On February 8th, 2017 Eric Appleton and I hosted a webinar to explore the math content on the TASC test, focusing our attention on the most recent GHI forms of the exam. We were joined by content and TASC assessment experts Tim Jones (Associate for Instructional Services from the New York State Education Department) and Deedra Arvin (Test Development Program Manager from Data Recognition Corporation/CTB). Tim and Deedra also joined us last summer when we explored the TASC science item specifications.

The goal of the webinar was to connect adult education math teachers with representatives from DRC and NYSED to develop a better understanding of what is on the math section of the TASC exam. We specifically wanted to help teachers make informed choices and narrow the scope of what they teach, so they can teach fewer topics with the depth that the TASC requires. We also wanted to examine the kinds of questions on the TASC, to help teachers plan the problems they give and the questions they ask their students.

*Please note, the webinar begins 5 minutes and 30 seconds in. *

In case you want to share it with other teachers, here are the presentation slides we used.

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### Teaching Ideas and Implications

A lot of information about the math on the TASC came out during our webinar and readers should begin by watching it. Building off of that, we wanted to address a few of the teaching implications that came out and point teachers towards some resources that might be helpful.

###### Notice/Wonder

This was the instructional routine Eric used during the launch. The basic steps are simple: (1) Find a problem, (2) Remove the question, (3) Ask students what they notice and what they wonder. As was stated in the webinar, the TASC requires students to make sense or situations and to be close readers. Notice/Wonder is a great way to slow students down and help them build up strategies for sense-making. Below are two resources to help teachers think about how to bring notice/wonder into their classrooms.

**Beginning to Problem-Solve with “I Notice, I Wonder”**– A brief two-page outline created by the Math Forum on teaching ideas for using the practice with students

**Ever Wonder What They’d Notice?**– A brief video (5 minutes) of a talk by Annie Fetter (of the Math Forum) discussing the positive impact of using sense-making activities like Notice/Wonder in math.

###### “Two Truths and a Lie”

One of the problems we looked at during the webinar ask students, “Which of the following statements is true”? The TASC readiness assessments have several problems that use this question. During the webinar, Tim made the point that students are being assessed on their ability to make sense of the situation and do *close reading* in the math section. Since over half of the math section has mathematics woven into a context, this isn’t something we can ignore or just assume our students will pick up. Here’s an instructional routine that you can use to prepare students to evaluate mathematical statements.

Give students a graph, chart, or situation with no question (like we saw during the first section of the webinar). Ask students to write two true statements and one false statement.

I like this activity because, like Notice/Wonder, it is accessible to all students and roots their problem-solving in sense-making. It also gives teachers a much needed opportunity to talk to math students about audience, precision and revision when communicating their ideas in math class. You can do it at the end of class, collect them and choose a few for students to evaluate next time. Or you can do it at the beginning of class and have a few volunteers share a few statements. Then let the class evaluate whether the statement is true or not. The important thing is for teachers not to indicate if a statement is true or false, but to ask students, ‘How can you figure out if this is true?”

When students have to write their own statements and evaluate those statements, they start to really appreciate the impact of words like “always”, “almost”, “constantly”, “each”, etc.

###### Using Incorrect Answers to Teach (“Distractors”)

Another thing that came out of the webinar was the fact that answer choices often include “distractors”, which are compelling incorrect answers. For many problems, students are not only being assessed on whether they can choose the correct answer – they are also being assessed on whether they can ignore the plausible incorrect answers.

In class, one of my goals is to get students to make interesting mistakes. I draw them out, because the road to learning something is lined with good, important and revealing mistakes. Everyone learns from them, because anyone can make them – we treat them like a canary in a mineshaft. For more ideas on how to use student mistakes in class, look here and here.

In case it is helpful for teachers trying to create compelling incorrect answers, here is my thinking in designing the incorrect choices in the “Which one of these statements is true?” taxi service problem from the webinar:

**Choice A: The ride costs $4.50 for each mile**– This is about the idea that a function has to work for more than one input and output. This might catch a student who only looks at the first input of 1 mile with the output of $4.50. Even if it doesn’t, I’ll often pretend that I think it is the answer (as I did during the webinar), so students have to explain to me why it*isn’t*correct. I want them to be able to do the same for themselves when I am not around.

