessment, Curriculum Reform, Elementary School Math, High school mathematics, Number Sense

Assessing conceptual understanding of integers

Assessing students’ understanding of operations involving integers should not just include assessing their skill in adding, subtracting, multiplying and dividing integers. Equally important is their conceptual understanding of the process itself and thus need assessing as well. Even more important is to make the assessment process  a context where students are given opportunity to connect previously learned concepts (this is the essence of assessment for learning). Because the study of integers is a pre-algebra topic, the tasks should also give opportunity to engage students in reasoning, number sense  and algebraic thinking. The tasks below meet these criteria. These tasks can also be used to teach mathematics through problem solving.

The purpose of Task 1 is to encourage students to reason in more general way. That is why the cells are not visible. Of course students can solve this problem by making a table first but that is not the most ideal solution.

adding integers
Task 1 – gridless addition table of integers

A standard way of assessing operations involving integers is to ask the students to perform the operation. Task 2 is different. it is more interested in engaging students in reasoning and in developing their number and operation sense.

subtracting integers
Task 2 – algebraic thinking and reasoning in numbers

Task 3 is an example of a task with many possible solutions.  Asking students to find a relation between the values in Box A and Box B links operations with integers to the study of varying quantities or quantitative relationship which are fundamental concepts in algebra.

Task 3 – Integers and Variables

More readings about algebraic thinking:

If you find this article helpful, please share. Thanks.

Posted in Elementary School Math, Number Sense

Algebraic thinking and subtracting integers – Part 2

Algebraic thinking is about recognizing, analyzing, and developing generalizations about patterns in numbers, number operations, and relationships among quantities and their representations.  It doesn’t have to involve working with the x‘s and other stuff of algebra. In this post I propose a way of scaffolding learning of operations with integers and some properties of the set of integers by engaging students in algebraic thinking.  I will focus on subtracting of integers because it difficult for students to learn and for teachers to teach conceptually. I hope you find this useful in your teaching.

The following subtraction table of operation can be generated by the students using the activity from my algebraic thinking and subtracting integers -part 1.

subtraction table of integers

Now, what can you do with this? You can use the following questions and tasks to scaffold learning using the table as tool.

Q1. List down at least five observation you can make from this table.

Q2. Which of the generalizations you made with addition of table of operation of integers still hold true here?

Q3.  Which of the statement that is true with whole numbers, still hold true  in the set of integers under subtraction?

Examples:

1. You make a number smaller if you take away a number from it.

2. You cannot take away a bigger number from a smaller number.

3. The smaller the number you take away, the bigger the result.

Make sure you ask students similar questions when you facilitate the lessons about the addition of integers. See also: Assessment tasks for addition and subtraction of integers.

Posted in Curriculum Reform, Mathematics education

My issues with Understanding by Design (UbD)

Everybody is jumping into this new education bandwagon like it is something that is new indeed. Here are some issues I want to raise about UbD.  I am quoting Wikipedia in this post but this is also how UbD is explained  in other sites.

Understanding by Design, or UbD, is an increasingly popular tool for educational planning focused on “teaching for understanding”.

Is not teaching for understanding been the focus of all curricular reforms, then and now? No curriculum reformer wants to be caught in the company of rote learning, never mind that it’s how curricula are implemented, regardless of its form, kind and  substance in many classes. Teaching for understanding is not something new.

UbD expands on “six facets of understanding”, which include students being able to explain, interpret, apply, have perspective, empathize, and have self-knowledge.

I wonder which of these facets has not been a part of what it means to understand then. I’m not sure in other subject areas but these facets of understanding such as explain, interpret, and apply does not capture what it means to understand mathematics.

To facilitate student understanding, teachers must explain the “big ideas” and “essential questions” as well as the requirements and evaluative criteria at the start of the class.

Back in college we attribute it to Ausubel who promoted the idea of using advance organizers.  Of course, you don’t tell your students right away how they will be assessed. They don’t have those rights, then. Also, this method only works for some topics. In mathematics if the approach is Teaching through Problem Solving or Discovery method, this is a no-no as it might limit the students thinking in exploring their own ways of working with the task at hand.

