*
Candidates should not only know
*

- how to do elementary mathematics,
- but should understand and
- be able to explain to students, in multiple ways,
*why the mathematics makes sense*.

- Number Systems (41%) -
*Math 153: Foundations of Number Systems* - Functions and Algebra (21%) -
*Math 250: Foundations of Reasoning* - Geometry and Measurement (18%) -
*Math 251: Foundations of Geometry* - Statistics and Probability (9%) -
*Math 252: Foundations of Statistics and Probability*

As you start preparing for the mathematics subtest of the General Curriculum MTEL (03), here are a few points to keep in mind.

- Take as many of the "Foundations of Mathematics" courses as you can fit into your schedule
(Math 153, 250, 251, 252). Make sure you take
*Foundations of Number Systems*(it is now a required course in the program). - If you are anxious about taking the mathematics test (or testing in general), seek support for working with testing anxiety. Make a pact with yourself not to abandon the test half-way. When you notice anxiety, put down your pen and give yourself a few minutes to check in with your body and to calm yourself.

- Create a study guide for yourself. The General Curriculum test objectives are available online. These objectives give you a great framework to build your study guide around. We encourage you to purchase a binder and divider tabs. Each divider tab should represent one objective. The contents within each section should answer to the terms outlined in an objective area. Setting up a binder like this may take some time, but with advanced planning will be well worth it!
- Collect and organize your resources: e.g. your class portfolio, text books, handouts, the
*Van de Walle*book, high school textbooks, online resources etc. - ALEKS or some other foundational mathematics course of study (e.g. Mary DeSouza's book). While not aligned with the MTEL, you will have a chance to gain practice and confidence with basic arithmetic and computational techniques.
- Practice Exams. One of the best ways to prepare for the General Curriculum test is to work with the practice exam, or at similar practice exams for other states (for details see Section ).
- Almost all of the problems on the Practice Exam are multi-step problems. A single reading is not enough to unravel what you need to do. There is no obvious formula to solve any of these problems. You will need to grapple with the material to make sense of it, and you will need to give yourself the time and focus to do that. That's normal and expected.
- Read carefully: it may help to underline or mark the important concepts listed in the problem. Write down whatever you know about that concept. As you prepare for the exam, write down the resources you have to fill in details about this concept.
- The commented solutions below are intended to show you a framework how to approach using MTEL-like problems in order (a) to identify mathematical concepts that you understand, (b) to identify mathematical concepts that you do not understand, (c) to assemble information about your resources, and (d) to observe a step-by-step process that might help you in problem-solving more generally.
- Below are sites that offer practice exams which may prove to be helpful study tools:
- The General Curriculum practice test. This is the closest thing to the MTEL and should be at the top of the list for everyone who is preparing for this exam.
- Arizona's Elementary Education exam:
- California's CSET: Multiple Subjects exam:
- Colorado's Elementary Education exam:
- Illinois' Elementary/Middle Grades exam:
- New York State's Elementary Assessment of Teaching Skills exam:

- Additional Resources. Please note that some of these suggestions are based on student recommendations and are not designed to ensure passage on this test, but to help provide you with supplemental information.
- Massachusetts Comprehensive Assessment System (MCAS). Previous test items are available online at
- What Your Fourth Grader Should Know by E.D. Hirsch.
- What Your Sixth Grader Should Know by E.D. Hirsch.

Van de Walle, *Elementary and Middle School Mathematics-Teaching Developmentally* is
the text book required for all *Mathematics Foundations* courses at Westfield State College.
The references given here are for the seventh edition. If you have a different edition, use the Index
of your book with your key words in order to guide your search.

**Have a game plan for how you are going to approach the test.**- The great thing about the MTEL is that you are not required to take the test in chronological order-how you take it is entirely up to you. We encourage you to look at the open response items first, set-up an outline about how you are going to approach answering these questions, and then go back to get started on multiple choice items. Approaching the test this way may help to alleviate some test anxiety and set a good pace for yourself with the rest of the test.
**Be clear and consistent with your responses.**- One of the goals of the MTEL is to determine whether or not you will be able to communicate clearly with students and parents alike. This being the case, it is essential that your handwriting on open response items is legible, your spelling and grammar are exceptional, and your answers are well thought out.
**Get focused!**- To calm your nerves once the test begins, it is best to give yourself a few minutes to relax and look over the test in its entirety before getting started. Given that you have a full four hours to take the test, take advantage of a restroom break. Getting up from the test for a few minutes and throwing some cold water on your face will work wonders.

Sample MTEL Problems

(b) Maybe you remember that squaring a positive integer will always give you another positive integer. (Note that for this question it's enough to find one answer you know is correct.)

(c) You might also remember the fact that the *positive* integers are "closed" under
addition and multiplication (e.g. squaring), but not under division or taking roots ("closed":
e.g. when you divide one integer by another you do not usually get an integer result).

If you consider all of the integers (positve or negative), these are "closed" under addition,
multiplication *and subtraction*.

As the numbers are written, it's not clear how many zeros each of them has, so let's write them out fully: earnings of 3,850,000 for the year and an estimate 30,000 per month. We could try to divide 3,850,000 by twelve but that will take time (it also looks like the answer will not require those details). Remembering that there is a related multiplication problem, we can look at 30,000 times 12: as a first guess, 30,000 times 10 would be 300,000 (so we see already that the estimate is likely too low compared to 3,850,000); more precisely , which is about a factor 10 too low.

Let's try the other point
: turns into

Consider the piece of the ribbon that runs diagonally up the left-hand - -face of the box. What is the length of that piece of ribbon? Use the Pythagorean Theorem.

If we now reflect that image across the -axis, the hand will end up in the upper-left quadrant, with the fingers pointing up and the thumb against the positive -axis.

It's not a triangle because you get four sides. It's not a rectangle because the angle at is equal to and not a right angle. The shape is not 3D (so not a prism).

Now we have two of the three interior angles of the triangle, namely and , so the remaining angle must be .

Finally, the angle is supplementary to this angle of , so it's equal to .

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