Aligning our Instructional Decisions
& Educational Goals:
Analyzing the decisions of two very different mathematics teaching approaches
I have been thinking a lot about why we teach mathematics in school. Every teacher wants the best for their students, but have we really thought about what “the best” means? Do we have an agreed upon set of goals that we are all aiming toward? Many teachers might believe that having students being able to do the math is our goal, but I think it might be worth taking a few minutes and thinking about the underlying goals of the curriculum, and even broader goals beyond the curriculum.
There seems to be 2 very distinct approaches to teaching mathematics. I believe the differences are based on a number of factors that might lead teachers to choose one of these paths (past experiences, personal successes…), however, it might be worthwhile for all of us to question our own assumptions about what good math instruction is, and what our underlying goals are. Below are 2 very different examples of classrooms.
| Teacher A | Teacher B | |
| What does a typical lesson look like? | Traditional teacher-led lesson.
Gradual Release of Responsibility model of teaching: |
3-Part problem solving lesson Lesson starts off with an activation. Students are provided with a problem that they need to reason through to explain their thinking. Students make decisions about what is important, where to start, how to communicate their thinking… Lesson ends with a discussion about what was learned, different models, strategies… are explored. Students learn from each other and connections are made between students’ thinking. The lesson might be followed up with individual practice to consolidate learning. |
| Role of teacher while students are working | Helper. Students who have listened to the lesson and have learned how to “do” the math are ready to complete the follow-up assignment. Other students will need more help so the teacher gives more guidance or examples to students (who have hands up, or form a line at the teacher’s desk) until the students can do the math independently. |
Observer, Questioner, Assesser… Teacher observes students working, attending to their thinking, strategies used, models for understanding, and other processes. The teacher knows when and if they should intervene by posing questions that may illicit deeper thinking. The teacher uses a variety of formative assessment strategies to know who understands what. The teacher contemplates how to consolidate the learning after the lesson is complete. |
| Role of student during working | Do the work Students repeat thinking of the teacher by completing a set of related questions. |
Think, plan, reflect, reason, justify, explain… The students use this time to build their own understanding of the material. |
| What does a typical unit look like? | The teacher ensures all curriculum content is “covered.” Skills are taught in order. Unit begins with mostly teacher-led instruction and may gradually move to some problem-solving experiences (now that the teacher believes that they are ready). New skills are continually being introduced. | Skills, concepts and thinking strategies are connected throughout the unit. Progress in student thinking is what moves the unit along. Thinking is continually deepened. Practice and consolidation are an integral part of the formative assessment process. |
| Role of problems | Teaching for problem solving Lessons are taught by teacher so students will know what/how to think before they begin problem solving. The teacher wants/expects students to get correct answers because they can evaluate their own teaching based on their students’ successes. |
Teaching through problem solving Problems take on the role of diagnostic assessments, formative assessments throughout the unit, and summative assessments. Students learn by being challenged (productive struggle), through listening to others’ strategies, models and justifications, and by making connections between others’ ideas and their own. |
| Role of context | Story problems Real-life scenarios are used to show when the specific skills will be needed later in life. The context does not actually help the student understand the math. The teacher might actually show the student how to ignore the context by looking for key words…). |
Problem solving When contexts are used they are used to help students make sense of the mathematics. The contexts actually lead students to think differently and learn about the mathematics more deeply. |
| Diagnostic assessments | Diagnostic assessments typically track right/wrong responses. They find out what students can do, and can’t do. Purpose is for teacher to know what they learned last year, and who is “good” at math or might need extra help. |
Diagnostic assessments aim to figure out how well students understand specific concepts. They find out how students think, which concepts they have mastery over, and look at possible reasons for misconceptions. Purpose is for teacher to know what to do next. |
| Formative assessments | Students raising hands provide teacher with information about how to pace lessons (even though the majority of students do not raise hands). Numbers of questions correct on assignments gives teacher feedback about who can “do” the math. | Formative assessments are used regularly to give feedback to the teacher about how to make instructional decisions. |
| Summative assessments | Summative assessments again focus on right/wrong responses. They find out what students can do, and can’t do. | Summative assessments aim to figure out how well students understand specific concepts. They find out how students think and which concepts they have mastery over. |
| Role of the teacher in this class | Holds all the knowledge. Makes all the decisions. Primary source of examples, sharing ideas, providing thinking strategies and models. Leads all conversations. | Finds the best opportunities for their students to grow and develop as mathematicians. Asking effective questions. Facilitates discussions after problem solving. |
| Role of the student in this class | Listener. Passive. Prove to the teacher that they can “do” the math. | Thinker, justifier, reason maker, analyzer, communicator, listener… Active. Prove to themselves that they can think mathematically. |
| Demonstration of “Understanding” | Questions tend to heavily favor knowledge with some application of math concepts later. Procedures are checked for accuracy. Students are assessed based on their products. | A balance of open questions, problems and other strategies are used. Students are assessed on their products, teacher observations, and conversations with the teacher. |
| Mindset | Fixed Some students are really good at getting the answers, others aren’t.Some students learn math quickly, some have a difficult time.Students rate themselves as good at math or not. Speed and accuracy are valued. The teacher might even put students in ability grouping (under the premise of differentiation). |
Growth Every student is learning and growing. Students and teachers view all students as capable and expect high standards.Mindset beliefs of teacher match the messages they send students. |
| Answer to, “when will we ever need to know this?” | This question is asked by a few, but thought regularly by many students because students don’t connect with their work. The teacher attempts to make the concepts meaningful by adding contexts from . Answer: “You use math all the time…jobs…at home…” |
