Mathematical reasoning is essential to bridging the gap between basic skills and higher-order thinking. In fact, research has shown that students who are taught reasoning skills early on ultimately become more confident, independent learners; they have a deeper understanding of how a concept can be applied in a variety of situations and are willing to take risks to see what works and what doesn’t. What follows are some ways you can actively engage your students in the reasoning process.

### Multiple Ways to Solve a Problem

Since every student learns in a different way, they generally problem solve in unique ways as well. That is why it’s important to create and encourage multiple ways for students to approach topics. Here is a sample problem: Robin has $5. How many more dollars does she need to buy a journal that costs $12? The teacher can lay out plastic cubes and play money, and encourage students to count on their fingers, write down their calculations, or use mental math to figure out the answer. Students can then take turns explaining their reasoning and justifying their explanations. By using this approach, students will then be able to compare the different strategies they used for solving the same problem. A problem like this also allows students to solve problems according to their developmental level. I’ve found that by allowing students to develop their own strategies for solving addition problems, they intuitively start to use the commutative and associative properties.

### Thinking It Through

Students who are successful problem solvers take a learned skill and apply it in different ways to a variety of situations. For this reason, you should reassure children that it’s ok to make mistakes, as what you’re really interested in is how they think through a problem, not whether they get the answer right. Here’s another reasoning problem you can use to help students build on their subtraction skills.

- Pose the following question: How can you weigh a suitcase that is too large to fit on a bathroom scale?
- Be sure to listen carefully to the students’ answers, try out some of their suggestions, if possible, and praise them for thinking through the problem.
- If they don’t come up with a solution, tell them that one way to find the weight of the suitcase is for one person to stand on the scales while holding it. Ask another student to write the total weight.
- Then put the suitcase aside and let the child weigh himself again, while another child notes his weight. If the student subtracts his weight from the total weight, the number remaining is the weight of the suitcase.

### Hands-On Problem Solving

There are many ways to provide older students with hands-on, manipulative-based reasoning activities. Here is one example in which students explore the relationships between different sized balls. Begin by showing the class a set of up to ten sports balls and invite them to talk about what they notice. They are likely to suggest all sorts of things, perhaps related to the sport that each is used for, or the material each is made of, in addition to their spherical shape.

- Draw a line on the ground and ask students the following questions: If the balls are put on the line, with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?
- Which arrangement makes the longest line of balls?
- Let students try many different arrangements of balls, then ask them to share their results with the class.
- Hopefully they’ll discover that to get a shorter line you have to put the smaller balls between the big balls because the bigger balls tower over the smaller balls with their curve allowing the smaller to fit in under resulting in a smaller distance used.
- By changing the order of the balls, students will naturally come to a better understanding of the properties of circles.
- You can extend the activity for older students by printing out copies of the balls to scale, then asking them to place the equipment in lines that make up an equilateral triangle or a square.

Here are some more mathematical reasoning lessons and activities you can use to challenge your students.

## Math Reasoning Lesson Plans

Math Sleuths

Students use their problem-solving skills to complete an online Math Hunt. Their share their answers and justify their thought processes with the class.

Math in Everyday Life

Students work with a partner to come up a with word problems, involving time, money, and simple fractions. They then share their word problems with other students.

Lift the Math Curse

Students read the book Math Curse by Jon Scieszka. Then they write story about a day in their life that includes up to fifteen math problems with an answer key. Students share their books with the class.