In my gifted classroom I do not use math textbooks. I know some parents and even some students find this strange and confusing, but in my experience I have not found that textbooks are engaging to most students, nor do I find the "average" math textbook to be appealing to gifted students And, I must confess, I hate to teach from the standard teacher guide. I much prefer to tailor my teaching to meet the needs of the students. I do some diagnostic assessments at the beginning of the year. I also do "Daily Math" each day, which consists of 5 standard "textbook" type questions. I use these to help me determine what my student needs are. If they already know a concept there is no point in spending hours "practicing" by answering questions in a textbook.
|Day One...students created a cover|
|"You're a Winner!" Activity from the Guide to Effective Instruction|
Some of my "go to" resources are the Guides to Effective Instruction in Mathematics, Grades 4 to 6 put out my the Ministry of Education for Ontario. I have used all of the guides at one point or another over the past few years. I particularly like to use the 3 part lessons from the guide to Patterning and Algebra . The use of interactive notebooks to teach using these guides seemed like a natural "fit" to me. I don't like giving out "worksheets" to students to keep in their three ring binders if I can avoid it. I prefer for their work to be kept in folders or duo tangs, and this idea of an interactive notebook seemed even better!
Above you can see an example of student notebook. On the first day I had the students create a cover, paste the rules for the notebook inside the front cover and start a table of contents. All pretty much the stand things I have seen for creating an interactive notebook. The next day we started using the lesson from the Guide to Effective Instruction in Mathematics. We talked about Blaise Pascal and the idea of "Triangular Numbers". I have read the book "Here's Looking at Euclid" by Alex Bellos and I thought my students would intrigued if I read them the section on Pascal and the "discovery" of Pascal's Triangle.
After our class discussion, I put the following on words on the blackboard: SEQUENTIAL, SPATIAL, TEMPORAL and LINGUISTIC. I had students add these words to the "Patterning and Algebra" section of their "Math Survival Guide". I allowed the students to use record what they thought the meanings of the words were, and to try and think of examples of each they could add to their definitions. After that,
I handed out a smaller, photocopied version of the anticipation guide and paste it into their notebooks. They then complete the “Before” side of the Anticipation Guide as a way of activating prior knowledge and identifying common misconceptions. I then had them brainstorm applications of patterns in the real world. Patterns may be categorized as sequential (playing cards), spatial (buildings), temporal (calendars), and
linguistic (spelling rules). I gave each student 4 different coloured post-it notes and asked them to think of an example of each and write it on the post-it. I then had students read out their examples and then come up to the board and place the post-it under the word had posted earlier as part of the vocabulary lesson. Students then added more examples to their Math Survival Guides.
|An Anticipation Guide was completed before we started.|
|Students used the rules from the "About the Math" section to complete the Pascal's Triangle Handout|
|Pascal's Triangle Page from Effective Guide to Instruction|
|Overview of activity for teacher|
|List of Curriculum Expectations and Materials list|
|About the Math Section from the You Be the Winner activity for Patterning and Algebra Grade 6 |
|About the math section...students put this on the RIGHT side of the page.|
|I gave each student a copy of the problem to glue into their math notebook|
As the students read through the problem we looked at each section of the KWC chart and filled it out BEFORE we even start to try to SOLVE the problem. This procedure is NOT in the Guide to Effective Instruction, nor is the vocabulary idea that used for the Math Survival Guide. These ideas are my own adaptations of the lessons in the guide. I know the majority of the students in my class would be able to to give you the correct answer to the problem from the guide, but they would not be able to REPRESENT their learning and fully explain how they know what they do. This activity is intended to model the type of thinking and communication skills expected from them in the mathematics program.
