Here are six lessons designed by Heather Taylor which incorporate Critical Thinking principles into math instruction:
Purpose:
1) To encourage geometry students to think of "prompts" or "tools" to help them when they get stuck on a proof.
2) To make proofs seem manageable instead of overwhelming.
When to use:
I used this after we had worked with proofs for awhile&endash;after about the second test on proofs. I did it after passing back the test, so they could use it for their test corrections. It took about 15 minutes, more if I gave them time to do the extra credit part in class.
Procedure:
1) Individually, I had them brainstorm and write down tactics they could do when they got to a point in a proof where they said, "I don’t know." I gave them a couple of examples, like "mark picture" or "look back at given." (3-5 min.)
2) I had them share their tactics with their homework group (3-4 people.) They could add tactics that another person thought of if they thought it also might be helpful for them. I had some volunteers share some with the whole class. (3-5 min.)
3) When everyone had a list, I passed out an index card and had them write "I don’t know Card" on top. Then they listed the tactics they had come up with. I encouraged them to try to use this card as they did their test corrections. (3-5 min.)
4) As an extra credit opportunity, I told them that they could write down their steps from their card on a piece of paper, and then correct a proof from the test that they had missed, and show how certain steps from their card helped them figure out the proof. (See attached sheet.)
Evaluation:
I think this was most helpful for the students who wrote specific tactics based on their individual mistakes. For example, if they always forgot to use a certain theorem, one of their tactics could be to look for opportunities to use it. For the students who just wrote down my suggestions, it probably didn’t help much. Although, maybe by thinking of these tactics as tools they could use, they felt a little more empowered. Had I called on them to use this card more often, it might have been more effective.
This was an effective critical thinking activity because it encouraged students to evaluate their mistakes and then figure out ways to try to avoid them in the future. This is a key component of critical thinking&endash;thinking about your thinking, and in this case, then trying to find ways to improve it.
Purpose:
1) To help students realize that there is often more than one way to complete a proof.
2) To give students more practice with proofs.
When to use:
I did this lesson twice, once for a book problem, and once for a commonly missed test problem. You can incorporate it into any part of the class&endash;start, middle or end. It takes about 15-20 minutes, depending on how long you want them to think about it.
Procedure:
1) Turn into homework groups. (I had them arranged in groups of 3-4 students of varying ability, with whom they regularly discussed homework questions, so they were used to working together as a group.)
2) The first time, I told them I wanted them to prove p. 121 #1 in three different ways. The second time, I told them I wanted them to prove #24-28 from their test in at least three ways. They needed to write their proofs on one page that they would turn in to me. You can also tell them it is for extra credit homework points&endash;this sometimes really motivates the groups to work together.
3) Then I wandered around, observed, and looked for correct proofs. Some groups were working together, while others were writing individual proofs and then comparing. I did not give any help or hints, but encouraged them to discuss ideas with each other.
4) I chose three different groups to write up a particular (correct) proof on the board. As a class, we discussed variations and similarities on the three proofs, and talked about extra steps that could have been added or omitted.
5) Then they had to write on their paper what they thought or felt as they did this activity. Was it hard? Fun? Frustrating? Easy? Boring? Why? See attached sheets for student comments.
Evaluation:
I think it was good that I did this lesson twice. Several students commented that it was easier the second time around, for different reasons. By the second time, they were more used to challenging themselves to approach the problem in different ways.
I was impressed at the conclusions they were drawing. For example, one person wrote, "Not only does it help us understand the problem better, but it will help us later in other proofs, when we’re stuck and think there’s only 1 way to solve it." This was exactly the point of the activity! So often in math, students are frustrated because since there is only one right answer, they think there is only one way to get there. This is not always the case, and proofs are a prime example of how there are several paths to the correct conclusion. (Interestingly though, this frustrates other students who like that there is only one answer, and become frustrated when there are several ways to solve the problem. They want to know which one is correct, when in reality, they all are.)
Another student recognized value in this process that applies to more than just this activity. She wrote, "We needed to have flexible minds to find different paths to the same answer." This is one of the main aspects of critical thinking&endash;being able to think from different viewpoints.
This was a successful critical thinking activity, because it encouraged students to think about different ways to solve a problem. It was challenging because as one student wrote, "It was difficult to totally redo the problem in a different way, because I still had the old answer in my head." The more we challenge them to think of alternate answers, the more we allow them to think critically.
Purpose:
1) Have geometry students derive the geometric mean so that they have a better understanding of where it comes from.
2) Relate geometric mean to ratios of similar triangles, so they do not think it is some isolated theorem.
When to use:
I used this to teach geometric mean, after similar triangles. It took pretty much the whole class period (45-50 min.)
Procedure:
1) I gave them the definition of geometric mean and wrote it on the board. (3-5 min.)
2) Then I drew and labeled a right triangle with an altitude to the hypotenuse. Individually I had them list three similar triangles, in the correct order. I wrote the theorem on the board that says three similar triangles are created in a right triangle with an altitude to the hypotenuse. (5 min.)