The emphasis of UbD on “big ideas” is welcome development but shouldn’t this be contained in the curriculum framework? The “essential questions”, those elusive questions that teachers have difficulty formulating since probably the time the  education community was talking about “art of questioning” are also good reminders to all of us that ‘hello, processing questions before or after any activity are what make and unmake a lesson’. But isn’t it that one can only identify the enduring understanding required and formulate good questions if he/she has a very good content knowledge (CK) and pedagogical content knowledge (PCK)?. Shouldn’t the money and time for training teachers how to design a lesson using UbD be spent instead on deepening their understanding of CK and PCK? Shouldn’t we make sure first that we have a good curriculum framework that articulates what are important for students to know and understand in each subject area and in each content topic?

The emphasis of UbD is on “backward design”, the practice of looking at the outcomes in order to design curriculum units, performance assessments, and classroom instruction.

In my part of the globe, there is a national curriculum which is a collection of SMART objectives. These learning objectives have always been stated in terms of outcomes. Weren’t they called competencies? Aren’t these competencies tell what to assess? The trouble is, our list of competencies consist of factual and procedural knowledge and very little on problem solving and reasoning which never really get taught because they are all found at the end of each chapter!

According to Wiggins, “The potential of UbD for curricular improvement has struck a chord in American education. Over 250,000 educators own the book. Over 30,000 Handbooks are in use. More than 150 University education classes use the book as a text.”

That explains everything. Everybody is hooked on the book that no one found time to do research if it works or not. Of course, on this part of the world where I come from I could not possibly have full access to current studies in educational planning and curriculum conducted elsewhere. I’m pretty sure though that we don’t have a study here yet. This is actually my issue. We’re jumping on a bandwagon created elsewhere without checking first if it will run on our roads.

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Understanding by Design recommends a structure for curriculum planning, for designing instruction. It is not surprising that this is a welcome development because of lack of the same when it comes to this area. College education and in-service programs have failed to equip teachers the knowledge and skills to identify the important ideas in their major field of study.

Click here for the proposed stages of lesson development by UbD (thanks Jimmy Wysocki). Imagine it in the hands of our classroom teachers. Imagine how their faces will look like if you tell them “these elements should be in your written lesson plans”! And when they look for resources, all they have is an anemic curriculum framework and textbooks teaching facts that can be Googled. They will follow the directives, of course, as they have always done in the past in this part of the globe. They won’t just have time anymore to study and prepare  for the actual teaching of the lesson, especially in examining how their students learn specific topic. Surely, they will have a very neat plans complete with the elements. But lest we forget, learning is still more a function of the experiences students engages in, that is the lesson, and not in the lesson plans format.

Lastly, UbD is a one size fits all for all subject areas. That’s what make it highly suspect. Click here and here for sequels of this post.

Posted in Mathematics education

What is scaffolding in education?

Scaffolding is a metaphor for describing a type of facilitating a teacher does to support students learning. Some educational paper lists some of these scaffolding like “breaking the task into smaller, more manageable parts; using ‘think alouds’, or verbalizing thinking processes when completing a task; cooperative learning, which promotes teamwork and dialogue among peers; concrete prompts, questioning; coaching; cue cards or modeling”. Visual scaffolding is also popular in teaching mathematics.

Scaffolding is the latest buzzword in education community. In an international conference I attended recently for instance, I heard the word in almost all the parallel paper presentations.

There was a demonstration lesson for teaching English during the conference. I am not an English teacher so I asked the person seated beside me, who happens to be an English teacher, to tell me what the teacher was doing as she hopped from one group of students to the other. She said with authority that the teacher was doing a lot of scaffolding. I didn’t know what to make of her statement. Was it a positive or a negative comment? Is it a good idea to do a lot of scaffolding or is it something that should be given sparingly? Where do you draw the line?

Scaffolding can be traced back to Lev Vygotsky’s idea of ZONE of PROXIMAL DEVELOPMENT (ZPD). Vygotsky suggests that there are two parts of learner’s developmental level: 1) the Actual developmental level 2) the Potential developmental level

The ZPD is “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance”. This is where scaffolding is crucial.

scaffolding in education
Note that the activity students should be engaging in is problem solving. A problem is a problem only when you do not how to solve it right away. So when scaffolding deprives the students from thinking and working on their own way of solving the problem then scaffolding has not helped learn how to solve problem. It only helped them to solve problems using the teacher’s method.

You may want to read the different interpretations of zone of proximal development in research.