Letting students struggle, persevere and even fail is integral to helping them learn problem solving skills.
Helping students improve their problem solving abilities can be done by learning how to question and guide students through problem solving activities that help them to develop better understanding. We also need to give them the time to experience the intensity of struggle, the frustration of failure and finally the jubilation of pushing through to completion. These experiences develop work habits and habits of mind that will improve their success in mathematics.
This post shares a real life experience of what this looks like in the classroom and tips to overcome the frequent concerns and questions I get from teachers learning to teach this way. The approach I describe below can be intimidating and scary for teachers at first. However, with perseverance, you will be successful and the rewards of having your students engaged, solving problems and learning are immensely gratifying!
Today in a grade 6 class we used an open task which resulted in such a wide variety of approaches. What really stood out to me was the need to guide and facilitate our students by seeing what they are doing and leading them towards deeper and better understanding. This is something that an educator must develop when teaching for meaning and understanding because, as the teacher, you need to react to what your students are doing and, inevitably, they all do things a little bit differently! Keeping the goal and the big ideas in mind will help you to determine what questions to ask your students and how to guide them. This post is all about how we (the classroom teacher and I) helped to guide our students to meet the learning goals. I will also give you the observations we noted and how differently the students responded to the task we gave them.
First, a little background on the students. This is a grade 6 classroom and we have already studied fractions and percentages. We are currently working with decimals by using the base 10 blocks. We have explored addition and subtraction (review) using the concept of place value: adding subtracting “like terms”. We have also examined multiplying, both whole numbers and decimals, by 10, 100, 1000 etc.
Note: We avoided rules such as “add a zero, two zeros, three zeros” because this rule does not help us to understand what is happening with the numbers and it leads to errors when multiplying decimal numbers by 10, 100, 1000 (student error example: “1.2 x 10 = 1.20 because you add a zero when timseing by 10”).
Instead we focused on what is happening with place value and through the use of many examples, students could identify a pattern: multiplying by powers of 10 changes the place value but the digits remain the same. We had also already explored multiplying by: 0.1, 0.01, 0.001 etc. and students could generalize the following: when multiplying by a number greater than one the answer gets larger and when multiplying by number less than one the answer gets smaller. We felt like this was an important generalization because, up until this point, when students have multiplied, the answer has always been larger and we wanted them to understand why, when we multiply by number less than one, the answer shrinks.
We decided to do a short mini lesson before giving them the task on what it means to multiply a whole number by decimal numbers. Using the base 10 blocks, we asked our students to show us the following: three groups of two tenths (3x.2), three groups of 0.02 (3 x 0.02), and three groups of 0.002 (3×0.002). Using this decimal language helped the students to see the pattern! We discussed with the class what was similar and different about multiplying with decimals and with whole numbers (3×2).
Differentiated Task #1
As a way to explore the concept of multiplying decimal numbers by whole numbers we gave them the following open task: “two numbers have a product of 2.4, what could those two numbers be? Find as many solutions as you can.”
Teacher Guidance #1
Immediately a student asks “is 1×2.4 the same as 2.4 ×1?” We addressed this to the whole class to let the students know we would count this only as one solution. Because our goal was for students to understand what it means to multiply a whole number by a decimal and to find the patterns and similarities to multiplying whole numbers, we didn’t want them to get bogged down by writing each equation twice (knowing that our goals were: to deepen their understanding of what it means to multiply a whole number by a decimal and to identify the relationship between whole number multiplication and decimal multiplication, helped us to determine what we wanted our students to focus on).
Some students started by adjusting the digits in the solution 1×2.4 and created solutions such as:
0.1 x 24
0.01 x 240
0.001 x 2400
10 x 0.24
100 x 0.024 etc.
Teacher Guidance #2
They were quickly making lists of numbers that were going into in millions and millionths. How we redirected them was to tell them to find other solutions using different digits than 1, 2, and 4. This redirected them to start looking at other solutions such as 0.3 x 8 and 0.4 x 6. Many students had already discovered that if they found the factors of 24 they could find many solutions to this problem. Some students adjusted the numbers in different ways such as:
1 x 2.4
0.5 x 4.8
0.25 x 9.6
One student figured it out that if he halved one number and doubled the other it would give the same product. Although this concept is not multiplying a whole number by a decimal, this student is exploring a connected concept of multiplying decimal numbers and is showing great understanding of how numbers can be multiplied to achieve the same product.
Teacher Guidance #3
Because the students started creating patterns similar to:
0.1 x 24
0.01 x 240
0.001 x 2400
10 x 0.24
100 x 0.024
we challenged them to find as many different “digit groups” as possible:
1 x 2.4 is in a different digit group than 2 x 1.2
Most students were able to find at least four different digit groups, with multiple solutions within each digit group.
Key Learning Task
We also asked our students to explain their strategies. We had this written on the board underneath the original task so that they would remember to be aware of their thinking when it came time to reflect.
Teacher Guidance #4
The way the students approach the problem was as different as the solutions themselves! Some of the students simply went to the patterns and because our goal was for students to really understand what it means, I asked them why these patterns work and most were able to explain clearly. Others used the base 10 blocks and found their solutions concretely. With these students we asked them; “can you use what you found using the blocks to extend the pattern to numbers that are too small or too large to show with the blocks?” Some of them were able to do this, while others were not.
Student Reaction to the Lesson:
What we noticed while students were working on this task:
All of the students were on task!
They were engaged–they liked the challenge of finding many solutions and especially finding unique solutions or unique digit groups.
Lesson Wrap up and Next Steps:
At the end of the lesson we asked them to come up to the board and write one or two of their solutions and the whole class could then see the wide range of possible solutions.
We will continue to focus on both goals of understanding what it means to multiply a whole number by decimal number and how we can use patterns to do this multiplication efficiently.
When I work with classroom teachers, one of the most common questions they ask me is “how do I know what questions to ask or how to respond to what they’ve done or said?”. Because there are unlimited student outputs, there is no list of perfect questions. However, here are some sample questions that will lead to deeper conceptual understanding and meaning (as well as evoke and develop communication and mathematical reasoning):
1.) Why does that work?
2.) How do you know you are correct?
3.) Can you explain it in another way or use a model?
4.) What do you notice that is similar or different?(in this case to whole number multiplication)
5.) Do you see any patterns?
Be Clear on the Lesson’s Goal
Because you will also need to be able to respond to what your students are doing and redirect them when needed, ensure you are clear about the goal of the lesson. For example, when we saw that some students were immediately creating these equations that were all in the same digit group we needed to redirect them to find other solutions and to think about how this relates to the factors of 24.
Extending Students’ Thinking
For those students that were working with different digit groups but were not expanding them into different place values, we urged them to try to use the same digits but with different place values and see what they could discover.
Note: Not once in our lesson did we talk about moving the decimal places, but rather changing the place value as this helps them to build number sense and conceptual understanding rather than just procedural fluency.
One of the reasons that I love doing these open type questions is to see the diverse ways that our students are thinking about an understanding mathematics. I also see a higher level of engagement and at the end of the lesson all of the students felt successful because they were! Every single student had multiple solutions and could explain why their solutions worked. Some students extended the task to multiplying far more complex decimal numbers than we were expecting and were excited for the opportunity to go above and beyond because there were no limits on what they could do to solve this problem.
Educating Now was created due to teacher requests to have Nikki as their daily math coach. The site has lesson by lesson video tutorials for teachers to help them prep for their next math class and incorporate manipulatives, differentiated tasks, games and specific language into their class. Teachers who use the site can improve student engagement and understanding, in addition to saving prep time, by watching a 10 minute video tutorial and downloading a detailed lesson plan.