## Recycling exercises

In a recent article, we discussed the nature of problems and their fundamental difference with exercises—more routine tasks.

Since many of the tasks you know or have may be traditional exercises, today I bring you **four strategies to enrich** and recycle some of them, through examples, to begin with.

Let’s go!

## 1. Less is more

Usually, teachers are **afraid of open questions**, because they may create an ambiguous situation for students. This is why we tend to specify the data as much as we can, so everyone knows what to do. Let’s see an example:

This is a rather interesting exercise, because it sets an uncommon context that demands a conversion of years into minutes. Nevertheless, we easily see a direct solution:

That is a huge number, but it didn’t take a huge effort, right? What can we do to enrich it? Less is more. Get rid of the data:

**Setting only the context, students will engage in a conversation, guided by ourselves as teachers**. Then, we can:

- Ask for an
**on-hands measurement**of their heart rate. Next, ask the students to calculate the class average, and therefore introducing this content in a meaningful context, if needed. - Ask for
**estimations to develop number sense**. For instance, “What is the life expectancy in the US?” And let our students look for information to foster autonomy. - Ask them to
**make decisions**on what data is to be considered: “Are we considering leap years?” And connections with other disciplines: “Why is there a different life expectancy between men and women?” - Etc.

Try it with the sandwich example from the previous article, and you’ll find your students arguing about how many sandwiches they would eat for lunch, about the size of the sandwiches, about the ingredients, even about how their answer would change depending on how many people are coming for lunch—good idea for introducing algebraic concepts!

Of course, this takes much more time than just multiplying 7 times 2, but from a problem-solving perspective in elementary, it is** much richer**.

## 2. How would you solve it?

To recycle a worksheet full of operations, we can be inspired by the lifelines of *Who Wants to Be a Millionaire?* Do you remember them? Once you chose to use the call, that lifeline was consumed.

So, what can we do with those worksheets? Ask our students to **choose wisely how they would solve each operation**, either using a calculator, any written strategy or by mental calculation. Once they choose a given strategy, they consume it, and they have to be consistent.

This approach aims for the students to **develop a strong sense of what strategy is best** depending on the situation. Even myself, a proud math teacher, would use the calculator when splitting up the bill in a restaurant, I confess.

At Innovamat, we love this approach so much that we created a virtual activity within our practice app that explicitly fosters such criteria for selecting the best strategy. You can click on the image to try it!

## 3. Make up a short story

This one is quite simple—for you, not for the students! Just write an operation on the board and ask your students to make up their own short story, **a context that makes sense**, with a question that might be answered by such operation. Speaking of quiz TV shows, *Jeopardize it!*

Let’s see an example:

Students have to find a realistic context. Consider this one, to begin with:

Do you like it? The figures seem correct, but think about it: this word problem would not be answered with a decimal result, since pencils are indivisible. If the operation was 412 25 =16 R12 , then the statement could talk about pencils: 16 per student and 12 are left over. Since our operation has decimals instead of a remainder, though, such a story makes no sense.

Therefore, a context where decimals make sense is needed. For instance, money:

Although here the figures are correct, this would be also wrong from an in-context number sense perspective, because the average price of a watermelon is far from $16.48. Furthermore, nobody would buy 25 watermelons in the market on a normal day…

This last option would be much better, depending on the size of your school:

Ultimately, creating short stories, or contexts, is a **creative and interconnected way of developing literacy** within the math class.

## 4. What is wrong?

This last strategy consists on presenting the students with a word problem along some right and wrong answers. It would be better to use answers from the students themselves or given by students from previous years, but you can also make them up conveniently, to **target common misconceptions**. What is important here is to give students the chance to discuss and explain why some of those answers are wrong (or right). They love to correct! Let’s see an example:

## In conclusion…

Eventually, **most of our class time should be devoted to ‘problem-solving’**, to build thinking classrooms, as Liljedahl puts it in his last bestseller (2020). Such transformation is not easy, though. Do not feel frustration if you can’t achieve it overnight: it is an ongoing journey many teachers are doing, sharing, and learning, as we try with our program. In the meantime, you might find our **strategies to recycle exercises very useful**. Try them and see what happens. And let me warn you: once our students get used to *think*, they won’t stop.

Liljedahl, P. (2020). Building thinking classrooms in mathematics, grades K-12: 14 Teaching Practices for Enhancing Learning. Corwin Mathematics.