Math journals, or problem solving notebooks as they are sometimes referred to, are books in which students record their math work and thinking. They can be used to:

- record the solutions to math problems, along with the strategy and thought processes used to arrive at the solution

- write about learning: At times students may be asked to reflect on their math learning. For example, students may be asked to write about "what you already know about ......" at the beginning of a unit or "what you did today, what your learned, and any questions you have", or "the three most important things you learned in this unit."

**By dating entries the journal provides a chronological record of the development of a student’s mathematical thinking throughout the year.**

While students learn how to "do" math, they must also learn how to articulate what they are learning. It is important to provide many opportunities for students to organize and record their work without the structure of a worksheet. Math journals support students' learning because, in order to get their ideas on paper, children must organize, clarify, and reflect on their thinking. Initially many students will need support and encouragement in order to communicate their ideas and thinking clearly on paper but, as with any skill, the more they practice the easier it will become.

Journals also serve as invaluable
assessment resources that can inform classroom instruction. Reviewing a
student’s math journal provides a useful insight into what a child
understands, how s/he approaches ideas and what misconceptions s/he has.

A good question ….

- builds in differentiation by allowing for multiple entry points and recording techniques, thereby allowing all students to work at their individual level of thinking,
- provides the opportunity for students to learn by answering the question, and the teacher to learn about each student from the attempt,
- may have more than one solution or a variety of possible solution paths that range from simple to complex,
- requires more than just remembering a fact or reproducing a skill,
- provides opportunities for students to represent their mathematical ideas using models and written language,
- provides opportunities for students to justify their reasoning and evaluate the reasoning of others,
- has clear, concise directions,
- provides opportunities for group work and discussion.

The most important thing to consider when developing a journal question is whether the question involves significant mathematics. Closed questions such as "Ben had 5 apples. He ate 2. How many apples did Ben have left?", often seen in early years classrooms, do little to develop a child's mathematical thinking if the child can answer the question before even getting back to his/her seat. The child may spend 15 minutes drawing and coloring apples but mathematical thinking is limited. Changing the question from a closed to an open format such as, 'Ben had 5 apples. He ate some of them. How many did he eat? How many did he have left?' creates greater potential to stimulate mathematical thinking and reasoning when a child is asked to show as many different solutions to the problem as s/he can.

Definitely!
Good tasks are open ended to allow for different strategies and products
to emerge. Many tasks have multiple solutions and students should be
encouraged to choose their own method of solving problems and
representing their findings. Repeating, or revisiting tasks, allows
students to engage with tasks at a deeper level. On the first occasion
the student may be focused on ‘how to do’ the task. Subsequent visits provide an opportunity for students to communicate their mathematical thinking and reasoning more clearly.

The methods
that children use for representing their thinking will also change over the course of a
year. Repeating a task provides a record of this growth for teachers,
parents and students. For example, in Kindergarten an open ended
addition task (see work samples below) may be explored early in the year
before children begin to write number sentences. Early in the year most
kindergarten students will record their thinking in relation to this
problem pictorially and may only record one or two solutions to the
problem. As the year progresses symbolic representations will gradually begin
to appear and representations will become more detailed. Making slight variations to a task (e.g. changing the numbers, context, or materials used) will help to maintain interest while students further
develop skills and concepts. Some teachers like to introduce tasks whole class and then place tasks in centers for children
to revisit at other times throughout the year. Other teachers choose one
journal task and repeat it, with slight variations, several times
throughout the year as a record of the development of math skills and understandings for student portfolios.

The work sample above on the left shows a Kindergarten students' attempt to record her thinking early in the school year in response to the task: Vanessa had 5 cupcakes. Some were chocolate. Some were vanilla. How many were chocolate? How many were vanilla? Three months later this student completed a similar task: Cameron had 6 buttons. Some were green. Some were purple. How many were green? How many were purple? On this occasion the child's written representation (above right) is more detailed and clearly demonstrates her developing understanding of addition. Although she repeats some number sentences, her drawings show all possible combinations of the six buttons.

Some teachers use several tasks a week as a warm up to the math lesson.
Other teachers set aside one period per week for journals, select a task that correlates with the current unit of study and allow more time for students to share their thinking with one another. Tasks may also be used for assessment purposes, or as homework. The important thing is to ensure that students are being given regular
opportunities throughout the year to represent their mathematical thinking
in ways which makes sense to them.

Our experiences in numerous K-5 classrooms have shown that a notebook with blank pages produces the best results. Although these are not always as readily available as ruled notebooks (and are often more expensive) they have a distinct advantage in that students are not restricted by lines and have the space to choose whether to use pictures, numbers, words or a combination of these to record their thinking. Click on the links below to visit our gallery pages to see examples of the types of written responses made by Kindergarten - 5th Grade students when encouraged to make their own decisions about how to record their thinking.