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## This is probably the most frequent question I get about mid-way through a professional development session on teaching math conceptually.

Not only are we tasked with teaching math differently, but also with assessing math differently. Did I mention we first have to actually learn the math differently? Yikes! Don’t panic. We are here to support you – Educating Now’s videos help you, the educator, to first understand math conceptually. I do want to point out that you were likely never taught math this way, nor were you instructed on how to teach math conceptually, so be patient with yourself as you learn math anew. Our videos also provide you with lesson plans that incorporate many of the curricular competencies so that you are teaching the new curriculum.

Last year I wrote a detailed blog on formative assessment, which you may want to read because this is the most VALUABLE type of assessment in terms of how it affects achievement and learning.

And I am going to delve a bit deeper into Assessment of learning in the next blog. So this post is really dedicated to summative assessment. We will answer the questions:

1. How do I know what to assess?

2. How often do I assess?

In the next post we will explore:

1. How are my students involved in the process (besides being test takers)?

2. How do I assign a letter grade to the assessment?

This used to be the easy part right? If the chapter was on square numbers and square roots then we used to give something that looked like this:

The problem with this test is that it is mostly procedural and if I hadn’t taught this concept visually, the word problems would be very difficult for students to solve. The other problem is that even if a student gets all the questions correct, I don’t know what they actually understand, but rather I only know what they can do. These are not the same thing. Students can perform operations without any understanding at all. This happens all too often in fact, and then they forget the procedure shortly after the test is done as they’ve moved on to memorizing a new procedure. Also, with respect to the curriculum, all that we are assessing here is the content area of square and square roots (knowing) and a little of the doing (with the word problems), and again we are not testing for real understanding.

If we want a chance of having this concept retained then we need to teach the concept (not just the procedure), use visuals, use contexts, and use words to make sure they fully understand the concept and then practice all of these, including the procedures.

Here is an example of an assessment that would encapsulate all of the above suggestions:

Can you see the significant difference in the information about your students’ understanding you will gather between these two assessments? Can you also see all of the competencies we are assessing here (visualizing, explaining, justifying, demonstrating understanding pictorially, solving problems, etc.)?

### How Often do I Assess?

The short answer is: all the time! Most of my assessments are formative, as this drives my instruction. Assessment should be so embedded into your teaching that the two are almost indistinguishable.

Below is an example of formative assessment: if I were teaching squares and square roots, I would ask students to use individual whiteboards to write on as I gave them pieces of the puzzle, they would give me the other pieces. This is how it happened in a grade 8 class recently. I wrote either the words, symbols or pictures on the board and they wrote the

other 2 pieces on their whiteboards. So, when I wrote: three squared or the square of three, they showed me: 32 =9 &

This formative assessment in my lessons tells me where my students are at with their understanding. This informs what I will do in the next lesson for example:

1. Do I need to use another model, like a number line or multiplication table to help them to understand this further?

2. Do I need to allow for some students to continue developing understanding while challenging some others to solve problems involving these concepts because they have demonstrated a firm grasp of the concept so far?

Formative assessment is NOT GRADED nor is it “for marks”. It is for you and for the students as a way of identifying how close they are to the learning targets or goals.

In terms of how often do I do a summative assessment, that totally depends on the formative assessment, the concept and the students in front of me. Although formative assessment happens every day throughout the entire lesson, a summative assessment ONLY happens when the learning is mastered and the students are now demonstrating their understanding.

I allow re-tests for this reason. I believe it is unrealistic to expect all students to have fully mastered a concept by 10:00 am on Thursday (or whenever your test is scheduled for). Our goal is not to compare them with each other at any given time, it is to help them fully understand the concepts and this will not happen at the same time for all students. Thus re-tests are there to allow for students to continue persevering and working towards full mastery. So, in terms of how often do you assess – there is no structured answer like ‘every week’. Unit tests or concept tests will happen after lots of formative assessments and opportunities for learning and will depend on the concept. I do them before we move onto a new concept.

One thing I see A LOT, especially in high school math classes, are quizzes each day or once a week. These are awesome if you are using them for FORMATIVE assessment but if students are getting marked on these and this is becoming part of their grade I would ask: “where is the opportunity to think, learn and really understand the concepts?”. Too often math becomes a performance subject where students are always being graded and so don’t engage in the thinking we want because they will be penalized for mistakes and students respond by just memorizing so that they can perform well on their quizzes. The same applies for marking homework – this sends a very clear fixed mindset message that mistakes will be penalized and there is no room or time for actually developing understanding (which happens through making mistakes).

Dylan William gives a nice example of how we often assess (especially in math when it comes to unit or chapter tests) in this 2-minute video:

When rethinking your assessments, I encourage you to do the following:

1. Start by looking at the curriculum and decide upon the content, competencies and big ideas you are focusing on for the assessment and create questions that incorporate those competencies and big ideas (where applicable)

2. Ask yourself: is this going to show me what they know, understand and can do with this concept?

3. Include visuals and written explanations

4. Ask yourself if I am promoting a message that I have high expectations for all students and that the goal of the assessment is to determine the depth and breadth of their understanding?

5. Ensure you have provided plenty of opportunities for students to develop a deep understanding of this concept.

6. Is this formative or summative?

Next post I will include some rubrics and suggestions for engaging students in the process of assessment (including some assessment as learning) as well as discuss how to put a grade on assessments using these new-fangled assessments.