So the good news is that I successfully defended my dissertation and am now a PhD; the bad news is I haven’t blogged since February! This seems to be the recurring theme for the last year; writing a blog post and then becoming busy again with work and completing my degree requirements. Hopefully, now that I have completed my PhD, I will be able to blog once a week.
This week I’d like to talk about a topic that has been a focus in my role as an Instructional Coach this past year – that of mathematical thinking, reasoning and proving. I supported teachers in my schools in these areas and it has been great to see students develop and demonstrate their mathematical thinking, aligned with learning goals and success criteria.
When students reason and prove in mathematics, they are explaining their thinking and providing the evidence in a systematic manner using a variety of representations.
Many times, it is difficult to understand what a student is showing in his/her work. Before I provide some suggestions, we need to understand what reasoning actually is; what is reasoning?
The Ontario curriculum states:
“The reasoning process supports a deeper understanding of mathematics by enabling students to make sense of the mathematics they are learning. The process involves exploring phenomena, developing ideas, making mathematical conjectures, and justifying results. Teachers draw on students’ natural ability to reason to help them learn to reason mathematically. Initially, students may rely on the viewpoints of others to justify a choice or an approach. Students should be encouraged to reason from the evidence they find in their explorations and investigations or from what they already know to be true, and to recognize the characteristics of an acceptable argument in the mathematics classroom. Teachers help students revisit conjectures that they have found to be true in one context to see if they are always true. For example, when teaching students in the junior grades about decimals, teachers may guide students to revisit the conjecture that multiplication always makes things bigger.” (Ontario Ministry of Education, p. 14).
What key words stand out in this definition? I’ve bolded and underlined what I feel are the important terms and statements when it comes to reasoning. Students need to be able to make conjectures based on their prior knowledge and then provide proof/disproof of that conjecture depending on the question being posed. They need multiple opportunities to practice demonstrating their thinking in a variety of ways and developing their ideas in group settings and individually.
So, when do we reason? I know I rely on my reasoning skills when I am presented with a new problem or when logical thinking is required. For example, I was presented with this problem recently:
You are in a cabin and it is pitch black. You have one match on you. Which do you light first; the newspaper, the lamp, the candle or the fire?
I used logical reasoning and decided to light any of the items, you had to light the match first!
Now this might not be a math problem but you get the idea; here’s one for you to try:
So, what number is the car park on? How do you know? (Feel free to post your answer in the Comment Section).
We also have to look at the context when we are reasoning; think of the “bus problem” that I am sure several of you might be familiar with. I don’t remember the numbers that were used in the original problem but it goes something like this:
“The Grade 6 classes were going on a school trip, there were 78 students altogether and one bus can hold 22 students. How many buses needed to be ordered?”
Many students perform the standard algorithm and divide 78 by 22 to get 3.5 and their answer is 3.5 buses without using reasoning to understand that you can’t have half a bus, so we really need 4 buses. In this case, the context helps us to reason through our answer and explain our thinking.
When students are given a problem, whether it is open- routed or open-ended, they have to deconstruct and understand the problem in order to determine which strategy to use; this requires reasoning since they need to use the information given in order to select a strategy to answer the question. Sometimes, there is information missing from the problem and students need to realize what information is missing and how to fill those gaps in order to solve the problem.
Last but not least, students will use their reasoning skills when dealing with open-ended questions. These types of questions have multiple answers and multiple pathways to get to the answer. They will need to use their reasoning skills to determine the best path to get to an answer.
As I said previously, sometimes when we are looking at student work, it is hard to decipher their thinking, evidence, and reasoning. This is where conversations and observations are powerful. As educators, we recognize the importance of triangulation of data, so when we don’t understand what a student has written, follow it up by a conversation with that student with some probing questions or better yet, observe students while they are working and ask those clarifying questions in order to better understand their thinking process.
Here are some probing questions you can ask:
- How did you do this step?
- How did you know to…..?
- Tell me more about…….?
- What happened when you……?
- Why did you…..?
- How is this like….?
- How is this different from….?
- Can you find another way to solve the problem?
- Does your solution make sense? How do you know?
- Can you explain…?
Finally, in order to track and support students’ mathematical thinking and make it more visible, here are some suggestions:
- Use Markers. When students use a pencil, they erase their mistakes and start over. This deprives us of seeing their thinking and how it evolved as they solved the problem. I have my students use markers and if they make a mistake, they place a line through it and continue underneath. This allows me to clearly see their thinking and have richer conversations with them, when looking at their work. This also supports growth mindset because students begin to see mistakes as valuable.
- Give them the answer. Math can be stressful for many kids. This is why I would regularly (not every day) provide the answer to the question (e.g., the answer is “47”) and tell them to “prove it.” This allowed students to explore and investigate more freely and without a lot of stress. It has worked really well.
- Modify the Question. Sometimes, I would modify the question for students in a “yes” or “no” format, (e.g., “Shelly says the answer is 2.5. Is Shelly right?” OR “Shelly says the answer is 2.5 but Jason says the answer is 3.5. Who is right and why?”). This also allows students to explore, investigate, and develop ideas but in a different way, especially with the latter example because students have to prove why either Shelly or Jason is wrong.
- Convince a friend. Can they explain their reasoning to a friend? Can they answer their friend’s questions effectively?
- Convince a skeptic. Can they explain their reasoning and thinking to someone who is not quite sure? Can they answer his/her questions effectively? Do they manage to convince the skeptic their line of thinking is correct?
- Trade with a classmate. I’d often have students exchange their solutions with a partner and see if the other person can follow their line of thinking. Can Student A understand what Student B has done on paper? If not, what can/does Student B need to do in order to make their thinking more visible and explicit? The same goes for Student A. This is another great way for students to communicate as well as seeing the different strategies used to solve the same problem.
- Group by strategy. Students can also be grouped by strategy and work together to show their thinking. After presenting a problem to students, let them get started individually; walk around and make a note of which strategy each student is using and then stop the class after a few minutes and pair them up (or groups of three; I don’t recommend groups of 4 since one or two students will end up taking a backseat), have these small groups continue working on the problem using their shared strategy and then share solutions as a class.
- Take the paper away. There are many ways to solve a problem on paper, yet we often don’t emphasize other ways to solve the problem, without the use of paper. My suggestion is, take the paper away and challenge them to show their thinking in another way; what about acting it out? Going outside? Using manipulatives? Can you think of other ways students can demonstrate their thinking without the use of paper?
- Marker for each student. In my first suggestion, I talked about having students using a marker (or pen/pencil crayon) to do their work; if students are working in small groups, give each student a different coloured marker so you can track individual thinking (a placemat can work well here).
I hope you find some of these suggestions helpful and effective!
I also recommend ThinkFun games for students to practice their reasoning, logical, and spatial thinking skills. These are fun and engaging games to support mathematics in any classroom. My 5 year old nephew is hooked on Rush Hour!
Here are a few titles:
- Rush Hour
- Gravity Maze
- Shape by Shape
- Block by Block
- Circuit Game
- Laser Maze
- Robot Turtles
- Brick by Brick
Visit their website at: http://www.thinkfun.com/
Next week, the plan is to continue talking about mathematical thinking and reasoning from the lens of assessment; in particular learning goals and success criteria. It is important that we assess what we are supposed to be assessing and this is where the difference between success criteria and task requirements come into play.
Until next week!