3) In their homework groups (groups of 3-4), I had them compare their triangles to make sure the vertices were in the correct order. I had someone tell me which three triangles they said, and wrote them on the board. As a class, we made sure they were in the correct order and went over how we got that. (3-5 min.)
4) Then I had them individually list three sets of ratios, comparing the first and second triangles, the second and third triangles, and the first and third triangles. When they were finished, they compared them with their group. (10 min.)
5) I had someone read me their ratios for the first and second triangles. After making sure everyone had this set, I had them ignore one of the ratios. I had them compare the resulting proportion and tell me how it related to the definition of geometric mean. I had them put it into words, and then we looked at our picture and substituted in the parts of the triangle. Then I wrote the theorem that they just proved on the board. (10-15 min.)
6) We repeated step 5 for the second and third triangles, and the first and third triangles. (10 min.)
7) I had the students evaluate how they liked this lesson as compared to normal note taking. Was it harder? Easier? More interesting? Boring? Frustrating?
Evaluation
This was an effective critical thinking lesson because the students were actively engaged in the material. They were figuring out the relationships and connecting the concepts instead of me telling them the connections. By connecting it to previous material, even though it took longer, I think they learned it better and hopefully will remember it better.
In addition, because I related it to similar triangles, they had two ways to solve this type of problem&endash;either by using similar triangles, or by using the geometric mean theorems. The more ways they see to solve a problem, the more chances there are of them getting it correct. And since students see and understand the material in different ways, by showing them several ways to do a problem, there is more likelihood that more students will grasp the concept.
A lesson like this does take a lot of time, and cannot be done for every topic. For example, this lesson depended on the fact that they had done similar triangles, so could list the ratios. By varying classroom activities and lesson formats, students will (hopefully) stay more engaged in their learning.
Inverses, Converses and Contrapositives
Purpose:
1) To help geometry students form inverses, converses and contrapositives.
2) To help geometry students recognize and verbalize the differences and similarities between the three statements.
3) To encourage students to remember how to differentiate between the three different forms.
4) To allow students to come up with their own examples and justifications for why they are true or false, thus enabling them to interact more with the material, and hopefully internalize it better.
When to use:
Use to introduce converses, inverses and contrapositives to students. Instead of me telling them what they are and how to form them, I let them figure out what they are and how to form them in groups. It took about the whole class period (45-50 minutes.)
Procedure:
1) I had students read individually the two pages that covered the topic in the book. (5 min.)
2) On a piece of paper, they were to write either, what they understood about what they read, or what questions they had about what they read. (5 min.)
3) In their homework group (groups of 3-4 students), they were to answer each other’s questions and come up with a way to remember how to form each sentence. (10 min.)
4) As a group, they were to come up with a statement, and then write its converse, inverse and contrapositive. Then they were supposed to decide which statements were true, which were false and why. (10 min.)
5) I had a member from each group write their four sentences on the board, and then as a class, we compared them with each other. We looked for similarities and differences dealing with which ones were true and false. Then we discussed how that related to "logically equivalent," which was a phrase used in their book. I also had people share how their group had figured out how to remember how to form each sentence. (15 min.)
6) I then asked them to write an evaluation about how they felt about this lesson compared to a normal lecture. Was it easier? Harder? More or less confusing? Interesting or boring? (See attached sheets for student evaluations.)
Evaluation:
For the most part, the students seemed to like this lesson. They were actively engaged and although there was some confusion and chaos, by straightening that out, they learned the topics better. For example, one student wrote, "It is better (than a lecture) because we have to think harder and so if we reach the conclusion we won’t forget it." Another student wrote, "This process helped me understand the chapter deeper, but we had to work harder in order to understand it. We did more thinking instead of just copying notes." Asking students to think is harder and takes more time than normal lectures. Critical thinking is not easy and does not just happen. We have to work at it and provide opportunities for our students to practice it.
A couple of students commented on the fact that it took more time than a lecture. "I like this way better because I had time to discuss things with other people and discover new ways to think about things." Another student wrote, "This project was better than normal note-taking because we taught ourselves the lesson. But it took a lot longer, so a lecture is probably better." There were a couple of other students who also said they liked lectures better. This highlights the point that students learn in different ways. The more we can vary our classroom activities, the more likely we are to reach students with varied ways of learning.
I think this lesson was a successful critical thinking lesson because it gave the students the chance to immerse themselves in the material. They had to think about their statements and why they were true or false and then justify it to others. They challenged each other to think about their statements, which hopefully in turn made the concept more memorable and concrete.
Purpose:
1) To encourage Algebra II / Trigonometry students to figure out which method is easiest to solve which type of quadratic equation.
2) To give them practice solving quadratic equations in three different ways.
When to use:
Use after reviewing factoring, completing the square and the quadratic formula. It takes about 10-15 minutes. Because they have learned these three methods in Algebra I, you can spend more time on when to use them, rather than on how to use them.
Procedure:
1) I wrote three quadratic equations on the board and told them to solve each of them by factoring, completing the square and the quadratic formula. I had chosen one equation that factored easily, one that lent itself well to completing the square (and the numbers were big, so the quadratic formula was more difficult) and the third that fit easily into the quadratic formula (but was more complicated by completing the square because you ended up with fractions.) All three of them were factorable, but the second and third ones were not obviously, easily factorable. As they did the problems, I wandered around and answered questions if they were stuck on a certain method.