Finding the answers to questions using specific skills isn’t the goal. Thinking mathematically is! This question is rarely asked because students are more engaged. Real world contexts are present, but there are also many highly engaging problems used that have no context… they are mathematically relevant. |
| Resources used | Resources promote parrot learning, rote learning, kill-and-drill…
Thinking questions in text books are skipped (“you don’t need to estimate, just give me the answer”). |
Resources promote contextual learning that is based on student development, deep conceptual understanding… |
| Role of technology | Technology may be used by the teacher to teach students a concept (i.e., Smartboards, Khan Academy…), or for students to practice skills that have been taught in class (kill and drill websites). | Technology may be used by students to interact with their mathematics. Many programs are designed to meet students’ developmental needs (i.e., Dreambox…). |
| Role of Manipulatives | Teacher shows students how a manipulative can help them get answers. Teacher demonstrates, students try to show understanding by following the teacher’s thinking. Goal is for students to use these tools as long as needed, until they master a skill or concept. |
Students are encouraged to select and use concrete learning tools to make models of mathematical ideas. Students learn that making their own models is a powerful means of building understanding and explaining their thinking to others. Manipulatives help students see patterns and relationships; make connections between the concrete and the abstract; test, revise and confirm their reasoning; remember how they solved a problem; and communicate their reasoning to others. Students see manipulatives as meaningful, not just for those who need it. |
| What is valued in this classroom? | Calculating, skills…
Correct answers. |
Thinking
Deepening Understanding, thinking, reasoning… |
| Who is doing the majority of the thinking in this classroom? | The Teacher. The teacher plans the lesson, the teacher thinks of the best way to explain the concept to their students, the teacher thinks of the context, models and shows students how to do it. If students help in any of this process, it is typically only a few who are involved in the lesson, and the teacher is the one who approves or disapproves of the ways students should think. Students then practice their teacher’s thinking. |
The Students. The teacher plans a rich opportunity for students to think through the math. Students reason, communicate, select manipulatives, models and/or strategies on their own or in a small group. Students are expected to think their way through the problem before any instruction is given. Process expectations are stressed. |
| What does the word Understanding mean? | Instrumental understanding: Teacher wants students to be able to “do” the math. If a student can follow the procedures required to get an answer, they “understand” their math (with or without knowing why the procedures work). |
Relational understanding:
Teacher views understanding not as black-and-white, has-it or doesn’t-have-it. Instead understanding is constantly deepening. Understanding is thought of as developmental (i.e., Doug Clement’s Learning Trajectories, Cathy Fosnot’s Landscape of Learning, Mathematics Continuum) |
| What are the goals in this classroom? | Whether the teacher is aware of this or not, their goals are very short-sighted (doing well on the test next week, passing the exam, achieving on high-stakes testing). In attempting to have their students do well on these assessments, the teacher inadvertently cuts out deep understanding of mathematics. The teacher may believe that their goal is to provide students with a long list of math skills that they will need in later life and that instrumental understanding is good enough, but the problem is that most students who will learn in this class won’t use these math skills after they take their last math class – all that hard work is lost. | The goals for this teacher are to develop life-long mathematicians who have the knowledge, thinking skills, confidence and perseverance to solve problems in their current and future lives.
Students begin to see math around them because they can connect with it. They question things and make sense of their world because they are thinking mathematically. Developing relational understanding will help students continue to think mathematically well beyond the day they take their last math class. |
The example of Teacher A shows a traditional path to teaching mathematics. Many students will learn math in this classroom, but not all. High-stakes test scores (provincial testing) might show that many students are on track, but as we know, many of these students will start to dislike mathematics as it gets “harder” (i.e., highschool or sooner). When students do not have a relational understanding of the mathematics they have learned, they will find mathematics increasingly more complicated and disconnected from them. Many of the students in classroom A, even those who did well that year, will lose their learning because the learning was procedural and was not truly understood.
Students in Teacher B’s classroom will be able to meet the same standards as classroom A on high-stakes test scores, but will be able to retain their learning later into highschool and beyond. Teacher B has taught their students to think mathematically, and that all students can learn math at the highest levels.
At the beginning of this article I suggested that we all question our assumptions about what good mathematics instruction is, and what our goals are for teaching mathematics. I hope that you have taken a deep look into how our instructional decisions and our goals are linked. Most teachers would fit somewhere in the middle, having traits of both the “traditional” teacher and that of a “modern” (for lack of a better term) teacher, however, it is important to understand that the decisions we make on a daily basis are aimed at achieving one ultimate goal. So it is of the utmost importance for us to know what that goal is and to align our instructional decisions toward reaching that goal. If I believe that I need to teach the concepts first, before problem solving begins then I am not developing mathematical thinkers, I am teaching for instrumental understanding. I will see progress throughout each unit and throughout the year, however, it will lead to students who stop taking mathematics courses in high school as soon as they can, and adults who simply don’t use math. If our goals are short-sighted like wanting our students to do well on a test, quiz or even a high-stakes test, then we might take many shortcuts that undermine our students’ thinking and development. Don’t get me wrong, we all want our students to do well on their tests, quizzes and high-stakes testing, but students achieving on these shouldn’t be our primary goal. These should simply be markers to determine if our students are learning and understanding math. Our goals need to meet the real purpose of why we are all here – to develop students who can think critically, question and make sense of their world independently, who understand the importance of thinking and learning!
As a final thought, I want you to ask yourself, how can we better align our instructional decisions with our educational goals?
Thinking Mathematically 