I also deviated a bit from the guide instructions when asking the students to work on the problem. Having done this problem for several years, I have observed that rarely do students demonstrate the range of solutions that the guide seems to suggest they should or will. This year, I decided to put the students into groups of four and "assign" them a specific strategy to work on with their group. This also put the onus on communication and representing skills rather than on solution finding. I gave each strategy listed in the guide to two groups of students. I chose the groups based on strengths and needs I had observed over multiple different mathematics and problem solving tasks completed so far during the year.
|I had the students create a KWC chart in the section above the problem. Here they write what they KNOW from the problem, WHAT they have to DO, FIND OUT, or FIGURE OUT, and if there are any special CONDITIONS|
|Description of the 3 Part Lesson|
|What the student page looks like with both the KWC chart above and the problem below.|
Create a table of values. The problem can be clarified for some students if they create a table
of values such as the one below.
2 chipmunk, blue jay, blue jay,
3 chipmunk, blue jay, blue jay, puppy, puppy, puppy
Model using concrete materials. Students may choose to use beads, tiles, cubes, and so on, to build a day-by day model on their desks. They might also use the letters A through J to represent the 10 types of toys. Thus, the toys received on the fifth day would be ABBCCCDDDDEEEEE.
Use numerical representation. Some students might develop an algorithm, such as:
The first 10 triangular numbers have 10 ones, 9 twos, 8 threes, and so on. Therefore, the sum
of the first 10 triangular numbers is equal to 10 X 1 + 9 X 2 + 8 X 3 + 7 X 4 + … + 3 X 8+
2 X 9 + 1 X 10,
Draw a diagram. Some students may draw symbols to represent their solution. Other students
may use graph paper to draw their representation. Encourage groups to clearly explain their
thought processes on chart paper. They should use numbers, pictures, and words.
|Looking Back section|
The students worked in their groups to not only "solve" the problem but to also accurately represent what they had determined to be the correct answer. I had them use another page in their interactive math notebook and create an APEC graphic organizer (Answer, Proof, Explain and Connect). As they solved the problem cooperatively, they had to individually record their work in the graphic organizer. Once they had reached consensus they recorded a group representation on a large piece of chart paper.
In the next class, we completed the"LOOKING BACK" section of the lesson.
After students had solved the problem and decided what choice the winner should make, I gathered the students together for a whole class discussion. Each group of students shared their findings with one other group before presenting to the whole class. Each of the different representations or strategies were asked to explain their chart and their thinking.
|Reflecting and Connecting section|
During students’ presentations, I encouraged students to consider the range of strategies and to try and make sense of each representation.
I asked students to reflect upon the following:
• How easy is your strategy to explain?
• What other strategies did you try?
• If you were to increase the number of days, would your strategy still work?
• What would you do differently if you were to solve a similar problem?
After the groups have presented their work,I illustrated the pattern by saying: “I’ve noticed
that many of your solutions are similar. Many of you recorded the number of toys per day and added up the results. Let’s look at the pattern on the board.”
Write the following numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55. Ask the students: “Which choice would give the winner the most toys?” (Solution: The winner received 220 toys in the 10 days.) I then illustrated the pattern by drawing dots on chart paper in the shape of triangles for the first 5 numbers in the sequence. I asked the students: “What do the shapes look like?” The students recognized that these numbers are called triangular numbers and recalled that we had discussed that they were studied by mathematicians in ancient Greece and in China long before Blaise Pascal was given credit for "Pascal's Triangle". We then brainstormed the uses of triangular numbers in real-life applications. Examples included the set-up for 10-pin bowling, cheerleaders’ pyramids, and displays of cans and other materials in supermarkets.
At the end of the lesson I asked the students to complete the second part of the anticipation guide and think about how their thinking about patterns and numbers had changed.
When they were finished completing the second part of the anticipation guide independently I asked the class to discuss the reasons for their choices on the “after” section of
the anticipation guide, and to come to a class consensus. I was able to take note of incorrect answers from individual students or large groups and I will use this information to determine student understanding and to identify areas of need.
I found using interactive math notebooks to be a great way to structure the delivery of the lessons and to have students record their thinking. I feel I have deviated somewhat from what I have seen about "Right" and "Left" hand use of the pages in an interactive notebook, but I think that I am adapting it to work for the way I teach and the way my students learn, and that is what the gift of teaching is all about!