2) After they finished, I had them answer the following three questions on a piece of paper to turn in. #1: Which was the easiest way to solve each problem and why? #2: Which was the hardest way to solve each problem and why? #3: When is it best to use each method and why?
3) At the start of the next class, I shared some of the conclusions that their classmates had reached about when to use each method and summarized them on the board. (See attached sheets for student observations.)
Evaluation:
For the students who took the time to think about the pros and cons of the different methods, they were able to figure out when it is preferable to use each method, or when it is preferable to not use a certain method. This is a key to effective critical thinking&endash;it requires time. For the students who rushed through it and answered factoring for each one, they did not evaluate their processes carefully. There was one student who wrote and said that he would never use the quadratic formula, because it was too hard and complicated, although for one problem, that was the only method by which he got the correct answer!
I think this was an effective activity for the students that took it seriously. This is an obstacle we face in every class&endash;some students put more thought and effort into activities than other students, and therefore, get more meaning out of them.
Because I did review with them the following day when it was best to use each method, they all received the same information. But the students who discovered it on their own, probably (hopefully) internalized it more effectively than the others.
Purpose:
1) To help students learn from their mistakes.
When to use:
I have students do test corrections after every test, although sometimes they are "optional extra credit" homework points, rather than required.
Procedure:
1) The student must first redo the entire problem correctly. This helps them practice the concept correctly.
2) Then they must write what they now understand about the problem. I don’t want them focusing on what they did wrong, but rather on what they learned.
3) At the end, they need to answer four "overall questions." #1: What do you feel you know well? #2: What do you feel you still need to work on? #3: Do you have any questions? #4: Do you have any comments? (In the attached test correction, there are not overall questions because I only started doing these overall questions second semester, and the test corrections I xeroxed were from first semester.)
Evaluation:
I think this helps with their critical thinking skills because it asks them to look at a problem and derive a lesson from it. For the students who spend time and do a good job on their test corrections, I think they are reinforcing concepts and learning them better. They are also thinking about the correct way to approach that type of problem.
I have contemplated adding in a section where they say what they did wrong. This would allow them to think and reflect on their past thinking; an integral part of critical thinking. The reason I have not included that part is because I really don’t want them to focus on what they did wrong because I don’t want that part to be the part that sticks in their head. Some students do this intuitively on their own, as they say what they now understand. For example, one student wrote, "I wrote that they would be comp. instead of congruent. The reason they are congruent is if you have an angle that is 40 and 2 angles that are comp. to it, they would be 50 but 50 isn’t comp to 50, it’s only congruent." This is a perfect example of how she is showing that she understands the concept and has learned from her mistake.
Purpose:
1) To help students focus on the important topics from the semester.
2) To make students think about the important parts of a problem and what information they need to include/provide so that the problem can be solved.
3) To allow students to generate review problems based on a variety of sources&endash;book, notes, and old tests
When to use:
I used this on a review day before the Alg. II / Trig. first semester exam. It can also be used before a quarter test, or even before a cycle test. It requires one and a half classes however&endash;they generate questions on one day (about 20 minutes), and then you compile them and pass them out on the next day.
Procedure:
1) I wrote up on the board five main categories of topics from the semester and the sub-topics contained within them. (This requires previous planning - I made five main categories because I had five homework groups.) I gave them a couple of minutes to copy it all down.
2) I then assigned each group a category and told them they had 20 minutes to come up with 2-3 questions about each sub-topic. They also had to solve each question, and include the answer on the sheet they were going to turn in to me. I wandered around and observed as they did this. Most groups chose to divide up the sub-topics and each member was responsible for generating their own questions.
3) I collected each groups’ questions, typed them up and passed them out with the answers the next day as a review sheet.
Evaluation:
I thought this was an effective critical thinking activity, because they had to think about what the problems needed to include, and they had to make them so the numbers worked out and were realistic. This required a different level of thinking. Instead of just solving problems, they first had to set them up so that they would work out. They needed to make sure they had all of the pertinent pieces and put them together in the correct order so that the problem made sense. This required a concrete understanding of how the topics are organized and put together, rather than just knowing how to solve a particular type of problem. Some students worked backwards from the answer to develop the problem. For example, a girl that was working on word problems kept asking if she was doing it right and if it made sense. I just kept asking questions back to her ("What do you think you are doing wrong? Why? What do you think you are missing?"), until she figured out what information she wanted to provide and what she was asking for in her problem.
The result was a creative, comprehensive review worksheet. (See attached worksheet.) Only one of the problems ended up not working out. And that just provided us with something to talk about on the next day when they were doing the worksheet. We talked about why it didn’t work, and what would need to change in order to make it work out.
I think this gave them a new appreciation of what needs to go into a problem. Also, they will now be more accustomed to looking for the information they need (in word problems especially) and ignoring the extraneous information.
It maybe would have been a better activity if I had let them come up with the five main categories and sub-topics. This would have taken more time, but would have allowed them to evaluate what the important topics were that we covered and decide how they should be grouped together.