This is a break from the traditional Math Monday format in order to focus on another of our curriculum’s processes – problem solving. (A past post focused on communication.) Problem solving is not only a process that is to be embedded throughout, it is an area to be reported on for our grades 1-8 provincial report cards. This is one of my most-requested areas for support.
Our curriculum guide states that one of the goals for students in mathematics education is to “use mathematics confidently, accurately, and efficiently to solve problems” (p.5). Page 12 of the guide says that “learning through problem solving should be the focus of mathematics at all grade levels.”
Tied to problem solving is the practice of communication (both listening to others’ approaches, strategies, and methods to understand the math behind their thinking as well as articulating your own most efficient strategy for solving a problem).
“In order for an activity to be problem-solving based, it must ask students to determine a way to get from what is known to what is sought.” (p.13, emphasis mine).
If students are given a pattern for “solving problems”, then asked to answer 10-15 questions with a similar pattern in the answer, this is not true problem solving. HOWEVER, problem solving is not about a loosely planned activity. On the contrary, good problem solving activities are carefully planned. How can we as mathematics teachers provide true problem solving opportunities for students?
Problem Solving Ideas and Approaches:
Manitoba has support documents for grade 5, grade 7, and grade 8. There are many activities within these documents beyond problem solving, but here are some examples of what’s available:
1. Grade 5:
a. p.138 of the Number Strand section (“party planning”)
b. p.14 of the Patterns section (“How Tall”)
c. p.12 of the Shape and Space section (“design a clubhouse”)
d. p.52 of the Measurement section (“Plan a Healthy Meal”)
2. Grade 6 (no Manitoba document available – suggestions below are from other online sources):
a. The Million Dollar Mission (number strand problem solving example)
b. Pack It In (volume/Shape and Space problem solving example)
c. Ah, The Days are Getting Longer (Measurement problem solving example)
d. Big Bucks (Patterns and Relations problem solving example)
3. Grade 7:
a. Pp.41-44 of the Number Strand section (“school supply kits”)
b. Pp.65-71 of the Patterns section (“Relations stories”)
c. Pp.67-68 of the Measurement section (“Maps, Floor Plans, Design Projects”)
d. Pp.21-22 of the Statistics section (“Defining the average potato”)
4. Grade 8:
a. Pp.22-23 of the Number Strand section (“Pythagorean Theorem in real life”)
b. Pp.53-55 of the Number Strand section (“gr.8 farewell”)
c. Pp.28-29 of the Patterns section (“business and marketing analyst”)
d. Pp.29-30 of the Shape and Space section (“connecting surface area and volume in real life”)
Sample Problem Solving Prompts:
Prior to choosing a problem solving prompt, teachers should determine which strand and outcome(s) you wish to focus on. “Planning with the end in mind”, or backwards design, or UdL, is an important step prior to any problem solving activity. See the Classroom Based Assessment documents mentioned below (in the Assessment section of this email) for further details on planning. That being said, here are three generic approaches to problem solving prompts or activities:
1. Choose any word problem (from a textbook or online source), delete the question portion of the word problem. Present the remainder of the text to your students and ask “what question do you think a textbook author would ask, given this information?
2. Choose any word problem (from a textbook or online source), delete all the given information from the problem and leave ONLY the question asked. Present the question to your students and ask “what information do you need in order to answer this question?”
3. Present your students with one of the videos from a Three Act Math Tasks video collection. Ask “what questions do you have?” Choose as a class to focus on answering one of those questions together. Then allow students to choose one more question to answer individually.
Problem Solving Assessment:
Manitoba has “Classroom Based Assessment” documents for grades 5-8 and grades 9-12.
In grades 5-8 as well as grades 9-12, teachers need to look at four main areas in problem solving and assess students’ progress within each area (from the document):
1. Understanding the problem;
a. rephrase the question in their own words
b. identify the question and draw a diagram
c. highlight the relevant information in one colour and the question in another
d. sort sentence strips from a problem into three groups: needed information, extra information, and the question
2. Using appropriate strategies;
a. the chosen strategy is appropriate for the question
b. the strategy is applied correctly
c. the answer is correct
d. the explanation is clear
e. the student states strategy used
3. Verifying the solution;
a. Self-assessment of their own solution
b. Assessment of an alternate solution
4. Formulating their own problems;
a. Create a problem, provide a solution, have another student (or have the teacher) assess the problem and solution.
(The above is taken from pp.28-30 of the gr.5-8 document, and pp.33-35 of the gr.9-12 document).