**Choice B : The ride costs $3.00 for each mile driven plus $1.50**– The function concepts of rate of change and starting amount are important in this problem. It is not just that the function is c = 1.5m + 3. It is also that it is $1.50 for every mile after you pay a flat fee of $3. This answer has the right numbers, but they are switched. Again, these are great mistakes to make, because it is valuable for students to have to think about how to explain why it is wrong.

**Choice C: For each 10 miles driven, the cab ride costs $18.00 –**This is all about close reading. The challenge/appeal of this answer is that it is almost true. If we remove the “each”, the statement is true – If you drive 10 miles, the cab ride will cost $18. But the word each means a 20 miles drive would cost $36, which is not the case.

As with “Two Truths and a Lie”, the goal is to engage students in the evaluation – “What are things I can do to determine if a reasonable answer is actually correct?”

###### The Many Faces of Functions

Another point the presenters made during the webinar was that when it comes to functions, students need to be comfortable with many different representations of functions: (a) tables, (b) graphs, (c) function equations, (d) rules written as statements, (e) formal function notation – i.e. *f(x)*. Some of the HSE math books out there start students with the formal notation first. The CUNY HSE Math Curriculum Framework models a progression that begins with students building a conceptual understanding of linear functions. It moves from the concrete and representational to the abstract, exposing students to tables, rules, graphs, equations before introducing formal function notation in Unit Three.

###### Teaching Functions and Algebra from the start

I wanted to go a little deeper into other important idea that came out of the webinar. The TASC math section is 52% functions and algebra. The panelists both made the point that teachers need not wait until students have “mastered the basics” before exploring this content. Teachers who do, often find themselves at the end of a semester with not much time to teach the most pressing and challenging material that makes up more than half of the test. Both Tim and Deedra’s informed reasoning and valuable advice is in the webinar, but I wanted to give an example from the CUNY math framework on how we can work on “the basics” and build students fluency with operations, while working on important function concepts.

Unit 1 of the framework introduces function machine. There are blank machines for teachers to fill in with any content, but here are two examples that are given:

In terms of functions and algebra, both have students practicing the idea that a function has an input that is put into a function rule to come up with an output. It also allows for students to be developing different strategies for finding an input when given an output (which the framework connects to the solving for x). In terms of operations, the first has students working on multiplication with decimals and the second has them working on benchmark percentages and percent change. Consider these as just one alternative to worksheets focused on repeated practice with operations. Students are doing the same calculations, but they are controlled and also serve a larger purpose of building a conceptual understanding of functions.

### Teaching Resources

These are all free and available resources that we have developed for teachers looking for materials that align with the content and problem-solving recommendations that came out of the webinar.

- Referred to several times during the webinar, The CUNY HSE Math Curriculum Framework focuses on problem-solving in functions and algebra. It integrates problem-solving strategies, productive struggle, perseverance and mathematical discussion into content learning. It models an approach for teaching functions without first having to master things like operations with fractions or long division. The framework includes a curriculum map, model lessons, rich engaging math problems, samples of student work, and more.

- CUNY Framework Posts are lessons and activities focused on high-emphasis content on the TASC. Some extend the units in the framework as well as provide additional support and practice to students wherever they need it. Framework Posts also has teaching materials for topics not covered in the framework (especially in geometry).

- CollectEDNY features reviews of high-quality, free teaching resources, including lesson plans, videos, activities, and more. The reviews are written by adult education teachers with adult education students and classrooms in mind. Each review describes the resource and offers advice for how to get started. Here are all of the math resources reviewed.

- MathMemos is a math teacher space within CollectEdNY where adult educators share rich math problems across a wide range of content areas, as well as samples of student work, and practical suggestions for bringing the problems to life in the classroom.

- More Information on the Content of the TASC Math Test – On Framework Posts we develop (and collect) resources to help math teachers understand the TASC. Here you’ll find things like an analysis of the TASC Readiness Assessments, a Student Calculator Guide, etc.