Properties of good tasks

Consider the cognitive demand. 

There might be good reasons to have students engage in low-level tasks like memorization, but good problem-solving tasks are often "procedures with connections" or "doing mathematics"

  • Memorization
  • Procedures without connections
  • Procedures with connections
  • Doing mathematics: 
    • Students are required to explore, analyze, reflect on previous knowledge, and self-regulate. These are the tasks in which students are require to come up with their own "pathway" for solving the problem. 
    • Doing math is exploring relationships.  It can be different representations, making connections, complex thinking, self-monitoring, and using their prior and new knowledge.

Create intellectual need/"if math is the aspirin, what is the headache?"
  • Tasks should introduce a big (probably complex) question right off the bat, and the rest of the task should revolve around that big question. This gives students a purpose for doing calculations, solving equations, etc. even though that purpose may not be completely real life. For example, students may write a quadratic equation and solve it because they see the purpose-- they are trying to answer “the big question” that the task revolves around.
  • If I want my students to learn how to solve an exponential equation, I need to give them a task that shows them there is a need for that new skill. Giving students a reason for learning something motivates them, and provides a context for connecting the new skill learned.

Delay feedback/Be less helpful
As a teacher one of my biggest flaws is trying to be too helpful, it is important to let the kids struggle with material and make connections on their own before giving them the answers, or verifying their responses.

Tasks should help students make connections: 
Traditionally, school curricula have too often presented mathematics as isolated bits of information and procedures to be memorized.  As a result, students are often limited in their ability to approach new problems for which they cannot recall a previously taught algorithm. Instead, tasks should help students make connections: 
  • Between the different representations we use in math class (graph, table, equation, etc.)  
  • Between the conceptual and procedural mathematics we do in our classes.
  • Among mathematical concepts.  
    • Math doesn't exist in discrete subsets, no matter what we title our courses. Connecting curricular ideas builds a deeper understanding of math as a discipline. Also, connecting to previous tasks can help develop student knowledge that "sticks" because it helps them to create a frame of reference for later developments.
    • Appropriate launch/conclusion activities will help to build connection between the understandings of the new concepts with previous mathematical concepts

Have a clear educational purpose
  • “For example, if the goal is to increase students' fluency in retrieving basic facts, definitions, and rules, then tasks that focus on memorization may be appropriate. If the goal is to increase students' speed and accuracy in solving routine problems, then tasks that focus on procedures without connections may be appropriate” (Stein et al., 2000).
  • start with the mathematics when selecting a problem solving task.  The teacher should think about the mathematics that they want the students to focus on, and then find tasks that build upon that math topic.
  • If you want students to investigate the area of polygons for example, then you need to select a task that is deep and rich enough to allow this to happen.  If you just want students to memorize the area formulas, then you would use a completely different task.
  • A teacher may choose, for example, to use a problem solving task as a way to introduce a mathematical concept, to help students develop intuition about a concept, or to build upon an already mastered concept. To me, it seems important to understand what the specific goal for student learning is before choosing or creating a task
  • Use "special" tasks in the beginning of the class to set norms and expectations, and then use content-focused tasks for the rest of the year.

Tasks should be problematic yet accessible
"problems cannot be too difficult, if they are students will give up before any learning has taken place. They also cannot be too straight forward or students will just do the work individually." 
  • Problematic activities focus on the big picture and the way to get there, as opposed to solving little problems with no idea why we are doing it.  I
  • Don’t give the students too much information in which the problem becomes merely procedures with connections.  Let the students struggle a little in finding their own work, but not enough that it will stump them. 
Yet accessible:
  • Encourage students to informally analyze tasks before formally analyzing them
  • Make the math accessible to the kids!  Three act tasks do a great job at this by showing a real world situation that kids can understand and work with before giving an equation or a graph.  By showing an informal representation you not only let them know that they can understand the content, but you don't scare them away with mathematical jargon!
  • although students should see challenging problems, they shouldn’t be so challenging that the students can’t approach them.
  • easy to start and difficult to finish

Create ambiguity. 

  • If the method for solving a problem is immediately apparent in the formulation of the task, it is not going to lead to high-level thinking. Multiple approaches should be possible, and students should have questions beyond what is given. 
  • Math can be subjective in real life, and that’s a scary thought for some students, who have been brought up to think of math as this precise, one-answer deity.  Math can be precise, to a point, but when thinking of things such as how many people are in the world, we can only make a guess!  And we need to emphasize this thought that math is not always going to give us an exact answer.  And that is okay.
  • One algorithm isn't the only way they students could solve something.  By allowing the students to be right or wrong in multiple ways instead of just one way, all of my students would be able to participate in some way that is meaningful not just "my way".
  • No specific pathway is hinted at in the task or question. This is a principle that separates high level “doing mathematics” tasks from lower level “procedural” tasks. This is important because it forces students to decide for themselves what route to take to solve a problem, instead of just filling in values in a formula or performing a standard procedure.

Involve explanation and argumentation, encourage fights.

  • Discourse is critical to building deep understanding, and it is better if it is student-centered rather than teacher-student interactions. Students should challenge each other's reasoning and explanations, and be asked often to justify why they feel they have the "best" solution. This can be in pairs, groups, or even in journals. 
  • When students are able to articulate what they are doing to someone else and why, it is a good indication that they aren't just applying an algorithm and moving on. When a student hears my explanation, it doesn't necessarily stick.  Even the clearest teacher explanations leave many students with incomplete understanding and shaky confidence (NCTM, p. 61).   Through explanation and conversation, a teacher can help guide a student to an area they are still shaky on or needs development.
  • Explanations and justifications are part of the task. Creating a task that revolves more around student communication of work instead of just a correct answer allows students to see that math is not about doing the correct operation, it is about communicating mathematical reasoning. This principle cannot stand alone though; it is hard to get students to communicate about a simple practice problem with one right answer. The question students are exploring has to be rich enough for there to be something to talk about.

Ask questions, not imperative statements

  • In order to engage their brains, students have to be pondering something - that comes from a question, not an imperative statement. I once read that 90% of the instructions in a typical algebra text are "solve", "find x", or "graph". Those verbs first of all, tell you what to do and basically how to do it. Secondly, and I think more egregiously, they tell the student to stop considering the problem and in the words of Nike, Just Do It. If we want higher thinking to occur, there need to be questions that students are having to chew on. 

Consider a range of verbs/engage students in a variety of mathematical practices

So often our books say: solve, simplify, multiply... Over and over.   If you ask students the exact same question in the exact same way, you will get the same type of response.  If you change the question just a little even if it is asking the same thing students will look at it differently and have a better understanding of the material. Many times we could switch it up:

  • Justify
  • Explain
  • explore
  • conjecture
  • prove
  • represent
  • communicate
  • analyze
  • validate