Posted on December 23rd, 2014

From the Institute for Learning, this is Laurie Speranzo.

We know that Accountable Talk^{®} discussions can be effective in all disciplines and for all age groups of students. When mathematics teachers incorporate talk into their classrooms, that talk will only be productive if they organize it around a high-level task. But what is a high-level task?

Consider the following two tasks and think about how they are similar and how they are different:

- Task 1 is a calculation task, which may sound like something with which you are familiar: Students are asked find 10% of $68.79, 20% of $68.79, and 30% of $68.79.
- Now, Task 2: One store is having a 30% off sale. Another store has a 20% discount, with an additional 10% off of the sale price. Which sale should you take advantage of if you want the best reduction on a sweater that costs $68.79? Does each sale result in the same reduction off the original cost? Explain what you would pay in each sale situation and the similarities or differences between the two sales.

In the simple conversion task, can't we almost hear students saying, "I multiplied the 68.79 by point ten to find 10%"? Are there other ways that students may solve the task? Students may divide the 68.79 into ten equal groups to find 10%. Students may know that moving the decimal point one place to the left yields 10% of a number. The responses are all going to be procedural and, why wouldn't they be? All the task requires is an answer and work being asked for is merely calculations, thus there is no reasoning about mathematics.

The second task about the sweater feels incredibly different. By referencing a real life context, the students are meant to draw connections between the situation and their calculations. The percentage of the whole is meant to be removed from the whole in order to find the final cost of the sweater, so you have a multi-step problem. The students are more likely to be invested in this problem, as it is something they may experience personally! Students could even find entry into the problem through reasoning before calculating—30% off of a number is not the same as 20% off plus an additional 10% off. Let's try this with a convenient number like 100: 30% off 100 is 30 off, leaving 70. But if I take 20% off of 100, it is 20 off, leaving us with 80. An additional 10% off of that new whole of 80 is only 8 off, leaving us with 72. 30% is not the same thing as 20% and then 10% more off because when you do the two-step problem, you change the whole after the first percentage taken!

Both of these tasks require students to find the percent of a number. However, the task with the context asks students to think and reason; children have to explain their thinking, not simply list their calculations. Could students actually just calculate the 70% remaining instead of calculating the 30% off and subtracting it from the original sweater cost? So students may attack this problem by calculating the percentage off, by calculating the final cost, or by reasoning about the discounts.

What a difference in the level of discussions that can be had in the classroom when students have to make mathematical connections between context and numbers as well as reason about the answer, rather than just calculate answers.

In order to classify tasks as high-level or low-level, we use the Mathematical Task Analysis Guide by Smith and Stein. This tool cites the research-based characteristics of high-level and low-level math tasks. If a task is high-level, it allows for use of multiple pathways or representations. High-level tasks ask students to make connections between representations and to explore mathematical relationships. Students are expected to put forth cognitive effort—yes, students are expected to experience productive struggle! In low-level tasks, students are not expected to grapple with mathematical concepts in order to build deep understanding.

Think about how many of the math practice standards are incorporated in high-level tasks! Students have to problem-solve because there is a context or model to interpret, which is Math Practice 1. There is modeling involved, which is Math Practice 4. Students are required to justify their answers, which is MP3. In looking for connections, students explore structures of mathematics, which is Math Practice 7.

So when choosing tasks in which to engage your students, I would challenge you to identify high-level tasks that provide opportunity for productive discussions about mathematical ideas. When you are selecting tasks, ask yourself:

- Are students asked for mathematical reasoning, not just calculations?
- Will students be grappling with making connections between representations?
- Are there several ways to solve the problem, so that the students are provided the chance to compare solution paths?

If the answer to each of these questions is yes, then the high-level task in front of you will be a springboard for a rich mathematical discussion.

**Laurie Speranzo** is a Fellow of the Institute for Learning's mathematics Disciplinary Literacy Team.

More information on Laurie Speranzo.