A balanced mathematics program should include the following three parts:

1. Conceptual understanding
2. Computational fluency
3. Problem-solving skills.

The combination of understanding the math taking place, being able to use math rules accurately and efficiently and the use of ingenuity, creativity and flexible thinking to solve mathematical problems at hand, are “mutually reinforcing.” According to Van De Walle, author of Elementary and Middle School Mathematics, introducing mathematically intriguing problems for students to solve as the main mathematical task of your lesson will inherently address conceptual understanding and computational fluency in the process. The carefully chosen problem you introduce to your class should build on previous concepts. Maybe you are teaching a procedure  or maybe you are trying to give students subtraction practice or maybe you are trying to stimulate their spatial reasoning. Whatever skill you are targeting can be addressed by the mathematical task you choose.

The class should be divided into 3 equal slots of time, say 20 minutes each for the before, during and after portions. There should be one problem for the whole class that focuses on a mathematical dilemma, with multiple entry points to the problem. This will help to differentiate for various learning types. Another way to differentiate is to have a challenging problem that builds on the first one for those students who finish early. They should also have an ongoing, long-term math project to work on.

Before the lesson, you must remember to:

1. Activate any prior knowledge the students may have. This will engage those parts of the child’s brain so they are prepared to make new connections
2. You must make sure there are a variety of tools and manipulatives available to all students
3. Make sure the students understand what is expected of them during work and at the end (in this case, a written explanation of work) and finally,
4. Make sure they understand what the problem is asking.

Once they get to work independently or with partners, begin the “during” part of the lesson. The teacher’s job here is not to help or teach, but to listen, engage and assess. Let the student struggle, make mistakes and explain their thinking without your interference. The best discussions come from disagreement, as does the best learning in my opinion and the best satisfaction comes from figuring out a problem on your own without help. During the lesson, you may give hints such as suggesting a strategy to try  but it is most important to first understand what the student has tried already and what their thinking is. Students should be expected to write their 3-part explanation of:
A. What they did to solve the problem
B. Why they did it that way and
C. How they know they have the right answer. This is to prepare for the class discussion in the after phase.

Even if students haven’t completed the problem, when the time is up, don’t cinch on the after discussion. Allow it the full 20 minutes. Most importantly, encourage student to student discussion, have one student answer each others questions or respond with questions of their own to a fellow student. Use equality. Ask every student how they got their answer, why they did it that way and how they know they have the correct answer, whether the student obtained the correct results or not. Asking how a student got their answer should not be a trigger to students to infer an incorrect response. The more varied the results, the more exciting the discussion and often, disagreement. Additionally, you are helping build a mathematical community of learners who take risks and respect each other’s ideas. Encourage  and model respectful disagreement. To be sure every child gets a chance to share, have them partner up and share their results with a classmate as in a think-pair-share model. They are prepared for this because they have written down their strategies and they knew the expectations from the beginning. Take mental or written notes during this discussion time to figure out what tomorrow’s problem will be, based on what the students grasp and what still eludes them.

Some of the positive aspects of this way of teaching math are that it differentiates for learners, it gives the teacher time to assess because you aren’t occupied during the entire math class with a needy child, it gives students who finish quickly multiple options for meaningful work to do that isn’t a punishment or a reward, the multiple entry points of the well-thought-out problem allow all students to solve the problem based on their current understanding of math, the writing encourages the reflective process which reinforces learning, the discussion can be had with confidence because everyone has a frame of reference in which to share; their journal entry. This gives students more self-confidence, students learn to respond respectfully to each other and really listen to their ideas and be able to agree or disagree with another student. The discussion additionally allows for social interaction which improves a student’s own ideas and understandings and it builds a strong math community. When the student tells their family what they did at school that day, you can be sure that it’s easier to remember the depth of that one problem tackled than a page or two of raw math equations.

I hope you can read Van De Walle’s book and adopt these practices into your math curriculum whether you are a classroom teacher, a student, or a parent trying to improve your understanding of how kids think and what they need to be great mathematicians.

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