What Does A “High” Mathematics Student Look Like and Does This Mean They Have a Growth Mindset about Mathematics?
How do you know they are ‘high’ in their mathematics achievement/learning?
Help young children 'contextualize' problems
Reflect on your assessment of your students' mathematical ability
When talking with teachers I often hear about their ‘high’ students in mathematics.
In conversation with these teachers I ask, how do you know they are ‘high’ in their mathematics achievement/learning? The majority of responses I get is that these students scored well on a paper and pencil assessment or did well on a worksheet or questions from a textbook. When I ask to see these assessments (or this work) often they are simply asking for answers ONLY, not an understanding of how they got to the answer. Another common response is that these ‘high’ achievers can complete number work with speed.
Here are questions I would like to ask:
- Does getting the right answers mean that students have made and/or are making meaningful connections?
- Does getting the right answers mean students are able to show and/or explain their thinking of how they got to this answer?
- Does getting the right answers mean students have a growth mindset towards mathematics and are able to look at mistakes (if made) as opportunities for their brain to grow and make meaningful connections?
- Is speed in completing a number task(s) mean students have conceptual understanding and procedural fluency?
Jo Boaler comments in her book that there seems to be a societal belief that those who can calculate number work quickly are both a ‘true’ and ‘smart’ math person. She also talks about her connections with MANY mathematicians and she defines these individuals as not fast math thinkers, but rather as slow, careful and deep thinkers.
Within her book she also talks about students who have been called ‘smart’ in math tend to dislike math and more than often, do not continue with math after high school. These ‘smart’ kids tend to have a fixed mindset and are usually crushed when their answers are not right.
With this said, it is of the utmost importance to change our misconceptions that math is a subject that is about calculating numbers fast, following rules and procedures learned and memorized through traditional methods of teaching.
Let me turn your attention to Mike Flynn’s viewpoint. I was first introduced to his work through a free webinar a colleague had sent to me back in the Spring. After watching this webinar I ordered his book, Beyond Answers.
In the webinar and in his book as well, he speaks to the fact that teachers of primary students need to help young children ‘contextualize’ problems. What this means is that if we are to write a mathematical equation, eg. 3 + 2 = 5, students need to be able to make connections to these symbols through storytelling. A story for this equation, might be, there were 2 birds on a branch. 3 more came to join them and together there are 5 birds on the branch.
Teachers of primary students need to help young children ‘contextualize’ problems
When I work with my grade three students I always have them tell me stories for all the operations, addition, subtraction, multiplication and division. I will share a story with you about my Gr. 4 colleague who shared the webinar with me. After watching the webinar she went back to her classroom and gave a basic fact multiplication equation to her students and asked them to give her a story.
She was in disbelief that many could not do this, yet they could provide the correct answer. This, undoubtedly, demonstrated to my colleague. that a deep understanding of the process of multiplication was not understood. If students are challenged with understanding multiplication, then division will also be a challenge, as well as fractions, factors, divisibility.
If students are challenged with understanding multiplication, then division will also be a challenge, as well as fractions, factors, divisibility.
If you are one who is doing timed basic facts drill and practice and/or teaching the steps of the traditional algorithms, do your students have a deep understanding of the operations or are they simply ‘fast’ at calculating and/or have memorized the facts and/or the steps of the algorithms?
I will also share another story with you that I recently experienced in a Grade 5/6 classroom. I asked the students how they knew 15 was divisible by 5. Many were unable to explain this to me. I kept pressing for understanding through guiding questions and finally one student responded with 15 divided by 5 is 3, three 5’s make 15.
I then asked them if 5 and 3 are factors of 15 and if there were any other factors. One of the students told me 7 and 8 so I asked the class, if I times/multiply 7 and 8 will that leave me with a product of 15?
As I looked at the students I could tell there was much confusion. So I asked what are factors and what happens if we multiply two factors, such as 3 and 5? Finally a student was able to tell me that 1 and 15 were also factors of 15.
Perhaps you are not seeing my connection to Mike Flynn’s view with ‘contextualization’. These grade 5/6 students were not contextualizing the equation of 15 divided by 3 = 5.
With all this to contemplate, I ask you to reflect on your assessment of your students’ mathematical ability with the four questions I put forth at the beginning of this blog. I would love to hear your thoughts, comments and questions with regards to this.
Let’s engage in a conversation! This, I believe will help you with your own mindset around the teaching and learning of mathematics. And if teachers have a growth mindset and understand the need to change the way mathematics is taught, we can most definitely help ALL children make connections, think logically, interpret data, use space and work flexibly and creatively with number.